# Neumann problems for nonlinear elliptic equations with lower order terms

M.F. Betta, O. Guibé, A. Mercaldo

► To cite this version:

M.F. Betta, O. Guibé, A. Mercaldo. Neumann problems for nonlinear elliptic equations with lower order terms. Nonlinear Analysis: Theory, Methods and Applications, 2024, 250, pp.113626. 10.1016/j.na.2024.113626. hal-05442736

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# Neumann problems for nonlinear elliptic equations with lower order terms

M.F. Betta<sup>a</sup>, O. Guibé<sup>b</sup>, A. Mercaldo<sup>c,*</sup>

## ARTICLE INFO

Communicated by Tobias Weth

MSC:

2000

35J25

Keywords:

Nonlinear elliptic equations

Neumann problems

Existence results

Stability result

## ABSTRACT

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is

$$ egin{cases} \lambda|u|^{p-2}u - \Delta_{\rho}u - \operatorname{div}(c(x)|u|^{p-2}u) + b(x)|\nabla u|^{p-2}\nabla u = f & \text{in } \Omega, \\ (|\nabla u|^{p-2}\nabla u + c(x)|u|^{p-2}u) \cdot \underline{n} = 0 & \text{on } \partial\Omega \end{cases} $$

where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \ge 2$, with Lipschitz boundary, $1 < p < N$, $\underline{n}$ is the outer unit normal to $\partial\Omega$, $\lambda > 0$, the datum $f$ belongs to the dual space of $W^{1,p}(\Omega)$ or to Lebesgue space $L^1(\Omega)$. Finally the coefficients $b, c$ belong to appropriate Lebesgue spaces or Lorentz spaces.

Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients $b$ and $c$.

## 1. Introduction

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is

$$ egin{cases} \lambda|u|^{p-2}u - \Delta_{\rho}u - \operatorname{div}(c(x)|u|^{p-2}u) + b(x)|\nabla u|^{p-2}\nabla u = f & \text{in } \Omega, \\ (|\nabla u|^{p-2}\nabla u + c(x)|u|^{p-2}u) \cdot \underline{n} = 0 & \text{on } \partial\Omega, \end{cases} \quad (1.1) $$

where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \ge 2$, with Lipschitz boundary, $1 < p < N$, $\underline{n}$ is the outer unit normal to $\partial\Omega$, $\lambda > 0$, the datum $f$ belongs to the dual space of the classical Sobolev space $W^{1,p}(\Omega)$ or it is a $L^1$-function. Finally the coefficients $b, c$ belong to appropriate Lebesgue spaces or Lorentz spaces.

When the nonlinear elliptic operator has no lower order terms and $f$ is an element of the dual space of the Sobolev space $W^{1,p}(\Omega)$, the existence and uniqueness of weak solutions to problem (1.1) is a consequence of the classical theory of pseudomonotone operators (cfr. [1,2]). If $f$ is just an $L^1$-function, and no longer an element of the dual space of $W^{1,p}(\Omega)$, one has to give a meaning to the notion of solution. Various definitions of solution to nonlinear elliptic equations with right-hand side in $L^1$ or measure have been introduced in literature, such as renormalized solutions, entropy solutions and solutions obtained as a limit of approximations (see, for example, [3–6] for Dirichlet boundary value problems). In the present paper we refer to the renormalized solutions (see [5–7]).

Existence results for Neumann boundary value problem (1.1) with $L^1$-data when $c = b = 0$ has been treated in various context. In [8–12] the existence of a distributional solution which belongs to a suitable Sobolev space and which has null mean value is

<footnote><sup>a</sup> Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola, C4 80143 Napoli, Italy</footnote>

<footnote><sup>b</sup> Univ Rouen Normandie, CNRS, LMRS UMR 6085, F-76000 Rouen, France</footnote>

<footnote><sup>c</sup> Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy</footnote>

<footnote>* Corresponding author.
E-mail addresses: mfrancesca.betta@uniparthenope.it (M.F. Betta), olivier.guibee@univ-rouen.fr (O. Guibé), mercaldo@unina.it (A. Mercaldo).</footnote>
proved. Nevertheless when $p$ is close to 1, i.e. $1 < p \le 2 - 1/N$, the distributional solution to problem (1.1) does not belong to a Sobolev space and in general is not a summable function; this implies that its mean value has no meaning. This difficulty is overcome in [13] by considering solutions $u$ which are not in $L^1(\Omega)$, but for which $\Phi(u)$ is in $L^1(\Omega)$, where $\Phi(t) = \int_0^t \frac{ds}{(1+|s|)^\alpha}$ with appropriate $0 < \alpha < 1$. In [14] the case where both the datum $f$ and the domain $\Omega$ are not regular is studied and existence of generalized solutions whose median is equal to zero are obtained with a natural process of approximations (see also [15]). Existence and uniqueness of renormalized solution for problem (1.1) with $\lambda = 0$ and $b = 0$ have been studied in [16,17] respectively.

Neumann problems have been studied by a different point of view in [18,19]. Finally we recall that there is a wide range of literature for elliptic equations similar to (1.1) with Dirichlet boundary conditions in the variational case and in the $L^1$ case, among them [20–25].

In this paper we prove the existence of weak solutions or renormalized solutions for problems whose prototype is (1.1) and we face the difficulties due to the presence of two lower order terms.

We firstly deal with weak solutions. Our main results are given by Theorems 3.1 and 3.2 which assert the existence of a weak solution to (1.1) when the norm of one of the two coefficients is small enough and the datum is an element of the dual space. Then in Theorems 4.4 and 4.5 we state the existence of a renormalized solution to (1.1) when the norm of one of the two coefficients is small enough and the datum is an $L^1$-function; in this case we also face the difficulties due to the low integrability of the datum.

In both cases the proofs of the existence results are based on a classical procedure of approximation which consists in considering problems of type (1.1) having truncated lower order terms. The crucial point of this procedure is to prove suitable a priori estimates. These estimates allow to find a limit function $u$ which is proved to be the desired solution of (1.1).

Considering Neumann boundary conditions rather than Dirichlet boundary conditions adds difficulties to obtain a priori estimates due, for example, to the lack of Poincaré inequality. The novelty of this paper is the use of different techniques to prove a priori estimates based on a specific version of J.-L. Lions Lemma, given by Lemma 2.1, and the choice of suitable test functions.

Once a priori estimates have been proved, then the classical procedure of passage to the limit in the approximated problems allows to prove that the limit function $u$ is a weak solution when the datum $f$ is in the dual space $W^{-1,p'}(\Omega)$. Moreover a stability result provides such a procedure for proving that the limit function $u$ is a renormalized solution when $f$ is an $L^1$ function [16]; see also [7,23].

The paper is organized as follows. In Section 2 we precise our assumptions, we give a version of J.-L. Lions Lemma adapted to our needs and we recall a few facts about Lorentz spaces. Section 3 is devoted to existence results for weak solution, given by Theorems 3.1 and 3.2, where $f$ is a function belonging to the dual space of $W^{1,p}(\Omega)$. The case of $L^1$ data is dealt with Section 4. After giving the definition of renormalized solution of Eq. (1.1) (see Definition 4.1), we establish in Theorems 4.4 and 4.5 the existence results. Finally Appendix is devoted to the proof of the stability of renormalized solutions which is dealt with Section 4.

## 2. Assumptions and preliminary results

In this section we firstly give the assumptions under which we prove our existence results. Then we prove an extension of the classical J.-L. Lions Lemma, given by Lemma 2.1 below, and we also recall a few facts about the Lorentz spaces.

### 2.1. Assumptions

Let us consider the following nonlinear elliptic Neumann problem

$$ egin{cases} 
\lambda |u|^{p-2}u - \operatorname{div}(a(x, u, \nabla u) + \Phi(x, u)) + H(x, \nabla u) = f & \text{in } \Omega, \\
(a(x, u, \nabla u) + \Phi(x, u)) \cdot \eta = 0 & \text{on } \partial\Omega,
\end{cases} 
\quad (2.1) $$

where $\Omega$ is a connected open subset of $\mathbb{R}^N$, $N \ge 2$, having finite Lebesgue measure and Lipschitz boundary, $\eta$ is the outer unit normal to $\partial\Omega$. We assume that $\lambda > 0$, $p$ is a real number such that $1 < p < N$ and

$$ \mathbf{a}: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N, $$

$$ \Phi: \Omega \times \mathbb{R} \to \mathbb{R}^N, $$

$$ H: \Omega \times \mathbb{R}^N \to \mathbb{R}, $$

are Carathéodory functions. Moreover $\mathbf{a}$ satisfies:

$$ \mathbf{a}(x, s, \xi) \cdot \xi \ge \alpha |\xi|^p, \quad \forall s \in \mathbb{R}, \forall \xi \in \mathbb{R}^N, \text{ a.e. in } \Omega \quad (2.2) $$

where $\alpha > 0$ is a given real number;

$$ (\mathbf{a}(x, s, \xi) - \mathbf{a}(x, s, \eta)) \cdot (\xi - \eta) > 0 \quad (2.3) $$

$\forall s \in \mathbb{R}, \forall \xi, \eta \in \mathbb{R}^N$ with $\xi \ne \eta$ and a.e. in $\Omega$; for any $k > 0$ there exist $a_k > 0$ and $b_k$ belonging to $L^{p'}(\Omega)$ such that

$$ |\mathbf{a}(x, s, \xi)| \le a_k |\xi|^{p-1} + b_k(x), \quad \forall |s| < k, \forall \xi \in \mathbb{R}^N, \text{ a.e. in } \Omega. \quad (2.4) $$
Actually in the proof of existence results for weak solutions we will assume the stronger growth condition, that is there exist $a_0 > 0$ and $b_0 \in L^{p'}(\Omega)$ such that

$$|a(x, s, \xi)| \le a_0(|\xi|^{p-1} + |s|^{p-1}) + b_0(x), \quad \forall s \in \mathbb{R}, \forall \xi \in \mathbb{R}^N, \text{ a.e. in } \Omega. \quad (2.5)$$

We assume that $\Phi$ and $H$ satisfy the following growth conditions respectively

$$|\Phi(x, s)| \le c(x)(1 + |s|^{p-1}) \quad \forall s \in \mathbb{R}, \text{ a.e. in } \Omega, \quad (2.6)$$

$$|H(x, \xi)| \le b(x)(1 + |\xi|^{p-1}) \quad \forall \xi \in \mathbb{R}^N, \text{ a.e. in } \Omega, \quad (2.7)$$

with $c \in L^{\frac{N}{p-1}}(\Omega)$ and $b \in L^N(\Omega)$.

Finally we assume that the datum $f$ is a measurable function in a Lebesgue space $L^r(\Omega)$, $1 \le r \le +\infty$, which can be or not an element of the dual space of the classical Sobolev space $W^{1,p}(\Omega)$.

If we assume that $f \in L^{(p^*)'}(\Omega) \cap (W^{1,p}(\Omega))'$ we deal with a weak solution $u$ to problem (2.1), this means that $u \in W^{1,p}(\Omega)$ and

$$\lambda \int_{\Omega} |u|^{p-2} u \varphi dx + \int_{\Omega} a(x, u, \nabla u) \nabla \varphi dx + \int_{\Omega} \Phi(x, u) \nabla \varphi dx \quad (2.8) \\ + \int_{\Omega} H(x, \nabla u) \varphi dx = \int_{\Omega} f \varphi dx,$$

for every $\varphi \in W^{1,p}(\Omega)$. It is worth noting that in view of growth assumptions (2.5), (2.6) and (2.7) all the terms in the previous equation are well defined.

If we assume that $f \in L^1(\Omega)$ we refer to renormalized solution to problem (2.1), whose definition is given by Definition 4.1 in Section 4.

## 2.2. Preliminary results

The next lemma is a specific version of J.-L. Lions Lemma and will be used to deal with existence in the case where $b$ is small.

**Lemma 2.1.** Let $c \in L^{N/(p-1)}(\Omega)$. For any $\varepsilon > 0$ there exists $C(c, \varepsilon) > 0$ such that, for every $u \in W^{1,p}(\Omega)$,

$$\left( \int_{\Omega} |c|^{p'} |u|^{p} dx \right)^{1/p} \le \varepsilon \| \nabla u \|_{L^p(\Omega)} + C(c, \varepsilon) \left( \int_{\Omega} |u|^{p-1} dx \right)^{1/(p-1)}. \quad (2.9)$$

**Remark 2.2.** If $p-1 \ge 1$ and $|c(x)| \ge 1$ a.e. in $\Omega$, inequality (2.9) is a consequence of J.-L. Lions Lemma applied to the Banach spaces $X = W^{1,p}(\Omega)$, $Y = \{v : \Omega \to \mathbb{R}\}$ measurable, $\int_{\Omega} |c|^{p'} |u|^{p} dx < \infty$ and $Z = L^{p-1}(\Omega)$.

**Proof.** Without loss of generality we can assume that $|c| \ge 1$ and, as in the classical case, we give a short proof by a contradiction argument. Let $\varepsilon > 0$ and for any $n \ge 1$ let $u_n \in W^{1,p}(\Omega)$ such that

$$1 = \left( \int_{\Omega} |c|^{p'} |u_n|^{p} dx \right)^{1/p} > \varepsilon \| \nabla u_n \|_{L^p(\Omega)} + n \left( \int_{\Omega} |u_n|^{p-1} dx \right)^{1/(p-1)}. \quad (2.10)$$

Since $|c(x)| \ge 1$ a.e. in $\Omega$ we obtain that $u_n$ is bounded in $W^{1,p}(\Omega)$. By Rellich-Kondrachov compact embedding theorem we deduce, up to a subsequence still indexed by $n$, that $u_n$ converges strongly to $u$ in $L^p(\Omega)$ and a.e. in $\Omega$. Since $c \in L^{N/(p-1)}(\Omega)$, by Sobolev embedding theorem and equi-integrability argument we obtain that

$$|c|^{p'} |u_n|^{p} \to |c|^{p'} |u|^{p} \quad \text{strongly in } L^1(\Omega) \quad (2.11)$$

so that

$$\int_{\Omega} |c|^{p'} |u|^{p} dx = 1.$$

The contradiction then comes from (2.10) which implies that

$$\lim_{n \to \infty} \left( \int_{\Omega} |u_n|^{p-1} dx \right)^{1/(p-1)} = 0.$$

This implies $u = 0$. $\square$

Now we recall the definition and a few properties of Lorentz spaces, which we will use in the following. For references about Lorentz spaces see, for example, [26].

Let us denote by $f^*$ the decreasing rearrangement of $f$, i.e. the decreasing function defined by

$$f^*(t) = \inf \{s \ge 0 : \text{meas}\{x \in \Omega : |f(x)| > s\} < t\}, \quad t \in [0, |\Omega|].$$
Moreover for $1 < q < \infty$ and $1 < s \le \infty$, denote

$$ \|f\|_{L^{q,s}(\Omega)} = \begin{cases} \left( \int_0^{|\Omega|} [f^*(t) t^{\frac{1}{q}}]^{\frac{1}{s}} \frac{dt}{t} \right)^{1/s}, & \text{if } s < \infty, \\ \sup_{t>0} t [\text{meas}\{x \in \Omega : |f(x)| > t\}]^{1/r}, & \text{if } s = \infty. \end{cases} \qquad (2.12) $$

The Lorentz space $L^{q,s}(\Omega)$ is the space of Lebesgue measurable functions such that

$$ \|f\|_{L^{q,s}(\Omega)} < +\infty, \qquad (2.13) $$

endowed with the norm defined by (2.12).

