Let Ω be a bounded domain in ${\mathbb{R}}^{n}$, $n\ge 3$, with a smooth boundary $\partial \mathrm{\Omega}$, and let $Y={(-\frac{1}{2},\frac{1}{2})}^{n}$.
Denote by ${G}_{0}={B}_{1}(0)$ the unit ball centered at the origin.
This plays a crucial role in the proof.
As far as we known, no results are known in the critical cases if ${G}_{0}$ is not a ball.
For $\delta >0$ and $\epsilon >0$, we define sets $\delta B=\{x:{\delta}^{-1}x\in B\}$ and ${\stackrel{~}{\mathrm{\Omega}}}_{\epsilon}=\{x\in \mathrm{\Omega}:\rho (x,\partial \mathrm{\Omega})>2\epsilon \}$.
Let

${a}_{\epsilon}={C}_{0}{\epsilon}^{\alpha},$

where $\alpha >1$ and ${C}_{0}$ is a given positive number.
Define

${G}_{\epsilon}=\bigcup _{j\in {\mathrm{{\rm Y}}}_{\epsilon}}({a}_{\epsilon}{G}_{0}+\epsilon j)=\bigcup _{j\in {\mathrm{{\rm Y}}}_{\epsilon}}{G}_{\epsilon}^{j},$

where ${\mathrm{{\rm Y}}}_{\epsilon}=\{j\in {\mathbb{Z}}^{n}:({a}_{\epsilon}{G}_{0}+\epsilon j)\cap {\overline{\stackrel{~}{\mathrm{\Omega}}}}_{\epsilon}\ne \mathrm{\varnothing}\}$, $N(\epsilon )=|{\mathrm{{\rm Y}}}_{\epsilon}|\cong {\epsilon}^{-n}$, and ${\mathbb{Z}}^{n}$ denotes the set of vectors *z* with integer
coordinates.
Define ${Y}_{\epsilon}^{j}=\epsilon Y+\epsilon j$, where $j\in {\mathrm{{\rm Y}}}_{\epsilon}$ and note that ${\overline{G}}_{\epsilon}^{j}\subset {\overline{Y}}_{\epsilon}^{j}$ and center of the ball ${G}_{\epsilon}^{j}$ coincides with the center of the cube ${Y}_{\epsilon}^{j}$.
Our “microscopic domain” is defined as

${\mathrm{\Omega}}_{\epsilon}=\mathrm{\Omega}\setminus \overline{{G}_{\epsilon}},{S}_{\epsilon}=\partial {G}_{\epsilon},\partial {\mathrm{\Omega}}_{\epsilon}=\partial \mathrm{\Omega}\cup {S}_{\epsilon}.$

We define the space ${W}_{0}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$ as the
completion, with respect to the norm of ${W}^{1,p}({\mathrm{\Omega}}_{\epsilon})$, of
the set of infinitely differentiable functions in ${\overline{\mathrm{\Omega}}}_{\epsilon}$ equal to zero in a neighborhood of $\partial \mathrm{\Omega}$, that is,

${W}_{0}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})=\{u\in {W}^{1,p}({\mathrm{\Omega}}_{\epsilon}):u=0\text{on}\partial \mathrm{\Omega}\}.$

Concerning the solvability of problem (1.1), we start by introducing the notion of weak solution.
Since we assume that $\sigma :\mathbb{R}\to \mathcal{\mathcal{P}}(\mathbb{R})$, where $\mathcal{\mathcal{P}}(\mathbb{R})$ denotes the set of subsets of $\mathbb{R}$, we recall, by well-known results (see, e.g., [2]), that

$\sigma \text{is a maximal monotone graph of}{\mathbb{R}}^{2}\text{,}0\in \sigma (0),$(2.1)

and that there exists a convex lower semicontinuous function $\mathrm{\Psi}:\mathbb{R}\to (-\mathrm{\infty},+\mathrm{\infty}]$, with $\mathrm{\Psi}(0)=0$, such that $\sigma =\partial \mathrm{\Psi}$ is its subdifferential.
We also know that if we define

$D(\sigma )=\{r\in \mathbb{R}\text{such that}\sigma (r)\ne \mathrm{\varnothing}\},$

where $\mathrm{\varnothing}$ denotes the empty set, and

$D(\mathrm{\Psi})=\{r\in \mathbb{R}\text{such that}\mathrm{\Psi}(r)+\mathrm{\infty}\},$

then $D(\sigma )\subset D(\mathrm{\Psi})\subset \overline{D(\mathrm{\Psi})}=\overline{D(\sigma )}$.

