REMOVABLE SINGULARITIES FOR
SUB-SOLUTIONS OF ELLIPTIC EQUATIONS
INVOLVING WEIGHTED VARIABLE EXPONENT
SOBOLEV SPACES

Juan Apaza

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Juan Apaza. REMOVABLE SINGULARITIES FOR SUB-SOLUTIONS OF ELLIPTIC EQUATIONS INVOLVING WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES. 2022. hal-03671853v5

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Distributed under a Creative Commons Attribution 4.0 International License
REMOVABLE SINGULARITIES FOR SUB-SOLUTIONS OF ELLIPTIC EQUATIONS INVOLVING WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES

JUAN ALCON APAZA

ABSTRACT. We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces $W^{1,p(\cdot)}(\Omega, \vartheta)$.

# 1. INTRODUCTION

This paper is devoted to the study of conditions guaranteeing the removability of singular set for non-negative weak sub-solutions, in the weighted variable exponent Sobolev spaces, for nonlinear elliptic equations of the form

$$-\operatorname{div} A(x, u, \nabla u) + B(x, u) = 0.$$

For the exacts requirements on the mappings A and B we refer to Section 3. In this work we employ the arguments used to prove the following results.

**Theorem A.** (see [13, Theorem 1]) Let $\Omega \subset \mathbb{R}^n$, $n \ge 2$, be an open set, and $E \subset \Omega$ be closed in $\Omega$. Denote by $M^\alpha(E)$ the $\alpha$-dimensional upper Minkowski content of E. Suppose that $\alpha \in [0, n-2]$ and $M^\alpha(E) = 0$. If f is subharmonic in $\Omega \setminus E$ and satisfies

$$f(x) \le C d(x, E)^{\alpha+2-n} \quad (x \in \Omega \setminus E)$$

for some positive constant C, then f has a subharmonic extension to $\Omega$.

**Theorem B.** (see [13, Theorem 3]) Let $\Omega \subset \mathbb{R}^n$, $n \ge 2$, be an open set, and $E \subset \Omega$ be closed in $\Omega$. Denote by $H^\alpha(E)$ the $\alpha$-dimensional Hausdorff measure of E. Suppose that $\alpha \in [0, n-2]$ and $H^\alpha(E) = 0$. If f is subharmonic in $\Omega \setminus E$ and satisfies

$$\frac{1}{|B(x, r)|} \int_{B(x, r)} \max\{f, 0\}dx \le Cr^{\alpha+2-n} \quad (\overline{B(x, r)} \subset \Omega)$$

for some positive constant C, then f has a subharmonic extension to $\Omega$.

Motivated by these theorems we give a generalization of the upper Minkowski content and the net measure involving weight and variable exponents, see the next section. In the case that certain smoothness, Hausdorff measure or uniform Minkowski condition is imposing on the singular set, a different approach can be found in [6, 8, 10].

# 2. PRELIMINARIES

Throughout the whole article $\Omega$ is a bounded open set in $\mathbb{R}^n$, $n \ge 3$. We will work on $\Omega$ with the Lebesgue measure $dx$. We first recall some facts on spaces $L^{p(\cdot)}(\Omega, \vartheta)$ and $W^{1,p(\cdot)}(\Omega, \vartheta)$. We write

$$L_+^\infty(\Omega) = \{p \in L^\infty(\Omega) \mid \operatorname{ess\,inf}_\Omega p > 1\}.$$

For $p \in L_+^\infty(\Omega)$, we define

$$W(\Omega) = \{\vartheta \in L_{\text{loc}}^1(\Omega) \mid \vartheta > 0 \text{ almost everywhere in } \Omega\}.$$

An element in $W(\Omega)$ is called a weight function. For $p \in L_+^\infty(\Omega)$, set

$$W_p(\Omega) = \{\vartheta \in W(\Omega) \mid \vartheta^{-\frac{1}{p-1}} \in L_{\text{loc}}^1(\Omega)\}.$$

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

<footnote>Key words and phrases. variable exponent, weight function, singular set, removable singularity.</footnote>
Let $p \in L_{+}^{\infty}(\Omega)$ and $\vartheta \in W(\Omega)$, we define the functional

$$ \rho_{p,\vartheta}(u) = \int_{\Omega} |u|^p \vartheta dx. $$

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

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

see [11, Theorem 2.5]. In the case that $\vartheta = 1$, we have $L^{p(\cdot)}(\Omega, \vartheta) = L^{p(\cdot)}(\Omega)$.

For any $p \in L_{+}^{\infty}(\Omega)$, we denote by $p' = \frac{p}{p-1}$ the conjugate function.

**Proposition 2.1.** (see [11, Theorem 2.1].) Let $p \in L_{+}^{\infty}(\Omega)$. For any $u \in L^{p(\cdot)}(\Omega)$ and $v \in L^{p'(\cdot)}(\Omega)$,

$$ \int_{\Omega} |uv| dx \le 2 \|u\|_{L^{p(\cdot)}(\Omega)} \|v\|_{L^{p'(\cdot)}(\Omega)}. \quad (2.1) $$

**Proposition 2.2.** (see [15, pp. 4], [5, Theorem 1.3].) Let $p \in L_{+}^{\infty}(\Omega)$. For any $u \in L^{p(\cdot)}(\Omega, \vartheta)$, we have

$$ \min \left\{ \rho_{p,\vartheta}(u)^{\frac{1}{p}}, \rho_{p,\vartheta}(u)^{\frac{1}{p^+}} \right\} \le \|u\|_{L^{p(\cdot)}(\Omega,\vartheta)} \le \max \left\{ \rho_{p,\vartheta}(u)^{\frac{1}{p}}, \rho_{p,\vartheta}(u)^{\frac{1}{p^+}} \right\}, \quad (2.2) $$

$$ \min \left\{ \|u\|_{L^{p(\cdot)}(\Omega, \vartheta)}^{p^{-}} , \|u\|_{L^{p(\cdot)}(\Omega, \vartheta)}^{p^{+}} \right\} \le \rho_{p, \vartheta}(u) \le \max \left\{ \|u\|_{L^{p(\cdot)}(\Omega, \vartheta)}^{p^{-}} , \|u\|_{L^{p(\cdot)}(\Omega, \vartheta)}^{p^{+}} \right\}, \quad (2.3) $$

where $p^{-} = \inf_{\Omega} p$ and $p^{+} = \sup_{\Omega} p$.

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

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

see [11, Theorem 3.1] and [15, Proposition 3.2].

In the proof our main results we employed the following type of partition of unity:

**Lemma 2.1.** (see [7, Lemma 3.1]) Let $\{Q_i\}_{i=1}^N$ be a finite disjoint collection of dyadic of length $l_i$. For each $i$, there is a non-negative function $\varphi_i \in C_0^\infty(\mathbb{R}^n)$ with $\text{spt} \varphi_i \subset \frac{3}{2}Q_i$ such that $\sum_{i=1}^N \varphi_i \le 1$ on $\mathbb{R}^n$, and $\sum_{i=1}^N \varphi_i = 1$ on $\cup_{i=1}^N Q_i$. Furthermore, for each multi-index $\lambda$, there is a constant $C_\lambda = C_\lambda(n, \lambda) > 0$ for which $|D^\lambda \varphi_i(x)| \le C_\lambda \ell_i^{-|\lambda|}$ for all $x$ and $i \in \{1, \dots, N\}$.