They are “intermediate spaces” between the Lebesgue spaces, in the sense that, for every $1 < s < r < \infty$, we have

$$ L^{r,1}(\Omega) \subset L^{r,r}(\Omega) = L^r(\Omega) \subset L^{r,\infty}(\Omega) \subset L^{s,1}(\Omega). \qquad (2.14) $$

The space $L^{r,\infty}(\Omega)$ is the dual space of $L^{r',1}(\Omega)$, where $\frac{1}{r} + \frac{1}{r'} = 1$, and one has the generalized Hölder inequality

$$ \left\{ \begin{array}{l} \forall f \in L^{r,\infty}(\Omega), \forall g \in L^{r',1}(\Omega), \\ \displaystyle \int_{\Omega} |fg| \le \|f\|_{L^{r,\infty}(\Omega)} \|g\|_{L^{r',1}(\Omega)}. \end{array} \right. \qquad (2.15) $$

## 2.3. Notation

In the whole paper, we denote $T_k(s)$, $k \ge 0$, the truncation at height $k$ that is $T_k(s) = \min(k, \max(s, -k))$, $\forall s \in \mathbb{R}$ and $G_m(s)$, $m \ge 0$, the function

$$ G_m(s) = (|s| - m)^+ \text{sign } s, \quad s \in \mathbb{R}. $$

Moreover $C$ denotes a positive constant which can vary from line to line and which depends only on the data.

## 3. Existence results for weak solutions

In the present section we prove the existence of a weak solution $u$ to (2.1) by assuming the stronger assumption (2.5) on the growth of $a$ which insures that $a(x, u, \nabla u)$ belongs to $(L^p'(\Omega))^N$ for any $u \in W^{1,p}(\Omega)$. Actually we prove that the weak solution $u$ is the limit of the sequence $(u_\epsilon)_{\epsilon>0}$ where $u_\epsilon$ is a weak solution belonging to $W^{1,p}(\Omega)$ of the approximated problem

$$ \left\{ \begin{array}{ll} \lambda|u_\epsilon|^{p-2}u - \text{div}(a(x, u_\epsilon, \nabla u_\epsilon) + \Phi_\epsilon(x, u_\epsilon)) + H_\epsilon(x, \nabla u_\epsilon) = f & \text{in } \Omega, \\ (a(x, u_\epsilon, \nabla u_\epsilon) + \Phi_\epsilon(x, u_\epsilon)) \cdot \underline{n} = 0 & \text{on } \partial\Omega, \end{array} \right. \qquad (3.1) $$

where for $\epsilon > 0$, we denote

$$ \Phi_{\epsilon}(x, s) = T_{\frac{1}{\epsilon}}(\Phi(x, s)), $$

$$ H_{\epsilon}(x, \xi) = T_{\frac{1}{\epsilon}}(H(x, \xi)), $$

i.e.

$$ \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u \varphi dx + \int_{\Omega} a(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla \varphi dx + \int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla \varphi dx \\ + \int_{\Omega} H_{\epsilon}(x, \nabla u_{\epsilon}) \varphi dx = \int_{\Omega} f \varphi dx \qquad (3.2) $$

for every $\varphi \in W^{1,p}(\Omega)$.

In view of assumptions on $a$, recalling that $\lambda > 0$ and since $\Phi_\epsilon$ and $H_\epsilon$ are bounded Carathéodory functions, classical results assure the existence of at least one weak solution $u_\epsilon$ to problem (3.1). In the following two subsections we prove a priori estimates and we pass to the limit as $\epsilon \to 0$. We distinguish the case where the norm of $b$ is small and the case where the norm of $c$ is small. Dealing separately with the two cases is mainly due to different test functions used to obtain the a priori estimates on $u_\epsilon$.

### 3.1. Case norm of b small

The main result of this section is the following:

**Theorem 3.1.** Under the assumptions (2.2), (2.3), (2.5)–(2.7) with

$$ f \in L^{(p')}(\Omega) \cap (W^{1,p}(\Omega))', $$
there exists a positive constant $C_1 = C_1(\lambda, \alpha, \Omega, p, f, c)$ depending only on the data but not on $b$ such that if the following smallness assumption holds true:

$$ \|b\|_{L^N(\Omega)} \le C_1, \qquad (3.3) $$

then there exists at least one weak solution $u \in W^{1,p}(\Omega)$ to problem (2.1).

**Proof.** We proceed by dividing the proof into three steps. In Step 1 we derive an estimate of $|u_\epsilon|^{p-1}$ in $L^1(\Omega)$ which depends on the data and on $\|\nabla u_\epsilon\|_{L^p}$. In Step 2, under a smallness condition on $b$, by using J.-L. Lions Lemma, Lemma 2.1, we are able to establish a priori estimates, independent of $\epsilon$, in $W^{1,p}(\Omega)$ of $u_\epsilon$. Finally Step 3 is devoted to pass to the limit in the approximate problem.

**Step 1.** A priori estimates of $|u_\epsilon|^{p-1}$ in $L^1(\Omega)$.

Let us consider $\varphi = T_1(u_\epsilon)$ as test function in (3.2). By ellipticity condition (2.2) and growth conditions on $\Phi$ and $H$, (2.6) and (2.7) respectively, since $|T_1(u_\epsilon)| \le 1$ a.e. in $\Omega$, we get

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^{p-1} |T_1(u_{\epsilon})| dx + \alpha \int_{\Omega} |\nabla T_1(u_{\epsilon})|^p dx \\ & \le \int_{\Omega} c(x)(1 + |T_1(u_{\epsilon})|^{p-1}) |\nabla T_1(u_{\epsilon})| dx + \int_{\Omega} b(x)(1 + |\nabla u_{\epsilon}|^{p-1}) dx + \|f\|_{L^1(\Omega)} \end{aligned} $$

By Hölder and Young inequalities we obtain

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^{p-1} |T_1(u_{\epsilon})| dx + \left( \alpha - \frac{\alpha}{p} \right) \int_{\Omega} |\nabla T_1(u_{\epsilon})|^p dx \\ & \le \left( \frac{2}{\alpha \frac{1}{p}} \right)^{p'} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} + \|b\|_{L^p(\Omega)} \|\nabla u_{\epsilon}\|_{L^p(\Omega)}^{p-1} \end{aligned} $$

We deduce the following estimates for $|T_1(u_\epsilon)| \|u_\epsilon\|^{p-1}$

$$ \begin{aligned} \int_{\Omega} |u_{\epsilon}|^{p-1} |T_1(u_{\epsilon})| dx & \le \left( \frac{2}{\alpha \frac{1}{p}} \right)^{p'} \frac{1}{\lambda p'} \|c\|_{L^{p'}(\Omega)}^{p'} + \frac{\|b\|_{L^1(\Omega)}}{\lambda} + \frac{\|f\|_{L^1(\Omega)}}{\lambda} \\ & \qquad + \frac{1}{\lambda} \|b\|_{L^p(\Omega)} \|\nabla u_{\epsilon}\|_{L^p(\Omega)}^{p-1} \end{aligned} \qquad (3.4) $$

By (3.4) we deduce the desired estimate on $|u_\epsilon|^{p-1}$:

$$ \begin{aligned} \int_{\Omega} |u_{\epsilon}|^{p-1} dx &= \int_{\{|u_{\epsilon}| \ge 1\}} |u_{\epsilon}|^{p-1} dx + \int_{\{|u_{\epsilon}| \le 1\}} |u_{\epsilon}|^{p-1} dx \\ & \le \int_{\{|u_{\epsilon}| \ge 1\}} |u_{\epsilon}|^{p-1} |T_1(u_{\epsilon})| dx + |\Omega| \\ & \le C(\Omega, p, \lambda) \left( 1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} + \|b\|_{L^p(\Omega)} \|\nabla u_{\epsilon}\|_{L^p(\Omega)}^{p-1} \right). \end{aligned} \qquad (3.5) $$

**Step 2.** $u_\epsilon$ is bounded in $W^{1,p}(\Omega)$.

Let us consider $u_\epsilon \in W^{1,p}(\Omega)$ as test function in (3.2). We get

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^p dx + \int_{\Omega} a(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon} dx + \int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon} dx \\ & + \int_{\Omega} H_{\epsilon}(x, \nabla u_{\epsilon}) u_{\epsilon} dx = \int_{\Omega} f u_{\epsilon} dx. \end{aligned} $$

By (2.2), (2.6) and (2.7) we obtain

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^p dx + \alpha \int_{\Omega} |\nabla u_{\epsilon}|^p dx \\ & \le \int_{\Omega} c(x)(1 + |u_{\epsilon}|^{p-1}) |\nabla u_{\epsilon}|^p dx + \int_{\Omega} b(x)(1 + |\nabla u_{\epsilon}|^{p-1}) |u_{\epsilon}|^p dx + \int_{\Omega} |f| |u_{\epsilon}|^p dx. \end{aligned} $$

Denote

$$ \beta = \min\{\lambda, \alpha\}. \qquad (3.6) $$

By Young inequality and Sobolev embedding we have

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^p dx + \alpha \int_{\Omega} |\nabla u_{\epsilon}|^p dx \\ & \le \frac{\alpha}{2p} \int_{\Omega} |\nabla u_{\epsilon}|^p dx + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} + \int_{\Omega} c(x) |u_{\epsilon}|^{p-1} |\nabla u_{\epsilon}|^p dx \\ & \qquad + \frac{\beta}{2p} \int_{\Omega} |u_{\epsilon}|^p dx + \left(\frac{2}{\beta}\right)^{p'/p} \frac{1}{p'} \|b\|_{L^{p'}(\Omega)}^{p'} + \int_{\Omega} b(x) |\nabla u_{\epsilon}|^{p-1} |u_{\epsilon}|^p dx \end{aligned} $$
$$ + \frac{\beta}{2p} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^{p} + \left(\frac{2}{\beta}\right)^{p/p'} \frac{1}{p'} \|f\|_{(W^{1,p}(\Omega))'}^{p'} $$

and then

$$ \int_{\Omega} |u_{\varepsilon}|^p dx + \int_{\Omega} |\nabla u_{\varepsilon}|^p dx \le \frac{p'}{\beta} I_1 + \frac{p'}{\beta} I_2 + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{1}{\beta} \|c\|_{L^{p'}(\Omega)}^{p'} \\ + \left(\frac{2}{\beta}\right)^{p'/p} \frac{1}{\beta} \left( \|b\|_{L^{p'}(\Omega)}^{p'} + \|f\|_{(W^{1,p}(\Omega))'}^{p'} \right), \quad (3.7) $$

with

$$ I_1 = \int_{\Omega} c(x) |u_{\varepsilon}|^{p-1} |\nabla u_{\varepsilon}| dx, \quad I_2 = \int_{\Omega} b(x) |\nabla u_{\varepsilon}|^{p-1} |u_{\varepsilon}| dx. $$

Now we estimate the two integrals $I_1$ and $I_2$.

Let us begin by evaluating $I_1$. By Hölder inequality and Lemma 2.1, we obtain for a parameter $\eta > 0$ (which will be chosen later)

$$ \begin{aligned} I_1 &\le \left( \int_{\Omega} c(x) p' |u_{\varepsilon}|^p dx \right)^{1/p'} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \\ &\le \left[ \eta \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} + C(c, \eta) \left( \int_{\Omega} |u_{\varepsilon}|^{p-1} dx \right)^{1/(p-1)} \right]^{p-1} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \\ &\le 2^{p-1} \left( \eta^{p-1} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^{p-1} + C(c, \eta)^{p-1} \int_{\Omega} |u_{\varepsilon}|^{p-1} dx \right) \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \\ &= 2^{p-1} \eta^{p-1} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^p + 2^{p-1} C(c, \eta)^{p-1} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \int_{\Omega} |u_{\varepsilon}|^{p-1} dx, \end{aligned} $$

where $C(c, \eta) > 0$ is the constant given by Lemma 2.1. Then, using estimate (3.5) proved in Step 1 and Young inequality, we get

$$ \begin{aligned} I_1 &\le 2^{p-1} \eta^{p-1} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^p + 2^{p-1} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \times \\ &\quad \times \left( 1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} + \|b\|_{L^p(\Omega)} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^{p-1} \right) \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \\ &\le (2^{p-1} \eta^{p-1} + 2^{p-1} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \|b\|_{L^p(\Omega)}) \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^p \\ &\quad + 2^{p-1} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \left( 1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} \right) \|\nabla u_{\varepsilon}\|_{L^p(\Omega)} \\ &\le (2^{p-1} \eta^{p-1} + 2^{p-1} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \|b\|_{L^p(\Omega)}) \|u_{\varepsilon}\|_{W^{1,p}}^p \\ &\quad + \frac{2^p}{p'} \left( \frac{p}{\beta} \right)^{p'/p} C(c, \eta)^p C(\Omega, p, \lambda)^p \left( 1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} \right)^{p'} \\ &\quad + \frac{\beta}{pp'} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^p. \end{aligned} $$

Now we evaluate the second integral $I_2$ in (3.7). By Hölder inequality and Sobolev imbedding theorem, we have

$$ \begin{aligned} I_2 &\le \|b\|_{L^N(\Omega)} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^{p-1} \|u_{\varepsilon}\|_{L^p(\Omega)} \\ &\le S \|b\|_{L^N(\Omega)} \|\nabla u_{\varepsilon}\|_{L^p(\Omega)}^{p-1} (\|\nabla u_{\varepsilon}\|_{L^p(\Omega)} + \|u_{\varepsilon}\|_{L^p(\Omega)}) \le S \|b\|_{L^N(\Omega)} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^p, \end{aligned} $$

where S is the best constant in the Sobolev imbedding theorem.

Collecting the estimates of $I_1$ and $I_2$ and (3.7) we deduce

$$ \begin{aligned} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^{p} &= \int_{\Omega} |u_{\varepsilon}|^{p} dx + \int_{\Omega} |\nabla u_{\varepsilon}|^{p} dx \\ &\le (2^{p-1} \eta^{p-1} + 2^{p-1} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \|b\|_{L^p(\Omega)}) \frac{p'}{\beta} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^{p} \\ &\quad + \frac{2^p}{\beta} \left(\frac{p}{\beta}\right)^{p'/p} C(c, \eta)^p C(\Omega, p, \lambda)^p \left(1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)}\right)^{p'} \\ &\quad + \frac{1}{p} \|u_{\varepsilon}\|_{W^{1,p}}^{p} + \frac{p'}{\beta} S \|b\|_{L^N(\Omega)} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^{p} \\ &\quad + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{1}{\beta} \|c\|_{L^{p'}(\Omega)}^{p'} + \left(\frac{2}{\beta}\right)^{p'/p} \frac{1}{\beta} \left(\|b\|_{L^{p'}(\Omega)}^{p'} + \|f\|_{(W^{1,p}(\Omega))'}^{p'}\right). \end{aligned} $$

Now let us choose $\eta > 0$ small enough, i.e. $0 < \frac{1}{p'} - 2^{p-1} \eta^{p-1} \frac{p'}{\beta}$, or equivalently,

$$ 0 < \eta < \frac{\beta^{1/(p-1)}}{2(p')^{2/(p-1)}}. $$
For such a choice of $\eta$ we then obtain using Hölder inequality

$$ \left(\frac{1}{p'} - 2^{p-1} \eta^{p-1} \frac{p'}{\beta}\right) \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^p \\ \le \left(2^{p-1}|\Omega|^{(N-p)/N} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \frac{p'}{\beta} + \frac{p'}{\beta} S\right) \|b\|_{L^N(\Omega)} \|u_{\varepsilon}\|_{W^{1,p}(\Omega)}^p \\ + \frac{2^p}{\beta} \left(\frac{p}{\beta}\right)^{p'/p} C(c, \eta)^p C(\Omega, p, \lambda)^p \left(1 + \|c\|_{L^{p'}(\Omega)}^{p'} + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)}\right)^{p'} \\ + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{1}{\beta} \|c\|_{L^{p'}(\Omega)}^{p'} + \left(\frac{2}{\beta}\right)^{p'/p} \frac{1}{\beta} \left(\|b\|_{L^{p'}(\Omega)}^{p'} + \|f\|_{(W^{1,p}(\Omega))'}^{p'}\right). $$

Finally, if the following smallness assumptions on $b$ holds true, namely

$$ \left( \frac{1}{p'} - 2^{p-1} \eta^{p-1} \frac{p'}{\beta} \right) - \left( 2^{p-1} |\Omega|^{(N-p)/N} C(c, \eta)^{p-1} C(\Omega, p, \lambda) \frac{p'}{\beta} + \frac{p'}{\beta} S \right) \|b\|_{L^N(\Omega)} > 0 $$

we obtain the a priori estimates

$$ \|u_{\varepsilon}\|_{W^{1,p}(\Omega)} \le C, \qquad (3.8) $$

where $C$ is a positive constant depending only on $\lambda, \alpha, \eta, \Omega, p, N, \|b\|_{L^N(\Omega)}$, $\|c\|_{L^{p'}(\Omega)}$ and $\|f\|_{(W^{1,p}(\Omega))'}$. Step 3. *Passage to the limit in the approximated problems*

