0$ in $\Omega^{\varepsilon}$, although $u^{\varepsilon}$ is not uniformly positive, except in the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$, as, for instance, in example (a). The existence and uniqueness of a weak solution of (1.1) can be settled by using the classical theory of semilinear monotone problems (see, for instance, \cite{Brezis}, \cite{Diaz1} and \cite{Lions}). As a result, we know that there exists a unique weak solution $u^{\varepsilon}\in V^{\varepsilon}\bigcap H^{2}(\Omega ^{\varepsilon})$, where \[ V^{\varepsilon}=\{v\in H^{1}(\Omega^{\varepsilon}) : v=0\text{ on } \partial \Omega\}. \] Moreover, if in the second model situation, which is in fact the most general case we treat here, with $\Omega ^{\varepsilon}$ we associate the following nonempty convex subset of $V^{\varepsilon}$: \begin{equation} K^{\varepsilon }=\{ v\in V^{\varepsilon }: G(v)\big|_{S^{\varepsilon }} \in L^{1}(S^{\varepsilon })\}, \end{equation} then $u^{\varepsilon}$ is also known to be characterized as being the unique solution of the following variational problem: \begin{quote} Find $u^{\varepsilon }\in K^{\varepsilon }$ such that \begin{equation} D_{f} \int_{\Omega ^{\varepsilon }}Du^{\varepsilon }D(v^{\varepsilon }-u^{\varepsilon })dx- \int_{\Omega ^{\varepsilon }}f(v^{\varepsilon }-u^{\varepsilon })dx+a\langle \mu ^{\varepsilon },G(v^{\varepsilon })-G(u^{\varepsilon })\rangle \geq 0 \end{equation} for all $v^{\varepsilon }\in K^{\varepsilon }$, where $\mu^{\varepsilon }$ is the linear form on $W_{0}^{1,1}(\Omega )$ defined by \[ \langle \mu ^{\varepsilon },\varphi \rangle =\varepsilon \int_{S^{\varepsilon }}\varphi d\sigma \quad \forall \varphi \in W_{0}^{1,1}(\Omega ). \] \end{quote} From a geometrical point of view, we shall just consider periodic structures obtained by removing periodically from $\Omega$, with period $\varepsilon Y$ (where $Y$ is a given hyper-rectangle in $\mathbb{R^n}$), an elementary reactive obstacle $T$ which has been appropriated rescaled and which is strictly included in $Y$, i.e. $\overline{T}\subset Y$. As usual in homogenization, we shall be interested in obtaining a suitable description of the asymptotic behavior, as $\varepsilon$ tends to zero, of the solution $u^{\varepsilon }$ in such domains. We will wonder, for example, whether the solution $u^{\varepsilon }$ converges to a limit $u$ as $\varepsilon \rightarrow 0$. And if this limit exists, can it be characterized? In the second model situation (in absence of any additional regularity on $g$), the solution $u^{\varepsilon}$, properly extended to the whole of $\Omega$, converges to the unique solution of the variational inequality: $u\in H^{1}_{0}(\Omega)$, \begin{equation} \int_{\Omega }QDuD(v-u)dx\geq \int_{\Omega }f(v-u)dx-a\frac{| \partial T|}{| Y\setminus T | } \int_{\Omega}(G(v)-G(u))dx, \end{equation} for all $v \in H^{1}_{0}(\Omega)$. Here, $Q=((q_{ij}))$ is the classical homogenized matrix, whose entries are \begin{equation} q_{ij}=D_{f}\Big( \delta _{ij}+\frac{1}{|Y\setminus T|} \int_{Y\setminus T}\frac{\partial \chi _{j}}{\partial y_{i}}dy\Big) \end{equation} in terms of the functions $\chi _{i}$, $i=1,\dots ,n$, solutions of the so-called cell problems \begin{equation} \begin{gathered} -\Delta \chi _{i}=0 \quad \text{in } Y\setminus T, \\ \frac{\partial (\chi _{i}+y_{i})}{\partial \nu }=0 \quad \text{on }\partial T, \\ \chi _{i}\quad\text{is $Y$-periodic.