In the rest of the paper we will always assume that
$f\in {L}^{{p}^{\prime}}(\mathrm{\Omega}),$
where, as usual, ${p}^{\prime}=\frac{p}{p-1}$.

Since ${u}_{\epsilon}$ is the minimizer of the following energy functional in
${W}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$ (see [26, 1]):

$E(u)={\int}_{{\mathrm{\Omega}}_{\epsilon}}|\nabla u{|}^{p}dx+{\epsilon}^{-\gamma}{\int}_{{S}_{\epsilon}}\mathrm{\Psi}(u)dS-{\int}_{{\mathrm{\Omega}}_{\epsilon}}fudx,$

we consider the following definition of weak solution.

#### Definition 2.1.

We will say that ${u}_{\epsilon}\in {W}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$ is a weak solution of problem (1.1) if ${u}_{\epsilon}(x)\in D(\mathrm{\Psi})$ for a.e. $x\in {S}_{\epsilon}$, and for all $v\in {W}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$, we have

${\int}_{{\mathrm{\Omega}}_{\epsilon}}|\nabla {u}_{\epsilon}{|}^{p-2}\nabla {u}_{\epsilon}\cdot \nabla (v-{u}_{\epsilon})dx+{\epsilon}^{-\gamma}{\int}_{{S}_{\epsilon}}(\mathrm{\Psi}(v)-\mathrm{\Psi}({u}_{\epsilon}))dS\ge {\int}_{{\mathrm{\Omega}}_{\epsilon}}f(v-{u}_{\epsilon})dx.$(2.2)

The existence and uniqueness of a weak solution to problem (2.2) is an easy consequence of well-known results:

#### Proposition 2.2.

*There exists a unique ${u}_{\epsilon}\mathrm{\in}{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\epsilon}\mathrm{,}\mathrm{\partial}\mathit{}\mathrm{\Omega}\mathrm{)}$ weak solution of (2.2).
Besides, there exists $K\mathrm{>}\mathrm{0}$ independent of ε such that*

${\parallel \nabla {u}_{\epsilon}\parallel}_{{L}^{p}({\mathrm{\Omega}}_{\epsilon})}+{\epsilon}^{-\gamma}{\parallel \mathrm{\Psi}({u}_{\epsilon})\parallel}_{{L}^{1}({S}_{\epsilon})}\le K.$(2.3)

The homogenized problem will involve the function $H:\mathbb{R}\to \mathbb{R}$ given by (1.5).
Let us present some of the properties satisfied by *H*.

#### Lemma 2.3.

*If σ satisfies (2.1), then the function **H* defined by (1.5) is a nondecreasing nonexpansion on $\mathrm{R}$ (i.e., a nondecreasing Lipschitz continuous function of Lipschitz constant $L\mathrm{\le}\mathrm{1}$).
Moreover, this function *H* is the unique function $H\mathrm{:}\mathrm{R}\mathrm{\to}\mathrm{R}$ satisfying the relation

${\mathcal{\mathcal{B}}}_{0}{|H(r)|}^{p-2}H(r)\in \sigma (r-H(r))\mathit{\hspace{1em}}\mathit{\text{for any}}r\in \mathbb{R}.$(2.4)