Since $u_{\varepsilon}$ satisfies (3.8) for every $\varepsilon > 0$, there exists a subsequence (still denoted by $u_{\varepsilon}$) and a measurable function $u$ such that

$$ u_{\varepsilon} \to u \quad \text{weakly in } W^{1,p}(\Omega), \qquad (3.9) $$

$$ u_{\varepsilon} \to u \quad \text{strongly in } L^q(\Omega), \quad p \le q < p, \qquad (3.10) $$

$$ u_{\varepsilon} \to u \quad \text{a.e. in } \Omega \qquad (3.11) $$

and in view of the growth assumption (2.5) on $a$, there exists a field $\sigma$ belonging to $(L^{p'}(\Omega))^N$ such that

$$ a(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \to \sigma \quad \text{weakly in } (L^{p'}(\Omega))^N. \qquad (3.12) $$

By the growth condition on $\Phi$ and (3.11) we have

$$ \Phi_{\varepsilon}(x, u_{\varepsilon}) \to \Phi(x, u) \quad \text{strongly in } (L^{p'}(\Omega))^N $$

and by the growth condition $H$ and (3.9) we have

$$ H_{\varepsilon}(x, \nabla u_{\varepsilon}) \text{ is bounded in } L^{(p^*)'}(\Omega). $$

Using the monotonicity (2.3) of $a$ and well-known compactness arguments (see [27,28]) we obtain that

$$ \nabla u_{\varepsilon} \to \nabla u \quad \text{a.e. in } \Omega, $$

so that $\sigma = a(x, u, \nabla u)$ a.e. in $\Omega$ and $H_{\varepsilon}(x, \nabla u_{\varepsilon}) \to H(x, \nabla u)$ a.e. in $\Omega$ and weakly in $L^{(p^*)'}(\Omega)$. Therefore Vitali's theorem allows to pass to the limit in the approximated problems and this allows to conclude that $u$ is a weak solution to (2.1). $\square$

### 3.2. Case norm of $c$ small

The main result of this section is the following:

**Theorem 3.2.** Under the assumptions (2.2), (2.3), (2.5)–(2.7) with

$$ f \in L^{(p^*)'}(\Omega) \cap (W^{1,p}(\Omega))', $$

there exists a positive constant $C_2 = C_2(\lambda, \alpha, \Omega, p, f, b)$ depending only on the data but not on $c$ such that if the following smallness assumption holds true:

$$ \|c\|_{L^{N/(p-1)}(\Omega)} \le C_2, \qquad (3.13) $$

then there exists at least one weak solution $u \in W^{1,p}(\Omega)$ to problem (2.1).

**Proof.** We proceed by dividing the proof into three steps. The main idea of the proof is to decompose $u_{\varepsilon}$ as $T_m(u_{\varepsilon}) + (u_{\varepsilon} - T_m(u_{\varepsilon}))$. In Step 1 we derive estimate for $u_{\varepsilon} - T_m(u_{\varepsilon})$ and we show, for well chosen $m = m_{\varepsilon}$, that $u_{\varepsilon} - T_m(u_{\varepsilon})$ is bounded in $W^{1,p}(\Omega)$. Using suitable test functions, Step 2 is devoted to prove that $m_{\varepsilon}$ is bounded and then in Step 3 we show that $T_{m_{\varepsilon}}(u_{\varepsilon})$ is bounded in $W^{1,p}(\Omega)$. This
allows to conclude that $u_{\varepsilon}$ is bounded in $W^{1,p}(\Omega)$. Finally we pass to the limit in the approximate problem (3.1) as in the proof of Theorem 3.1.

Step 1. A priori estimates of $G_m(u_\varepsilon)$ in $W^{1,p}(\Omega)$.

Let us consider $G_m(u_\varepsilon)$ as test function in (3.2) and $\beta = \min\{\alpha, \lambda\}$ defined in (3.6). By ellipticity condition (2.2) and growth conditions on $\Phi$ and $H$, (2.6) and (2.7) respectively, we get:

$$ 
\begin{aligned}
& \beta \int_{\Omega} |u_{\varepsilon}|^{p-2} u_{\varepsilon} G_m(u_{\varepsilon}) dx + \beta \int_{\Omega} |\nabla G_m(u_{\varepsilon})|^p dx \\
& \leq \int_{\Omega} c(x)(1 + |u_{\varepsilon}|^{p-1}) |\nabla G_m(u_{\varepsilon})| dx + \int_{\Omega} b(x)(1 + |\nabla u_{\varepsilon}|^{p-1}) |G_m(u_{\varepsilon})| dx \\
& \qquad + \int_{\Omega} |f| |G_m(u_{\varepsilon})| dx \\
& = J_1 + J_2 + J_3.
\end{aligned} 
\quad (3.14) $$

Let us evaluate $J_1$. By Hölder inequality, Sobolev imbedding theorem and Young inequality, we get:

$$ 
\begin{aligned}
J_1 &= \int_{\Omega} c(x)(1 + |u_{\varepsilon}|^{p-1}) |\nabla G_m(u_{\varepsilon})| dx \\
&\leq \int_{\Omega} c(x) |\nabla G_m(u_{\varepsilon})| dx + \int_{\Omega} c(x) |u_{\varepsilon}|^{p-1} |\nabla G_m(u_{\varepsilon})| dx \\
&\leq |\Omega|^{\frac{(N-p)(p-1)}{Np}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)} \\
&\quad + 2^{p-1} \int_{\{|u_{\varepsilon}|>m\}} c(x) |G_m(u_{\varepsilon})|^{p-1} |\nabla G_m(u_{\varepsilon})| dx \\
&\quad + m^{p-1} 2^{p-1} \int_{\{|u_{\varepsilon}|>m\}} c(x) |\nabla G_m(u_{\varepsilon})| dx \\
&\leq |\Omega|^{\frac{(N-p)(p-1)}{Np}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)} \\
&\quad + 2^{p-1} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|G_m(u_{\varepsilon})\|_{L^{p*}(\Omega)}^{p-1} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)} \\
&\quad + m^{p-1} 2^{p-1} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} |\Omega|^{\frac{p-1}{p*}} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)} \\
&\leq \frac{1}{p'} |\Omega|^{\frac{N-p}{N}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} + \frac{1}{p} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)}^{p} \\
&\quad + 2^{p-1} S \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|G_m(u_{\varepsilon})\|_{W^{1,p}(\Omega)}^{p} \\
&\quad + \frac{m^{p} 2^{p}}{p'} \|\Omega|^{\frac{p}{p*}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} + \frac{1}{p} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)}^{p} \\
&\leq \frac{1}{p'} |\Omega|^{\frac{N-p}{N}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} + \left( \frac{2}{p} + 2^{p-1} S \right) \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} \|G_m(u_{\varepsilon})\|_{W^{1,p}(\Omega)}^{p} \\
&\quad + \frac{m^{p} 2^{p}}{p'} \|\Omega|^{\frac{p}{p*}} \|c\|_{L^{p-1}(\Omega)}^{\frac{N}{N-p}} .
\end{aligned} 
$$

Now we evaluate $J_2$. As in [23] (see also [25]), we define the following set $Z_{\varepsilon}$. As $|\Omega|$ is finite, the set of the constants $k$ such that $|\{x \in \Omega, |u_{\varepsilon}(x)| = k\}| > 0$ is at most countable. Let $Z_{\varepsilon}^c$ be the (countable) union of all those sets. Its complementary $Z_{\varepsilon} = \Omega \setminus Z_{\varepsilon}^c$ is therefore the union of the sets such that $|\{x \in \Omega, |u_{\varepsilon}(x)| = k\}| = 0$. Since for every $k$

$$ \nabla u_{\varepsilon} = 0 \quad \text{a.e. on } \{x \in \Omega, |u_{\varepsilon}(x)| = k\}, $$

and since $Z_{\varepsilon}^c$ is at most a countable union, we obtain that

$$ \nabla u_{\varepsilon} = 0 \quad \text{a.e. on } Z_{\varepsilon}^c. \quad (3.15) $$

By Hölder inequality, Sobolev imbedding theorem and Young inequality, we get:

$$ 
\begin{aligned}
J_2 &= \int_{\Omega} b(x)(1 + |\nabla u_{\varepsilon}|^{p-1}) |G_m(u_{\varepsilon})| dx \\
&= \int_{\{|u_{\varepsilon}|>m\}} b(x) |G_m(u_{\varepsilon})| dx + \int_{\{|u_{\varepsilon}|>m\} \cap Z_{\varepsilon}} b(x) |\nabla u_{\varepsilon}|^{p-1} |G_m(u_{\varepsilon})| dx \\
&\leq |\Omega|^{\frac{p-1}{p}} \|b\|_{L^N(\Omega)} \|G_m(u_{\varepsilon})\|_{W^{1,p}}^{p*} + \|b\|_{L^N(\{|u_{\varepsilon}|>m\} \cap Z_{\varepsilon})} \|G_m(u_{\varepsilon})\|_{W^{1,p}}^{p*} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)}^{p-1} \\
&\leq S |\Omega|^{\frac{p-1}{p}} \|b\|_{L^N(\Omega)} \|G_m(u_{\varepsilon})\|_{W^{1,p}} + S \|b\|_{L^N(\{|u_{\varepsilon}|>m\} \cap Z_{\varepsilon})} \|G_m(u_{\varepsilon})\|_{W^{1,p}(\Omega)}^{p} \\
&\leq \frac{S^{p'}}{p'} \|\Omega\| \|b\|_{L^N(\Omega)} + \left( \frac{1}{p} + S \right) \|b\|_{L^N(\{|u_{\varepsilon}|>m\} \cap Z_{\varepsilon})} \|G_m(u_{\varepsilon})\|_{W^{1,p}(\Omega)}^{p}.
\end{aligned} 
\quad (3.16) $$
Finally we evaluate $J_3$. By Hölder inequality, Sobolev imbedding theorem and Young inequality, we get

$$ 
\begin{aligned}
J_3 &= \int_{\Omega} |f| |G_m(u_\epsilon)| dx \\
&\le \|f\|_{L^{(p^*)'}(\Omega)} \|G_m(u_\epsilon)\|_{p^*} \le S \|f\|_{L^{(p^*)'}(\Omega)} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)} \\
&\le \frac{S^{p'}}{p' \beta^{p'}} \|f\|_{L^{(p^*)'}(\Omega)}^{p'} + \frac{\beta}{p} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)}^{p'}.
\end{aligned} 
\quad (3.17) $$

Combining (3.14)–(3.17) we obtain

$$ 
\begin{aligned}
& \frac{\beta}{p'} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)}^p \\
& \le \frac{1}{p'} |\Omega|^{\frac{N-p}{N}} \|c\|_{L^{p-1}(\Omega)} + \left(\frac{2}{p} + 2^{p-1} S\right) \|c\|_{L^{p-1}(\Omega)} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)}^p \\
& \quad + \frac{2^p}{p'} \|c\|_{L^{p-1}(\Omega)} |\Omega|^{\frac{p}{p^*}} m^p \\
& \quad + \frac{S^{p'}}{p'} |\Omega| \|b\|_{L^N(\{|u_\epsilon|>m|\cap Z_\epsilon\})} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)}^p \\
& \quad + \frac{S^{p'}}{p' \beta^{p'}} \|f\|_{L^{(p^*)'}(\Omega)}^{p'}.
\end{aligned} 
\quad (3.18) $$

By equi-integrability of $b$, for $\eta > 0$ (independent on $\epsilon > 0$) we can choose $m = m_\epsilon$ such that

$$ |\{|u_\epsilon| > m_\epsilon\} \cap Z_\epsilon| = \eta $$

and

$$ \left( \int_{\{|u_\epsilon| > m_\epsilon\} \cap Z_\epsilon} |b|^N dx \right)^{\frac{1}{N}} \le \frac{1}{4} \frac{\beta}{p'} \frac{p}{Sp + 1}. $$

Therefore if $c$ verifies the following smallness assumption

$$ \|c\|_{L^{p-1}(\Omega)} \le \frac{1}{4} \frac{\beta}{p'} \frac{p}{2^{p-1} Sp + 2} $$

we deduce that, for $m = m_\epsilon$, we get

$$ 
\begin{aligned}
& \frac{\beta}{2p'} \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)}^p \\
& \le \frac{1}{p'} |\Omega|^{\frac{N-p}{N}} \|c\|_{L^{p-1}(\Omega)} + \frac{2^p}{p'} \|c\|_{L^{p-1}(\Omega)} |\Omega|^{\frac{p}{p^*}} m^p \\
& \quad + \frac{S^{p'}}{p'} |\Omega| \|b\|_{L^N(\{|u_\epsilon|>m|\cap Z_\epsilon\})} + \frac{S^{p'}}{p' \beta^{p'}} \|f\|_{L^{(p^*)'}(\Omega)}^{p'}.
\end{aligned} 
\quad (3.19) $$

**Step 2.** $m_\epsilon$ is bounded independently of $\epsilon$ and uniform estimate on $G_m(u_\epsilon)$.

Let us begin this step by proving that $m = m_\epsilon$ is bounded. To this aim we consider $|T_m(u_\epsilon)|^{\theta-1} T_m(u_\epsilon)$ as test function in (3.1), with $\theta \ge 1$ to be chosen sufficiently large.

By ellipticity condition (2.2) and growth conditions on $\Phi$ and $H$, (2.6) and (2.7) respectively, we get

$$ 
\begin{aligned}
& \beta \int_{\Omega} |u_\epsilon|^{p-1} |T_m(u_\epsilon)|^{\theta} dx + \beta \theta \int_{\Omega} |T_m(u_\epsilon)|^{\theta-1} |\nabla T_m(u_\epsilon)|^p dx \\
& \le \theta \int_{\Omega} c(x)(1 + |u_\epsilon|^{p-1}) |T_m(u_\epsilon)|^{\theta-1} |\nabla T_m(u_\epsilon)| dx \\
& \quad + \int_{\Omega} b(x)(1 + |\nabla u_\epsilon|^{p-1}) |T_m(u_\epsilon)|^{\theta} dx + \int_{\Omega} |f| |T_m(u_\epsilon)|^{\theta} dx \\
& = H_1 + H_2 + H_3.
\end{aligned} 
\quad (3.20) $$

Let us evaluate $H_1$. By Young inequality

$$ 
\begin{aligned}
& H_1 = \theta \int_{\Omega} |c(x)| |T_m(u_\epsilon)|^{\theta-1} |\nabla T_m(u_\epsilon)| dx \\
& \quad + \theta \int_{\Omega} |c(x)| |T_m(u_\epsilon)|^{p-1} |T_m(u_\epsilon)|^{\theta-1} |\nabla T_m(u_\epsilon)| dx \\
& \le \frac{\beta \theta}{4p} \int_{\Omega} |T_m(u_\epsilon)|^{\theta-1} |\nabla T_m(u_\epsilon)|^p dx \\
& \quad + \left(\frac{4}{\beta}\right)^{p'/p} \frac{\theta}{p'} \int_{\Omega} |c(x)|^p |T_m(u_\epsilon)|^{\theta-1} dx
\end{aligned} 
\quad (3.21) $$
$$
\begin{aligned}
&+ \frac{\beta\theta}{4p} \int_{\Omega} |T_m(u_{\varepsilon})|^{\theta-1} |\nabla T_m(u_{\varepsilon})|^p dx \\
&+ \left(\frac{4}{\beta}\right)^{p'/p} \frac{\theta}{p'} \int_{\Omega} |c(x)|^{p'} |T_m(u_{\varepsilon})|^{\theta-1+p} dx \\
\leq & \frac{\beta\theta}{2p} \int_{\Omega} |T_m(u_{\varepsilon})|^{\theta-1} |\nabla T_m(u_{\varepsilon})|^p dx \\
&+ \left(\frac{4}{\beta}\right)^{p'/p} \frac{\theta}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} (m^{\theta-1} + m^{\theta-1+p}).
\end{aligned}
$$