} \end{gathered} \end{equation} We remark that if $g$ is smooth, then $g$ is the classical derivative of $G$. The chemical situation behind the second nonlinear problem that we will treat in this paper is slightly different from the previous one; it also involves a chemical reactor containing reactive grains, but we assume that now there is an internal reaction inside the grains, instead just on their boundaries. In fact, it is therefore a transmission problem with an unknown flux on the boundary of each grain. To simplify matters, we shall just focus on the case of a function $g$ which is continuous, monotone increasing and such that $g(0)=0$; examples (a) and (b) are both covered by this class of functions $g's$ and, of course, both are still our main practical examples. A simplified setting of this kind of models is as follows: \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{p}\Delta v^{\varepsilon }+ag(v^{\varepsilon })=0,\quad \text{in }\Omega \setminus \overline{\Omega ^{\varepsilon }} \\ -D_{f}{\frac{\partial u^{\varepsilon }}{\partial \nu }}=D_{p} {\frac{\partial v^{\varepsilon }}{\partial \nu }}\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=v^{\varepsilon }\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $D_p$ is a second diffusion coefficient characterizing the granular material filling the reactive obstacles. As in the previous case, the classical semilinear theory guarantees the well-posedness of this problem. When we define $\theta^{\varepsilon}$ as \[ \theta ^{\varepsilon }(x)=\begin{cases} u^{\varepsilon }(x)& x\in \Omega ^{\varepsilon }, \\ v^{\varepsilon }(x)& x\in \Omega \setminus \overline{\Omega^{\varepsilon }}, \end{cases} \] and we introduce \[ A=\begin{cases} D_{f}Id & \text{in }Y\setminus T \\ D_{p}Id & \text{in }T,% \end{cases} \] then our main result of convergence for this model shows that $\theta^{\varepsilon}$ converges weakly in $H^{1}_{0}(\Omega)$ to the unique solution of the homogenized problem \begin{equation} \begin{gathered} -{\sum_{i,j=1}^{n}a_{ij}^{0}}{\frac{\partial ^{2}u}{\partial x_{i} \partial x_{j}}}+a{\frac{|T|}{|Y\setminus T |}}g(u)=f\quad \text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega. \end{gathered} \end{equation} Here, $A^{0}=((a_{ij}^{0}))$ is the homogenized matrix, whose entries are \begin{equation} a_{ij}^{0}=\frac{1}{|Y|}\int_{Y}\big( a_{ij}+a_{ik} \frac{\partial \chi_{j}}{\partial y_{k}}\big)dy, \end{equation} in terms of the functions $\chi _{j}$, $j=1,\dots ,n$, solutions of the so-called cell problems \begin{equation} \begin{gathered} -\mathop{\rm div}(AD(y_{j}+\chi _{j}))=0\quad \text{in }Y, \\ \chi_{j} \quad\mbox{is $Y$-periodic}. \end{gathered} \end{equation} Note that the two reactive flows studied in this paper, namely (1.1) and (1.7), lead to completely different effective behavior. The macroscopic problem (1.4) arises from the homogenization of a boundary-value problem in the exterior of some periodically distributed obstacles and the zero-order term occurring in (1.4) has its origin in this particular structure of the model. The influence of the chemical reactions taking place on the boundaries of the reactive obstacles is reflected in the appearance of this zero-order extra-term. On the other hand, the second model is again a boundary-value problem, but this time in the whole domain $\Omega$, with discontinuous coefficients. Its macroscopic behavior (see (1.8)) also involves a zero-order term, but of a completely different nature; it is originated in the chemical reactions occurring inside the grains. The approach we used is the so-called energy method introduced by Tartar \cite{Tartar1}, \cite{Tartar2} for studying homogenization problems. It consists of constructing suitable test functions that are used in our variational problems. However, it is worth mentioning that the $\Gamma $-convergence of integral functionals involving oscillating obstacles could be a successful alternative. Extensive references