To define the Minkowski content, let $\alpha : \Omega \to [0, n]$ be a measurable function and $F \subset \mathbb{R}^n$. For $\varepsilon > 0$, we write

$$ F_{\varepsilon} = \{x \in \mathbb{R}^n \mid d(x, F) < \varepsilon\}, $$

where $d(x, F)$ denote the Euclidean distance from a point $x \in \mathbb{R}^n$ to $F$.

**Definition 2.1.** The $(\alpha, \vartheta)$-upper Minkowski content of $F$ is defined by

$$ \mathcal{M}_{\vartheta}^{\alpha}(F) = \limsup_{\varepsilon \to 0^+} \int_{F_{\varepsilon}} \varepsilon^{\alpha-n} d\vartheta, $$

where $\vartheta \in W(\Omega)$ and $d\vartheta = \vartheta dx$.

Now, we proceed to define the *net measure*. For $x \in \mathbb{R}^n$ and $\ell > 0$, we denote by

$$ Q(x, \ell) = \{y \in \mathbb{R}^n \mid |x_i - y_i| \le \ell/2, i = 1, \dots, n\}, $$

the cube of center $x$ and length $\ell$. Recall that a dyadic cube is represented as

$$ [m_1 2^{-k}, (m_1 + 1) 2^{-k}] \times \dots \times [m_n 2^{-k}, (m_n + 1) 2^{-k}], $$

where $k, m_1, \dots, m_n$ are integers.

For a cube $Q$ of length $\ell(Q) = \ell$, we let $\frac{3}{2}Q$ denote the cube with the same center and length $3\ell/2$. For any integer $k$ let $G_k$ be a collection of cubes represented by a form

$$ [m_1 2^{-k}, (m_1 + 1) 2^{-k}] \times \dots \times [m_n 2^{-k}, (m_n + 1) 2^{-k}], $$

for some integers $m_1, \dots, m_n$.
Let $\gamma > 0$ be fixed. For each $\varepsilon > 0$ we write

$$ \mathcal{L}_{\alpha, \gamma, \vartheta}^{\varepsilon}(F) = \inf \left\{ \sum_j \left[ \int_{Q_j} (2^{-kj})^{\alpha-n} d\vartheta \right]^{\gamma} \mid F \subset \text{int}(\cup_j Q_j), Q_j \in G_{kj}, 2^{-kj} < \varepsilon \right\}. $$

**Definition 2.2.** The $(\alpha, \gamma, \vartheta)$-net measure of $F$ is defined by

$$ \mathcal{L}_{\alpha, \gamma, \vartheta}(F) = \lim_{\varepsilon \to 0^+} \mathcal{L}_{\alpha, \gamma, \vartheta}^{\varepsilon}(F). $$

We finish this section, introducing the following condition used in this work.

**Assumption 2.1.** Assume $p \in L_{+}^{\infty}(\Omega)$ and $\vartheta \in W_p(\Omega)$. Suppose $\varphi \in W^{1,p(\cdot)}(\Omega, \vartheta)$ has compact support spt $\varphi = K$. Then there exist functions $\varphi_m \in C^{\infty}(\Omega)$, with spt $\varphi_m \subset K_{d(K,\partial\Omega)/2}$, such that

$$ \varphi_m \to \varphi \quad \text{in } W^{1,p(\cdot)}(\Omega, \vartheta). $$

**Remark.** In the case that $\vartheta \equiv 1$ and $p \in \mathcal{P}^{\log}(\Omega)$ is bounded, the Assumption 2.1 is verified, see [2, Corollary 11.2.4]. On the other hand, if $p \in \mathcal{P}^{\log}(\Omega)$ is bounded and $\vartheta$ satisfies the Muckenhoupt condition, see [1, 3], following the same lines of the proof of [12, Lemma 2.4] we obtain the Assumption 2.1; for auxiliary results see [14, Theorem 2.1.4], [4, Appendix C.5], [1, Theorem 1.1] and [3, Theorem 1.1]. Finally, if $p \equiv$ constant and $\vartheta$ belongs to Muckenhoupt's $A_p$-class, the Assumption 2.1 is satisfied; see [9, Theorem 2.5].

### 3. MAIN RESULTS

In the sequel $E \subset \Omega$ is a compact set. For us a function is integrable if it has an integral (which may equal $\pm\infty$).

**Definition 3.1.** Let $\vartheta \in W_p(\Omega)$ and $p \in L_{+}^{\infty}(\Omega)$. We define

$$ W_{E, \text{loc}}^{1, p(\cdot)}(\Omega, \vartheta) = \left\{ u : \Omega \to \bar{\mathbb{R}} \mid \begin{array}{l} u \text{ is integrable, } u|_{\Omega \setminus E} \in W_{\text{loc}}^{1, p(\cdot)}(\Omega \setminus E, \vartheta), \\ \text{and there is an integrable function } \tilde{\nabla} u : \Omega \to \bar{\mathbb{R}}^n \text{ so that } \tilde{\nabla} u = \nabla u \text{ a.e. on } \Omega \setminus E \end{array} \right\}. $$

Now, we will divide our main results into two sections.

#### 3.1. Singular set satisfying $M_{\vartheta}^{\alpha}(E) = 0$. First, we will work with nonlinear elliptic equations of the form

$$ -\operatorname{div} A(x, u, \nabla u) + B(x, u) = 0. \quad (3.4) $$

We assume that $A : \Omega \times [0, \infty) \times \mathbb{R}^n \to \mathbb{R}^n$ and $B : \Omega \times [0, \infty) \to \mathbb{R}$ are measurable functions. Furthermore, there exists $\mu > 0$ such that the following conditions are satisfied almost everywhere:

$$ \langle A(x, u, \eta), \eta \rangle \geq (\mu|\eta|^{p(x)} - \mu^{-1}u^{p(x)} + a_1(x)) \vartheta(x), \quad (3.5) $$

$$ |A(x, u, \eta)| \leq (\mu^{-1}|\eta|^{p(x)-1} + \mu^{-1}u^{p(x)-1} + a_2(x)) \vartheta(x), \quad (3.6) $$

$$ |B(x, u)| \leq (\mu^{-1}u^{p(x)-1} + a_3(x)) \vartheta(x), \quad (3.7) $$

where $\vartheta \in W_p(\Omega)$, $a_1 \in L^{q_1}(\Omega, \vartheta)$, $a_2 \in L^{q_2}(\Omega, \vartheta)$, $a_3 \geq 0$, $a_3 \in L^{q_3}(\Omega, \vartheta)$, $a_3 \geq 0$, $\alpha, \beta : \Omega \to [0, n]$ are measurable functions, and $p, q_1, q_2, q_3 \in L_{+}^{\infty}(\Omega)$. Furthermore,

$$ (p\beta)^+ \leq n - \alpha^+ - p^+ - 1, \quad (3.8) $$

$$ q_1^{-}, q_3^{-} \geq \left( \frac{p}{p-1} \right)^{+}, q_2^{-} \geq \max \left\{ \left( \frac{p}{p-1} \right)^{+}, \frac{n - \alpha^{+}}{n - \alpha^{+} - \beta^{+} - 2} \right\}. \quad (3.9) $$