Let us evaluate $H_2$. By Young and Hölder inequalities and estimates (3.19), since $\nabla u_{\varepsilon} = \nabla T_m(u_{\varepsilon}) + \nabla G_m(u_{\varepsilon})$, we have

$$
\begin{aligned}
H_2 &= \int_{\Omega} b(x)(1 + |\nabla u_{\varepsilon}|^{p-1}) |T_m(u_{\varepsilon})|^{\theta} dx \\
&= \int_{\Omega} b(x)|T_m(u_{\varepsilon})|^{\theta} dx + \int_{\Omega} b(x)|\nabla T_m(u_{\varepsilon})|^{p-1} |T_m(u_{\varepsilon})|^{\theta} dx \\
&\quad + \int_{\Omega} b(x)|\nabla G_m(u_{\varepsilon})|^{p-1} |T_m(u_{\varepsilon})|^{\theta} dx \\
&\leq m^{\theta} |\Omega|^{1-\frac{1}{N}} \|b\|_{L^N(\Omega)} + \frac{\beta\theta}{4p'} \int_{\Omega} |\nabla T_m(u_{\varepsilon})|^p |T_m(u_{\varepsilon})|^{\theta-1} dx \\
&\quad + \left(\frac{4}{\beta\theta}\right)^{p/p'} \frac{1}{p} \int_{\Omega} |b|^p |T_m(u_{\varepsilon})|^{\theta+p-1} dx \\
&\quad + m^{\theta} |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} \|\nabla G_m(u_{\varepsilon})\|_{L^p(\Omega)}^{p-1} \\
&\leq m^{\theta} |\Omega|^{1-\frac{1}{N}} \|b\|_{L^N} + \frac{\beta\theta}{4p'} \int_{\Omega} |\nabla T_m(u_{\varepsilon})|^p |T_m(u_{\varepsilon})|^{\theta-1} dx \\
&\quad + \left(\frac{4}{\beta\theta}\right)^{p/p'} \frac{1}{p} |\Omega|^{\frac{N-p}{N}} \|b\|_{L^N(\Omega)}^p m^{\theta+p-1} \\
&\quad + m^{\theta} |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} C(\Omega, p, N, \lambda, \alpha) \\
&\quad \times \left( \|c\|_{L^{\frac{N}{p-1}}(\Omega)}^{1/p'} + \|b\|_{L^N(\Omega)}^{1/p'} + \|f\|_{L^{(p*)'}(\Omega)} + \|c\|_{L^{\frac{N}{p-1}}(\Omega)}^{1/p'} m^{p-1} \right).
\end{aligned}
\quad (3.22)
$$

Now we evaluate $H_3$. By the definition of $T_m$ we have

$$H_3 = \int_{\Omega} |f| |T_m(u_{\varepsilon})|^\theta dx \leq m^\theta |\Omega|^{\frac{N-p}{Np}} \|f\|_{L^{(p*)'}(\Omega)} \quad (3.23)$$

Combining (3.20)–(3.23), we get

$$
\begin{aligned}
& \beta \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_m(u_{\varepsilon})|^{\theta} dx + \beta \theta \left(1 - \frac{1}{2p} - \frac{1}{4p'}\right) \int_{\Omega} |T_m(u_{\varepsilon})|^{\theta-1} |\nabla T_m(u_{\varepsilon})|^p dx \\
& \leq \left(\frac{4}{\beta}\right)^{p/p'} \frac{\theta}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} (m^{\theta-1} + m^{\theta-1+p}) + \left(\frac{4}{\beta\theta}\right)^{p/p'} \frac{|\Omega|^{\frac{N-p}{N}}}{p} \|b\|_{L^N(\Omega)}^p m^{\theta+p-1} \\
& \quad + |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} C(\Omega, p, N, \lambda, \alpha) \left( \|c\|_{L^{\frac{N}{p-1}}(\Omega)}^{1/p'} + \|f\|_{L^{(p*)'}(\Omega)} \right) m^{\theta} \\
& \quad + |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} C(\Omega, p, N, \lambda, \alpha) \|c\|_{L^{\frac{N}{p-1}}(\Omega)}^{1/p'} m^{p-1+\theta} + |\Omega|^{\frac{N-p}{Np}} \|f\|_{L^{(p*)'}(\Omega)} m^{\theta} \\
& \quad + m^{\theta} |\Omega|^{1-\frac{1}{N}} \|b\|_{L^N(\Omega)} + m^{\theta} |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} C(\Omega, p, N, \lambda, \alpha) \|b\|_{L^N(\Omega)}^{1/p'}.
\end{aligned}
\quad (3.24)
$$

Since $|u_{\varepsilon}|^{p-1} \geq |T_m(u_{\varepsilon})|^{p-1} = m$ on $\{|u_{\varepsilon}| > m\}$ and $(1 - \frac{1}{2p} - \frac{1}{4p'}) > 0$, by recalling that $|\{|u_{\varepsilon}| > m\}| = \eta$ we deduce from (3.24) that

$$
\begin{aligned}
\beta m^{p-1+\theta} \eta &\leq \left(\frac{4}{\beta}\right)^{p/p'} \frac{\theta}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} m^{\theta-1} \\
&\quad + \left[ |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N(\Omega)} C(\Omega, p, N, \lambda, \alpha) \left( \|c\|_{L^{\frac{N}{p-1}}(\Omega)}^{1/p'} + \|f\|_{L^{(p*)'}(\Omega)} \right) + |\Omega|^{\frac{N-p}{Np}} \|f\|_{L^{(p*)'}(\Omega)} \right] m^{\theta} \\
&\quad + \left[ \left(\frac{4}{\beta}\right)^{p/p'} \frac{\theta}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} + \left(\frac{4}{\beta\theta}\right)^{p/p'} \frac{|\Omega|^{\frac{N-p}{N}}}{p} \|b\|_{L^N(\Omega)}^{p'} \right] m^{\theta}.
\end{aligned}
\quad (3.25)
$$
$$ + C(\Omega, p, N, \lambda, \alpha) |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N} \|c\|^{\frac{1}{Np}} \left[ \frac{1}{L^{\frac{1}{p-1}}} \right] m^{\theta+p-1}. $$

Let us choose $\theta \ge 1$ in such a way that

$$ \left(\frac{4}{\beta\theta}\right)^{p/p'} \frac{|\Omega|^{\frac{N-p}{N}}}{p} \|b\|_{L^N}^p < \frac{\eta}{2}\beta, $$

that is

$$ \theta > \frac{4}{\beta} \left( \frac{2 \|b\|_{L^N}^p |\Omega|^{\frac{N-p}{N}}}{\eta \beta p} \right)^{\frac{p'}{p}}. \qquad (3.26) $$

Since $\theta - 1 < \theta + p - 1$ and $\theta < \theta + p - 1$, the choice of $\theta$, the following smallness condition on $c$

$$ \frac{\beta\eta}{2} - \left[ \left( \frac{4}{\beta\theta} \right)^{p/p'} \frac{\theta}{p'} \|c\|_{L^{\frac{N}{p-1}}}^{p'} + C(\Omega, p, N, \lambda, \alpha) |\Omega|^{\frac{N-p}{Np}} \|b\|_{L^N} \|c\|^{\frac{1}{Np}} \right] > 0 $$

and (3.25) give the boundedness of $m$ by a constant $C_m$ which depends on $\lambda, \alpha, \eta, \theta, \Omega, p, N, b, \|c\|_{L^{\frac{N}{p-1}}(\Omega)}$ and $\|f\|_{L^{(p')}(\Omega)}$.

Therefore by (3.19) we deduce the a priori estimates for $G_m(u_\epsilon)$ in $W^{1,p}(\Omega)$.

$$ \|G_m(u_\epsilon)\|_{W^{1,p}(\Omega)} \le C'_0, \qquad (3.27) $$

where $C'_0$ is a positive constant depending only on the data, the smallness condition on $c$ but not depending on $\epsilon$.

**Step 3.** A priori estimates of $T_m(u_\epsilon)$ in $W^{1,p}(\Omega)$.

Let us consider $T_m(u_\epsilon)$ as test function in the approximated problem (3.1). By ellipticity condition (2.2) and growth conditions on $\Phi$ and on $H$, (2.6) and (2.7) respectively, Step 1 and Step 2, we obtain

$$ \begin{aligned} & \beta \int_{\Omega} (|u_{\epsilon}|^{p-2} u_{\epsilon}) T_m(u_{\epsilon}) dx + \beta \int_{\Omega} |\nabla T_m(u_{\epsilon})|^p dx \\ & \le \int_{\Omega} c(x)(1 + |T_m(u_{\epsilon})|^{p-1}) |\nabla T_m(u_{\epsilon})| dx + \int_{\Omega} b(x)(1 + |\nabla u_{\epsilon}|^{p-1}) |T_m(u_{\epsilon})| dx \\ & \qquad + \int_{\Omega} |f| |T_m(u_{\epsilon})| dx \\ & \le (1 + m^{p-1}) \|c\|_{L^p'(\Omega)} \|\nabla T_m(u_{\epsilon})\|_{L^p(\Omega)} \\ & \qquad + m \int_{\Omega} b(x)(1 + |\nabla G_m(u_{\epsilon})|^{p-1} + |\nabla T_m(u_{\epsilon})|^{p-1}) dx + m \|f\|_{L^1(\Omega)} \\ & \le (1 + m^{p-1}) |\Omega|^{\frac{N-p}{N}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|\nabla T_m(u_{\epsilon})\|_{L^p(\Omega)} \\ & \qquad + m (\|b\|_{L^1(\Omega)} + \|b\|_{L^p(\Omega)} \|\nabla G_m(u_{\epsilon})\|_{L^p(\Omega)}^{p-1} + \|b\|_{L^p} \|\nabla T_m(u_{\epsilon})\|_{L^p(\Omega)}^{p-1}) + m \|f\|_{L^1(\Omega)} \\ & \le \frac{(1 + m^{p-1})'}{\beta^{p/p'} p'} |\Omega|^{\frac{N-p}{p}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} + \frac{\beta}{p} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|\nabla T_m(u_{\epsilon})\|_{L^p(\Omega)}^p \\ & \qquad + m \|b\|_{L^1(\Omega)} + m \|b\|_{L^p(\Omega)} (C'_0)^{p-1} \\ & \qquad + \frac{1}{p \beta^{p/p'}} \|b\|_{L^p(\Omega)}^{p'} + \frac{\beta}{p'} \|\nabla T_m(u_{\epsilon})\|_{L^p(\Omega)}^p + m \|f\|_{L^1(\Omega)}. \end{aligned} \qquad (3.28) $$

By smallness assumption on $\|c\|_{L^{\frac{N}{p}}}$ we deduce that

$$ \|T_m(u_\epsilon)\|_{W^{1,p}(\Omega)} \le C'_1 \qquad (3.29) $$

where $C'_1$ is a positive constant only on $\lambda, \alpha, \Omega, p, N, m, b, \|c\|_{L^{\frac{N}{p-1}}(\Omega)}$, $\|f\|_{L^{(p')}(\Omega)}$ but not on $\epsilon$. Since $u_\epsilon = T_m(u_\epsilon) + G_m(u_\epsilon)$, by (3.27)

and (3.29), we deduce the a priori estimate

$$ \|u_\epsilon\|_{W^{1,p}(\Omega)} \le C_2 $$

where $C_2$ is a positive constant depending only on $\lambda, \alpha, \Omega, p, N, b, \|c\|_{L^{\frac{N}{p-1}}(\Omega)}$ and $\|f\|_{L^{(p')}(\Omega)}$. The proof proceeds by passing to the limit as in Step 3 of previous theorem. $\square$

### 4. Existence result for renormalized solutions

In the present section we consider a data $f$ in $L^1$. As explain in Introduction, it requires a specific notion of solution. Here we use the framework of renormalized solution. In the spirit of [6,7,16,23] and since we consider here elliptic equation with $L^1$ data and Neumann boundary conditions we now give the definition of a renormalized solution to (2.1).
**Definition 4.1.** A real function $u$ defined in $\Omega$ is a renormalized solution to (2.1) if

$$ u \text{ is measurable and finite almost everywhere in } \Omega, \qquad (4.1) $$

$$ T_k(u) \in W^{1,p}(\Omega), \text{ for any } k > 0, \qquad (4.2) $$

$$ \lim_{n \to +\infty} \frac{1}{n} \left( \int_{\Omega} |T_n(u)|^p dx + \int_{\{x \in \Omega: |u(x)| < n\}} \mathbf{a}(x, u, \nabla u) \nabla u dx \right) = 0 \qquad (4.3) $$

and if for every function $h$ belonging to $W^{1,\infty}(\mathbb{R})$ with compact support and for every $\varphi \in L^{\infty}(\Omega) \cap W^{1,p}(\Omega)$ we have

$$ \begin{aligned} & \lambda \int_{\Omega} h(u) |u|^{p-2} u \varphi dx + \int_{\Omega} h(u) \mathbf{a}(x, u, \nabla u) \nabla \varphi dx \\ & + \int_{\Omega} h'(u) \mathbf{a}(x, u, \nabla u) \nabla u \varphi dx + \int_{\Omega} h(u) \Phi(x, u) \nabla \varphi dx \\ & + \int_{\Omega} h'(u) \Phi(x, u) \nabla u \varphi dx + \int_{\Omega} H(x, \nabla u) \varphi h(u) dx = \int_{\Omega} f \varphi h(u) dx. \end{aligned} \qquad (4.4) $$

**Remark 4.2.** A renormalized solution is not an $L_{\text{loc}}^1(\Omega)$-function and therefore it has not a distributional gradient. Condition (4.2) allows to define a generalized gradient of $u$ according to Lemma 2.1 of [3], which asserts the existence of a unique measurable function $\nu$ defined in $\Omega$ such that $\nabla T_k(u) = \chi_{|u|<k} \nu$ a.e. in $\Omega$, $\forall k > 0$. This function $\nu$ is the generalized gradient of $u$ and it is denoted by $\nabla u$.

Equality (4.5) is formally obtained by using in (2.1) the test function $\varphi h(u)$ and by taking into account Neumann boundary conditions. Actually in a standard way one can check that every term in (4.5) is well-defined under the structural assumptions on the elliptic operator and on the function $\Phi$ and $H$.

**Remark 4.3.** As usual condition (4.3) is called the decay of the truncated energy and is crucial to obtain stability or uniqueness results. Compared to elliptic equations with Dirichlet boundary conditions which is in general

$$ \lim_{n \to +\infty} \frac{1}{n} \int_{\{x \in \Omega: |u(x)| < n\}} \mathbf{a}(x, u, \nabla u) \nabla u dx = 0, $$

condition (4.3) is more complete and is natural due to the lack of Poincaré inequality in the case of Neumann boundary conditions. Using the ellipticity condition it allows us to get

$$ \lim_{n \to +\infty} \frac{1}{n} \|T_n(u)\|_{W^{1,p}(\Omega)}^p = 0 $$

and then we have, for any $k > 0$,

$$ \|T_k(u)\|_{W^{1,p}(\Omega)}^p \leq M(k+1), $$

where $M > 0$ does not depend on $k$. As a consequence the Boccardo-Gallouët estimates (see [3,29] and [23, Lemma A.1]) yield

$$ |u|^{p-1} \in L^{p^*/p, \infty}(\Omega), \quad |\nabla u|^{p-1} \in L^{N^*/\infty}(\Omega). \qquad (4.5) $$

As in the variational case the strategy is to pass to the limit in the approximate problems. More precisely we denote by $u_\epsilon$ a weak solution belonging to $W^{1,p}(\Omega)$ of the following problem

$$ \left\{ \begin{array}{ll} \lambda |u_\epsilon|^{p-2} u - \text{div} \left( \mathbf{a}_\epsilon(x, u_\epsilon, \nabla u_\epsilon) + \Phi_\epsilon(x, u_\epsilon) \right) + H_\epsilon(x, \nabla u_\epsilon) = f_\epsilon & \text{in } \Omega, \\ (\mathbf{a}_\epsilon(x, u_\epsilon, \nabla u_\epsilon) + \Phi_\epsilon(x, u_\epsilon)) \cdot \underline{n} = 0 & \text{on } \partial\Omega, \end{array} \right. \qquad (4.6) $$

where for $\epsilon > 0$, we denote

$$ \mathbf{a}_{\epsilon}(x, s, \xi) = \mathbf{a}(x, T_{\frac{1}{\epsilon}}(s), \xi) $$

$$ \Phi_{\epsilon}(x, s) = T_{\frac{1}{\epsilon}}(\Phi(x, s)), $$

$$ H_{\epsilon}(x, \xi) = T_{\frac{1}{\epsilon}}(H(x, \xi)), $$

and $f_\epsilon \in L^{p'}(\Omega)$ such that

$$ f_{\epsilon} \to f \quad \text{strongly in } L^1(\Omega), $$

$$ \|f_{\epsilon}\|_{L^1(\Omega)} \leq \|f\|_{L^1(\Omega)}, \quad \forall \epsilon > 0. $$

Therefore $u_\epsilon$ satisfies the following equality:

$$ \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} \varphi dx + \int_{\Omega} \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla \varphi dx + \int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla \varphi dx \qquad (4.7) $$
$$+ \int_{\Omega} H_{\varepsilon}(x, \nabla u_{\varepsilon}) \varphi dx = \int_{\Omega} f_{\varepsilon} \varphi dx,$$

for every $\varphi \in W^{1,p}(\Omega)$.