on this topic can be found in the monographs of Dal Maso \cite{DalMaso} and of Braides and Defranceschi \cite{Braides-Defranceschi}. For example, our main result in Chapter 2 (cf. Theorem \ref{thm2.6}) can also be interpreted as a $\Gamma$-convergence-type result for the functionals \[ v\mapsto \frac {1}{2} D_{f}\int_{\Omega ^{\varepsilon }}DvDvdx+a\langle \mu ^{\varepsilon },G(v)\rangle-\int_{\Omega ^{\varepsilon }}fvdx +I_{K^{\varepsilon }}(v) \] (where $I_{K^{\varepsilon }}$ is the indicator function of the set $% K^{\varepsilon }$, i.e. $I_{K^{\varepsilon }}$ is equal to zero if v belongs to $K^{\varepsilon }$ and $+\infty $ otherwise) to the limit functional \[ v\mapsto \frac{1}{2}\int_{\Omega}QDvDvdx+a \frac{|\partial T|}{|Y\setminus T |}% \int_{\Omega }G(v)dx-\int_{\Omega}fvdx, \] which is the energy functional associated to (1.3). Also, let us mention that another possible way to get the limit problem (1.8) could be to use the two-scale convergence technique, coupled with periodic modulation, as in \cite{Bourgeat-Luckhaus-Mikelic}. Regarding our second problem, i.e. chemical reactive flows through periodic array of cells, a related work was completed by {Hornung et al.} \cite{Hornung-Jager-Mikelic} using nonlinearities which are essentially different from the ones we consider in the present paper. The proof of these authors is also different, since it is mainly based on the technique of two-scale convergence, which, as already mentioned, proves to be a successful alternative for this kind of problems. However, we have decided to use the energy method, coupled with monotonicity methods and results from the theory of semilinear problems, because it offered us the possibility to cover the nonlinear cases of practical importance mentioned above. The structure of our paper is as follows: first, let us mention that we shall just focus on the case $n\geq 3$, which will be treated explicitly. The case $n=2$ is much more simpler and we shall omit to treat it. In Section 2 we start by analyzing the first nonlinear problem, namely (1.1). We begin with the case of a monotone smooth function $g$ and we prove the convergence result using the energy method. Next, we treat the case of a maximal monotone graph, by writing our microscopic problem in the form of a variational inequality. The case of a reactive flow penetrating a periodical structure of grains is addressed in Section 3. Finally, notice that throughout the paper, by $C$ we shall denote a generic fixed strictly positive constant, whose value can change from line to line. \section{Chemical reactions on the walls of a chemical reactor} In this section, we will be concerned with the stationary reactive flow of a fluid confined in the exterior of some periodically distributed obstacles, reacting on the boundaries of the obstacles. We will treat separately the situation in which the nonlinear function $g$ in (1.1) is a monotone smooth function satisfying the condition $g(0)=0$ and the situation in which $g$ is a maximal monotone graph with $g(0)=0$. Let $\Omega $ be a smooth bounded connected open subset of $\mathbb{R}^{n}$ $% (n\geq 3)$ and let $Y$ $=[0,l_{1}[\times \dots [0,l_{n}[$ be the representative cell in $\mathbb{R}^{n}$. Denote by $T$ an open subset of $Y$ with smooth boundary $\partial T$ such that $\overline{T}\subset Y$. We shall refer to $% T $ as being \textit{the elementary obstacle}. Let $\varepsilon $ be a real parameter taking values in a sequence of positive numbers converging to zero. For each $\varepsilon $ and for any integer vector $k\in \mathbb{Z}^{n}$, set $T_{k}^{\varepsilon }$ the translated image of$\ \varepsilon T$ by the vector $% kl=(k_{1}l_{1},\dots ,k_{n}l_{n}):$% \[ T_{k}^{\varepsilon }=\varepsilon (kl+T). \] The set $T_{k}^{\varepsilon }$ represents the obstacles in $\mathbb{R}^{n}$. Also, let us denote by $T^{\varepsilon }$ the set of all the obstacles contained in $\Omega $, i.e. \[ T^{\varepsilon }=\bigcup \left\{ T_{k}^{\varepsilon } : \overline{T_{k}^{\varepsilon }}\mathbf{\subset }\Omega , k\in \mathbb{% Z}^{n}\right\} . \] Set \[ \Omega ^{\varepsilon }=\Omega \setminus {\overline{T^{\varepsilon }}}. \] Hence, $\Omega ^{\varepsilon }$ is a periodical domain with periodically distributed obstacles of size of the same order as the period. Remark that the obstacles do not intersect the boundary $\partial \Omega $. Let \[ S^{\varepsilon }=\cup \{\partial T_{k}^{\varepsilon }\mid \overline{% T_{k}^{\varepsilon }}\mathbf{\subset }\Omega , k\in \mathbb{Z}^{n}\}. \] So \[ \partial \Omega ^{\varepsilon }=\partial \Omega \cup S^{\varepsilon }. \] We shall also use the following notation: $|\omega |$ is the Lebesgue measure of any measurable subset $\omega$ of $\mathbb{R}^{n}$, $\chi _{\omega }$ is the characteristic function of the set $\omega$, $Y^{*}=Y\setminus \overline{T}$, and \begin{equation} \rho =\frac{|Y^{*}|}{|Y|}. \end{equation} Moreover, for an arbitrary function $\psi \in L^{2}(\Omega ^{\varepsilon })$, we shall denote by $\widetilde{\psi }$ its extension by zero inside the obstacles: \[ \widetilde{\psi }=\begin{cases} \psi & \text{in } \Omega ^{\varepsilon }, \\ 0 & \text{in } \Omega \setminus \overline{\Omega ^{\varepsilon }}. \end{cases} \] Also, for any open subset $D\subset \mathbb{R}^{n}$ and any function $g\in L^{1}(D)$, we set \begin{equation} \mathcal{M}_{D}(g)=\frac{1}{|D|}\int_{D}gdx. \end{equation} In the sequel we reserve the symbol $\#$ to denote periodicity properties. \subsection{Setting of the problem} As already mentioned, we are interested in studying the behavior of the solution, in such a periodical domain, of the problem \begin{equation} \begin{gathered} -D_{f}\Delta u^{\varepsilon }=f\quad \text{in }\Omega ^{\varepsilon }, \\ -D_{f} {\frac{\partial u^{\varepsilon }}{\partial \nu }} =a\varepsilon g(u^{\varepsilon })\quad \text{on }S^{\varepsilon }, \\ u^{\varepsilon }=0\quad \text{on }\partial \Omega . \end{gathered} \end{equation} Here, $\nu $ is the exterior unit normal to $\Omega ^{\varepsilon }$, $a>0$, $f\in L^{2}(\Omega )$ and $g$ is assumed to be given. Two model situations will be considered; the case in which $g$ is a monotone smooth function satisfying the condition $g(0)=0$ and the case of a maximal monotone graph with $g(0)=0$, i.e. the case in which $g$ is the subdifferential of a convex lower semicontinuous function $G$. These two general situations are well illustrated by the following important practical examples: \begin{itemize} \item[(a)] $g(v)=\dfrac{\alpha v}{1+\beta v}$, $\alpha, \beta>0$ (Langmuir kinetics) \item[(b)] $g(v)=|v|^{p-1}v$, $0

0$ in $\Omega^{\varepsilon}$,
although $u^{\varepsilon}$ is not uniformly positive except in the case in
which $g$ is a monotone smooth function satisfying the condition $g(0)=0$,
as, for instance, in example $a)$. Moreover, since $u$ represents a
concentration, it could be natural to assume that $f\leq 1$, and again one
can prove that, in this case, $u \leq 1$. Without loss of generality, in
what follows we shall assume that $D_{f}=1$.
\subsection{First model situation: $g$ smooth}
Let $g$ be a continuously differentiable function, monotonously
non-decreasing and such that $g(v)=0$ if and only if $v=0$. We shall suppose that there
exist a positive constant $C$ and an exponent $q$, with $0\leq q< n/(n-2)$,
such that
\begin{equation}
|\frac{\partial g}{\partial v}|\leq C(1+|v|^{q}).
