# ON REMOVABLE SINGULARITIES FOR SOLUTIONS OF NEUMANN PROBLEM FOR ELLIPTIC EQUATIONS INVOLVING VARIABLE EXPONENT

Juan Alcon Apaza

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Juan Alcon Apaza. ON REMOVABLE SINGULARITIES FOR SOLUTIONS OF NEUMANN PROBLEM FOR ELLIPTIC EQUATIONS INVOLVING VARIABLE EXPONENT. 2022. hal-03599196v7

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Distributed under a Creative Commons Attribution 4.0 International License
ON REMOVABLE SINGULARITIES FOR SOLUTIONS OF NEUMANN PROBLEM FOR ELLIPTIC
EQUATIONS INVOLVING VARIABLE EXPONENT

JUAN ALCON APAZA

ABSTRACT. We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this singularity for equations in the variable exponent Sobolev space W¹,p(·)(Ω).

1. INTRODUCTION

This paper is devoted to the study of conditions guaranteeing the removability of singular set for solutions of
nonlinear elliptic equations with Neumann boundary conditions of the form:

$$
\begin{equation}
\begin{cases}
-\operatorname{div} A(x, u, \nabla u) + a(x, u) + g(x, u) = 0 & \text{in } \Omega, \\
A(x, u, \nabla u) \cdot \nu + b(x, u) + h(x, u) = 0 & \text{on } \partial\Omega.
\end{cases}
\tag{1.1}
\end{equation}
$$

Throughout the whole article Ω is a bounded open set in Rⁿ, n ≥ 2, with Lipschitz boundary ∂Ω, and Γ ⊂ ∂Ω is a
compact set. We always equip ∂Ω with the (n − 1)-dimensional Hausdorff measure.

We assume that $A : \Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$, $a : \Omega \times \mathbb{R} \to \mathbb{R}$, $g : \bar{\Omega} \times \mathbb{R} \to \mathbb{R}$, $b, h : \partial\Omega \times \mathbb{R} \to \mathbb{R}$ are measurable functions, $g$ and $h$ are locally bounded. Furthermore, there exists $\mu > 0$ such that the following conditions are satisfied almost everywhere:

$$
\langle A(x, u, \eta), \eta \rangle \geq \mu |\eta|^{p_1(x)}, \quad (1.2)
$$

$$
|A(x, u, \eta)| \le \mu^{-1} \left( |\eta|^{p_1(x)-1} + |u|^{p_1(x)-1} + 1 \right), \quad (1.3)
$$

$$
|a(x, u)| \le \mu^{-1} (|u|^{p_1(x)-1} + 1), \quad (1.4)
$$

$$
g(x, u) \operatorname{sign} u \ge \mu |u|^{p_2(x)} - \mu^{-1}, \quad (1.5)
$$

$$
|b(x, u)| \le \mu^{-1} (|u|^{q_1(x)-1} + 1), \quad (1.6)
$$

$$
h(x, u) \operatorname{sign} u \geq \mu |u|^{q_2(x)} - \mu^{-1}, \quad (1.7)
$$

$$
A(x, u, -\eta) = -A(x, u, \eta), \tag{1.8}
$$

where $p_1, p_2 : \Omega \to \mathbb{R}$, $q_1, q_2 : \partial\Omega \to \mathbb{R}$ are measurable functions satisfying

$$
p_1^- , p_1^+ \in (1, n) , \bar{q}_1^- > 1 , \min \{ \bar{p}_2 , \bar{q}_2^- \} - \max \{ p_1^+ , q_1^+ \} + 1 > 0 . \quad (1.9)
$$

Also, for some $d \ge 0$,

$$
\mathrm{ess} \sup_{\Omega} \frac{p_1 p_2}{p_2 - p_1 + 1} < n - d, \qquad (1.10)
$$

where $p_i^- = \operatorname{ess\,inf}_\Omega p_i$, $p_i^+ = \operatorname{ess\,sup}_\Omega p_i$, $q_i^- = \operatorname{ess\,inf}_{\partial\Omega} q_i$ and $q_i^+ = \operatorname{ess\,sup}_{\partial\Omega} q_i; i = 1, 2$.

Concerning the singular set Γ ⊂ ∂Ω, we suppose that |Γ| = 0, and there is a small enough r₀ ∈ (0, 1) so that
U = {x ∈ Ω | dist(x, Γ) < 2r₀} is an open set with Lipschitz boundary ∂U, {x ∈ Ω | dist(x, Γ) = 2r₀} ≠ ∅, and
for a suitable positive constant C > 0:

$$
|\{\operatorname{dist}(\cdot, \Gamma) - \ell| < r\}| \leq Cr^{n-d}, \quad |\{\operatorname{dist}(\cdot, \Gamma) = r\}| \leq Cr^{n-d-1} \quad \text{and} \quad |\partial\Omega \cap \{\operatorname{dist}(\cdot, \Gamma) - \ell| < r\}| \leq Cr^{n-d-1}, \tag{1.11}
$$

where $0 < r \le \ell < r_0$.

The main result of this paper is the following theorem.

<footnote n="2020">2020 Mathematics Subject Classification. 35A21, 35D30, 35J57, 35J60.</footnote>

<footnote>Key words and phrases. variable exponent, singular set, removable singularity, Neumann problem.</footnote>
**Theorem 1.1.** Suppose that the conditions (1.2) - (1.11) are satisfied. Let $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ be a solution of equation (1.1) in $\bar{\Omega}\setminus\Gamma$. Then, the singularity of $u$ at $\Gamma$ is removable.

We follow the same lines as in [5] and [7]; which handle the case when the singular set is an interior point of $\Omega$, and the solution of an elliptic equation $u$ has no boundary conditions.

## 2. PRELIMINARIES

We first recall some facts on spaces $L^{p(\cdot)}(\Omega)$ and $W^{1,p(\cdot)}(\Omega)$. Denote by $\mathbf{P}(\Omega)$ the set of all Lebesgue measurable functions $p: \Omega \to [1, \infty]$. For the details see [4, 6, 8, 1]. Let $p \in \mathbf{P}(\Omega)$, we define the functional

$$ \rho_p(u) = \int_{\Omega \setminus \Omega_\infty} |u|^p dx + \operatorname{ess\,sup}_{\Omega_\infty} |u|, $$

where $\Omega_\infty = \{x \in \Omega \mid p(x) = \infty\}$.

The variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ is the class of all functions $u$ such that $\rho_p(tu) < \infty$, for some $t > 0$. $L^{p(\cdot)}(\Omega)$ is a Banach space equipped with the norm

$$ \|u\|_{L^{p(\cdot)}(\Omega)} = \inf\{\lambda > 0 \mid \rho_p\left(\frac{u}{\lambda}\right) \le 1\}; $$

see [6, Theorem 2.5].