As explain in Section 3, classical results give the existence of at least one weak solution $u_{\varepsilon}$. We explicitly remark that this weak solution is also a renormalized solution.

As in the variational framework of Section 3 we distinguish two cases, the case where the norm of $b$ is small and the case where the norm of $c$ is small. However due to the lack of regularity of $u$ and since we expect to have $H(x, \nabla u) \in L^1(\Omega)$, we assume a more restrictive condition on the summability of $b$. Instead of $b \in L^N(\Omega)$, we now assume that

$$b \in L^{N,1}(\Omega).$$

Indeed under such an assumption on $b$, the growth condition (2.7) and the Boccardo-Gallouët estimates (4.5) imply that $H(x, \nabla u) \in L^1(\Omega)$.

### 4.1. Case norm of b small

The main result of this section is the following:

**Theorem 4.4.** Under the assumptions (2.2), (2.3), (2.5), (2.6) and (2.7) with

$$b \in L^{N,1}(\Omega), \quad f \in L^1(\Omega),$$

there exists a positive constant $C_3 = C_3(\lambda, \alpha, \Omega, p, f, c)$ depending only on the data but not on $b$ such that if the following smallness assumption holds true:

$$\|b\|_{L^{N,1}(\Omega)} \le C_3, \quad (4.8)$$

then there exists at least one renormalized solution $u$ to problem (2.1).

**Proof.** We proceed by dividing the proof into three steps. In Step 1 we derive some a priori estimates on $u_{\varepsilon}$ which depend on $\varepsilon$. Step 2 is devoted to prove a uniform estimate on $\nabla u_{\varepsilon}$ in $L^{N',\infty}(\Omega)$ with the help of J.-L. Lions Lemma, Lemma 2.1. Finally in Step 3 we pass into the limit in the approximate problems by using the stability result for renormalized solutions given by Proposition A.1 in Appendix below.

**Step 1.** A priori estimates of $|u_{\varepsilon}|^{p-1}$ in $L^1(\Omega)$.

Let us denote

$$M_{\varepsilon} = \int_{\Omega} b(x)(1 + |\nabla u_{\varepsilon}|^{p-1})dx + \|f_{\varepsilon}\|_{L^1(\Omega)} \quad (4.9)$$

Then arguing as in Step 1 of Theorem 3.1, we consider $\varphi = T_1(u_{\varepsilon})$ as test function in (4.7) and we use ellipticity condition (2.2), growth conditions on $\Phi$, $H$, (2.6) and (2.7) respectively. Moreover since $|T_1(u_{\varepsilon})| \le 1$ a.e. in $\Omega$, we get

$$\int_{\Omega} |u_{\varepsilon}|^{p-1} dx \le C(\Omega, p, \lambda, \alpha)(1 + \|c\|_{L^{p'}(\Omega)}^{p'}) + \frac{M_{\varepsilon}}{\lambda}. \quad (4.10)$$

**Step 2.** $|\nabla u_{\varepsilon}|$ is bounded in $L^{N',\infty}(\Omega)$.

Let us consider $T_k(u_{\varepsilon})$ as test function in (4.7). We get

$$\lambda \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_k(u_{\varepsilon})| dx + \int_{\Omega} \mathbf{a}_{\varepsilon}(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \nabla T_k(u_{\varepsilon}) dx + \int_{\Omega} \Phi_{\varepsilon}(x, u_{\varepsilon}) \nabla T_k(u_{\varepsilon}) dx \\ + \int_{\Omega} H_{\varepsilon}(x, \nabla u_{\varepsilon}) T_k(u_{\varepsilon}) dx = \int_{\Omega} f_{\varepsilon} T_k(u_{\varepsilon}) dx.$$

By (2.2), (2.6) and (2.7) we get:

$$\lambda \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_k(u_{\varepsilon})| dx + \alpha \int_{\Omega} |\nabla T_k(u_{\varepsilon})|^p dx \\ \le \int_{\Omega} c(x)(1 + |T_k(u_{\varepsilon})|^{p-1}) |\nabla T_k(u_{\varepsilon})| dx + k M_{\varepsilon} \\ = \int_{\Omega} c(x) |\nabla T_k(u_{\varepsilon})| dx + \int_{\Omega} c(x) |T_k(u_{\varepsilon})|^{p-1} |\nabla T_k(u_{\varepsilon})| dx + k M_{\varepsilon}.$$

By Young inequality we get

$$\lambda \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_k(u_{\varepsilon})| dx + \alpha \int_{\Omega} |\nabla T_k(u_{\varepsilon})|^p dx \\ \le \frac{\alpha}{2p} \int_{\Omega} |\nabla T_k(u_{\varepsilon})|^p dx + \left(\frac{2}{\alpha}\right)^{p/p} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} + \frac{\alpha}{2p} \int_{\Omega} |\nabla T_k(u_{\varepsilon})|^p dx$$
$$ 
\begin{aligned}
& + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{1}{p'} \int_{\Omega} c(x)^{p'} |T_k(u_{\epsilon})|^p dx + k M_{\epsilon} \\
\leq & \frac{\alpha}{p} \int_{\Omega} |\nabla T_k(u_{\epsilon})|^p dx + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} \\
& + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{1}{p'} \int_{\Omega} c(x)^{p'} |T_k(u_{\epsilon})|^p dx + k M_{\epsilon}.
\end{aligned}
$$

By Lemma 2.1 we deduce

$$ 
\begin{aligned}
& \lambda \int_{\Omega} |u_{\epsilon}|^{p-1} |T_k(u_{\epsilon})| dx + \alpha \int_{\Omega} |\nabla T_k(u_{\epsilon})|^p dx \\
\leq & \frac{\alpha}{p} \int_{\Omega} |\nabla T_k(u_{\epsilon})|^p dx + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} + k M_{\epsilon} \\
& + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{2^{p-1}}{p'} \left[ \eta^p \|\nabla T_k(u_{\epsilon})\|_{L^{p}(\Omega)}^{p} + C(c, \eta) \left( \int_{\Omega} |T_k(u_{\epsilon})|^{p-1} dx \right)^{\frac{p'}{p}} \right].
\end{aligned}
$$

Therefore by choosing $\eta > 0$ such that

$$ \alpha - \frac{\alpha}{p} - \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{2^{p-1}}{p'} \eta^p > 0, $$

we deduce the following estimates

$$ 
\begin{aligned}
\|T_k(u_{\epsilon})\|_{W^{1,p}(\Omega)}^p \leq & C(\alpha, p, \lambda) \left[ \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} \right. \\
& \left. + \left(\frac{2}{\alpha}\right)^{\frac{p'}{p}} \frac{2^{p-1}}{p'} C(c, \eta) \left( \int_{\Omega} |u_{\epsilon}|^{p-1} dx \right)^{\frac{p'}{p}} + k M_{\epsilon} \right],
\end{aligned}
$$

for any $k > 0$. By Lemma A.1 in [23] which is a slight generalization of the Boccardo-Gallouët estimates (see [3,29]) in the $W_{0}^{1,p}(\Omega)$ case and which can be easily adapted to the case where $T_k(u_\epsilon)$ belongs to the Sobolev space $W^{1,p}(\Omega)$ we obtain

$$ 
\begin{aligned}
\| |\nabla(u_{\epsilon})|^{p-1} \|_{L^{N',\infty}} \leq & C(N, \alpha, p, \lambda, \eta) |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} \left[ \|c\|_{L^{p'}(\Omega)} + \int_{\Omega} |u_{\epsilon}|^{p-1} dx \right] \\
& + C(N, \alpha, p, \lambda) M_{\epsilon}.
\end{aligned}
$$

By Young inequality and Step 1, we conclude

$$ 
\begin{aligned}
\| |\nabla u_{\epsilon}|^{p-1} \|_{L^{N',\infty}} \leq & C(N, p, \alpha, \lambda, \eta, |\Omega|) \left[ \|c\|_{L^{p'}(\Omega)} + \int_{\Omega} |u_{\epsilon}|^{p-1} dx + M_{\epsilon} \right] \\
\leq & C(N, p, \alpha, \lambda, \eta, |\Omega|) \left[ \|c\|_{L^{p'}(\Omega)} + \|c\|_{L^{p'}(\Omega)}^{p'} + 1 + M_{\epsilon} \right] \\
\leq & C(N, p, \alpha, \lambda, \eta, |\Omega|) \left[ \|c\|_{L^{p'}(\Omega)}^{p'} + 1 + \|b\|_{L^1(\Omega)} \right. \\
& \left. + \int_{\Omega} b(x) |\nabla u_{\epsilon}|^{p-1} dx + \|f_{\epsilon}\|_{L^1(\Omega)} \right].
\end{aligned}
$$

Now by Hölder inequality for Lorentz spaces, we get

$$ 
\begin{aligned}
\| |\nabla u_{\epsilon}|^{p-1} \|_{L^{N',\infty}} \leq & C(N, p, \alpha, \lambda, \eta, |\Omega|) \left[ \|c\|_{L^{p'}(\Omega)}^{p'} + 1 + \|b\|_{L^1(\Omega)} \right. \\
& \left. + \|b\|_{L^{N,1}(\Omega)} \| |\nabla u_{\epsilon}|^{p-1} \|_{L^{N',\infty}} + \|f\|_{L^1(\Omega)} \right].
\end{aligned}
$$

where $C(N, p, \alpha, \lambda, \eta, |\Omega|)$ is a constant depending on $N, p, \alpha, \lambda, \eta$ (and then on the function $c$ via J.-L. Lions Lemma) and $\Omega$ but not on depending $b$ nor $\epsilon$. It follows that if $b$ verifies the following smallness condition

$$ 1 - C(N, p, \alpha, \lambda, |\Omega|) \|b\|_{L^{N,1}(\Omega)} > 0, $$

then we deduce that there exists $C_0 > 0$ (independent of $\epsilon$) such that

$$ \| |\nabla u_{\epsilon}|^{p-1} \|_{L^{N',\infty}} \leq C_0 \left( \|c\|_{L^{p'}(\Omega)}^{p'} + 1 + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} \right) \quad (4.11) $$

and by (4.10)

$$ \int_{\Omega} |u_{\epsilon}|^{p-1} dx \leq C_0 \left( \|c\|_{L^{p'}(\Omega)}^{p'} + 1 + \|b\|_{L^1(\Omega)} + \|f\|_{L^1(\Omega)} \right). \quad (4.12) $$

### Step 3. Passage to the limit in the approximated problem.

To conclude the proof of Theorem 4.4, we use Eq. (A.1) which establishes the stability of the renormalized solutions with respect to the right-hand side in order to pass into the limit in problem (4.6) (cf. Section 5 of [25]; see also [23]).
The solution $u_{\epsilon}$ of (4.6) satisfies

$$ \left\{ \begin{array}{ll} \lambda|u_{\epsilon}|^{p-2}u_{\epsilon} - \operatorname{div}(\mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) + \mathbf{\Phi}_{\epsilon}(x, u_{\epsilon})) = \Psi_{\epsilon} & \text{in } \Omega, \\ (\mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) + \mathbf{\Phi}_{\epsilon}(x, u_{\epsilon})) \cdot \underline{n} = 0 & \text{on } \partial\Omega, \end{array} \right. \quad (4.13) $$

where

$$ \left\{ \begin{array}{l} \Psi_{\epsilon} = f_{\epsilon} - H_{\epsilon}(x, \nabla u_{\epsilon}) \\ \text{is equi-integrable in } L^{1}(\Omega). \end{array} \right. $$

Indeed using the growth condition (2.7) on $H_{\epsilon}$ we have, a.e. $x \in \Omega$,

$$ |H_{\epsilon}(x, \nabla u_{\epsilon})| \le b(x)(1 + |\nabla u_{\epsilon}|^{p-1}). \quad (4.14) $$

Since $b$ belongs to the Lorentz space $L^{N,1}(\Omega)$ and since $\nabla u_{\epsilon}$ is bounded in the Lorentz space $L^{N',\infty}(\Omega)$, we deduce that $H_{\epsilon}(x, \nabla u_{\epsilon})$ is equi-integrable in $L^1(\Omega)$. Recalling that $f_{\epsilon}$ converges strongly to $f$ in $L^1$ we obtain that $\Psi_{\epsilon}$ is equi-integrable in $L^1(\Omega)$ and, up to a subsequence still indexed by $\epsilon$, there exists $\Psi \in L^1(\Omega)$ such that

$$ \Psi_{\epsilon} \to \Psi \text{ weakly in } L^1(\Omega). \quad (4.15) $$

We are now in a position to apply Proposition A.1: up to a subsequence (still indexed by $\epsilon$)

$$ u_{\epsilon} \to u \text{ a.e. in } \Omega, $$

$$ \nabla u_{\epsilon} \to \nabla u \text{ a.e. in } \Omega, $$

$$ T_k(u_{\epsilon}) \to T_k(u) \text{ strongly in } W^{1,p}(\Omega), \forall k > 0, $$

where $u$ is a renormalized solution of

$$ \left\{ \begin{array}{ll} \lambda|u|^{p-2}u - \operatorname{div}(\mathbf{a}(x, u, \nabla u) + \mathbf{\Phi}(x, u)) = \Psi & \text{in } \Omega, \\ (\mathbf{a}(x, u, \nabla u) + \mathbf{\Phi}(x, u)) \cdot \underline{n} = 0 & \text{on } \partial\Omega. \end{array} \right. $$

It remains to identify $\Psi$ with $f - H(x, \nabla u)$ which is a straightforward consequence of the pointwise convergence of $\nabla u_{\epsilon}$. $\square$

## 4.2. Case norm of c small

The main result of this section is the following:

**Theorem 4.5.** Under the assumptions (2.2), (2.3), (2.5), (2.6) and (2.7) with

$$ b \in L^{N,1}(\Omega), \quad f \in L^1(\Omega), $$

there exists a positive constant $C_4 = C_4(\lambda, \alpha, \Omega, p, f, b)$ depending only on the data but not on $c$ such that if the following smallness assumption holds true:

$$ \|c\|_{L^{p-1}(\Omega)} \le C_4, \quad (4.16) $$

then there exists at least one renormalized solution $u$ to problem (2.1).

**Proof.** We proceed by dividing the proof into three steps. In Step 1 we derive some a priori estimates on $|\nabla G_m(u_{\epsilon})|^{p-1}$ in $L^{N',\infty}(\Omega)$ for well chosen $m = m_1$. Step 2 is devoted to prove that $m_{\epsilon}$ is bounded. This allows to prove that $T_{m_1}(u_{\epsilon})$ is bounded in $W^{1,p}(\Omega)$. Therefore we deduce the a priori estimates of $|\nabla u_{\epsilon}|^{p-1}$ in $L^{N',\infty}(\Omega)$. Finally in Step 3 we pass to the limit in the approximate problem (3.1) as in the proof of Theorem 4.4.