**Definition 3.2.** We will say that $u \in W_{E, \text{loc}}^{1, p(\cdot)}(\Omega, \vartheta)$ is a (weak) sub-solution of equation (3.4) in $\Omega \setminus E$ if

$$ \int_{\Omega \setminus E} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + B(\cdot, u) \varphi dx \leq 0 \quad (3.10) $$

for all $\varphi \in W^{1, p(\cdot)}(\Omega, \vartheta)$, $\varphi \geq 0$, with compact support in $\Omega \setminus E$.
**Definition 3.3.** We will say that the sub-solution $u$ of equation (3.4) in $\Omega \setminus E$ has a removable singularity at $E$: if $u \in W_{E, \text{loc}}^{1,p(\cdot)}(\Omega, \vartheta)$ implies $u \in W_{\text{loc}}^{1,p(\cdot)}(\Omega, \vartheta)$ and the equality (3.10) is fulfilled for all $\varphi \in W^{1,p(\cdot)}(\Omega, \vartheta)$, $\varphi \ge 0$, with compact support.

Inspired by Theorem A we have our first main result.

**Theorem 3.1.** Suppose that (3.5) - (3.9) are satisfied, furthermore $M_{\vartheta}^{\alpha}(E) = 0$. If $u$ is a non-negative sub-solution of (3.4) in $\Omega \setminus E$, and

$$ u \le C d(\cdot, E)^{-\beta} \quad \text{a.e. on } \Omega \setminus E, \qquad (3.11) $$

where $C$ is a suitable positive constant, then the singularity of $u$ at $E$ is removable.

*Proof.* 1. Let $\varepsilon \in (0, 1)$ be arbitrary. Since $M_{\vartheta}^{\alpha}(E) = 0$, there is $r_0 \in (0, \varepsilon)$ such that $\overline{E_{r_0}} \subset \Omega$ and

$$ \int_{E_r} r^{\alpha-n} d\vartheta < \varepsilon \quad \text{for all } r \in [0, r_0]. \qquad (3.12) $$

For $r \in [0, r_0]$ and $j = 0, 1, \dots$, we write

$$ K_j = \{x \in \mathbb{R}^n \mid d(x, E) < 2^{-j}r \}, $$

then

$$ E_r = \bigcup_{j=0}^{\infty} (K_j \setminus K_{j+1}), $$

and, by (3.11), (3.12) and (3.8),

$$ \begin{aligned} \int_{K_j \setminus K_{j+1}} u^p d\vartheta &\le \int_{K_j \setminus K_{j+1}} C^p d(\cdot, E)^{-\beta p} d\vartheta \le C_1(p) \int_{K_j \setminus K_{j+1}} (2^{-(j+1)}r)^{-\beta p} d\vartheta \\ &= C_1 \int_{K_j \setminus K_{j+1}} 2^{\beta p} (2^{-j}r)^{n-\alpha-\beta p} (2^{-j}r)^{\alpha-n} d\vartheta \le C_1 2^{(\beta p)^+} (2^{-j}r)^{n-\alpha^{+}-(\beta p)^+} \varepsilon \\ &\le C_2(p, \beta) 2^{-j-1} r^{p^{++}+1} \varepsilon. \end{aligned} $$

Therefore, considering $|E| = 0$,

$$ \int_{E_r} u^p d\vartheta = \sum_{j=0}^{\infty} \int_{K_j \setminus K_{j+1}} u^p d\vartheta \le C_2 \varepsilon r^{p^{++}+1}, \qquad (3.13) $$

for all $r \in [0, r_0]$. Thus $u \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$.

2. Now, we prove $|\tilde{\nabla} u| \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$. 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 $|\psi'_r| \le C_3/r$, where $C_3$ is a positive constant. Set

$$ \varphi = (\psi_r^{p^+} \circ d(\cdot, E)) u. $$

We have $\varphi \in W^{1,p(\cdot)}(\Omega, \vartheta)$, $\varphi \ge 0$, with compact support in $\Omega \setminus E$. For simplicity we write $\psi_r = \psi_r \circ d(\cdot, E)$ and $\psi'_r = \psi'_r \circ d(\cdot, E)$. Substituting $\varphi$ into (3.10),

$$ \int_{\Omega \setminus E} \left\langle A(\cdot, u, \nabla u), \psi_r^{p^+} \nabla u + p^+ \psi_r^{p^+ - 1} \psi_r' u \nabla d(\cdot, E) \right\rangle + B(\cdot, u) \psi_r^{p^+} u dx \le 0. $$

By conditions (3.5) - (3.9),

$$ q_1^- \ge \frac{n - \alpha^+}{n - \alpha^+ - 1}, \quad q_2^- \ge \frac{n - \alpha^+}{n - \alpha^+ - \beta^+ - 2}, \quad q_3^- \ge \frac{n - \alpha^+}{n - \alpha^+ - \beta^+ - 1} \qquad (3.14) $$
and

$$
\begin{aligned}
& \int_{\{\frac{r}{2} \le d(\cdot, E) \le \frac{5r}{2}\}} \psi_r^{p^+} (\mu|\nabla u|^p - \mu^{-1}u^p + a_1) \, d\vartheta \\
& \le \int_{\{\frac{r}{2} \le d(\cdot, E) \le \frac{5r}{2}\}} \frac{C_3}{r} p^+ \psi_r^{p^+ - 1} (\mu^{-1}|\nabla u|^{p-1} + \mu^{-1}u^{p-1} + a_2) \, ud\vartheta \\
& \qquad + \int_{\{\frac{r}{2} \le d(\cdot, E) \le \frac{5r}{2}\}} \psi_r^{p^+} (\mu^{-1}u^{p-1} + a_3) \, ud\vartheta
\end{aligned}
$$

Consequently, by (3.13), (2.1), (2.2), Young's inequality and (3.11),

$$
\begin{aligned}
& \int_{\{r \le d(\cdot, E) \le 2r\}} |\nabla u|^p d\vartheta \le C_4 \varepsilon \frac{5r}{2} \\
& \quad + C_4 r^{(n-\alpha^+)} \binom{q_1^-}{q_1^-} \|a_1\|_{L^{q_1(\cdot)}(\Omega, \vartheta)} \left[ \int_{E_{5r/2}} \left(\frac{5r}{2}\right)^{\alpha-n} d\vartheta \right]^{\frac{q_1-1}{q_1}} \\
& \quad + C_4 \int_{\{\frac{r}{2} \le d(\cdot, E) \le \frac{5r}{2}\}} C_5 (\varepsilon_1, p) \left(\frac{u}{r}\right)^p + \varepsilon_1 \psi_r^{\frac{(p+1)p}{p-1}} |\nabla u|^p d\vartheta \\
& \quad + C_4 r^{(n-\alpha^+)} \binom{q_2^-}{q_2^-} - \beta^{+-1} \|a_2\|_{L^{q_2(\cdot)}(\Omega, \vartheta)} \left[ \int_{E_{5r/2}} \left(\frac{5r}{2}\right)^{\alpha-n} d\vartheta \right]^{\frac{q_2-1}{q_2}} \\
& \quad + C_4 r^{(n-\alpha^+)} \binom{q_3^-}{q_3^-} - \beta^+ \|a_3\|_{L^{q_3(\cdot)}(\Omega, \vartheta)} \left[ \int_{E_{5r/2}} \left(\frac{5r}{2}\right)^{\alpha-n} d\vartheta \right]^{\frac{q_3-1}{q_3}},
\end{aligned}
$$

where $C_4 = C_4(\mu, \alpha, p, C_2) > 0$. Take $\varepsilon_1 = \frac{1}{2C_4}$. From (3.14), (3.11), (3.12), (3.8), we have

$$
\frac{1}{2} \int_{\{r \le d(\cdot, E) \le 2r\}} |\nabla u|^p d\vartheta \le C_6 \varepsilon^{\gamma_1} r + C_6 \varepsilon r^{n-\alpha^+-(p\beta)^+-1} \le 2C_6 \varepsilon^{\gamma_1} r.
$$