Concerning the homogenized problem (1.3), we point out that since *H* is a nondecreasing nonexpansion on $\mathbb{R}$, for the parameters $\mathcal{\mathcal{A}}$ and ${\mathcal{\mathcal{B}}}_{0}$ given by (1.4) and (1.6), and for $f\in {L}^{{p}^{\prime}}(\mathrm{\Omega})$, there exists a unique weak solution $u\in {W}_{0}^{1,p}(\mathrm{\Omega})$ of problem (1.3).
Moreover, ${|H(u)|}^{p-2}H(u)\in {L}^{{p}^{\prime}}(\mathrm{\Omega})$.
For the proof it is enough to set $V={W}_{0}^{1,p}(\mathrm{\Omega})$ and define the operator $A:V\to {V}^{\prime}$ by

$\u3008Av,w\u3009={\int}_{\mathrm{\Omega}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla wdx+{\int}_{\mathrm{\Omega}}\mathcal{\mathcal{A}}|H(v){|}^{p-2}H(v)wdx\mathit{\hspace{1em}}\text{for any}w\in V\text{.}$(2.5)

Notice that, since *H* is Lipschitz, $H(v)\in {L}^{p}(\mathrm{\Omega})$ for any $v\in {L}^{p}(\mathrm{\Omega})$.
Then *A* is a hemicontinuous strictly monotone coercive operator, and the existence and uniqueness of a weak solution *u* is standard (see, e.g., [26]).

We will make fundamental use of the following reformulation of a weak solution.
Since the *limit operator*
$A:V\to {V}^{\prime}$, with $V={W}_{0}^{1,p}(\mathrm{\Omega})$, given by (2.5), is hemicontinuous and monotone, we can use the Brezis–Sibony characterization (see [3, Lemma 1.1] or [26, Chapter 2, Theorem 2.2]), that is, $u\in {W}_{0}^{1,p}(\mathrm{\Omega})$ is a weak solution of (1.3) if and only if

${\int}_{\mathrm{\Omega}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla (v-u)dx+{\int}_{\mathrm{\Omega}}{\mathcal{\mathcal{B}}}_{0}|H(v){|}^{p-2}H(v)(v-u)dx\ge {\int}_{\mathrm{\Omega}}f(v-u)dx\mathit{\hspace{1em}}\text{for any}v\in {W}_{0}^{1,p}(\mathrm{\Omega})\text{.}$(2.6)

The main result of this paper is the following convergence result.

#### Theorem 2.4.

*Let $n\mathrm{\ge}\mathrm{3}$, $\mathrm{1}\mathrm{<}p\mathrm{<}n$, $\alpha \mathrm{=}\frac{n}{n\mathrm{-}p}$ and $\gamma \mathrm{=}\alpha \mathit{}\mathrm{(}p\mathrm{-}\mathrm{1}\mathrm{)}$.
Let σ be any maximal monotone graph of ${\mathrm{R}}^{\mathrm{2}}$, with $\mathrm{0}\mathrm{\in}\sigma \mathit{}\mathrm{(}\mathrm{0}\mathrm{)}$, and let $f\mathrm{\in}{L}^{{p}^{\mathrm{\prime}}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$.
Let ${u}_{\epsilon}\mathrm{\in}{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\epsilon}\mathrm{,}\mathrm{\partial}\mathit{}\mathrm{\Omega}\mathrm{)}$ be the (unique) weak solution of problem (1.1).
Then there exists an extension ${\stackrel{\mathrm{~}}{u}}_{\epsilon}$ of ${u}_{\epsilon}$ such that ${\stackrel{\mathrm{~}}{u}}_{\epsilon}\mathrm{\rightharpoonup}u$ in ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ as $\epsilon \mathrm{\to}\mathrm{0}$,
where $u\mathrm{\in}{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ is the (unique) weak solution of problem (1.3) associated to the function **H*, defined by (1.5).

The other key result we will prove in this paper is Theorem 2.9 below, the statement of which requires some preliminary lemmas.
The extension ${\stackrel{~}{u}}_{\epsilon}$ of solutions ${u}_{\epsilon}$ can be obtained by applying the methods of [30].