**Proposition 2.1.** (see [6, Theorem 2.1].) Let $p \in \mathbf{P}(\Omega)$. If $u \in L^{p(\cdot)}(\Omega)$ and $v \in L^{p'(\cdot)}(\Omega)$, then

$$ \int_{\Omega} |uv| dx \le 2 \|u\|_{L^{p(\cdot)}(\Omega)} \|v\|_{L^{p'(\cdot)}(\Omega)}, $$

where

$$ p'(x) = \begin{cases} \infty & \text{if } p(x) = 1, \\ 1 & \text{if } p(x) = \infty, \\ \frac{p(x)}{p(x)-1} & \text{if } p(x) \neq 1 \text{ and } p(x) \neq \infty. \end{cases} $$

**Proposition 2.2.** (see [4, Theorem 1.3].) Let $p \in \mathbf{P}(\Omega)$ with $p^+ < \infty$. For any $u \in L^{p(\cdot)}(\Omega)$, we have

$$ (1) \quad \text{if } \|u\|_{L^{p(\cdot)}(\Omega)} \ge 1, \text{ then } \|u\|_{L^{p(\cdot)}(\Omega)}^{p^-} \le \int_{\Omega} |u|^p dx \le \|u\|_{L^{p(\cdot)}(\Omega)}^{p^+}, $$

$$ (2) \quad \text{if } \|u\|_{L^{p(\cdot)}(\Omega)} < 1, \text{ then } \|u\|_{L^{p(\cdot)}(\Omega)}^{p^+} \le \int_{\Omega} |u|^p dx \le \|u\|_{L^{p(\cdot)}(\Omega)}^{p^-}. $$

The variable exponent Sobolev space $W^{1,p(\cdot)}(\Omega)$ is the class of all functions $u \in L^{p(\cdot)}(\Omega)$ which have the property $|\nabla u| \in L^{p(\cdot)}(\Omega)$. The space $W^{1,p(\cdot)}(\Omega)$ is a Banach space equipped with the norm

$$ \|u\|_{W^{1,p(\cdot)}(\Omega)} = \|u\|_{L^{p(\cdot)}(\Omega)} + \|\nabla u\|_{L^{p(\cdot)}(\Omega)}. $$

More precisely, we have

**Proposition 2.3.** (see [6, Theorem 3.1].) Let $p \in \mathbf{P}(\Omega)$. The space $W^{1,p(\cdot)}(\Omega)$ is a Banach space, which is separable if $p \in L^\infty(\Omega)$ and reflexive if $1 < p^- \le p^+ < \infty$.

Next we will see the definitions that we use in this work. Firstly we will make some observations regarding to the trace. Let $p \in \mathbf{P}(\Omega)$. Obviously $W^{1,p(\cdot)}(\Omega) \subset W^{1,1}(\Omega)$ because $p^- \ge 1$. From $W^{1,1}(\Omega) \to L^1(\partial\Omega)$ we know that for all $u \in W^{1,p(\cdot)}(\Omega)$ there already holds $u|_{\partial\Omega} \in L^1(\partial\Omega)$. Thus for $W^{1,p(\cdot)}(\Omega)$, the trace $u|_{\partial\Omega}$ has definite meaning; see [3, p. 1398].

Define

$$ L_{\text{loc}}^{p(\cdot)}(\bar{\Omega}\setminus\Gamma) := \{u : \Omega \to \mathbb{R} \mid u \in L^{p(\cdot)}(U) \text{ for all open subset } U \subset \Omega \text{ with } \bar{U} \cap \Gamma = \emptyset\} $$

and

$$ W_{\text{loc}}^{1,p(\cdot)}(\bar{\Omega}\setminus\Gamma) := \{u : \Omega \to \mathbb{R} \mid u \in W^{1,p(\cdot)}(U) \text{ for all open subset } U \subset \Omega \text{ with } \bar{U} \cap \Gamma = \emptyset\}. $$

Similarly we define $L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$.

Let $v: \Omega \to \mathbb{R}$ be a function, we call $\text{supp } v = \{x \in \Omega \mid v(x) \neq 0\}$ the support of $v$. For $E \subset \mathbb{R}^n$ and $x \in \mathbb{R}^n$, we denote by $d(x, E)$ the Euclidean distance from $x$ to $E$.
**Definition 2.4.** We will say that $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ is a (weak) solution of equation (1.1) in $\bar{\Omega}\setminus\Gamma$ if

$$ \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + a(\cdot, u)\varphi + g(\cdot, u)\varphi \,dx + \int_{\partial\Omega} b(\cdot, u)\varphi + h(\cdot, u)\varphi \,d\sigma = 0 \quad (2.12) $$

for all $\varphi \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$, with $\text{supp} \varphi \subset \bar{\Omega}\setminus\Gamma$.

Let us observe that the trace of $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ possibly is not defined on $\partial\Omega$, however it is defined on $\{x \in \partial\Omega | d(x, \Gamma) > r\}$, and is essentially bounded, for small enough values $r > 0$.

**Definition 2.5.** We will say that the solution $u$ of equation (1.1) in $\bar{\Omega}\setminus\Gamma$ has a removable singularity at $\Gamma$: if $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ implies $u \in W^{1,p_1(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$ and the equality (2.12) is fulfilled for all $\varphi \in W^{1,p_1(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$.

### 3. THE BEHAVIOR OF SOLUTIONS NEAR THE SINGULAR SET

The main result of this section is Theorem 3.4, which will be used in the proof of Theorem 1.1. We begin with the following results.

**Lemma 3.1.** (see [2, p. 1004]). Let $0 < \theta < 1$, $\sigma > 0$, $\xi(h)$ be a nonnegative function on the interval $[1/2, 1]$, and let

$$ \xi(k) \le C_0(h-k)^{-\sigma} (\xi(h))^{\theta}, \quad 1/2 \le k < h \le 1, $$

for some positive constant $C_0$. Then, there exists $C_1(\sigma, \theta) > 0$ such that

$$ \xi(1/2) \le C_1 C_0^{\frac{1}{1-\theta}}. $$

**Lemma 3.2.** If $p \in (1, n)$, $u \in W^{1,p}(\mathcal{U})$ and $u = 0$ on $\{x \in \bar{\Omega} | d(x, \Gamma) = 2r_0\}$, then

$$ \left( \int_{\mathcal{U}} |u|^q dx \right)^{\frac{1}{q}} \le C \left( \int_{\mathcal{U}} |\nabla u|^p dx \right)^{\frac{1}{p}} \quad (3.13) $$

for each $q \in [p, \frac{np}{n-p})$, where $C = C(n, p, q, \mathcal{U})$ is a positive constant.

*Proof.* The proof is by contradiction, considering that $W^{1,p}(\mathcal{U})$ is compactly embedded in $L^q(\mathcal{U})$. $\square$

<figure><img src="image_2.png" /></figure>

We define,

$$ V_{\ell,r} = \{x \in \Omega \mid |d(x, \Gamma) - \ell| < r\}. $$

**Proposition 3.3.** Assume that the conditions (1.2) - (1.7), (1.9) and (1.11) are satisfied. Suppose that $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ satisfies

$$ \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + a(\cdot, u)\varphi + g(\cdot, u)\varphi \,dx + \int_{\partial\Omega} b(\cdot, u)\varphi + h(\cdot, u)\varphi \,d\sigma \le 0, \quad (3.14) $$

for all $\varphi \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$, $\varphi \ge 0$, with $\text{supp} \varphi \subset \bar{\Omega}\setminus\Gamma$. Then, if $0 < r < \ell < r_0$ we have the estimate

$$ \|\max\{u, 0\}\|_{L^\infty(V_{\ell,r/2})} \le C r^{-\tau}, \quad (3.15) $$

where $C = C(n, \mu, p_1, p_2, q_1, q_2, \mathcal{U}) > 0$ and $\tau = \tau(n, \mu, p_1, p_2, q_1, q_2, \mathcal{U}) > \frac{\max\{p_1^+, q_1^+\}}{\min\{p_2^-, q_2^-\} - \max\{p_1^+, q_1^+\} + 1}$.