Therefore,

$$
\int_{E_r} |\widetilde{\nabla} u|^p d\vartheta = \sum_{i=0}^{\infty} \int_{\{2^{-(i+1)}r \le d(\cdot, E) \le 2^{-ir}\}} |\nabla u|^p d\vartheta \le C_7 \varepsilon^{\gamma_1} r, \quad (3.15)
$$

where $C_i = C_i(\mu, \alpha, \beta, p, a_1, a_2, a_3) > 0$, $i = 6, 7$, $\gamma_1 = \gamma_1(q_1, q_2, q_3) \in (0, 1)$. Hence $|\widetilde{\nabla} u| \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$.

3. Recall from step 1 and 2 that $u \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$ and $|\widetilde{\nabla} u| \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$. We now claim $u \in W_{\text{loc}}^{1,p(\cdot)}(\Omega, \vartheta)$. To verify this assertion, fix $\varphi \in C_0^\infty(\Omega)$, we assume that $|\varphi| \le 1$ and $|\nabla \varphi| \le 1$. For $r \in (0, r_0/2)$, let $\zeta_r : \mathbb{R} \to \mathbb{R}$ be a smooth function such that

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

$0 \le \zeta_r \le 1$ and $|\zeta'_r| \le C_8/r$, where $C_8$ is a positive constant. For simplicity we write $\zeta_r = \zeta_r \circ d(\cdot, E)$ and $\zeta'_r = \zeta'_r \circ d(\cdot, E)$. We have

$$
\int_{\Omega \setminus E} u [(1 - \zeta_r) \varphi]_{x_i} \, dx = - \int_{\Omega \setminus E} u_{x_i} (1 - \zeta_r) \varphi \, dx,
$$

where $i = 1, \dots, n$. If $r \to 0^+$:

$$
\int_{\Omega} u \varphi_{x_i} \, dx = - \int_{\Omega} u_{x_i} \varphi \, dx. \qquad (3.16)
$$

Indeed, (2.1), (2.2) and (3.13) implies

$$
\begin{aligned}
\left| \int_{\Omega \setminus E} u (\zeta_r)_{x_i} \varphi \, dx \right| &\le 2 \left\| \frac{C_8 u}{r} \vartheta^{1/p} \right\|_{L^{p(\cdot)}(E_{2r})} \left\| \vartheta^{-1/p} \right\|_{L^{p'(\cdot)}(E_{2r})} \\
&\le C_9(p, \beta) \left\| \vartheta^{-1/p} \right\|_{L^{p'(\cdot)}(E_{r_0})} (\varepsilon r)^{1/p^+} \to 0 \quad \text{as } r \to 0^+,
\end{aligned}
$$
since $\vartheta \in W_p(\Omega)$. So (3.16) is proved.

4. To complete the proof of the theorem. Let $\varphi \in W^{1,p(\cdot)}(\Omega, \vartheta)$ be with compact support and $\varphi \ge 0$. By Assumption 2.1, we consider the case $\varphi \in C^\infty(\Omega)$, $\varphi \le 1$ and $|\nabla \varphi| \le 1$. We show

$$ \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + B(\cdot, u) \varphi dx \le 0. $$

Let $K = \text{spt } \varphi$. We may suppose that $\overline{K}_{r_0} \subset \Omega$. Choose $\ell = 2^{-k}$ so small that $3\ell\sqrt{n} < r_0$. Cover $K$ by a finite collection of dyadic cubes $Q_i$ with same length $\ell(Q_i) = \ell$, $i = 1, \dots, N$. Moreover, we can assume that

$$ \frac{3}{2} Q_i \cap E \neq \emptyset \quad \text{for } i = 1, \dots, N_*, $$

$$ \frac{3}{2} Q_i \cap E = \emptyset \quad \text{for } i = N^* + 1, \dots, N, $$

for some $N_* \in \mathbb{N}$, $1 \le N^* \le N$. By Lemma 2.1, we have there exist non-negative functions $\varphi_i \in C_0^\infty(\mathbb{R}^n)$ with $\text{spt } \varphi_i \subset \frac{3}{2}Q_i$ such that $\sum_{i=1}^N \varphi_i = 1$ on $\cup_{i=1}^N Q_i \supset K$. Furthermore, $|\nabla \varphi_i| \le C_{10}(n)\ell^{-1}$, $i = 1, \dots, N$, on $\mathbb{R}^n$. Since $u$ is a sub-solution of (3.4) in $\Omega \setminus E$, and $\varphi\varphi_i \in W^{1,p(\cdot)}(\Omega, \vartheta)$, $\varphi\varphi_i \ge 0$, $i = N_* + 1, \dots, N$ has compact support in $\Omega \setminus E$, we have

$$ \int_{\Omega \setminus E} \langle A(\cdot, u, \nabla u), \nabla (\varphi \varphi_i) \rangle + B(\cdot, u) (\varphi \varphi_i) \, dx \le 0, \quad (3.17) $$

Then, by (3.6) and (3.7),

$$ \begin{align*} \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + B(\cdot, u) \varphi dx &= \sum_{i=1}^{N} \int_{\frac{3}{2}Q_i} \langle A(\cdot, u, \nabla u), \nabla (\varphi \varphi_i) \rangle + B(\cdot, u) (\varphi \varphi_i) dx \\ &\le \sum_{i=1}^{N_*} \int_{\frac{3}{2}Q_i} \langle A(\cdot, u, \nabla u), \nabla (\varphi \varphi_i) \rangle + B(\cdot, u) (\varphi \varphi_i) dx \\ &\le \sum_{i=1}^{N_*} \int_{\frac{3}{2}Q_i} \mu^{-1}(2+C_{10})\ell^{-1} (|\nabla u|^{p-1} + u^{p-1}) + (1+C_{10})\ell^{-1}a_2 + a_3 d\vartheta \\ &\le C_{11}(n, \mu) \sum_{i=1}^{N_*} \int_{\frac{3}{2}Q_i} \ell^{-1} (|\nabla u|^p + 1) + \ell^{-1}(u^p + 1) + \ell^{-1}a_2 + a_3 d\vartheta \\ &= C_{11} \int_{E_{3\ell\sqrt{n}}} \ell^{-1} \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) |\nabla u|^p + \ell^{-1} \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) u^p d\vartheta \\ &\quad + C_{11} \int_{E_{3\ell\sqrt{n}}} \ell^{-1} \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) + \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) d\vartheta \\ &\quad + C_{11} \int_{E_{3\ell\sqrt{n}}} \ell^{-1} \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) a_2 + \left( \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \right) a_3 d\vartheta. \tag{3.18} \end{align*} $$