**Step 1.** Estimates of $|\nabla G_m(u_{\epsilon})|^{p-1}$ in $L^{N',\infty}(\Omega)$.

Let us consider $T_k(G_m(u_{\epsilon}))$ as test function in (4.7). By ellipticity condition (2.2) and growth conditions on $\mathbf{\Phi}$ and $H$, (2.6) and (2.7) respectively, we get:

$$ \beta \int_{\Omega} (|u_{\epsilon}|^{p-2} u_{\epsilon}) T_k(G_m(u_{\epsilon})) dx + \beta \int_{\Omega} |\nabla T_k(G_m(u_{\epsilon}))|^p dx \quad (4.17) $$

$$ \le \int_{\Omega} c(x)(1 + |u_{\epsilon}|^{p-1}) |\nabla T_k(G_m(u_{\epsilon}))| dx + \int_{\Omega} b(x)(1 + |\nabla u_{\epsilon}|^{p-1}) |T_k(G_m(u_{\epsilon}))| dx \\ + \int_{\Omega} |f_{\epsilon}| |T_k(G_m(u_{\epsilon}))| dx. $$

$$ \le \int_{\Omega} c(x)(1 + |u_{\epsilon}|^{p-1}) |\nabla T_k(G_m(u_{\epsilon}))| dx + k \int_{\Omega} b(x)(1 + |\nabla G_m(u_{\epsilon})|^{p-1}) \\ + k \int_{\Omega} |f| dx, $$
where $\beta$ is defined in (3.6). Let us evaluate the first integral in the right hand side. Denote $\gamma_p = \max\{1, 2^{p-1}\}$. By Hölder and Young inequalities, we get:

$$ \int_{\Omega} c(x)(1 + |u_{\varepsilon}|^{p-1}) |\nabla T_k(G_m(u_{\varepsilon}))| dx \quad (4.18) $$

$$ \leq |\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|\nabla T_k(G_m(u_{\varepsilon}))\|_{L^p(\Omega)} $$

$$ + \gamma_p \int_{\Omega} c(x) |T_m(u_{\varepsilon})|^{p-1} |\nabla T_k(G_m(u_{\varepsilon}))| dx $$

$$ + \gamma_p \int_{\Omega} c(x) |G_m(u_{\varepsilon})|^{p-1} |\nabla T_k(G_m(u_{\varepsilon}))| dx $$

$$ \leq |\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|\nabla T_k(G_m(u_{\varepsilon}))\|_{L^p(\Omega)} + \frac{\gamma_p^{p'}}{\beta^{p'/p}} \frac{m^p}{p'} \|c\|_{L^{p'}(\Omega)} $$

$$ + \frac{\beta}{p} \|\nabla T_k(G_m(u_{\varepsilon}))\|_{L^p(\Omega)}^p $$

$$ + \gamma_p \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|\nabla T_k(G_m(u_{\varepsilon}))\|_{L^p(\Omega)} \|T_k(G_m(u_{\varepsilon}))\|_{L^{p'}}^{p-1} $$

We deduce by Young inequality and Sobolev imbedding theorem

$$ \int_{\Omega} c(x)(1 + |u_{\varepsilon}|^{p-1}) |\nabla T_k(G_m(u_{\varepsilon}))| dx \quad (4.19) $$

$$ \leq |\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)} + \frac{\gamma_p^{p'}}{\beta^{p'/p}} \frac{m^p}{p'} \|c\|_{L^{p'}(\Omega)} $$

$$ + \frac{\beta}{p} \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p $$

$$ + S\gamma_p \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p $$

$$ = \frac{\gamma_p^{p'}}{\beta^{p'/p}} \frac{m^p}{p'} \|c\|_{L^{p'}(\Omega)} + \frac{\beta}{p} \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p $$

$$ + \left( \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p} + \gamma_p S \right) \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p $$

$$ + \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p'} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} $$

Combining (4.17) and (4.19), since $|u_{\varepsilon}| \ge T_k(G_m(u_{\varepsilon}))$ a.e. in $\Omega$ we get

$$ \beta \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p \le \left[ \left( \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p} + \gamma_p S \right) \|c\|_{L^{\frac{N}{p-1}}(\Omega)} + \frac{\beta}{p} \right] \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p \\ + \frac{\gamma_p^{p'}}{\beta^{p'/p}} \frac{m^p}{p'} \|c\|_{L^{p'}(\Omega)} + \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p'} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} \\ + k \int_{\Omega} b(x)(1 + |\nabla G_m(u_{\varepsilon})|^{p-1}) + k \|f\|_{L^1(\Omega)} $$

Let us now assume the following smallness conditions on $c$,

$$ \bar{\beta} \stackrel{\text{def}}{=} \frac{\beta}{p'} - \left( \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p} + \gamma_p S \right) \|c\|_{L^{\frac{N}{p-1}}(\Omega)} > 0. \quad (4.20) $$

Then we get:

$$ \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p \le \frac{\gamma_p^{p'}}{\beta \bar{\beta}^{p'/p}} \frac{m^p}{p'} \|c\|_{L^{p'}(\Omega)} \\ + k \left[ \frac{\|f\|_{L^1(\Omega)} + \|b\|_{L^1(\Omega)}}{\bar{\beta}} + \frac{1}{\bar{\beta}} \int_{\Omega} b(x) |\nabla G_m(u_{\varepsilon})|^{p-1} dx \right] \\ + \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p'} \|c\|_{L^{\frac{N}{p-1}}(\Omega)} $$

i.e.

$$ \|T_k(G_m(u_{\varepsilon}))\|_{W^{1,p}(\Omega)}^p \le M k + L, \quad \forall k > 0 $$
where

$$M = \frac{1}{\bar{\beta}} (\|f\|_{L^1(\Omega)} + \|b\|_{L^1(\Omega)}) + \frac{1}{\bar{\beta}} \int_{\Omega} b(x) |\nabla G_m(u_{\epsilon})|^{p-1} dx, \quad (4.21)$$

$$L = \frac{\gamma_p^{p'}}{\bar{\beta} \beta^{p'/p}} \frac{m^p}{p^p} \|c\|_{L^{p'}(\Omega)} + \frac{|\Omega|^{1-\frac{p-1}{N}-\frac{1}{p}}}{p^{p'}} \|c\|_{L^{\frac{N}{p-1}}(\Omega)}. \quad (4.22)$$

By Lemma A1 in [23] (adapted to the case where $T_k(G_m(u_\epsilon))$ belongs to the Sobolev space $W^{1,p}(\Omega)$), we deduce

$$\| |\nabla G_m(u_\epsilon)|^{p-1} \|_{L^{N',\infty}(\Omega)} \le C(N, p) \left[ M + |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} L^{\frac{1}{p'}} \right]. \quad (4.23)$$

As in Theorem 3.2 we consider $Z_\epsilon^c$ and we obtain:

$$\int_{\Omega} b(x) |\nabla G_m(u_{\epsilon})|^{p-1} dx \le \|b\|_{L^{N,1}(|\{u_{\epsilon}|>m\} \cap Z_{\epsilon})} \| |\nabla G_m(u_{\epsilon})|^{p-1} \|_{L^{N',\infty}(\Omega)}.$$

By the equi-integrability of $b$ in $L^{N,1}(\Omega)$, for $\epsilon > 0$ (independent on $\epsilon > 0$) we can choose $m = m_1$ such that

$$|\{ |u_{\epsilon}| > m_1 \} \cap Z_{\epsilon} | = \eta$$

and hence

$$\frac{C(N, p)}{\bar{\beta}} \|b\|_{L^{N,1}(|\{u_{\epsilon}|>m\} \cap Z_{\epsilon})} \le \frac{1}{2}. \quad (4.24)$$

Therefore by (4.23) we deduce

$$\| |\nabla G_m(u_\epsilon)|^{p-1} \|_{L^{N',\infty}(\Omega)} \le \frac{2C(N, p)}{\bar{\beta}} (\|f\|_{L^1(\Omega)} + \|b\|_{L^1(\Omega)} + 2C(N, p) |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} L^{\frac{1}{p'}}). \quad (4.25)$$

Step 2. $m_1$ is bounded.

Let us consider the test function

$$|T_{m_1}(u_\epsilon)|^{\theta-1} T_{m_1}(u_\epsilon),$$

for some $\theta \ge 1$. By ellipticity condition (2.2), growth conditions (2.6) and (2.7), we get

$$\beta \int_{\Omega} |u_{\epsilon}|^{p-1} |T_{m_1}(u_{\epsilon})|^{\theta} dx + \beta \theta \int_{\Omega} |T_{m_1}(u_{\epsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\epsilon})|^{p} dx \quad (4.26)$$

$$\le \int_{\Omega} b(x)(1 + |\nabla u_{\epsilon}|^{p-1}) |T_{m_1}(u_{\epsilon})|^{\theta} dx + \int_{\Omega} |f| |T_{m_1}(u_{\epsilon})|^{\theta} dx \\ + \theta \int_{\Omega} c(x)(1 + |T_{m_1}(u_{\epsilon})|^{p-1}) |T_{m_1}(u_{\epsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\epsilon})| dx \\ \le m_1^{\theta} (\|b\|_{L^1} + \|f\|_{L^1}) + \int_{\Omega} b(x) |\nabla u_{\epsilon}|^{p-1} |T_{m_1}(u_{\epsilon})|^{\theta} dx \\ + \theta \int_{\Omega} c(x) |T_{m_1}(u_{\epsilon})|^{p+\theta-2} |\nabla T_{m_1}(u_{\epsilon})| dx + \theta \int_{\Omega} c(x) |T_{m_1}(u_{\epsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\epsilon})| dx.$$

Let us evaluate the first integral on the right-hand side:

$$\int_{\Omega} b(x) |\nabla u_{\epsilon}|^{p-1} |T_{m_1}(u_{\epsilon})|^{\theta} dx \le m_1^{\theta} \int_{\Omega} b(x) |\nabla G_{m_1}(u_{\epsilon})|^{p-1} \\ + \int_{\Omega} b(x) |\nabla T_{m_1}(u_{\epsilon})|^{p-1} |T_{m_1}(u_{\epsilon})|^{\theta} dx.$$

On the other hand, by Young inequality, we get:

$$b(x) |T_{m_1}(u_\epsilon)|^\theta |\nabla T_{m_1}(u_\epsilon)|^{p-1} = b(x) \left( \frac{\beta \theta}{4} |T_{m_1}(u_\epsilon)|^{\theta-1} |\nabla T_{m_1}(u_\epsilon)|^p \right)^{\frac{p-1}{p}} \times \\ \times \left( \frac{4}{\beta \theta} \right)^{\frac{p-1}{p}} |T_{m_1}(u_\epsilon)|^{\theta - \frac{(\theta-1)(p-1)}{p}} \\ \le \frac{\beta \theta}{4p} |T_{m_1}(u_\epsilon)|^{\theta-1} |\nabla T_{m_1}(u_\epsilon)|^p + \frac{1}{p} |b(x)|^p \left( \frac{4}{\beta \theta} \right)^{p-1} |T_{m_1}(u_\epsilon)|^{\theta+p-1}.$$

We deduce

$$\int_{\Omega} b(x) |\nabla u_{\epsilon}|^{p-1} |T_{m_1}(u_{\epsilon})|^{\theta} dx \quad (4.27) \\ \le m_1^{\theta} \int_{\Omega} b(x) |\nabla G_{m_1}(u_{\epsilon})|^{p-1} + \frac{1}{p} m_1^{\theta+p-1} \left( \frac{4}{\beta \theta} \right)^{p-1} \|b\|_{L^p}^p$$
$$ + \frac{\beta\theta}{4p'} \int_{\Omega} |\nabla T_{m_1}(u_{\varepsilon})|^p |T_{m_1}(u_{\varepsilon})|^{\theta-1} dx. $$

Now we evaluate the second integral on the right-hand side of (4.26). By using Young inequality, since it results,

$$ \theta c(x) |T_{m_1}(u_{\varepsilon})|^{p+\theta-2} |\nabla T_{m_1}(u_{\varepsilon})| = \theta \left(\frac{\beta\theta}{4}\right)^{1/p} |T_{m_1}(u_{\varepsilon})|^{\frac{\theta-1}{p}} |\nabla T_{m_1}(u_{\varepsilon})| \left(\frac{4}{\beta\theta}\right)^{1/p} c(x) |T_{m_1}(u_{\varepsilon})|^{p+\theta-2-\frac{\theta-1}{p}} $$

we get

$$ \begin{aligned} & \theta \int_{\Omega} c(x) |T_{m_1}(u_{\varepsilon})|^{p+\theta-2} |\nabla T_{m_1}(u_{\varepsilon})|^p dx \\ & \le \frac{\beta\theta}{4p} \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx + \theta^{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} \int_{\Omega} c(x)^{p'} |T_{m_1}(u_{\varepsilon})|^{\theta+p-1} dx \\ & \le \frac{\beta\theta}{4p} \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx + \theta^{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} m_1^{\theta+p-1} \|c\|_{L^{p'}}^{p'} . \end{aligned} \quad (4.28) $$

Let us evaluate the last term in (4.26). By Young inequality, since $\frac{1}{p} < 1$, we have

$$ \theta c(x) |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})| \le \frac{\beta\theta}{4p} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p + \frac{\theta^{p'}}{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} c(x)^{p'} m_1^{\theta-1}. $$

Therefore

$$ \begin{aligned} & \theta \int_{\Omega} c(x) |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})| dx \\ & \le \frac{\beta\theta}{4p} \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx + \frac{\theta^{p'}}{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} \|c\|_{L^{p'}}^{p'} m_1^{\theta-1}. \end{aligned} \quad (4.29) $$

Combining (4.26), (4.27), (4.28) and (4.29), we get

$$ \begin{aligned} & \beta \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_{m_1}(u_{\varepsilon})|^{\theta} dx + \beta\theta \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx \\ & \le m_1^{\theta} (\|b\|_{L^1} + \|f\|_{L^1}) + m_1^{\theta} \int_{\Omega} b(x) |\nabla G_{m_1}(u_{\varepsilon})|^{p-1} \\ & \qquad + \frac{\beta\theta}{4p'} \int_{\Omega} |\nabla T_{m_1}(u_{\varepsilon})|^p |T_{m_1}(u_{\varepsilon})|^{\theta-1} dx + \frac{1}{p} m_1^{\theta+p-1} \left(\frac{4}{\beta\theta}\right)^{p-1} \|b\|_{L^p}^p \\ & \qquad + \frac{\beta\theta}{4p} \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx + \theta^{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} m_1^{\theta+p-1} \|c\|_{L^{p'}}^{p'} \\ & \qquad + \frac{\beta\theta}{4p} \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx + \frac{\theta^{p'}}{p'} \left(\frac{4}{\beta\theta}\right)^{\frac{p'}{p}} \|c\|_{L^{p'}}^{p'} m_1^{\theta-1}. \end{aligned} \quad (4.30) $$

Therefore we get

$$ \begin{aligned} & \beta \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_{m_1}(u_{\varepsilon})|^{\theta} dx \\ & \qquad + \beta\theta \left(1 - \frac{1}{4p'} - \frac{1}{4p} - \frac{1}{4p}\right) \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta-1} |\nabla T_{m_1}(u_{\varepsilon})|^p dx \\ & \le m_1^{\theta} (\|b\|_{L^1} + \|f\|_{L^1}) + m_1^{\theta} \int_{\Omega} b(x) |\nabla G_{m_1}(u_{\varepsilon})|^{p-1} dx \\ & \qquad + \frac{1}{p} \left(\frac{4}{\beta\theta}\right)^{p-1} \|b\|_{L^1}^p m_1^{\theta+p-1} \\ & \qquad + \theta^{p'} \left(\frac{4}{\beta\theta}\right)^{p'/p} \|c\|_{L^{p'}}^{p'} m_1^{\theta+p-1} + \frac{\theta^{p'}}{p'} \left(\frac{4}{\beta\theta}\right)^{p'/p} \|c\|_{L^{p'}}^{p'} m_1^{\theta-1}. \end{aligned} \quad (4.31) $$