#### Lemma 2.6.

*Let ${\mathrm{\Omega}}_{\epsilon}$ be the domain defined above and let $\mathrm{1}\mathrm{<}p\mathrm{<}n$, $n\mathrm{\ge}\mathrm{3}$.
Then there exists an extension operator ${P}_{\epsilon}\mathrm{:}{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\epsilon}\mathrm{)}\mathrm{\to}{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ such that
*

${\parallel {P}_{\epsilon}u\parallel}_{{W}^{1,p}(\mathrm{\Omega})}\le {C}_{1}{\parallel u\parallel}_{{W}^{1,p}({\mathrm{\Omega}}_{\epsilon})},$${\parallel \nabla ({P}_{\epsilon}u)\parallel}_{{L}_{p}(\mathrm{\Omega})}\le {C}_{2}{\parallel \nabla u\parallel}_{{L}_{p}({\mathrm{\Omega}}_{\epsilon})}.$

Moreover, by applying this extension theorem and the methods introduced in [30], we can prove the following useful estimates.

#### Lemma 2.7.

(i)

*Let *
$u\in {W}_{0}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$,
$p>1$
* and *
$n\ge 3$
*.
Then there exists positive constant *
*C*
* such that
*

${\parallel u\parallel}_{{L}^{p}({\mathrm{\Omega}}_{\epsilon})}\le C{\parallel \nabla u\parallel}_{{L}^{p}({\mathrm{\Omega}}_{\epsilon})}.$

(ii)

*Let *
$u\in {W}^{1,p}({Y}_{\epsilon})$
* be such that *
${\int}_{{Y}_{\epsilon}}u=0$
*.
Then*

${\parallel u\parallel}_{{L}^{p}({Y}_{\epsilon})}\le {K}_{1}\epsilon {\parallel \nabla u\parallel}_{{L}^{p}({Y}_{\epsilon})},$

*where the constant *
${K}_{1}$
* is independent of *
$\epsilon $.

Thanks to the a priori estimate (2.3) and the properties of the extension operator ${P}_{\epsilon}:{W}_{0}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})\to {W}^{1,p}(\mathrm{\Omega})$, we know that and there exists $u\in {W}_{0}^{1,p}(\mathrm{\Omega})$ such that

${P}_{\epsilon}{u}_{\epsilon}\rightharpoonup u\mathit{\hspace{1em}}\text{in}{W}_{0}^{1,p}(\mathrm{\Omega}).$

The difficult task is to show that $u\in {W}_{0}^{1,p}(\mathrm{\Omega})$ is the weak
solution of problem (1.3) such as it is ensured in Theorem 2.4.

Motivated by this and (2.6), we will also use the fact that if ${u}_{\epsilon}\in {W}_{0}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$ is the weak solution of problem (1.1), then

${\int}_{{\mathrm{\Omega}}_{\epsilon}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla (v-{u}_{\epsilon})dx+{\epsilon}^{-\gamma}{\int}_{{S}_{\epsilon}}(\mathrm{\Psi}(v)-\mathrm{\Psi}({u}_{\epsilon}))dS\ge {\int}_{{\mathrm{\Omega}}_{\epsilon}}f(v-{u}_{\epsilon})dx$(2.7)

for any test function $v\in {W}^{1,p}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$.

The problematic term, in order to pass to the limit, is the boundary integrals over ${S}_{\epsilon}$.
Here we will follow a technique of proof introduced by the last author (Shaposhnikova) in collaboration with different co-authors (see, e.g., [28, 37, 34]), which can be applied in different frameworks.

#### Lemma 2.8.

*Let ${z}_{\epsilon}\mathrm{\in}{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ for some $p\mathrm{>}\mathrm{1}$, and assume that ${z}_{\epsilon}\mathrm{\rightharpoonup}{z}_{\mathrm{0}}$ in ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ as $\epsilon \mathrm{\to}\mathrm{0}$.
Then
*

$\left|{2}^{2(n-1)}\epsilon \sum _{j\in {\mathrm{{\rm Y}}}_{\epsilon}}{\int}_{\partial {T}_{\epsilon /4}^{j}}{z}_{\epsilon}\mathit{d}S-{\omega}_{n}{\int}_{\mathrm{\Omega}}{z}_{0}\mathit{d}x\right|\to 0\mathit{\hspace{1em}}\mathit{\text{as}}\epsilon \to 0,$