Here $\chi_{\frac{3}{2}Q_i}$ is the characteristic function of $\frac{3}{2}Q_i$, $i = 1, \dots, N_*$. For each cube $Q_i$, $i = 1, \dots, N_*$, there are at most $3^n$ cubes $Q_j$, $\ell(Q_j) = \ell$, $j = 1, \dots, N_i \le 3^n$ (adjacent cubes to $Q_i$ with equal length), such that $\frac{3}{2}Q_i \cap \frac{3}{2}Q_j \ne \emptyset$. Then

$$ \sum_{i=1}^{N_*} \chi_{\frac{3}{2}Q_i} \le 3^n \quad \text{on } E_{3\ell\sqrt{n}}. \quad (3.19) $$

By (3.18), (3.19), (3.15), (3.13), (3.12), (2.1) and (3.14), we have

$$ \begin{align*} \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + B(\cdot, u) \varphi dx &\le (C_7 + C_2) C_{11} \ell^{-1} 3^n (3\ell\sqrt{n}) \varepsilon^{\gamma_1} + C_{12} \ell^{-1} 3^n (3\ell\sqrt{n})^{n-\alpha^+} \varepsilon \\ &\quad + C_{11} \ell^{-1} 3^n 2 (3\ell\sqrt{n})^{(n-\alpha^+)} \left(\frac{q_2}{q_2-1}\right) \|a_2\|_{L^{q_2(\cdot)}(\Omega, \vartheta)} \varepsilon^{\gamma_1} + C_{11} 3^n 2 (3\ell\sqrt{n})^{(n-\alpha^+)} \left(\frac{q_3}{q_3-1}\right) \|a_3\|_{L^{q_3(\cdot)}(\Omega, \vartheta)} \varepsilon^{\gamma_1} \\ &\le C_{13} \varepsilon^{\gamma_1}, \end{align*} $$
where $C_{12}, C_{13}$ are positive constants independent of $\epsilon$. Since $\epsilon \in (0, 1)$ was arbitrary, it follows that

$$ \int_{\Omega} \langle A(\cdot, u, \nabla u), \nabla \varphi \rangle + B(\cdot, u) \varphi \,dx \le 0. $$

This completes the proof of Theorem 3.1. □

**3.2. Singular set satisfying $\mathcal{L}_{\alpha, \gamma, \vartheta}(E) = 0$.** In this section we will work with nonlinear elliptic equations of the form

$$ - \operatorname{div} A(x, \nabla u) = 0. \qquad (3.20) $$

We assume that $A: \Omega \times \mathbb{R}^n \to \mathbb{R}^n$ is a measurable function. Furthermore, there exists $\mu > 0$ such that the following conditions are satisfied almost everywhere:

$$ \langle A(x, \eta), \eta \rangle \ge \mu |\eta|^{p(x)} \vartheta(x), \qquad (3.21) $$

$$ |A(x, \eta)| \le \mu^{-1} |\eta|^{p(x)-1} \vartheta(x), \qquad (3.22) $$

where $\vartheta \in W_p(\Omega)$, $\alpha, \tau: \Omega \to [0, n]$ are measurable functions, and $p \in L_+^\infty(\Omega)$. Moreover,

$$ \tau \le n - \alpha - p^{+} - \frac{p^{+}}{p^{-}} \quad \text{a.e. on } \Omega. \qquad (3.23) $$

**Definition 3.4.** We will say that $u \in W_{E, \text{loc}}^{1, p(\cdot)}(\Omega, \vartheta)$ is a (weak) sub-solution of equation (3.20) in $\Omega \setminus E$ if

$$ \int_{\Omega \setminus E} \langle A(\cdot, \nabla u), \nabla \varphi \rangle \,dx \le 0 \qquad (3.24) $$

for all $\varphi \in W^{1, p(\cdot)}(\Omega, \vartheta)$, $\varphi \ge 0$, with compact support in $\Omega \setminus E$.

**Definition 3.5.** We will say that the sub-solution $u$ of equation (3.20) in $\Omega \setminus E$ has a removable singularity at $E$: if $u \in W_{E, \text{loc}}^{1, p(\cdot)}(\Omega, \vartheta)$ implies $u \in W_{\text{loc}}^{1, p(\cdot)}(\Omega, \vartheta)$ and the equality (3.24) is fulfilled for all $\varphi \in W^{1, p(\cdot)}(\Omega, \vartheta)$, $\varphi \ge 0$, with compact support.

**Lemma 3.1.** Suppose that there is a locally bounded function $\sigma: \Omega \times \Omega \to (0, \infty)$ such that

$$ \int_{Q(y, \ell)} \ell^{-\tau} d\vartheta \le \sigma(y, z) \int_{Q(z, \ell)} \ell^{-\tau} d\vartheta, \qquad (3.25) $$

for all adjacent dyadic cubes $Q(y, \ell), Q(z, \ell) \subset \Omega$. If $K \subset \Omega$ is a compact set, then

$$ \int_{\frac{3}{2}Q(y, \ell)} \ell^{-\tau} d\vartheta \le C(n, \sigma, K) \int_{Q(z, \ell)} \ell^{-\tau} d\vartheta, \qquad (3.26) $$

$$ \int_{B(y, \ell)} \ell^{-\tau} d\vartheta \le C(n, \sigma, K) \int_{Q(y, \ell)} \ell^{-\tau} d\vartheta, \qquad (3.27) $$

where $y = z$ or $Q(y, \ell), Q(z, \ell) \subset \Omega$ are adjacent dyadic cubes, with $y, z \in K$ and $\ell\sqrt{n} < d(K, \partial\Omega)$.

**Lemma 3.2.** Assume that (3.21) - (3.23), (3.25) are satisfied, and $\mathcal{L}_{\alpha, \gamma, \vartheta}(E) = 0$ for some $\gamma \in (0, 1/p^+]$. If $u$ is a non-negative sub-solution of (3.20) in $\Omega \setminus E$ and satisfies

$$ \int_{B(y,r)} u^p d\vartheta \le C_* \int_{B(y,r/\sqrt{n})} r^{-\tau} d\vartheta, \qquad (3.28) $$

for all ball $B(y, r) \subset \subset \Omega$, where $C_*$ is a suitable positive constant. Then

$$ \min \left\{ \left( \int_{B(y, r_1)} |\widetilde{\nabla} u|^p d\vartheta \right)^{\frac{1}{p^{-}}} , \left( \int_{B(y, r_1)} |\widetilde{\nabla} u|^p d\vartheta \right)^{\frac{1}{p^{+}}} \right\} \\ \le C_0 \max \left\{ \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y, r_2/\sqrt{n})} r_2^{-\tau} d\vartheta \right]^{\frac{1}{p^{-}}} , \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y, r_2/\sqrt{n})} r_2^{-\tau} d\vartheta \right]^{\frac{1}{p^{+}}} \right\} \qquad (3.29) $$

for all ball $B(y, r_2) \subset \subset \Omega$ with $0 < r_2 - r_1 < 1$, where $C_0 = C_0(n, \mu, p, \sigma, E, C_*) > 0$.
Proof. 1. Let $r_2 > r_1 > 0$ and $y \in \Omega$ be such that $\overline{B(y, r_2)} \subset \Omega$ and $r_2 - r_1 < 1$. Choose $r_0 \in (0, 1)$ such that $\overline{E}_{r_0} \subset \Omega$. Take $\varepsilon \in (0, r_0)$ be arbitrarily. Since $\mathcal{L}_{\alpha, \gamma, \vartheta}(E) = 0$ there is a finite collection of mutually disjoint dyadic cubes $Q_i = Q(y_i, \ell_i)$ so that $E \subset \text{int}(\cup_{i=1}^N Q_i)$,

$$ \sum_{i=1}^{N} \left( \int_{Q_i} \ell_i^{\alpha-n} \mathrm{d}\vartheta \right)^{\gamma} < \varepsilon, \qquad (3.30) $$