Moreover

$$ \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_{m_1}(u_{\varepsilon})|^{\theta} dx \ge \int_{\Omega} |T_{m_1}(u_{\varepsilon})|^{\theta+p-1} dx \ge m_1^{\theta+p-1} |\Omega|. $$

Therefore, since by (4.24)

$$ \frac{C(N, p)}{\beta} \|b\|_{L^{N,1}(|\{u_{\varepsilon}|>m\}| \cap Z_{\varepsilon})} \le \frac{1}{2}, $$
by Hölder inequality in Lorentz spaces, we deduce

$$ \begin{aligned} |\Omega|\beta m_1^{\theta+p-1} &\le m_1^{\theta} (\|b\|_{L^1} + \|f\|_{L^1}) + \frac{\tilde{\beta}}{C(N,p)} \frac{1}{2} m_1^{\theta} \|\nabla G_{m_1}(u_{\epsilon})\|^{p-1} \|_{L^{N',\infty}} \\ &\quad + m_1^{\theta+p-1} \left(\frac{4}{\tilde{\beta}\theta}\right)^{p-1} \left[ \frac{1}{p} \|b\|_{L^1}^p + \theta^{p'} \|c\|_{L^{p'}}^{p'} \right] \\ &\quad + m_1^{\theta-1} \left(\frac{4}{\tilde{\beta}\theta}\right)^{p-1} \frac{\theta^{p'}}{p'} \|c\|_{L^{p'}}^{p'} \end{aligned} $$

By Step 1 we get

$$ \begin{aligned} |\Omega|\beta m_1^p &\le m_1^p \left( \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \frac{1}{p} \|b\|_{L^p}^p + \theta^{p'} \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \|c\|_{L^{p'}}^{p'} \right) \\ &\quad + m_1 \left( \|b\|_{L^1} + \|f\|_{L^1} + \tilde{\beta} |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} L^{\frac{1}{p'}} \right) + \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \frac{\theta^{p'}}{p'} \|c\|_{L^{p'}}^{p'} \end{aligned} $$

Choose $\theta$ such that

$$ \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \frac{1}{p} \|b\|_{L^p}^p \le \frac{\tilde{\beta}|\Omega|}{4}. $$

Moreover by the smallness assumption (4.20) on $\|c\|$ we get

$$ \theta^{p'} \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \|c\|_{L^{p'}}^{p'} \le \frac{\tilde{\beta}|\Omega|}{4}. $$

We deduce by Young inequality

$$ \begin{aligned} \frac{|\Omega|\beta m_1^p}{2} &\le m_1 \left( \|b\|_{L^1} + \|f\|_{L^1} + \tilde{\beta} |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} L^{\frac{1}{p'}} \right) \\ &\quad + \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \frac{\theta^{p'}}{p'} \|c\|_{L^{p'}}^{p'} \\ &\le \frac{|\Omega|\beta m_1^p}{4} + \frac{1}{p'} \left( \frac{4}{p} \frac{1}{|\Omega|\tilde{\beta}} \right)^{\frac{p'}{p}} \left( \|b\|_{L^1} + \|f\|_{L^1} + \tilde{\beta} |\Omega|^{\frac{1}{N'} - \frac{1}{p'}} L^{\frac{1}{p'}} \right) \\ &\quad + \left( \frac{4}{\tilde{\beta}\theta} \right)^{p-1} \frac{\theta^{p'}}{p'} \|c\|_{L^{p'}}^{p'} \end{aligned} $$

Therefore, by the definition of L given in (4.22), we conclude

$$ \frac{|\Omega|\beta m_1^p}{2} \le C''(N, p, |\Omega|, \theta, \|b\|_{L^1}, \|f\|_{L^1}, \|c\|_{L^{p'}}). $$

This implies that $m_1$ is bounded by a positive constant which does not depend on $\epsilon$, but depend only on the data. Therefore (4.25) allows to conclude that, under the smallness assumption (4.20),

$$ \|\nabla G_{m_1}(u_{\epsilon})\|_{L^{N',\infty}(\Omega)} \le C $$

where $C > 0$ does not depend on $\epsilon$.

From inequality (4.32) for $\theta = 1$ it follows that

$$ \nabla T_{m_1}(u_{\epsilon}) \text{ is bounded in } (L^p(\Omega))^N $$

so that

$$ \|\nabla u_{\epsilon}\|_{L^{N',\infty}(\Omega)}^{p-1} \le C \qquad (4.32) $$

where $C > 0$ does not depend on $\epsilon$.

**Step 3.** *Passage to the limit in the approximate problems.* To conclude the proof we have to pass to the limit in the approximate problems by proving that the approximate solutions $u_{\epsilon}$ converge to a function $u$ which is a renormalized solution to problem (2.1). Due to (4.32), this is done exactly as in the Step 3 of Theorem 4.4 and here it is omitted. □

## Data availability

No data was used for the research described in the article.
Acknowledgements

This work was done during the visits made by the third authors to Laboratoire de Mathématiques “Raphaël Salem” de l’Université de Rouen and by the second author to Dipartimento di Matematica e Applicazioni “R. Caccioppoli” dell’Università degli Studi di Napoli Federico II. The authors gratefully thank these institution for their warm hospitality. The research of the third author was partially supported by Italian MIUR through research projects PRIN 2017 Direct and inverse problems for partial differential equations: theoretical aspects and applications, PRIN 2022: PRIN20229M52AS Partial differential equations and related geometric-functional inequalities, PRIN PNRR 2022 - P2022YFAJH - Linear and Nonlinear PDE's: New directions and Applications and by GNAMPA of INDAM. The first and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Appendix. The proof of the stability result

In this Appendix we state and prove a stability result for renormalized solutions.

**Proposition A.1.** For $\epsilon > 0$ let $u_{\epsilon}$ be a renormalized solution of

$$ egin{cases} \lambda|u_{\epsilon}|^{p-2}u_{\epsilon} - \operatorname{div}(a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) + \Phi_{\epsilon}(x, u_{\epsilon})) = \Psi_{\epsilon} & \text{in } \Omega, \\ (a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) + \Phi_{\epsilon}(x, u_{\epsilon})) \cdot \underline{n} = 0 & \text{on } \partial\Omega, \end{cases} \quad (A.1) $$

where $a_{\epsilon} : \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N$ and $\Phi_{\epsilon} : \Omega \times \mathbb{R} \to \mathbb{R}^N$ are Carathédory functions and where $\Psi_{\epsilon} \in L^1(\Omega)$. Let us assume that $a_{\epsilon}$ verifies uniformly in $\epsilon$ conditions (2.2), (2.3), (2.4) and that $\Phi_{\epsilon}$ verifies uniformly in $\epsilon$ condition (2.6). Let us denote by $a : \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N$, $\Phi : \Omega \times \mathbb{R} \to \mathbb{R}^N$ two Carathédory functions. Assume that (2.3) holds true and

$$ egin{cases} a_{\epsilon}(x, s_{\epsilon}, \eta_{\epsilon}) \to a(x, s, \eta) \\ \text{for every sequences } s_{\epsilon} \in \mathbb{R}, \eta_{\epsilon} \in \mathbb{R}^N \text{ such that } \\ s_{\epsilon} \to s, \eta_{\epsilon} \to \eta, \text{ as } \epsilon \to 0, \end{cases} \\ 
\begin{cases} \Phi_{\epsilon}(x, s_{\epsilon}) \to \Phi(x, s) \\ \text{for every sequence } s_{\epsilon} \in \mathbb{R} \text{ such that } s_{\epsilon} \to s, \text{ as } \epsilon \to 0. \end{cases} $$

Assume that $\Psi$ belongs to $L^1(\Omega)$ is such that

$$ \Psi_{\epsilon} \to \Psi \text{ weakly in } L^1(\Omega), \text{ as } \epsilon \to 0. \quad (A.2) $$

Then, up to a subsequence still indexed by $\epsilon$,

$$ u_{\epsilon} \to u \text{ a.e. in } \Omega, \quad (A.3) $$

$$ \nabla u_{\epsilon} \to \nabla u \text{ a.e. in } \Omega, \quad (A.4) $$

$$ T_k(u_{\epsilon}) \to T_k(u) \text{ strongly in } W^{1,p}(\Omega), \forall k > 0, \quad (A.5) $$

where $u$ is a renormalized solution of

$$ egin{cases} \lambda|u|^{p-2}u - \operatorname{div}(a(x, u, \nabla u) + \Phi(x, u)) = \Psi & \text{in } \Omega, \\ (a(x, u, \nabla u) + \Phi(x, u)) \cdot \underline{n} = 0 & \text{on } \partial\Omega. \end{cases} \quad (A.6) $$

**Remark A.2.** The almost everywhere convergence (A.4) and the strong convergence (A.5) are due to the strict monotonicity of the operator $a$. If we replace assumption (2.3) by

$$ (a(x, s, \xi) - a(x, s, \eta)) \cdot (\xi - \eta) \ge 0 $$

$\forall s \in \mathbb{R}, \forall \xi, \eta \in \mathbb{R}^N$ with $\xi \neq \eta$ and a.e. in $\Omega$, then Proposition A.1 remains true in a weaker form, that is without (A.4) and (A.5).

**Proof.** We proceed by dividing the proof into five steps. In Step 1 we derive a priori estimates on $u_{\epsilon}$ and we extract a subsequence to define $u$, the limit of $u_{\epsilon}$ as $\epsilon$ goes to zero. Step 2 is devoted to prove the uniform decay of the truncated energy. The key point of proof is Step 3 in which we establish the convergence of the truncated energy. In Step 4 convergences (A.4) and (A.5) are obtained using the strict monotonicity of $a$. Finally, this allows us in Step 5 to pass to the limit as $\epsilon$ tends to 0 in the renormalized formulation.

Step 1. A priori estimates

To obtain a priori estimates on $T_k(u_{\epsilon})$ we first have to prove

$$ egin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} T_k(u_{\epsilon}) dx + \int_{\Omega} a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx \\ & + \int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx = \int_{\Omega} \Psi_{\epsilon} T_k(u_{\epsilon}) dx. \end{aligned} \quad (A.7) $$
As usual $T_k(u_\epsilon)$ is not an admissible test function to (A.1) and to show (A.7) we use the function $h_n$ defined for $n \ge 1$, by

$$h_n(s) = \begin{cases} 0 & \text{if } |s| > 2n, \\ \frac{2n - |s|}{n} & \text{if } n < |s| \le 2n, \\ 1 & \text{if } |s| \le n. \end{cases} \qquad (A.8)$$

Using the admissible test function $h_n(u_\epsilon)T_k(u_\epsilon)$ in the renormalized formulation of (A.1) we have for any $n \ge 1$

$$\begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} T_k(u_{\epsilon}) h_n(u_{\epsilon}) dx + \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx \\ & + \int_{\Omega} T_k(u_{\epsilon}) h'_{n}(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon} dx + \int_{\Omega} h_n(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx \\ & + \int_{\Omega} T_k(u_{\epsilon}) h'_{n}(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon} dx \\ & = \int_{\Omega} \Psi_{\epsilon} h_n(u_{\epsilon}) T_k(u_{\epsilon}) dx. \end{aligned} \qquad (A.9)$$

We now pass to the limit as $n$ goes to $+\infty$. In view of the definition of $h_n$ we have, for $n > k$

$$\begin{aligned} & \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx = \int_{\Omega} \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx, \\ & \int_{\Omega} h_n(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx = \int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx. \end{aligned}$$

The decay of the truncate energy (4.3) implies that

$$\left| \int_{\Omega} T_k(u_{\epsilon}) h'_{n}(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon} dx \right| \le \frac{k}{n} \int_{\{|u_{\epsilon}| < n\}} \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon} dx \to 0$$

and the Lebesgue dominated convergence theorem leads to

$$\lim_{n \to +\infty} \int_{\Omega} \Psi_{\epsilon} h_n(u_{\epsilon}) T_k(u_{\epsilon}) dx = \int_{\Omega} \Psi'_{\epsilon} T_k(u_{\epsilon}) dx.$$

Using Hölder and Sobolev inequality, the growth control (2.6) on $\Phi_\epsilon$ gives

$$\begin{aligned} \left| \int_{\Omega} T_k(u_{\epsilon}) h'_{n}(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon} dx \right| & \le \frac{k}{n} \int_{\{|u_{\epsilon}| < 2n\}} c(x)(1 + |T_{2n}(u_{\epsilon})|^{p-1}) |\nabla u_{\epsilon}| dx \\ & \le \frac{k}{n} \|c\|_{L^{p-1}(\Omega)} \left( |\Omega|^{\frac{(N-p)(p-1)}{Np}} + \|T_n(u_{\epsilon})\|_{W^{1,p}(\Omega)}^{p-1} \right) \|\nabla T_n(u_{\epsilon})\|_{L^p(\Omega)}. \end{aligned}$$

Again the decay of the truncate energy (4.3) allows one to conclude that the limit of the term above is zero.

Finally the Beppo–Levi Theorem gives that

$$\lim_{n \to +\infty} \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} T_k(u_{\epsilon}) h_n(u_{\epsilon}) dx = \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} T_k(u_{\epsilon}) dx.$$

Then passing to the limit in (A.9) leads to (A.7).

It is worth noting that (A.7) gives the a priori estimate

$$\int_{\Omega} |u_{\epsilon}|^{p-1} dx \le C, \qquad (A.10)$$

where $C > 0$ does not depend on $\epsilon$. Indeed using (A.7) for $k = 1$ together with the ellipticity condition (2.2) on a and the growth condition (2.6) gives

$$\begin{aligned} \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} T_1(u_{\epsilon}) dx + \alpha \int_{\Omega} |\nabla T_1(u_{\epsilon})|^p dx & \\ & \le \|\Psi_{\epsilon}\|_{L^1(\Omega)} + 2 \int_{\Omega} c(x) |\nabla T_1(u_{\epsilon})| dx. \end{aligned}$$

Since $\Psi_{\epsilon}$ weakly converges to $\Psi$ in $L^1(\Omega)$ and since $c$ belongs to $L^{\frac{N}{p-1}}(\Omega) \subset L^p'(\Omega)$, Hölder's inequality and the basic inequality $|r|^{p-1} \le 1 + |r|^{p-2} r T_1(r)$, for any $r \in \mathbb{R}$, allow us to deduce that (A.10) holds true where $C$ depends on $\lambda$, $\alpha$, $c$, $N$, $\Omega$, and $\sup_{\epsilon} \|\Psi'_{\epsilon}\|_{L^1(\Omega)}$. We now use (A.7) and (A.10) to prove that for some $M$ (independent of $\epsilon$) we have

$$\begin{aligned} \|T_k(u_\epsilon)\|_{W^{1,p}(\Omega)}^p &= \int_{\Omega} |T_k(u_\epsilon)|^p dx + \int_{\Omega} |\nabla T_k(u_\epsilon)|^p dx \qquad (A.11) \\ &\le M \left[ \int_{\Omega} \Psi_{\epsilon} T_k(u_\epsilon) dx + 1 \right]. \end{aligned}$$

In view of assumptions (2.2) and (A.2) the only term to study in (A.7) is

$$\int_{\Omega} \Phi_{\epsilon}(x, u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx.$$
Since $\Phi_{\varepsilon}$ verifies uniformly in $\varepsilon$ the growth condition (2.6) we repeat the arguments developed in the proof of Theorem 4.4 (see Step 2) which use Lemma 2.1 and lead to, for $\eta > 0$,

$$ \left| \int_{\Omega} \Phi_{\varepsilon}(x, u_{\varepsilon}) \nabla T_k(u_{\varepsilon}) dx \right| \le \frac{\alpha}{p} \int_{\Omega} |\nabla T_k(u_{\varepsilon})|^p dx + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{1}{p'} \|c\|_{L^{p'}(\Omega)}^{p'} \right. \\ \left. + \left(\frac{2}{\alpha}\right)^{p'/p} \frac{2^{p-1}}{p'} \left[ \eta^p \|\nabla T_k(u_{\varepsilon})\|_{L^{p'}(\Omega)}^p + C(c, \eta) \left( \int_{\Omega} |T_k(u_{\varepsilon})|^{p-1} dx \right)^{p'} \right] \right. \\ \left. \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_k(u_{\varepsilon})| dx + \int_{\Omega} \mathbf{a}_{\varepsilon}(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \cdot \nabla T_k(u_{\varepsilon}) dx \right| \le C(\alpha, p, \lambda, \eta, \Omega, f, c) \left[ \int_{\Omega} \Psi_{\varepsilon} T_k(u_{\varepsilon}) dx + 1 \right], $$