\end{equation}
Let us introduce the functional space
\[
V^{\varepsilon }=\left\{ v\in H^{1}(\Omega ^{\varepsilon
}) : v=0\text{on }\partial \Omega \right\} ,
\]
with
$\| v\| _{V^{\varepsilon }}=\| \nabla v\|_{L^{2}(\Omega ^{\varepsilon })}$.
The weak formulation of problem (2.3) (written for $D_{f}=1$) is:\\
Find $u^{\varepsilon }\in V^{\varepsilon }$ such that
\begin{equation}
{\int_{\Omega ^{\varepsilon }}\nabla u^{\varepsilon
}\cdot \nabla \varphi dx+a\varepsilon \int_{S^{\varepsilon
}}g(u^{\varepsilon })\varphi d\sigma =\int_{\Omega ^{\varepsilon
}}f\varphi dx}\quad \forall \varphi \in V^{\varepsilon }.
\end{equation}
By classical existence results (see \cite{Brezis}), there exists a unique
weak solution $u^{\varepsilon }\in V^{\varepsilon }\cap H^{2}(\Omega
^{\varepsilon })$ of problem (2.3).
The solution $u^{\varepsilon }$ of problem (2.3) being defined only on $%
\Omega ^{\varepsilon }$, we need to extend it to the whole of $\Omega $ to
be able to state the convergence result. In order to do that, let us recall
the following well-known extension result (see \cite{Cioranescu-Paulin}).
\begin{lemma} \label{lm2.1}
There exists a linear continuous extension operator
$$
P^{\varepsilon }\in \mathcal{L}(L^{2}(\Omega ^{\varepsilon });L^{2}(\Omega ))
\cap \mathcal{L} (V^{\varepsilon}; H_{0}^{1}(\Omega ))
$$
and a positive constant $C$, independent of $\varepsilon $, such that
for any $v\in V^{\varepsilon }$,
\begin{gather*}
\| P^{\varepsilon }v\| _{L^{2}(\Omega )}\leq C\| v\|_{L^{2}
(\Omega ^{\varepsilon })},\\
\| \nabla P^{\varepsilon }v\|_{L^{2}(\Omega )}\leq C\|
\nabla v\|_{L^{2}(\Omega ^{\varepsilon })}\,.
\end{gather*}
\end{lemma}
An immediate consequence of the previous lemma is the following
Poincar\'{e}'s inequality in $V^{\varepsilon }$.
\begin{lemma} \label{lm2.2}
There exists a positive constant $C$, independent of $\varepsilon $, such
that for any $v\in V^{\varepsilon }$,
\[
\| v\|_{L^{2}(\Omega ^{\varepsilon })}\leq C\| \nabla
v\|_{L^{2}(\Omega ^{\varepsilon })}\,.
\]
\end{lemma}
The main result of this section is as follows.
\begin{theorem} \label{thm2.3}
One can construct an extension $P^{\varepsilon }u^{\varepsilon }$ of the
solution $u^{\varepsilon }$ of the variational problem (2.5) such that
$P^{\varepsilon }u^{\varepsilon }\rightharpoonup u$
weakly in $H_{0}^{1}(\Omega )$,
where $u$ is the unique solution of
\begin{equation}
\begin{gathered}
-{\sum_{i,j=1}^{n}q_{ij}}{\frac{\partial ^{2}u}{\partial x_{i}\partial
x_{j}}}+a{\frac{|\partial T|}{|Y^{*}|} }g(u)=f\quad \text{in }\Omega, \\
u=0\quad \text{on }\partial \Omega \,.