$3\ell_i\sqrt{n} < \varepsilon$ and $Q_i \cap E \neq \emptyset$, $i = 1, \dots, N$.

Now, we attach to each cube $Q_i$, $\ell(Q_i) = \ell_i$, $i = 1, \dots, N$, all adjacent dyadic cubes with the same length $\ell_i$. Since two dyadic cubes are either mutually disjoint or one is contained in the other, we may drop extra cubes away. Proceeding in this way we get a collection of mutually disjoint cubes $Q_{j,k} = Q(y_{j,k}, \ell_{j,k})$, $(j, k) \in \{1, \dots, N_1\} \times \{1, \dots, m_j\}$ satisfying

$$ \ell_{j,k} = \ell_{a_j}, \quad j = 1, \dots, N_1 \le N, \quad k = 1, \dots, m_j \le 3^n, \qquad (3.31) $$

where $\{1 \le a_1 < \dots < a_{N_1} \le N\} \subset \mathbb{Z}$, and furthermore $Q_{j,k}$ is adjacent to $Q_{a_j}$. Additionally, by (3.28) and (3.26),

$$ \int_{\frac{3}{2}Q_{j,k}} u^p \mathrm{d}\vartheta \le C_1 \int_{Q_{a_j}} \ell_{a_j}^{-\tau} \mathrm{d}\vartheta, \qquad (3.32) $$

$j = 1, \dots, N_1$, $k = 1, \dots, m_j$, for some constant $C_1 = C_1(n, \sigma, E_{2r_0/3}, C_*) > 0$.

2. We next choose non-negative test functions $\varphi_i$, $\varphi_{j,k}$, $i = 1, \dots, N$, $j = 1, \dots, N_1$, $k = 1, \dots, m_j$, given by the Lemma 2.1, with the following properties:

$$ \text{spt } \varphi_i \subset \frac{3}{2}Q_i, \quad |\nabla \varphi_i| \le C_2(n) \ell_i^{-1}, \qquad (3.33) $$

$$ \text{spt } \varphi_{j,k} \subset \frac{3}{2}Q_{j,k}, \quad |\nabla \varphi_{j,k}| \le C_2(n) \ell_{j,k}^{-1}, \qquad (3.34) $$

$$ \sum_i \varphi_i + \sum_{j,k} \varphi_{j,k} = 1 \quad \text{on } (\cup_i Q_i) \cup (\cup_{j,k} Q_{j,k}). \qquad (3.35) $$

Let $\psi_0 : \mathbb{R}^n \to \mathbb{R}$ be a smooth function with $\text{spt } \psi_0 \subset B(y, r_2)$, $0 \le \psi_0 \le 1$, $|\nabla \psi_0| \le C_3(n)(r_2 - r_1)^{-1}$, and $\psi = 1$ on $B(y, r_1)$. Write

$$ \xi = \psi_0 \left[ 1 - \left( \sum_i \varphi_i + \sum_{j,k} \varphi_{j,k} \right) \right]. $$

Substituting $\xi^{p^+} u$ into (3.24), we see

$$ \int_{\Omega \setminus E} \langle A(\cdot, \nabla u), \nabla (\xi^{p^+} u) \rangle \mathrm{d}x \le 0, \\ \int_{\Omega \setminus E} \xi^{p^+} \langle A(\cdot, \nabla u), \nabla u \rangle \mathrm{d}x \le \int_{\Omega \setminus E} p^+ \xi^{p^+-1} |uA(\cdot, \nabla u)| |\nabla \xi| \mathrm{d}x. $$

Then, by (3.21), (3.22), (2.1) and (2.2),

$$ \int_{\Omega \setminus E} \xi^{p^+} |\nabla u|^p \mathrm{d}\vartheta \le \mu^{-2} \int_{\Omega \setminus E} p^+ \xi^{p^+-1} |\nabla \xi| |\nabla u|^{p-1} u \mathrm{d}\vartheta \\ \le 2\mu^{-2} p^+ \left\| \xi^{p^+-1} |\nabla u|^{p-1} \vartheta^{\frac{p-1}{p}} \right\|_{L^{\frac{p-1}{p}}(\Omega \setminus E)} \left\| |\nabla \xi| u \vartheta^{\frac{1}{p}} \right\|_{L^{p^+}(\Omega)} \\ \le 2\mu^{-2} p^+ \max \left\{ \left( \int_{\Omega \setminus E} \xi^{p^+} |\nabla u|^p \mathrm{d}\vartheta \right)^{\frac{p-1}{p}} + \left( \int_{\Omega \setminus E} \xi^{p^+} |\nabla u|^p \mathrm{d}\vartheta \right)^{\frac{p-1}{p}} \right\} \\ \cdot \left[ \sum_i \left\| |\nabla \varphi_i| u \vartheta^{\frac{1}{p}} \right\|_{L^{p^+}(\frac{3}{2}Q_i)} + \sum_{j,k} \left\| |\nabla \varphi_{j,k}| u \vartheta^{\frac{1}{p}} \right\|_{L^{p^+}(\frac{3}{2}Q_{j,k})} + \left\| |\nabla \psi_0| u \vartheta^{\frac{1}{p}} \right\|_{L^{p^+}(B(y,r_2))} \right]. $$
Consequently, using (2.2), (3.28), (3.26), (3.32), (3.23) and (3.30), we compute