*Proof.* 1. Let $u = C_*w$, where $C_* > 1$ is a number that will be determined below. We assume that $|\{x \in V_{\ell,r/2} | w(x) > 0\}| \ne 0$, otherwise, (3.15) is immediate. Set $\Omega' = \{x \in V_{\ell,r} | w(x) > 0\}$. Take

$$ m_t = \ess \sup \{w(x) \mid x \in V_{\ell, tr} \cap \Omega'\}, \quad 1/2 \le t \le 1. $$

Let $1/2 \le s < t \le 1$. Define the functions $z : \Omega \to \mathbb{R}$, $z_k : \Omega \to \mathbb{R}$ by

$$ \begin{aligned} z(x) &= w(x) - m_t \xi (|d(x, \Gamma) - \ell|), \\ z_k(x) &= \begin{cases} \max\{w(x) - m_t \xi (|d(x, \Gamma) - \ell|) - k, 0\} & \text{if } x \in V_{\ell, tr}, \\ 0 & \text{if } x \in \Omega \setminus V_{\ell, tr}, \end{cases} \end{aligned} $$

where $0 \le k \le \ess \sup_{\Omega'} z$, and $\xi : \mathbb{R} \to \mathbb{R}$ is a smooth function satisfying: $\xi \equiv 0$ on $(-\infty, sr]$, $\xi \equiv 1$ on $[\frac{s+t}{2}r, \infty)$,

$$ 0 \le \xi \le 1 \quad \text{and} \quad |\xi'| \le \frac{C_1}{r(t-s)} \quad \text{on } \mathbb{R}, $$
where $C_1$ is a suitable positive constant. Observe that $z_k \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ and $\text{supp } z_k \subset \bar{\Omega} \cap \{|d(\cdot, \Gamma) - \ell| \le \text{tr}\} \subset \bar{\Omega}\setminus\Gamma$. It is assumed that $m_{1/2} > 1$. The conclusion is obviously right for the case of $0 < m_{1/2} \le 1$. For simplicity we write $\xi(|d(x, \Gamma) - \ell|) = \xi(x)$ and $\xi'(|d(x, \Gamma) - \ell|) = \xi'(x)$.

Take $k \in [0, \mathcal{K})$, where $\mathcal{K} = \sup\{k \in [0, \text{ess sup}_{\Omega'} z] \mid \{|x \in V_{\ell, \text{tr}} \mid z_k(x) > 0\} \neq 0\}$. Observe that $\mathcal{K} \ge m_s \ge m_{1/2} > 1$. Substituting $\varphi = z_k$ into (3.14), we obtain

$$ \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla z_k \rangle + a(\cdot, u)z_k + g(\cdot, u)z_k \,dx + \int_{\partial\Omega} b(\cdot, u)z_k + h(\cdot, u)z_k \,d\sigma \le 0. $$

Denote $\Omega_k = \{x \in V_{\ell, \text{tr}} \mid z_k(x) > 0\}$. Then,

$$ \int_{\Omega_k} \left\langle A(\cdot, C_*w, C_*\nabla w), \nabla w - m_t \xi' \frac{d(\cdot, \Gamma) - \ell}{|d(\cdot, \Gamma) - \ell|} \nabla d(\cdot, \Gamma) \right\rangle \,dx \\ + \int_{\Omega_k} g(\cdot, u)z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} h(\cdot, u)z_k \,d\sigma \le \int_{\Omega_k} |a(\cdot, u)|z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} |b(\cdot, u)|z_k \,d\sigma. $$

By (1.2) - (1.7), we have

$$ \begin{aligned} & \int_{\Omega_k} \mu C_*^{p_1-1} |\nabla w|^{p_1} \,dx - \int_{\Omega_k} \frac{\mu^{-1} C_1 m_t}{r(t-s)} (|C_*\nabla w|^{p_1-1} + |C_*w|^{p_1-1} + 1) \,dx \\ & \quad + \int_{\Omega_k} (\mu|C_*w|^{p_2} - \mu^{-1}) z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} (\mu|C_*w|^{q_2} - \mu^{-1}) z_k \,d\sigma \\ & \le \int_{\Omega_k} \mu^{-1} (|C_*w|^{p_1-1} + 1) z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} \mu^{-1} (|C_*w|^{q_1-1} + 1) z_k \,d\sigma. \end{aligned} $$

Since $m_t > 1$, $C_* > 1$, and observing that $w \ge k$ and $m_t \ge w \ge z_k$ on $\Omega_k$, we have

$$ \begin{aligned} & \int_{\Omega_k} C_*^{p_1-1} |\nabla w|^{p_1} \,dx + \int_{\Omega_k} C_*^{p_2} k^{p_2} z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} C_*^{q_2} k^{q_2} z_k \,d\sigma \\ & \le C_2(\mu) \int_{\Omega_k} \frac{m_t C_*^{p_1-1}}{r(t-s)} |\nabla w|^{p_1-1} \,dx \tag{3.16} \\ & \quad + C_2 C_*^{\max\{p_1^+, q_1^+\}} - 1 \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|). \end{aligned} $$

On other hand, using Young's inequality,

$$ \int_{\Omega_k} \frac{m_t C_*^{p_1-1}}{r(t-s)} |\nabla w|^{p_1-1} \,dx \le \int_{\Omega_k} C_*^{p_1-1} \left\{ C_3(\varepsilon_1, p_1) \left[ \frac{m_t}{r(t-s)} \right]^{p_1} + \varepsilon_1 |\nabla w|^{p_1} \right\} \,dx. \quad (3.17) $$

Take $\varepsilon_1 = \frac{1}{2C_2}$. Then, by (3.16) and (3.17),

$$ \begin{aligned} & \frac{1}{2} \int_{\Omega_k} C_*^{p_1-1} |\nabla w|^{p_1} \,dx + \int_{\Omega_k} C_*^{p_2} k^{p_2} z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} C_*^{q_2} k^{q_2} z_k \,d\sigma \\ & \le C_4 (\mu, p_1) C_*^{\max\{p_1^+, q_1^+\}} - 1 \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|). \end{aligned} $$

Note that $\nabla z_k = \nabla w - m_t \xi' \frac{d(\cdot, \Gamma) - \ell}{|d(\cdot, \Gamma) - \ell|} \nabla d(\cdot, \Gamma)$ in $\Omega_k$, therefore

$$ \begin{aligned} & \frac{1}{2} \int_{\Omega_k} C_*^{p_1-1} \left( \frac{1}{2^{p_1-1}} |\nabla z_k|^{p_1} - m_t^{p_1} |\xi'|^{p_1} \right) \,dx + \int_{\Omega_k} C_*^{p_2} k^{p_2} z_k \,dx + \int_{\partial\Omega_k \cap \partial\Omega} C_*^{q_2} k^{q_2} z_k \,d\sigma \quad (3.18) \\ & \le C_4 C_*^{\max\{p_1^+, q_1^+\}} - 1 \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|). \end{aligned} $$

We have

$$ |\nabla z_k|^{p_1-1} \le 1 + |\nabla z_k|^{p_1}. \quad (3.19) $$
Then, from (3.18) and (3.19),

$$ \begin{aligned} & \frac{1}{2p_1^+} \int_{\Omega_k} C_*^{p_1-1} (|\nabla z_k|^{p_1^-} - 1) \, dx + \int_{\Omega_k} C_*^{p_2} k^{p_2} z_k \, dx + \int_{\partial\Omega_k \cap \partial\Omega} C_*^{q_2} k^{q_2} z_k \, d\sigma \\ & \le C_5 (\mu, p_1) C_*^{\max\{p_1^+, q_1^+\}-1} \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|). \end{aligned} \quad (3.20) $$