*where ${\omega}_{n}$ is the surface area of the unit sphere in ${\mathrm{R}}^{n}$.*

This lemma (which we remark is independent of α and γ, see the
proof in [37]) is the key point of the homogenization technique in the critical case.
It is based in the general idea that if ${P}_{\epsilon}^{j}$ is the center of the ball ${G}_{\epsilon}^{j}=\{x\in {Y}_{\epsilon}^{j}:|x-{P}_{\epsilon}^{j}|<{a}_{\epsilon}\}$ and if ${T}_{\epsilon}^{j}$ denotes the ball of radius $\epsilon /4$ centered at the point ${P}_{\epsilon}^{j}$, then we can get several explicit estimates on the solution ${w}_{\epsilon}^{j}(x)$ for $j=1,\mathrm{\dots},N(\epsilon )$ of the auxiliary cellular boundary value problem

$\{\begin{array}{cccc}& {\mathrm{\Delta}}_{p}{w}_{\epsilon}^{j}=0,\hfill & & \hfill x\in {T}_{\epsilon}^{j}\setminus \overline{{G}_{\epsilon}^{j}},\\ & {w}_{\epsilon}^{j}=1,\hfill & & \hfill x\in \partial {G}_{\epsilon}^{j},\\ & {w}_{\epsilon}^{j}=0,\hfill & & \hfill x\in \partial {T}_{\epsilon}^{j}.\end{array}$(2.8)

One of the many remarkable properties of this cellular problem is that its (unique) weak solution, ${w}_{\epsilon}^{j}$, is radially symmetric (recall that ${G}_{0}$ is a ball) and satisfies that ${\partial}_{{\nu}_{p}}{w}_{\epsilon}^{j}$ is constant on $\partial {T}_{\epsilon}^{j}$ and on $\partial {G}_{\epsilon}^{j}$.
Due to the divergence theorem,

${\int}_{{G}_{\epsilon}^{j}}|\nabla {w}_{\epsilon}^{j}{|}^{p-2}\nabla {w}_{\epsilon}^{j}\cdot \nabla zdx={\int}_{\partial {T}_{\epsilon}^{j}}z{\partial}_{{\nu}_{p}}{w}_{\epsilon}^{j}dS+{\int}_{\partial {G}_{\epsilon}^{j}}z{\partial}_{{\nu}_{p}}{w}_{\epsilon}^{j}dS\mathit{\hspace{1em}}\text{for any}z\in {W}^{1,p}({T}_{\epsilon}^{j}\setminus \overline{{G}_{\epsilon}^{j}})\text{.}$

Furthermore, we can make explicitly several computations.
Hence, we have an explicit way to compare the reaction term on ${S}_{\epsilon}$ with an auxiliary term on balls with radius $C\epsilon $, and Lemma 2.8 becomes very useful.

Another key idea of our proof is to relate a general test function $v\in {W}_{0}^{1,p}(\mathrm{\Omega})$, used to check the limit characterization (2.6), with some suitable *correction*
${v}_{\epsilon}$, which is a better fitted test function in the microscopic weak formulation (2.7).
In fact, by density, it will be enough to do that with a smooth test function $v\in {\mathcal{\mathcal{C}}}_{c}^{\mathrm{\infty}}(\mathrm{\Omega})$.
We will construct such adaptation among test functions in the form ${v}_{\epsilon}=v-h{W}_{\epsilon}$, where, for the moment, $h\in {W}^{1,\mathrm{\infty}}(\mathrm{\Omega})$ without any other property, and, which is crucial, ${W}_{\epsilon}\in {W}_{0}^{1,\mathrm{\infty}}(\mathrm{\Omega})$ defined as