$$
\begin{align*}
& \min \left\{ \left( \int_{\Omega \setminus E} \xi^{p^{+}} |\nabla u|^p d\nu \right)^{1-\left(\frac{p-1}{p}\right)^{+}} , \left( \int_{\Omega \setminus E} \xi^{p^{+}} |\nabla u|^p d\nu \right)^{1-\left(\frac{p-1}{p}\right)^{-}} \right\} \\
& \le C_4(n, \mu, p) \max \left\{ \left[ (r_2-r_1)^{-p^{+}} \int_{B(y,r_2)} u^p d\nu \right]^{\frac{1}{p^{-}}} , \left[ (r_2-r_1)^{-p^{+}} \int_{B(y,r_2)} u^p d\nu \right]^{\frac{1}{p^{+}}} \right\} \\
& \qquad + C_4 \sum_i \max \left\{ \left( \ell_i^{-p^{+}} \int_{\frac{3}{2}Q_i} u^p d\nu \right)^{\frac{1}{p^{-}}} , \left( \ell_i^{-p^{+}} \int_{\frac{3}{2}Q_i} u^p d\nu \right)^{\frac{1}{p^{+}}} \right\} \\
& \qquad + C_4 \sum_{j,k} \max \left\{ \left( \ell_{j,k}^{-p^{+}} \int_{\frac{3}{2}Q_{j,k}} u^p d\nu \right)^{\frac{1}{p^{-}}} , \left( \ell_{j,k}^{-p^{+}} \int_{\frac{3}{2}Q_{j,k}} u^p d\nu \right)^{\frac{1}{p^{+}}} \right\} \\
& \le C_5 \max \left\{ \left[ (r_2-r_1)^{-p^{+}} \int_{B(y,r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{-}}} , \left[ (r_2-r_1)^{-p^{+}} \int_{B(y,r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{+}}} \right\} + C_6 \varepsilon,
\end{align*}
$$

where $C_5 = C_5(n, \mu, p, \sigma, E_{2r_0/3}, C_*) > 0$ and $C_6 > 0$ is independent of $\varepsilon$. Hence,

$$
\begin{align*}
& \min \left\{ \left( \int_{\Omega \setminus E} \xi^{p^{+}} |\nabla u|^p d\nu \right)^{\frac{1}{p^{-}}}, \left( \int_{\Omega \setminus E} \xi^{p^{+}} |\nabla u|^p d\nu \right)^{\frac{1}{p^{+}}} \right\} \\
& \le C_5 \max \left\{ \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y, r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{-}}}, \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y, r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{+}}} \right\} + C_6 \varepsilon.
\end{align*}
$$

Since $\varepsilon \in (0, 1)$ was arbitrary:

$$
\begin{align*}
& \min \left\{ \left( \int_{B(y,r_1)} |\widetilde{\nabla} u|^p d\nu \right)^{\frac{1}{p^{-}}}, \left( \int_{B(y,r_1)} |\widetilde{\nabla} u|^p d\nu \right)^{\frac{1}{p^{+}}} \right\} \\
& \le C_5 \max \left\{ \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y,r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{-}}} , \left[ (r_2 - r_1)^{-p^{+}} \int_{B(y,r_2/\sqrt{n})} r_2^{-\tau} d\nu \right]^{\frac{1}{p^{+}}} \right\}.
\end{align*}
$$

Therefore, we conclude (3.29).

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

Motivated by Theorem B we have our last result.

**Theorem 3.2.** Assume that (3.21) - (3.23), (3.25) and (3.28) are satisfied, furthermore $\mathcal{L}_{\alpha,\gamma,\vartheta}(E) = 0$ for some $\gamma \in (0, 1/p^+)$. If $u$ is a non-negative sub-solution of (3.20) in $\Omega \setminus E$, then the singularity of $u$ at $E$ is removable.

*Proof.* 1. Let $\varphi \in W^{1,p(\cdot)}(\Omega, \vartheta)$ be with compact support. By Assumption 2.1 we can suppose that $\varphi \in C^\infty(\Omega)$ and $|\nabla \varphi| \le 1$. Write $K = \text{spt } \varphi$, choose $r_0 \in (0, 1)$ such that $\bar{K}_{r_0} \subset \Omega$. Let $\varepsilon \in (0, r_0)$ be arbitrarily given. Since $\mathcal{L}_{\alpha,\gamma,\vartheta}(E) = 0$, there is a finite collection of mutually disjoint dyadic cubes $Q_i(y_i, \ell_i)$, $3\sqrt{n}\ell_i < \varepsilon$, $i = 1, \dots, N$, so that $E \subset \text{int} (\cup_{i=1}^N Q_i)$ and

$$
\sum_{i=1}^{N} \left( \int_{Q_i} \ell_i^{\alpha-n} d\nu \right)^{\gamma} < \varepsilon. \qquad (3.36)
$$

We will cover $K$ by a suitable collection of mutually disjoint dyadic cubes:

We attach to each cube $Q_i$, $\ell(Q_i) = \ell_i$, $i = 1, \dots, N$, all adjacent dyadic cubes with the same length $\ell_i$. Since two dyadic cubes are either mutually disjoint or one is contained in the other, we may drop extra cubes away. Proceeding in this way we get a collection of mutually disjoint cubes $Q_{j,k} = Q(y_{j,k}, \ell_{j,k})$, $(j, k) \in \{1, \dots, N\} \times \{1, \dots, m_j\}$ satisfying

$$
\ell_{j,k} = \ell_{a_j}, \quad j = 1, \dots, N_1 \le N, \quad k = 1, \dots, m_j \le 3^n, \qquad (3.37)
$$

where $\{1 \le a_1 < \dots < a_{N_1} \le N\} \subset \mathbb{Z}$, and furthermore $Q_{j,k}$ is adjacent to $Q_{a_j}$.
Finally, we cover the remaining set $K \setminus [(\cup_i Q_i) \cup (\cup_{j,k} Q_{j,k})]$ by mutually disjoint dyadic cubes $\tilde{Q}_h$, all with the same length $\ell(\tilde{Q}_h) = \ell_0$, $h = 1, \dots, N_2$, where $\ell_0 = \min\{\ell_i | i = 1, \dots, N\}$. We see

$$ \frac{3}{2}\tilde{Q}_h \cap E = \emptyset \quad \text{for } h = 1, \dots, N_2. $$

2. From (3.29) we have $|\nabla u| \in L_{\text{loc}}^{p(\cdot)}(\Omega, \vartheta)$. We claim $u \in W_{\text{loc}}^{1,p(\cdot)}(\Omega, \vartheta)$. To verify this assertion, let $\psi_i, \psi_{j,k}$, $i = 1, \dots, N, j = 1, \dots, N_1, k = 1, \dots, m_j$ be non-negative test functions given by the Lemma 2.1, satisfying

$$ \begin{aligned} \text{spt } \psi_i &\subset \frac{3}{2}Q_i, & |\nabla \psi_i| &\le C_1(n)\ell_i^{-1}, \\ \text{spt } \psi_{j,k} &\subset \frac{3}{2}Q_{j,k}, & |\nabla \psi_{j,k}| &\le C_1(n)\ell_{j,k}^{-1}, \\ \sum_i \psi_i + \sum_{j,k} \psi_{j,k} &= 1 & \text{on } (\cup_i Q_i) \cup (\cup_{j,k} Q_{j,k}). \end{aligned} $$

Write

$$ \zeta = \sum_i \psi_i + \sum_{j,k} \psi_{j,k}. $$

We have

$$ \int_{\Omega \setminus E} u [(1 - \zeta) \varphi]_{x_b} \,dx = - \int_{\Omega \setminus E} u_{x_b} (1 - \zeta) \varphi \,dx, $$

where $b = 1, \dots, n$. If $\varepsilon \to 0^+$:

$$ \int_{\Omega} u \varphi_{x_b} \,dx = - \int_{\Omega} u_{x_b} \varphi \,dx. \quad (3.38) $$