On the other hand, let $\gamma \in (p_1^-, \frac{(n-1)p_1^-}{n-p_1^-})$ and $\alpha \in (p_1^-, \gamma)$ be arbitrary. Observe that $\text{supp } z_k \cap \Omega \subset \Omega \cap \{|d(\cdot, \Gamma) - \ell| \le \text{tr}\} \subset \mathcal{U}$. By (3.13), and since the trace $W^{1,p_1^-}(\mathcal{U}) \to L^\gamma(\partial\mathcal{U})$ is continuous,

$$ \left( \int_{V_{\ell, \mathrm{tr}}} z_k^\alpha \mathrm{d}x \right)^{\frac{1}{\alpha}} + \left( \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial \Omega} z_k^\alpha \mathrm{d}\sigma \right)^{\frac{1}{\alpha}} \le C_6 r^{\frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha}} \left( \int_{V_{\ell, \mathrm{tr}}} |\nabla z_k|^{p_1^-} \mathrm{d}x \right)^{\frac{1}{p_1^-}}, \quad (3.21) $$

where $C_6 = C_6(n, \mu, p_1, \gamma, \mathcal{U}) > 0$. Using Hölder's inequality, we get

$$ \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x \le \left( \int_{V_{\ell, \mathrm{tr}}} z_k^\alpha \mathrm{d}x \right)^{\frac{1}{\alpha}} |\Omega_k|^{\frac{\alpha-1}{\alpha}}, \quad (3.22) $$

$$ \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial \Omega} z_k \mathrm{d}\sigma \le \left( \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial \Omega} z_k^\alpha \mathrm{d}\sigma \right)^{\frac{1}{\alpha}} |\partial \Omega_k \cap \partial \Omega|^{\frac{\alpha-1}{\alpha}}. \quad (3.23) $$

From (3.20) - (3.23), we have

$$ \begin{aligned} & \frac{C_*^{p_1^- - 1}}{2p_1^+} \left[ C_6^{-1} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|)^{-\frac{\alpha-1}{\alpha}} r^{-\frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha}} \left( \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x + \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial\Omega} z_k \mathrm{d}\sigma \right) \right]^{p_1^-} \\ & + C_*^{\min\{p_2^-, q_2^-\}} \min\{k^{p_2^-, k^{p_2^+}, k^{q_2^-, q_2^+}\}} \left( \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x + \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial\Omega} z_k \mathrm{d}\sigma \right) \\ & \le C_5 C_*^{\max\{p_1^+, q_1^+\}-1} \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|) + \frac{C_*^{p_1^- - 1}}{2p_1^+} |\Omega_k|. \end{aligned} \quad (3.24) $$

2. Take $\varepsilon \in (0, 1)$. Then, (3.24) implies

$$ \begin{aligned} & \varepsilon C_*^{p_1^- - 1} \left( \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x + \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial \Omega} z_k \mathrm{d}\sigma \right)^{p_1^-} r^{-p_1^- \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha}} \\ & + (1-\varepsilon) C_*^{\min\{p_2^-, q_2^-\}} \min\{k^{p_2^-, k^{p_2^+}, k^{q_2^-, q_2^+}\}} \\ & \cdot (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|)^{p_1^- \frac{\alpha-1}{\alpha}} \left( \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x + \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial\Omega} z_k \mathrm{d}\sigma \right) \\ & \le C_7 C_*^{\max\{p_1^+, q_1^+\}-1} \left\{ \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} + 1 \right\} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|)^{1+p_1^- \frac{\alpha-1}{\alpha}}, \end{aligned} \quad (3.25) $$

where $C_7 = C_7(C_5, C_6, \mu, p_1) > 0$. Applying Young's inequality $a^\varepsilon b^{1-\varepsilon} \le \varepsilon a + (1-\varepsilon)b$ in the left-hand side of (3.25), we obtain

$$ \begin{aligned} & \min\{k^{p_2^- (1-\varepsilon)}, k^{p_2^+ (1-\varepsilon)}, k^{q_2^- (1-\varepsilon)}, k^{q_2^+ (1-\varepsilon)}\} C_*^{(p_1^- - 1)\varepsilon + \min\{p_2^-, q_2^-\}} (1-\varepsilon) \\ & \cdot r^{-p_1^- \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha}} \varepsilon \left( \int_{V_{\ell, \mathrm{tr}}} z_k \mathrm{d}x + \int_{\partial V_{\ell, \mathrm{tr}} \cap \partial \Omega} z_k \mathrm{d}\sigma \right)^{p_1^- \varepsilon + 1-\varepsilon} \\ & \le C_7 C_*^{\max\{p_1^+, q_1^+\}-1} \left\{ \left[ \frac{m_t}{r(t-s)} \right]^{\max\{p_1^+, q_1^+\}} + 1 \right\} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|)^{1+p_1^- \frac{\alpha-1}{\alpha}} \varepsilon. \end{aligned} $$
Therefore,

$$
\begin{align}
& \min \left\{ k^{\frac{p_2^{-}(1-\epsilon)}{\beta}}, k^{\frac{p_2^{+}(1-\epsilon)}{\beta}}, k^{\frac{q_2^{-}(1-\epsilon)}{\beta}}, k^{\frac{q_2^{+}(1-\epsilon)}{\beta}} \right\} C_* \frac{\min\{p_2^{-}, q_2^{-}\} - \max\{p_1^{+}, q_1^{+}\} + 1 - (\min\{p_2^{-}, q_2^{-}\} - p_1^{-} + 1)_{\frac{\epsilon}{\beta}}}{\beta} \nonumber \\
& \le (2C_7)^{\frac{1}{\beta}} \left( \int_{\Omega_k} z_k \, dx + \int_{\partial\Omega_k \cap \partial\Omega} z_k \, d\sigma \right)^{-\frac{p_1^{-}\epsilon+1-\epsilon}{\beta}} r^{p_1^{-}} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\epsilon}{\beta} \left[ \frac{m_t}{r(t-s)} \right]^{\frac{\max\{p_1^{+}, q_1^{+}\}}{\beta}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|), \tag{3.26}
\end{align}
$$

where $\beta = 1 + p_1^{-} \frac{\alpha-1}{\alpha} \epsilon$.

3. Integrating (3.26) with respect to $k$,

$$
\begin{align*}
& C_* \frac{\min\{p_2^{-}, q_2^{-}\} - \max\{p_1^{+}, q_1^{+}\} + 1 - (\min\{p_2^{-}, q_2^{-}\} - p_1^{-} + 1)_{\frac{\epsilon}{\beta}}}{\beta} \int_0^\kappa \min\left\{ k^{\frac{p_2^{-}}{\beta}}, k^{\frac{p_2^{+}}{\beta}}, k^{\frac{q_2^{-}}{\beta}}, k^{\frac{q_2^{+}}{\beta}} \right\} dk \\
& \le (2C_7)^{\frac{1}{\beta}} r^{p_1^{-}} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\epsilon}{\beta} \left[ \frac{m_t}{r(t-s)} \right]^{\frac{\max\{p_1^{+}, q_1^{+}\}}{\beta}} \\
& \quad \cdot \int_0^\kappa \left( \int_{\Omega_k} z_k \, dx + \int_{\partial\Omega_k \cap \partial\Omega} z_k \, d\sigma \right)^{-\frac{p_1^{-}\epsilon+1-\epsilon}{\beta}} (|\Omega_k| + |\partial\Omega_k \cap \partial\Omega|) dk.
\end{align*}
$$

Let us consider the equalities

$$
\frac{d}{dk} \left( \int_{\Omega_k} z_k \, dx \right) = -|\Omega_k| \quad \text{and} \quad \frac{d}{dk} \left( \int_{\partial \Omega_k \cap \partial \Omega} z_k \, d\sigma \right) = -|\partial \Omega_k \cap \partial \Omega|.
$$