${W}_{\epsilon}=\{\begin{array}{cc}{w}_{\epsilon}^{j},\hfill & x\in {T}_{\epsilon}^{j}\setminus \overline{{G}_{\epsilon}^{j}},j=1,\mathrm{\dots},N(\epsilon )=|{\mathrm{{\rm Y}}}_{\epsilon}|,\hfill \\ 1,\hfill & x\in {G}_{\epsilon},\hfill \\ 0,\hfill & x\in {\mathbb{R}}^{n}\setminus {\bigcup}_{j=1}^{N(\epsilon )}{T}_{\epsilon}^{j},\hfill \end{array}$(2.9)

with ${w}_{\epsilon}^{j}$ the solution of the auxiliary cellular boundary value problem (2.8).
The following technical result will explain why the function *H*, arising in the limit problem (1.3), was taken in this concrete form (more precisely, so that (2.4) holds), different from the boundary kinetics σ.

#### Theorem 2.9.

*Let ${u}_{\epsilon}\mathrm{\in}{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}{\mathrm{\Omega}}_{\epsilon}\mathrm{,}\mathrm{\partial}\mathit{}\mathrm{\Omega}\mathrm{)}$, $\mathrm{1}\mathrm{<}p\mathrm{<}n$, be a sequence of uniformly bounded norm, and let $v\mathrm{\in}{\mathcal{C}}_{c}^{\mathrm{\infty}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$, $h\mathrm{\in}{W}^{\mathrm{1}\mathrm{,}\mathrm{\infty}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ and
${v}_{\epsilon}\mathrm{=}v\mathrm{-}h\mathit{}{W}_{\epsilon}\mathrm{.}$
Then
*

$\underset{\epsilon \to 0}{lim}\left({\int}_{{\mathrm{\Omega}}_{\epsilon}}{|\nabla {v}_{\epsilon}|}^{p-2}\nabla {v}_{\epsilon}\cdot \nabla ({v}_{\epsilon}-{u}_{\epsilon})dx\right)=\underset{\epsilon \to 0}{lim}({I}_{1,\epsilon}+{I}_{2,\epsilon}+{I}_{3,\epsilon}),$

*where
*

${I}_{1,\epsilon}={\displaystyle {\int}_{{\mathrm{\Omega}}_{\epsilon}}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla (v-{u}_{\epsilon})dx,$(2.10)${I}_{2,\epsilon}=-{\epsilon}^{-\gamma}{\mathcal{\mathcal{B}}}_{0}{\displaystyle {\int}_{{S}_{\epsilon}}}|h{|}^{p-2}h(v-h-{u}_{\epsilon})dS,$${I}_{3,\epsilon}=-{A}_{\epsilon}\epsilon {\displaystyle \sum _{j\in {\mathrm{{\rm Y}}}_{\epsilon}}}{\displaystyle {\int}_{\partial {T}_{\epsilon}^{j}}}|h{|}^{p-2}h(v-{u}_{\epsilon})dS,$(2.11)

*with ${A}_{\epsilon}$ being a bounded sequence, see (5.1).
Besides, if ${\stackrel{\mathrm{~}}{u}}_{\epsilon}$ is an extension of ${u}_{\epsilon}$ and ${\stackrel{\mathrm{~}}{u}}_{\epsilon}\mathrm{\rightharpoonup}u$ in ${W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$, then, for
any $v\mathrm{\in}{W}_{\mathrm{0}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$,*

$\underset{\epsilon \to 0}{lim}{\int}_{{\mathrm{\Omega}}_{\epsilon}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla (v-h{W}_{\epsilon}-{u}_{\epsilon})dx={\int}_{\mathrm{\Omega}}|\nabla v{|}^{p-2}\nabla v\cdot \nabla (v-u)dx.$

The aforementioned corrector term in the form $h{W}_{\epsilon}$, where $h\in {W}^{1,\mathrm{\infty}}({\mathrm{\Omega}}_{\epsilon},\partial \mathrm{\Omega})$ will be taken to satisfy the condition $h(x)=H(v(x))$ for a.e. $x\in \mathrm{\Omega}$, with *H* given
by (2.4).
These conditions rise naturally so that the term ${I}_{2,\epsilon}$ above cancels out with the reaction term.

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