Indeed, (2.1), (2.2), (3.28), (3.26), (3.23) and (3.36) implies

$$ \begin{aligned} \left| \int_{\Omega \setminus E} u \zeta_{x_b} \varphi \,dx \right| &\le 2 \left\| u \zeta_{x_b} \vartheta^{1/p} \right\|_{L^{p(\cdot)}(\Omega)} \left\| \vartheta^{-1/p} \right\|_{L^{p'(\cdot)}(\Omega)} \\ &\le C_2(\vartheta) \sum_i \left\| u |\nabla \psi_i| \vartheta^{1/p} \right\|_{L^{p(\cdot)}(\frac{3}{2}Q_i)} + C_2 \sum_{j,k} \left\| u |\nabla \psi_{j,k}| \vartheta^{1/p} \right\|_{L^{p(\cdot)}(\frac{3}{2}Q_{j,k})} \\ &\le C_3(\vartheta, n) \sum_i \max \left\{ \left( \ell_i^{-p+} \int_{\frac{3}{2}Q_i} u^p \,d\vartheta \right)^{\frac{1}{p-}}, \left( \ell_i^{-p+} \int_{\frac{3}{2}Q_i} u^p \,d\vartheta \right)^{\frac{1}{p+}} \right\} \\ &\quad + C_3 \sum_{j,k} \max \left\{ \left( \ell_{j,k}^{-p+} \int_{\frac{3}{2}Q_{j,k}} u^p \,d\vartheta \right)^{\frac{1}{p-}}, \left( \ell_{j,k}^{-p+} \int_{\frac{3}{2}Q_{j,k}} u^p \,d\vartheta \right)^{\frac{1}{p+}} \right\} \\ &\le C_4(\vartheta, n, \sigma, K) \varepsilon \to 0 \quad \text{as } \varepsilon \to 0^+. \end{aligned} $$

So (3.38) is proved.

3. Additionally we assume that $0 \le \varphi \le 1$. We show

$$ \int_{\Omega} \langle A(\cdot, \nabla u), \nabla \varphi \rangle \,dx \le 0. $$

We next choose non-negative test functions $\varphi_i, \varphi_{j,k}, \tilde{\varphi}_h, i = 1, \dots, N, j = 1, \dots, N_1, k = 1, \dots, m_j, h = 1, \dots, N_2$, given by the Lemma 2.1, with the following properties:

$$ \begin{aligned} \text{spt } \varphi_i &\subset \frac{3}{2}Q_i, & |\nabla \varphi_i| &\le C_5(n)\ell_i^{-1}, \\ \text{spt } \varphi_{j,k} &\subset \frac{3}{2}Q_{j,k}, & |\nabla \varphi_{j,k}| &\le C_5(n)\ell_{j,k}^{-1}, \\ \text{spt } \tilde{\varphi}_h &\subset \frac{3}{2}\tilde{Q}_h, & |\nabla \tilde{\varphi}_h| &\le C_5(n)\ell_0^{-1}, \\ \sum_i \varphi_i + \sum_{j,k} \varphi_{j,k} + \sum_h \tilde{\varphi}_h &= 1 & \text{on } K. \end{aligned} $$
Write

$$ \xi = \sum_{i} \varphi_{i} + \sum_{j,k} \varphi_{j,k} + \sum_{h} \tilde{\varphi}_{h}. $$

Since $\frac{3}{2}\tilde{Q}_h \cap E = \emptyset$ and $\text{spt } \tilde{\varphi}_h \subset \frac{3}{2}\tilde{Q}_h$, substituting $\varphi\tilde{\varphi}_h$ into (3.24), we have

$$ \int_{\frac{3}{2}\tilde{Q}_h} \langle A(\cdot, \nabla u), \nabla (\varphi \tilde{\varphi}_h) \rangle dx \le 0, $$

for $h = 1, \dots, N_2$. From this,

$$ \begin{aligned} \int_{\Omega} \langle A(\cdot, \nabla u), \nabla \varphi \rangle dx &= \int_{\Omega} \langle A(\cdot, \nabla u), \nabla (\varphi \xi) \rangle dx \\ &\le \sum_{i} \int_{\frac{3}{2}Q_i} \langle A(\cdot, \nabla u), \nabla (\varphi \varphi_i) \rangle dx + \sum_{j,k} \int_{\frac{3}{2}Q_{j,k}} \langle A(\cdot, \nabla u), \nabla (\varphi \varphi_{j,k}) \rangle dx. \end{aligned} \quad (3.39) $$

We can assume that $\int_{E_{r_0}} |\nabla u|^p d\nu < 1$, since $|E| = 0$. By (3.39), (3.22), (3.26), (3.29) and (3.36),

$$ \begin{aligned} \int_{\Omega} \langle A(\cdot, \nabla u), \nabla \varphi \rangle dx &\le C_6 \sum_{i} \int_{\frac{3}{2}Q_i} \ell_i^{-1} (|\nabla u|^p + 1) d\nu + C_6 \sum_{j,k} \int_{\frac{3}{2}Q_{j,k}} \ell_{j,k}^{-1} (|\nabla u|^p + 1) d\nu \\ &\le C_7 \varepsilon + C_6 \sum_{i} \int_{B(y_i, 3\sqrt{n}\ell_i/4)} \ell_i^{-1} |\nabla u|^p d\nu + C_6 \sum_{j,k} \int_{B(y_{j,k}, 3\sqrt{n}\ell_{j,k}/4)} \ell_{j,k}^{-1} |\nabla u|^p d\nu \\ &\le C_7 \varepsilon + C_8 \sum_{i} \ell_i^{-1} \max \left\{ \left[ \left( \sqrt{n}\ell_i - \frac{3\sqrt{n}\ell_i}{4} \right)^{-p+} \int_{B(y_i, \ell_i)} (\sqrt{n}\ell_i)^{-\tau} d\nu \right] \right. \\ &\qquad \left. \left[ \left( \sqrt{n}\ell_i - \frac{3\sqrt{n}\ell_i}{4} \right)^{-p+} \int_{B(y_i, \ell_i)} (\sqrt{n}\ell_i)^{-\tau} d\nu \right]^{\frac{p-}{p+}} \right\} \\ &\quad + C_8 \sum_{j,k} \ell_{j,k}^{-1} \max \left\{ \left[ \left( \sqrt{n}\ell_{j,k} - \frac{3\sqrt{n}\ell_{j,k}}{4} \right)^{-p+} \int_{B(y_{j,k}, \ell_{j,k})} (\sqrt{n}\ell_{j,k})^{-\tau} d\nu \right] \right. \\ &\qquad \left. \left[ \left( \sqrt{n}\ell_{j,k} - \frac{3\sqrt{n}\ell_{j,k}}{4} \right)^{-p+} \int_{B(y_{j,k}, \ell_{j,k})} (\sqrt{n}\ell_{j,k})^{-\tau} d\nu \right]^{\frac{p-}{p+}} \right\}. \end{aligned} \quad (3.40) $$

Combining (3.40), (3.27), (3.23) and (3.36), we deduce that

$$ \int_{\Omega} \langle A(\cdot, \nabla u), \nabla \varphi \rangle dx \le C_9 \varepsilon, $$

here the $C_i$, $i = 6, \dots, 9$ are positive constants independent of $\varepsilon$. Since $\varepsilon \in (0, r_0)$ was arbitrary, the proof is complete. $\square$

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

Email address: jpablo@id.uff.br