Since $1 - \frac{p_1^{-}\epsilon+1-\epsilon}{\beta} > 0$ and $\mathcal{K} > 1$,

$$
\begin{align*}
& \mathcal{K}^{1+\frac{\min\{p_2^{-}, q_2^{-}\}(1-\epsilon)}{\beta}} C_* \frac{\min\{p_2^{-}, q_2^{-}\} - \max\{p_1^{+}, q_1^{+}\} + 1 - (\min\{p_2^{-}, q_2^{-}\} - p_1^{-} + 1)_{\frac{\epsilon}{\beta}}}{\beta} \\
& \le \epsilon^{-1} C_8 r^{p_1^{-}} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\epsilon}{\beta} \left[ \frac{m_t}{r(t-s)} \right]^{\frac{\max\{p_1^{+}, q_1^{+}\}}{\beta}} \left( \int_{\Omega_0} z_0 \, dx + \int_{\partial\Omega_0 \cap \partial\Omega} z_0 \, d\sigma \right)^{1-\frac{p_1^{-}\epsilon+1-\epsilon}{\beta}},
\end{align*}
$$

where $C_8 = C_8(n, \mu, \alpha, \gamma, p_1, p_2, q_2, \mathcal{U}) > 0$. We note $\mathcal{K} \ge m_s$. Apply the estimates

$$
\int_{\Omega_0} z_0 dx \le m_t |V_{\ell,r}| \quad \text{and} \quad \int_{\partial \Omega_0 \cap \partial \Omega} z_0 d\sigma \le m_t |\partial V_{\ell,r} \cap \partial \Omega|,
$$

we have

$$
\begin{align*}
& m_s^{1+\frac{\min\{p_2^{-}, q_2^{-}\}(1-\epsilon)}{\beta}} C_* \frac{\min\{p_2^{-}, q_2^{-}\} - \max\{p_1^{+}, q_1^{+}\} + 1 - (\min\{p_2^{-}, q_2^{-}\} - p_1^{-} + 1)_{\frac{\epsilon}{\beta}}}{\beta} \\
& \le C_9 r^{p_1^{-}} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\epsilon}{\beta} \left[ \frac{m_t}{r(t-s)} \right]^{\frac{\max\{p_1^{+}, q_1^{+}\}}{\beta}} m_t^{1-\frac{p_1^{-}\epsilon+1-\epsilon}{\beta}} r^{(n-d-1)\left(1-\frac{p_1^{-}\epsilon+1-\epsilon}{\beta}\right)},
\end{align*}
$$

where $C_9 = C_9(n, \mu, \varepsilon, \alpha, \gamma, p_1, p_2, q_2, \mathcal{U}) > 0$.

Now we take $C_* > 0$ such that

$$
C_* \frac{\min\{p_2^{-}, q_2^{-}\} - \max\{p_1^{+}, q_1^{+}\} + 1 - (\min\{p_2^{-}, q_2^{-}\} - p_1^{-} + 1)_{\frac{\epsilon}{\beta}}}{\beta} = r^{p_1^{-}} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\epsilon}{\beta} - \frac{\max\{p_1^{+}, q_1^{+}\}}{\beta} + (n-d-1) \left(1 - \frac{p_1^{-}\epsilon+1-\epsilon}{\beta}\right).
$$
Hence $C_* = r^{-\tau}$, where

$$
\begin{align*}
\tau &= - \left[ p_1^{-} \frac{(n-d-1)(\gamma-\alpha)}{\gamma\alpha} \frac{\varepsilon}{\beta} - \frac{\max\{p_1^+, q_1^+\}}{\beta} + (n-d-1) \left( 1 - \frac{p_1^-\varepsilon + 1 - \varepsilon}{\beta} \right) \right] \\
&\quad \cdot \left[ \frac{\min\{p_2^-, q_2^-\} - \max\{p_1^+, q_1^+\} + 1 - (\min\{p_2^-, q_2^-\} - p_1^- + 1)\varepsilon}{\beta} \right]^{-1} \\
&> \frac{\max\{p_1^+, q_1^+\}}{\min\{p_2^-, q_2^-\} - \max\{p_1^+, q_1^+\} + 1} > 0,
\end{align*}
$$

if $1 > (n-d-1)(1-\frac{p_1^-}{\gamma})$ and $\varepsilon \in (0, \frac{\min\{p_2^-, q_2^-\}-\max\{p_1^+, q_1^+\}+1}{\min\{p_2^-, q_2^-\}-p_1^-+1})$.

On the other hand,

$$
m_s \le C_9^{\frac{\beta}{\max\{p_1^+, q_1^+\}}} \sigma^{\frac{\theta}{(t-s)^\sigma}},
$$

where

$$
\theta = \left[ \frac{\max\{p_1^+, q_1^+\}}{\beta} + 1 - \frac{p_1^- \varepsilon + 1 - \varepsilon}{\beta} \right] \left[ 1 + \frac{\min\{p_2^-, q_2^-\} (1 - \varepsilon)}{\beta} \right]^{-1} < 1,
$$
$$
\sigma = \left[ \frac{\max\{p_1^+, q_1^+\}}{\beta} \right] \left[ 1 + \frac{\min\{p_2^-, q_2^-\} (1 - \varepsilon)}{\beta} \right]^{-1} .
$$

By virtue of Lemma 3.1, we derive

$$
m_{1/2} \le C_{10} (n, \mu, p_1, p_2, q_1, q_2, \mathcal{U}).
$$

From the substitution $u = C_*w$ we obtain

$$
\operatorname{ess \, sup} \{u(x) \mid x \in V_{\ell,r/2} \cap \Omega'\} = C_* m_{1/2} \le C_{10} C_* = C_{10} r^{-\tau}.
$$

Therefore, we conclude the proof of the Proposition 3.3.

<figure><img src="image_3.png" /></figure>

The next theorem follows easily from the Proposition 3.3.

**Theorem 3.4.** Suppose that the conditions (1.2) - (1.9) and (1.11) are satisfied. Let $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ be a solution of equation (1.1) in $\bar{\Omega}\setminus\Gamma$. Then, in $\{x \in \Omega \mid 0 < d(x, \Gamma) < r_0\}$, the following inequality holds almost everywhere:

$$
|u(x)| \leq C d(x, \Gamma)^{-\tau}, \quad (3.27)
$$

where $C = C(n, \mu, p_1, p_2, q_1, q_2, \mathcal{U}) > 0$ and $\tau = \tau(n, \mu, p_1, p_2, q_1, q_2, \mathcal{U}) > \frac{\max\{p_1^+, q_1^+\}}{\min\{p_2^-, q_2^-\}-\max\{p_1^+, q_1^+\}+1}$.

Proceeding in the same way as in the Proposition 3.3 and Theorem 3.4, we have the following results.

**Proposition 3.5.** Suppose that the conditions (1.2) - (1.5) and (1.11) are satisfied, additionally

$$
p_2^- - p_1^+ + 1 > 0. \tag{3.28}
$$

Assume that $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ satisfies

$$
\int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + a(\cdot, u)\varphi + g(\cdot, u)\varphi \,dx \le 0, \quad (3.29)
$$

for all $\varphi \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$, $\varphi \ge 0$, with $\text{supp}\varphi \subset \bar{\Omega}\setminus\Gamma$. Then, if $0 < r < \ell < r_0$ we have the estimate

$$
\|\max\{u, 0\}\|_{L^{\infty}(V_{\ell,r/2})} \le C r^{-\tau},
$$

where $C = C(n, \mu, p_1, p_2, \mathcal{U}) > 0$ and $\tau = \tau(n, \mu, p_1, p_2, \mathcal{U}) > \frac{p_1^+}{p_2^- - p_1^+ + 1}$.