By choosing $\eta > 0$ small enough and by using (2.2), (A.7) and (A.10) we then deduce the following estimates

$$ \int_{\Omega} |u_{\varepsilon}|^{p-1} |T_k(u_{\varepsilon})| dx + \int_{\Omega} \mathbf{a}_{\varepsilon}(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \cdot \nabla T_k(u_{\varepsilon}) dx \le C(\alpha, p, \lambda, \eta, \Omega, f, c) \left[ \int_{\Omega} \Psi_{\varepsilon} T_k(u_{\varepsilon}) dx + 1 \right], \quad (A.12) $$

where $C(\alpha, p, \lambda, \eta, \Omega, f, c)$ does not depend on $\varepsilon$. Recalling that $\Psi_{\varepsilon}$ is bounded in $L^1$ and the ellipticity of $\mathbf{a}_{\varepsilon}$ allow one to deduce A.2. Moreover taking into account (2.4) and (A.11), we obtain that for any $k > 0$

$$ \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) \text{ is bounded in } (L^{p'}(\Omega))^N $$

uniformly with respect to $\varepsilon$. Therefore there exists a measurable function $u : \Omega \to \overline{\mathbb{R}}$ and for any $k > 0$ ($k$ being in the countable set $\mathbb{N}$) there exists a function $\sigma_k$ belonging to $(L^{p'}(\Omega))^N$ such that, up to a subsequence still indexed by $\varepsilon$,

$$ u_{\varepsilon} \to u \text{ a.e. in } \Omega, \quad (A.13) $$

$$ T_k(u_{\varepsilon}) \to T_k(u) \text{ weakly in } W^{1,p}(\Omega), \quad (A.14) $$

$$ \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) \to \sigma_k \text{ weakly in } (L^{p'}(\Omega))^N \quad \forall k > 0. \quad (A.15) $$

Fatou Lemma and (A.10) imply that $u$ is finite almost everywhere in $\Omega$. Moreover writing for $k' > k$ that

$$ \chi_{\{|u_{\varepsilon}|<k|} \mathbf{a}_{\varepsilon}(x, T_{k'}(u_{\varepsilon}), \nabla T_{k'}(u_{\varepsilon})) = \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) \quad \text{a.e. in } \Omega $$

and using the point-wise convergence of $u_{\varepsilon}$ we obtain that

$$ \sigma_k = \sigma_{k'} \quad \text{a.e. in } \{|u| < k\}. \quad (A.16) $$

Step 2. We now prove the uniform decay of the truncated energy:

$$ \lim_{k \to +\infty} \limsup_{\varepsilon \to 0} \frac{1}{k} \int_{\Omega} |T_k(u_{\varepsilon})|^p dx = 0, \quad (A.17) $$

$$ \lim_{k \to +\infty} \limsup_{\varepsilon \to 0} \frac{1}{k} \int_{\{|u_{\varepsilon}|<k|} \mathbf{a}_{\varepsilon}(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \cdot \nabla u_{\varepsilon} dx = 0. \quad (A.18) $$

From A.2 we have

$$ \frac{1}{k} \left( \int_{\Omega} |T_k(u_{\varepsilon})|^p dx + \int_{\Omega} \mathbf{a}_{\varepsilon}(x, u_{\varepsilon}, \nabla u_{\varepsilon}) \cdot \nabla T_k(u_{\varepsilon}) dx \right) \le \frac{C(\alpha, p, \lambda, \eta, \Omega, f, c)}{k} \left[ \int_{\Omega} \Psi_{\varepsilon} T_k(u_{\varepsilon}) dx + 1 \right], $$

where $C(\alpha, p, \lambda, \eta, \Omega, f, c)$ does not depend on $\varepsilon$.

Since $\Phi_{\varepsilon}$ converges to $\Phi$ weakly in $L^1(\Omega)$ and since $T_k(u_{\varepsilon})$ is a sequence bounded in $L^{\infty}(\Omega)$ which converges to $T_k(u)$ almost everywhere in $\Omega$, an application of Egorov theorem leads to

$$ \lim_{\varepsilon \to +\infty} \int_{\Omega} \Psi_{\varepsilon} T_k(u_{\varepsilon}) dx = \int_{\Omega} \Psi T_k(u) dx. $$

Since $u$ is finite almost everywhere in $\Omega$, Lebesgue dominated convergence theorem yields that

$$ \lim_{k \to +\infty} \lim_{\varepsilon \to +\infty} \frac{1}{k} \int_{\Omega} \Psi_{\varepsilon} T_k(u_{\varepsilon}) dx = \lim_{k \to +\infty} \int_{\Omega} \Psi \frac{T_k(u)}{k} dx = 0 $$

and then (A.17) and (A.18) hold true.

Step 3. We prove in this step that for any $k > 0$

$$ \mathbf{a}(x, T_k(u), \nabla T_k(u)) = \sigma_k, \quad (A.19) $$

$$ \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) \nabla T_k(u_{\varepsilon}) \to \mathbf{a}(x, T_k(u), \nabla T_k(u)) \nabla T_k(u) \quad (A.20) $$

weakly in $L^1(\Omega)$ as $\varepsilon$ goes to zero.

Proving (A.19) and (A.20) relies on establish

$$ \lim_{\varepsilon \to 0} \int_{\Omega} \left( \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) - \mathbf{a}_{\varepsilon}(x, T_k(u_{\varepsilon}), \nabla T_k(u_{\varepsilon})) \right) \cdot (\nabla T_k(u_{\varepsilon}) - \nabla T_k(u)) dx = 0. \quad (A.21) $$
To show (A.21) we use Eq. (4.13) and the function $h_n$ defined in (A.8). Using the admissible test function $h_n(u_\epsilon)(T_k(u_\epsilon) - T_k(u))$ to (4.13) we have

$$ \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon})(\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx \\ = A_{k,n,\epsilon} + B_{k,n,\epsilon} + C_{k,n,\epsilon} + D_{k,n,\epsilon} + E_{k,n,\epsilon} \quad (A.22) $$

with

$$ A_{k,n,\epsilon} = \int_{\Omega} h_n(u_{\epsilon}) \Psi_{\epsilon}(T_k(u_{\epsilon}) - T_k(u))dx, $$

$$ B_{k,n,\epsilon} = - \int_{\Omega} h_n(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon})(\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx, $$

$$ C_{k,n,\epsilon} = - \int_{\Omega} h'_{n}(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon}(T_{k}(u_{\epsilon}) - T_{k}(u))dx, $$

$$ D_{k,n,\epsilon} = - \int_{\Omega} h'_{n}(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon}(T_{k}(u_{\epsilon}) - T_{k}(u))dx, $$

$$ E_{k,n,\epsilon} = -\lambda \int_{\Omega} h_n(u_{\epsilon})|u_{\epsilon}|^{p-2}u_{\epsilon}(T_k(u_{\epsilon}) - T_k(u))dx. $$

We now pass to the limit in (A.22) first as $\epsilon$ goes to zero and then as $n$ goes to infinity.

Due to the pointwise convergence of $u_\epsilon$ the sequence $T_k(u_\epsilon) - T_k(u)$ converges to zero almost everywhere in $\Omega$ and is bounded in $L^\infty(\Omega)$. Since $\Psi_\epsilon$ converges to $\Psi$ weakly in $L^1(\Omega)$ we obtain from Egorov theorem that

$$ \lim_{\epsilon \to 0} A_{k,n,\epsilon} = \lim_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon}) \Psi_{\epsilon}(T_k(u_{\epsilon}) - T_k(u))dx = 0. $$

For $\epsilon < 1/n$ we have $h_n(s)\Phi_\epsilon(x, s) = h_n(s)\Phi(x, s)$ for any $s \in \mathbb{R}$ and a.e. in $\Omega$. Using the point-wise convergence of $u_\epsilon$ the sequence $h_n(u_\epsilon)\Phi_\epsilon(x, u_\epsilon)$ converges to $h_n(u)\Phi(x, u)$ a.e. in $\Omega$ as $\epsilon$ goes to zero while by (2.6) we have $h_n(u_\epsilon)|\Phi_\epsilon(x, u_\epsilon)| \le (1 + (2n)^{p-1})c(x)$. It follows that $h_n(u_\epsilon)\Phi_\epsilon(x, u_\epsilon)$ converges to $h_n(u)\Phi(x, u)$ strongly in $(L^q(\Omega))^N$ with $q = N/(p-1)$ if $N > p$ and $q > N/(N-1)$ if $N = p$. Due to the weak convergence of $\nabla T_k(u_\epsilon)$, see (A.14), we deduce that

$$ \lim_{\epsilon \to 0} B_{k,n,\epsilon} = - \lim_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon})(\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx = 0. $$

With arguments already used we also have

$$ \lim_{\epsilon \to 0} C_{k,n,\epsilon} = - \lim_{\epsilon \to 0} \int_{\Omega} h'_{n}(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon}(T_{k}(u_{\epsilon}) - T_{k}(u))dx = 0. $$

Since

$$ |D_{k,n,\epsilon}| \le \frac{2k}{n} \int_{\{|u_\epsilon| \le 2n\}} \mathbf{a}_\epsilon(x, u_\epsilon, \nabla u_\epsilon) \nabla u_\epsilon dx $$

and due to (A.18) we obtain that

$$ \lim_{n \to 0} \limsup_{\epsilon \to 0} D_{k,n,\epsilon} = 0. $$

At last the point-wise convergence of $u_\epsilon$ and since $h_n$ has a compact support Lebesgue dominated Theorem gives that

$$ \lim_{\epsilon \to 0} E_{k,n,\epsilon} = 0. $$

As a consequence we obtain that for any $k > 0$

$$ \lim_{n \to \infty} \limsup_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon})(\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx = 0. $$

In view of the definition of $h_n$, for any $n > k$, we have

$$ h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) = \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) \quad \text{a.e. in } \Omega. $$

It follows that

$$ \limsup_{\epsilon \to 0} \int_{\Omega} \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u_{\epsilon}) dx \quad (A.23) $$

$$ \le \limsup_{n \to \infty} \limsup_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u) dx. $$

According to the definition of $h_n$ we have

$$ h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) = h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, T_{2n}(u_{\epsilon}), \nabla T_{2n}(u_{\epsilon})) \quad \text{a.e. in } \Omega. $$

so that (A.13) and (A.15) give

$$ \lim_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon}) \mathbf{a}_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla T_k(u) dx = \int_{\Omega} h_n(u) \sigma_{2n} \nabla T_k(u) dx. $$
From (A.16) it follows that if $n > k$ we have $\sigma_{2n} \nabla T_k(u) = \sigma_k \nabla T_k(u)$ a.e. in $\Omega$ so that, for any $n > k$,

$$ \lim_{\epsilon \to 0} \int_{\Omega} h_n(u_{\epsilon})a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon})\nabla T_k(u)dx = \int_{\Omega} \sigma_k \nabla T_k(u)dx. \quad (A.24) $$

Therefore (A.23) and (A.24) allow one to conclude that

$$ \limsup_{\epsilon \to 0} \int_{\Omega} a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon}))\nabla T_k(u_{\epsilon})dx \le \int_{\Omega} \sigma_k \nabla T_k(u)dx. \quad (A.25) $$

We are now in a position to prove (A.21). Indeed the monotone character of a implies that for any $\epsilon > 0$

$$ \begin{aligned} & 0 \le \int_{\Omega} (a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon})) - a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u))) \\ & \qquad \cdot (\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx. \end{aligned} \quad (A.26) $$

Moreover, using the point-wise convergence of $T_k(u_\epsilon)$ and growth assumption (2.4), the function $a_\epsilon(x, T_k(u_\epsilon), \nabla T_k(u))$ converges to $a(x, T_k(u), \nabla T_k(u))$ strongly in $(L^p(\Omega))^N$. Writing

$$ \begin{aligned} & \int_{\Omega} (a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon})) - a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u))) (\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx \\ & = \int_{\Omega} a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon})) (\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx \\ & \quad - \int_{\Omega} a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u)) (\nabla T_k(u_{\epsilon}) - \nabla T_k(u))dx, \end{aligned} $$

and using (A.25) and (A.26) allow to conclude that (A.21) holds for any $k > 0$.

From (A.26) we have for any $k > 0$

$$ \lim_{\epsilon \to 0} \int_{\Omega} a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon}))\nabla T_k(u_{\epsilon})dx = \int_{\Omega} \sigma_k \nabla T_k(u)dx. $$

The monotone character of a and the usual Minty argument (see [1]) imply (A.19). Moreover from (A.21) we get

$$ (a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u_{\epsilon})) - a_{\epsilon}(x, T_k(u_{\epsilon}), \nabla T_k(u))) (\nabla T_k(u_{\epsilon}) - \nabla T_k(u)) \to 0 $$

strongly in $L^1(\Omega)$ as $\epsilon$ goes to zero, so that the monotone character of a gives (A.20).

**Step 4.** The almost everywhere convergence of $\nabla u_\epsilon$ and the strong convergence of $T_k(u_\epsilon)$ in $W^{1,p}(\Omega)$.

Since a is strictly monotone, the almost everywhere convergence (up to a subsequence still indexed by $\epsilon$) of $\nabla T_k(u_\epsilon)$ is a consequence of (A.21) and of classical results for monotone operator (see [1,28]). The ellipticity condition (2.2), the growth condition (2.4), the pointwise convergence of $\nabla T_k(u_\epsilon)$, (A.21) and an equi-integrability argument then imply that $\nabla T_k(u_\epsilon)$ converges strongly to $\nabla T_k(u)$ in $(L^p(\Omega))^N$. Recalling that $u_\epsilon$ converges a.e. to $u$ which is finite a.e. in $\Omega$ allows us to conclude that (A.4) and (A.5) hold true.

**Step 5.** Passing to the limit in the approximated problem.

Let $h$ be a function in $W^{1,\infty}(\mathbb{R})$ with compact support, contained in the interval $[-k, k]$, $k > 0$ and let $\varphi \in W^{1,p}(\Omega) \cap L^\infty(\Omega)$. Using $\varphi h(u_\epsilon)$ as a test function in the approximated problem we have

$$ \begin{aligned} & \lambda \int_{\Omega} |u_{\epsilon}|^{p-2} u_{\epsilon} \varphi h(u_{\epsilon}) dx + \int_{\Omega} h(u_{\epsilon}) a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla \varphi dx \\ & + \int_{\Omega} h'(u_{\epsilon}) a_{\epsilon}(x, u_{\epsilon}, \nabla u_{\epsilon}) \nabla u_{\epsilon} \varphi dx + \int_{\Omega} h(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla \varphi dx \\ & + \int_{\Omega} h'(u_{\epsilon}) \Phi_{\epsilon}(x, u_{\epsilon}) \nabla u_{\epsilon} \varphi dx = \int_{\Omega} \Psi_{\epsilon} \varphi h(u_{\epsilon}) dx. \end{aligned} \quad (A.27) $$

Since the support of $h$ is compact, using arguments developed in previous Steps and (A.19), (A.20) it is an easy task to pass to the limit in the previous equality and to obtain that

$$ \begin{aligned} & \lambda \int_{\Omega} |u|^{p-2} u \varphi h(u) dx + \int_{\Omega} h(u) a(x, u, \nabla u) \nabla \varphi dx \\ & + \int_{\Omega} h'(u) a(x, u, \nabla u) \nabla u \varphi dx + \int_{\Omega} h(u) \Phi(x, u) \nabla \varphi dx \\ & + \int_{\Omega} h'(u) \Phi(x, u) \nabla u \varphi dx = \int_{\Omega} \Psi \varphi h(u) dx. \end{aligned} $$

At last Fatou Lemma, the point-wise convergence of $u_\epsilon$, (A.18) and (A.20) give the decay of the truncated energy (4.3). $\square$

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