**Theorem 3.6.** Suppose that the conditions (1.2) - (1.5), (1.8), (1.11) and (3.28) are satisfied. Let $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ be a solution of equation (1.1) in $\bar{\Omega}\setminus\Gamma$, with $b \equiv h \equiv 0$. Then, in $\{x \in \Omega \mid 0 < d(x, \Gamma) < r_0\}$, the following inequality holds almost everywhere:

$$
|u(x)| \leq C d(x, \Gamma)^{-\tau},
$$

where $C = C(n, \mu, p_1, p_2, \mathcal{U}) > 0$ and $\tau = \tau(n, \mu, p_1, p_2, \mathcal{U}) > \frac{p_1^+}{p_2^- - p_1^+ + 1}$.
## 4. THE REMOVABILITY OF SINGULAR SET

Next, we prove the main theorem of this paper. Before, we start with the following

**Lemma 4.1.** Suppose that the conditions (1.2) - (1.7), (1.9) - (1.11) are satisfied. If $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ satisfies (3.14), then

$$ \max\{u, 0\} \in L^{\infty}(\Omega). $$

*Proof.* We proceed by contradiction. For $r \in (0, r_0^2)$, we denote

$$ \Lambda(r) = \ess\sup \{\max\{u(x), 0\} \mid r \le d(x, \Gamma) \le r_0^2, x \in \Omega\}. $$

We have $\lim_{r \to 0^+} \Lambda(r) = \infty$. For sufficiently small values $r$ we define the function $\psi_r : \mathbb{R} \to \mathbb{R}$ as follows:

$$ \psi_r(t) = \begin{cases} 0 & \text{if } t < r, \\ 1 & \text{if } t > \sqrt{r}, \\ \frac{2}{\ln \frac{1}{r}} \ln \frac{t}{r} & \text{if } r \le t \le \sqrt{r}. \end{cases} $$

Choosing $\delta > 0$ such that $\Lambda(\delta) > 1$, set

$$ \varphi = \left( \ln \max \left\{ \frac{u}{\Lambda(\delta)}, 1 \right\} \right) \psi_r^{\gamma} \circ d(\cdot, \Gamma), $$

where $\gamma = \ess\sup_{\Omega} \frac{p_1 p_2}{p_2 - p_1 + 1}$. We have $\varphi \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$, with $\text{supp} \varphi \subset \bar{\Omega} \cap \{d(\cdot, \Gamma) \ge r\} \subset \bar{\Omega}\setminus\Gamma$.

For simplicity we write $\psi_r \circ d(\cdot, \Gamma) = \psi_r$ and $\psi'_r \circ d(\cdot, \Gamma) = \psi'_r$.

Denote $\Omega_\delta = \{x \in \Omega \mid u(x) > \Lambda(\delta)\}$. Substituting $\varphi$ into (3.14), we obtain

$$ \int_{\Omega_\delta} \left[ \frac{\psi_r^\gamma}{u} \langle A(\cdot, u, \nabla u), \nabla u \rangle + \gamma \psi_r' \psi_r^{\gamma-1} \left( \ln \frac{u}{\Lambda(\delta)} \right) \langle A(\cdot, u, \nabla u), \nabla d(\cdot, \Gamma) \rangle \right. \\ \left. + a(\cdot, u) \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) + g(\cdot, u) \psi_r' \left( \ln \frac{u}{\Lambda(\delta)} \right) \right] dx \\ + \int_{\partial \Omega_\delta \cap \partial \Omega} \left[ b(\cdot, u) \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) + h(\cdot, u) \psi_r' \left( \ln \frac{u}{\Lambda(\delta)} \right) \right] d\sigma \le 0. $$

Since $u > \Lambda(\delta) > 1$ in $\Omega_\delta$, and by virtue of the conditions (1.2) - (1.7), we have

$$ \begin{aligned} & \int_{\Omega_\delta} \mu \frac{\psi_r^\gamma}{u} |\nabla u|^{p_1} dx + \int_{\Omega_\delta} \left( \mu - \frac{3\mu^{-1}}{\Lambda(\delta)^{p_2-p_1+1}} \right) \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) u^{p_2} dx \\ & + \int_{\partial \Omega_\delta \cap \Omega} \left( \mu - \frac{3\mu^{-1}}{\Lambda(\delta)^{q_2-q_1+1}} \right) \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) u^{q_2} d\sigma \\ & \le \int_{\Omega_\delta} \mu^{-1} \gamma \psi_r' \psi_r^{\gamma-1} \left( \ln \frac{u}{\Lambda(\delta)} \right) (|\nabla u|^{p_1-1} + u^{p_1-1} + 1) dx \\ & \le \int_{\Omega_\delta} \mu^{-1} \gamma \psi_r^\gamma u^{-1} \left\{ C_1 (\varepsilon_1, p_1) \left[ \psi_r' \psi_r^{-1} u \left( \ln \frac{u}{\Lambda(\delta)} \right) \right]^{p_1} + \varepsilon_1 |\nabla u|^{p_1} \right\} dx \\ & + \int_{\Omega_\delta} \mu^{-1} \gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) \left[ (C_2 (\varepsilon_2, p_1, p_2) + 1) (\psi_r')^{\frac{p_2}{p_2-p_1+1}} + \varepsilon_2 \psi_r^{\frac{\gamma-1}{p_1-1}} u^{p_2} + \psi_r^{\frac{\gamma-1}{p_1-1}} \right] dx. \end{aligned} $$

We can assume that

$$ \varepsilon_3 = \mu - \frac{\mu^{-1}(3 + \gamma)}{\Lambda(\delta)^{\min\{p_2, q_2\} - \max\{p_1, q_1\} + 1}} > 0, $$

because $\lim_{r \to 0^+} \Lambda(r) = \infty$. Take $\varepsilon_1 = \frac{\mu^2}{2\gamma}$, $\varepsilon_2 = \frac{\mu\varepsilon_3}{2\gamma}$. Since $\frac{\gamma-1}{p_1-1} \ge \gamma$ and $\psi_r \le 1$,

$$ \begin{aligned} & \int_{\Omega_\delta} \frac{\psi_r^\gamma}{u} |\nabla u|^{p_1} dx + \int_{\Omega_\delta} \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) u^{p_2} dx + \int_{\partial \Omega_\delta \cap \Omega} \psi_r^\gamma \left( \ln \frac{u}{\Lambda(\delta)} \right) u^{q_2} d\sigma \\ & \le C_3 \int_{\Omega_\delta} (\psi_r')^{p_1} \psi_r^{\gamma-p_1} u^{p_1-1} \left( \ln \frac{u}{\Lambda(\delta)} \right)^{p_1} + \left( \ln \frac{u}{\Lambda(\delta)} \right) (\psi_r')^{\frac{p_2}{p_2-p_1+1}} dx, \end{aligned} \quad (4.30) $$
where $C_3 = C_3(\mu, \gamma, \varepsilon_3, p_1, p_2) > 0$.

Additionally, let us consider

$$ (\psi_r')^{p_1} \psi_r^{\gamma-p_1} u^{p_1-1} \left(\ln \frac{u}{\Lambda(\delta)}\right)^{p_1} \\ \le \left(\ln \frac{u}{\Lambda(\delta)}\right) \left\{ C_4 (\varepsilon_4, p_1, p_2) \left[ (\psi_r')^{p_1} \left(\ln \frac{u}{\Lambda(\delta)}\right)^{p_1-1} \right]^{\frac{p_2}{p_2-p_1+1}} + \varepsilon_4 \psi_r^{\frac{(\gamma-p_1)p_2}{p_1-1}} u^{p_2} \right\}, \\ \frac{(\gamma-p_1)p_2}{p_1-1} \ge \gamma \quad \text{and} \quad \psi_r'(t) = \frac{2}{t \ln \frac{1}{r}} > 1 \text{ for } t \in (r, \sqrt{r}). $$

Choose $\varepsilon_4 = \frac{1}{2C_3}$. By (4.30),

$$ \int_{\Omega_{\delta}} \frac{\psi_r^{\gamma}}{u} |\nabla u|^{p_1} dx + \int_{\Omega_{\delta}} \psi_r^{\gamma} \left(\ln \frac{u}{\Lambda(\delta)}\right) u^{p_2} dx + \int_{\partial \Omega_{\delta} \cap \Omega} \psi_r^{\gamma} \left(\ln \frac{u}{\Lambda(\delta)}\right) u^{q_2} d\sigma \\ \le C_5 (\mu, \gamma, \varepsilon_3, p_1, p_2) \int_{\Omega_{\delta} \cap \{r \le d(\cdot, \Gamma) \le \sqrt{r}\}} \left(\ln \frac{u}{\Lambda(\delta)}\right) \left[ \left(\ln \frac{u}{\Lambda(\delta)}\right)^{\frac{(p_1-1)p_2}{p_2-p_1+1}} + 1 \right] \left(\frac{2}{d(\cdot, \Gamma) \ln \frac{1}{r}}\right)^{\frac{p_1 p_2}{p_2-p_1+1}} dx. $$

We can assume that $1 < \ln \frac{1}{d(x,\Gamma)}$ if $0 < d(x,\Gamma) \le \sqrt{r}$. Using (3.27), we see

$$ \int_{\Omega_{\delta}} \frac{\psi_r^{\gamma}}{u} |\nabla u|^{p_1} dx + \int_{\Omega_{\delta}} \psi_r^{\gamma} \left(\ln \frac{u}{\Lambda(\delta)}\right) u^{p_2} dx \\ \le C_6 \left(\ln \frac{1}{r}\right)^{-\frac{p_1 p_2}{p_2-p_1+1}} \int_{\Omega_{\delta} \cap \{r \le d(\cdot, \Gamma) \le \sqrt{r}\}} \left(\ln \frac{1}{d(\cdot, \Gamma)}\right)^{1+\frac{(p_1-1)p_2}{p_2-p_1+1}} \left(\frac{1}{d(\cdot, \Gamma)}\right)^{\frac{p_1 p_2}{p_2-p_1+1}} dx \\ \le C_7 \left(\ln \frac{1}{r}\right)^{-\frac{p_1 p_2}{p_2-p_1+1}} \int_r^{\sqrt{r}} \left(\ln \frac{1}{r}\right)^{1+\frac{(p_1-1)p_2}{p_2-p_1+1}} \left(\frac{1}{t}\right)^{\gamma} t^{n-d-1} dt \\ = C_7 \left(\ln \frac{1}{r}\right)^{-\frac{p_1 p_2}{p_2-p_1+1}} \left(\ln \frac{1}{r}\right)^{1+\frac{(p_1-1)p_2}{p_2-p_1+1}} \frac{r^{\frac{n-d-\gamma}{2}}}{n-d-\gamma} \left(1-r^{\frac{n-d-\gamma}{2}}\right), $$

where $C_i = C_i(n, \tau, \mu, \gamma, \varepsilon_3, p_1, p_2, q_1, q_2, \mathcal{U}) > 0$, $i = 6, 7$. Since $\lim_{r \to 0^+} 1/r = \infty$, we can assume that $(\ln 1/r)^{1+\frac{(p_1-1)p_2}{p_2-p_1+1}} \le r^{-\frac{n-d-\gamma}{4}}$. Therefore, if $r \to 0^+$,

$$ \int_{\Omega_{\delta}} \frac{|\nabla u|^{p_1}}{u} dx + \int_{\Omega_{\delta}} \left(\ln \frac{u}{\Lambda(\delta)}\right) u^{p_2} dx = 0. $$

Hence, $u(x) = \Lambda(\delta)$ almost every in $\Omega_{\delta}$. Thus, we have a contradiction, and proves the Lemma 4.1. $\square$

<figure><img src="image_4.png" /></figure>

Now we are ready to prove the main Theorem 1.1.

**Proof of Theorem 1.1.** 1. First we prove $u \in W^{1,p_1(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$. As consequence of a Lemma 4.1, $u \in L^{\infty}(\Omega)$. Next, for $r < 2r_0/5$, let $\psi_r : \mathbb{R} \to \mathbb{R}$ be a smooth function such that

$$ \psi_r(t) = \begin{cases} 0 & \text{if } t < r/2 \text{ or } t > 5r/2, \\ 1 & \text{if } r < t < 2r, \end{cases} $$

$0 \le \psi_r \le 1$ and $|\psi_r'| \le C/r$, where $C$ is a suitable positive constant. Set

$$ \varphi = \left( \psi_r^{p_1} \circ d(\cdot, \Gamma) \right) u. $$

We have $\varphi \in W^{1,p_1(\cdot)}(\bar{\Omega} \setminus \Gamma) \cap L^{\infty}_{\text{loc}}(\bar{\Omega} \setminus \Gamma)$, with $\text{supp} \varphi \subset \bar{\Omega} \cap \{r/2 \le d(\cdot, \Gamma) \le 5r/2\}$. For simplicity we write $\psi_r = \psi_r \circ d(\cdot, \Gamma)$ and $\psi_r' = \psi_r' \circ d(\cdot, \Gamma)$. Substituting $\varphi$ into (2.12),

$$ \int_{\Omega} \left\langle A(\cdot, u, \nabla u), p_1^{+} \psi_r^{p_1^{+}-1} \psi_r' u \nabla d(\cdot, \Gamma) + \psi_r^{p_1^{+}} \nabla u \right\rangle \\ + a(\cdot, u) \psi_r^{p_1^{+}} u + g(\cdot, u) \psi_r^{p_1^{+}} u dx + \int_{\partial \Omega} b(\cdot, u) \psi_1^{p_1^{+}} u + h(\cdot, u) \psi_1^{p_1^{+}} u d\sigma \le 0 $$
By conditions (1.2) - (1.7), we have

$$
\begin{align*}
& \int_{\Omega} \mu |\nabla u|^{p_1} \psi_r^{p_1} dx + \int_{\Omega} \mu \psi_r^{p_1} |u|^{p_2+1} dx + \int_{\partial \Omega} \mu \psi_r^{p_1} |u|^{q_2+1} d\sigma \\
& \leq \int_{\Omega} \mu^{-1} p_1^+ \psi_r^{p_1-1} |\psi_r'| |u| \left( |\nabla u|^{p_1-1} + |u|^{p_1-1} + 1 \right) dx \\
& \quad + \int_{\Omega} \mu^{-1} \left( |u|^{p_1-1} + 1 \right) \psi_r^{p_1} |u| + \mu^{-1} \psi_r^{p_1} |u| dx \\
& \quad + \int_{\partial \Omega} \mu^{-1} \left( |u|^{q_1-1} + 1 \right) \psi_r^{p_1} |u| + \mu^{-1} \psi_r^{p_1} |u| d\sigma.
\end{align*}
$$

Since $|u|^{p_1} \le \max \left\{ \|u\|_{L^\infty(\Omega)}^{p_1^+}, \|u\|_{L^\infty(\Omega)}^{p_1^-} \right\}$ and $|u|^{q_1} \le \max \left\{ \|u\|_{L^\infty(\partial\Omega)}^{q_1^+}, \|u\|_{L^\infty(\partial\Omega)}^{q_1^-} \right\}$,

$$
\int_{\Omega} |\nabla u|^{p_1} \psi_r^{p_1} dx \\
\le \mu^{-2} p_1^+ \int_{\Omega} C_1 (\varepsilon_1, p_1) \left[ |\psi_r'| |u| \psi_r^{p_1+1-\frac{p_1^+(p_1-1)}{p_1}} \right]^{p_1} + \varepsilon_1 \left[ |\nabla u|^{p_1-1} \psi_r^{\frac{p_1^+(p_1-1)}{p_1}} \right]^{\frac{p_1}{p_1-1}} dx + C_2 r^{n-d-1},
$$

where $C_2$ is a positive constant independent of $r$. Choosing $\varepsilon_1 = \frac{\mu^2}{2p_1^+}$,

$$
\int_{\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}} |\nabla u|^{p_1} dx \le C_3 r^{n-d-p_1^+},
$$

where $C_3$ is a positive constant independent of $r$. Therefore,

$$
\int_{\Omega \cap \{d(\cdot, \Gamma) \le 2r\}} |\nabla u|^{p_1} dx \\
= \sum_{i=0}^{\infty} \int_{\Omega \cap \{2^{-i}r \le d(\cdot, \Gamma) \le 2^{-i+1}r\}} |\nabla u|^{p_1} dx \le C_3 \sum_{i=0}^{\infty} \left(\frac{r}{2^i}\right)^{n-d-p_1^+} < \infty.
$$

So $|\nabla u| \in L^{p_1(\cdot)}(\Omega)$, and thus we have proved that $u \in W^{1,p_1(\cdot)}(\Omega) \cap L^\infty(\Omega)$.

2. Now, we will show that $u$ is a solution of equation (1.1) in the domain $\Omega$. For $r \in (0, r_0)$, let $\xi_r : \mathbb{R} \to \mathbb{R}$ be a smooth function such that

$$
\xi_r(t) = \begin{cases} 1 & \text{if } |t| \le r, \\ 0 & \text{if } 2r \le |t|, \end{cases}
$$

$0 \le \xi \le 1$ and $|\xi'_r| \le C/r$, where $C$ is a suitable positive constant. For simplicity we write $\xi_r = \xi_r \circ d(\cdot, \Gamma)$ and $\xi'_r = \xi'_r \circ d(\cdot, \Gamma)$. Let $\varphi \in W^{1,p_1(\cdot)}(\Omega) \cap L^\infty(\Omega)$. We have $(1 - \xi_r \circ d(\cdot, \Gamma)) \varphi \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega} \setminus \Gamma) \cap L_{\text{loc}}^\infty(\bar{\Omega} \setminus \Gamma)$, with $\text{supp} \varphi \subset \bar{\Omega} \setminus \Gamma$. Then, (2.12) yields

$$
\int_{\Omega} \langle A(\cdot, u, \nabla u), (1 - \xi_r) \nabla \varphi - \varphi \xi_r' \nabla d(\cdot, \Gamma) \rangle + a(\cdot, u) (1 - \xi_r) \varphi + g(\cdot, u) (1 - \xi_r) \varphi dx \\
+ \int_{\partial \Omega} b(\cdot, u) (1 - \xi_r) \varphi + h(\cdot, u) (1 - \xi_r) \varphi d\sigma = 0. \tag{4.31}
$$

When $r \to 0^+$, for all $\varphi \in W^{1,p_1(\cdot)}(\Omega) \cap L^\infty(\Omega)$, the equality (4.31) implies (2.12). Indeed, we have

$$
\begin{align*}
& \lim_{r \to 0} \int_{\Omega} \langle A(\cdot, u, \nabla u), (1 - \xi_r) \nabla \varphi \rangle + a(\cdot, u) (1 - \xi_r) \varphi + g(\cdot, u) (1 - \xi_r) \varphi dx \\
& \quad + \int_{\partial \Omega} b(\cdot, u) (1 - \xi_r) \varphi + h(\cdot, u) (1 - \xi_r) \varphi d\sigma \\
& = \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + a(\cdot, u) \varphi + g(\cdot, u) \varphi dx + \int_{\partial \Omega} b(\cdot, u) \varphi + h(\cdot, u) \varphi d\sigma.
\end{align*}
$$
Additionally, by (1.3), Proposition 2.1 and 2.2:

$$
\begin{aligned}
& \left| \int_{\Omega} \langle A(\cdot, u, \nabla u), \varphi \xi_r' \nabla d(\cdot, \Gamma) \rangle \, dx \right| \\
& \le \frac{C_4}{r} \int_{\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}} (|\nabla u|^{p_1-1} + |u|^{p_1-1} + 1) \varphi \, dx \\
& \le \frac{C_5}{r} \int_{\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}} |\nabla u|^{p_1-1} + |u|^{p_1-1} + 1 \, dx \\
& \le \frac{C_6}{r} \left\| |\nabla u|^{p_1-1} \right\|_{L^{\frac{p_1}{p_1-1}}(\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\})} \|1\|_{L^{p_1}(\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\})} + C_6 r^{n-d-1} \\
& \le \frac{C_7}{r} \max \left\{ \left( \int_{\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}} |\nabla u|^{p_1} \, dx \right)^{\left(\frac{p_1-1}{p_1}\right)^+}, \left( \int_{\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}} |\nabla u|^{p_1} \, dx \right)^{\left(\frac{p_1-1}{p_1}\right)^-} \right\} \\
& \cdot |\Omega \cap \{r \le d(\cdot, \Gamma) \le 2r\}|^{\frac{1}{p_1}} + C_6 r^{n-d-1} \\
& \le C_8 r^{\frac{n-d-p_1}{p_1}} \to 0 \quad \text{as } r \to 0,
\end{aligned}
$$

where $C_i$, $i = 4, \dots, 8$, are positive constants independent of $r$. So, we have obtained that equality (2.12) is fulfilled for all $\varphi \in W^{1,p_1(\cdot)}(\Omega) \cap L^\infty(\Omega)$. Therefore, the singular set $\Gamma$ is removable for solutions of equation (1.1).

<figure><img src="image_5.png" /></figure>

Similarly as in the Lemma 4.1 and in the Theorem 1.1 we have the following results.

**Lemma 4.2.** Suppose that the conditions (1.2) - (1.5), (1.9) - (1.11) and (3.28) are satisfied. If $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ satisfies (3.29), then

$$ \max\{u, 0\} \in L^{\infty}(\Omega). $$

**Theorem 4.3.** Suppose that the conditions (1.2) - (1.5), (1.8) - (1.11) and (3.28) are satisfied. Let $u \in W_{\text{loc}}^{1,p_1(\cdot)}(\bar{\Omega}\setminus\Gamma) \cap L_{\text{loc}}^{\infty}(\bar{\Omega}\setminus\Gamma)$ be a solution of equation (1.1) in $\bar{\Omega}\setminus\Gamma$, with $b \equiv h \equiv 0$. Then, the singularity of $u$ at $\Gamma$ is removable.

**Acknowledgments:** I would like to thank Professor Sergio Almaraz for useful suggestions.

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September 12, 2022

UNIVERSIDADE FEDERAL FLUMINENSE, INSTITUTO DE MATEMÁTICA, CAMPUS DO GRAGOATÁ, RUA PROF. MARCOS WALDEMAR DE FREITAS, S/N, BLOCO H, NITERÓI, RJ 24210-201, BRAZIL

Email address: jpablo@id.uff.br