\end{gathered}
\end{equation}
Here, $Q=((q_{ij}))$ is the classical homogenized matrix, whose
entries are
\begin{equation}
q_{ij}=\delta _{ij}+\frac{1}{|Y^{*}|}
\int_{Y^{*}}\frac{\partial \chi _{j}}{\partial y_{i}}dy
\end{equation}
in terms of the functions $\chi _{i}$, $i=1,\dots ,n$,
solutions of the so-called cell problems
\begin{equation}
\begin{gathered}
-\Delta \chi _{i}=0 \quad \text{in } Y^{*}, \\
\frac{\partial (\chi _{i}+y_{i})}{\partial \nu }=0 \quad\text{on } \partial T, \\
\chi _{i}\quad\text{is $Y$-periodic.}
\end{gathered}
\end{equation}
The constant matrix $Q$ is symmetric and positive-definite.
\end{theorem}
\begin{proof} We divide the proof into four steps.
\noindent\textit{First step.} Let $u^{\varepsilon }\in V^{\varepsilon }$ be the
solution of the variational problem (2.5) and let $P^{\varepsilon
}u^{\varepsilon }$ be the extension of $u^{\varepsilon }$ inside the
obstacles given by Lemma \ref{lm2.1}. Taking $\varphi =u^{\varepsilon }$ as a test
function in (2.5), using Schwartz and Poincar\'{e}'s inequalities, we easily
get
\[
\| P^{\varepsilon }u^{\varepsilon }\|_{H_{0}^{1}(\Omega )}\leq C.
\]
Consequently, by passing to a subsequence, still denoted by $P^{\varepsilon
}u^{\varepsilon }$, we can assume that there exists $u\in H_{0}^{1}(\Omega )$
such that
\begin{equation}
P^{\varepsilon }u^{\varepsilon }\rightharpoonup u\quad \text{weakly in }%
H_{0}^{1}(\Omega ).
\end{equation}
It remains to identify the limit equation satisfied by $u$.
\noindent\textit{Second step}. In order to get the limit equation satisfied by $u$ we
have to pass to the limit in (2.5). For getting the limit of the second term
in the left hand side of (2.5), let us introduce, for any
$h\in L^{s'}(\partial T)$, $1\leq s'\leq \infty $, the linear form
$\mu_{h}^{\varepsilon }$ on $W_{0}^{1,s}(\Omega )$ defined by
\[
\langle \mu _{h}^{\varepsilon },\varphi \rangle
=\varepsilon \int_{S^{\varepsilon }}h(\frac{x}{\varepsilon
})\varphi d\sigma \quad \forall \varphi \in W_{0}^{1,s}(\Omega ),
\]
with $1/s+1/s'=1$. It is proved in \cite{Cioranescu-Donato} that
\begin{equation}
\mu _{h}^{\varepsilon }\rightarrow \mu _{h}\quad \text{strongly in }%
(W_{0}^{1,s}(\Omega ))',
\end{equation}
where $\langle \mu _{h},\varphi \rangle =\mu _{h}\int_{\Omega}\varphi dx$,
with
\[
\mu _{h}=\frac{1}{|Y|}\int_{\partial T}h(y)d\sigma .
\]
In the particular case in which $h\in L^{\infty }(\partial T)$ or even when $h$
is constant, we have
\[
\mu _{h}^{\varepsilon }\rightarrow \mu _{h}\quad \text{strongly in }%
W^{-1,\infty }(\Omega ).
\]
In what follows, we shall denote by $\mu ^{\varepsilon }$ the above
introduced measure in the particular case in which $h=1$. Notice that in
this case $\mu _{h}$ becomes $\mu _{1}=|\partial T|/|Y|$.
Let us prove now that for any $\varphi \in \mathcal{\ D}(\Omega )$ and for
any $v^{\varepsilon }\rightharpoonup v$ weakly in $H_{0}^{1}(\Omega )$, we
get
\begin{equation}
\varphi g(v^{\varepsilon })\rightharpoonup \varphi g(v)\quad
\text{weakly in }W_{0}^{1,\overline{q}}(\Omega ),
\end{equation}
where
\[
\overline{q}=\frac{2n}{q(n-2)+n}.
\]
To prove (2.11), let us first note that
\begin{equation}
\sup \| \nabla g(v^{\varepsilon })\|_{L^{\overline{q}}(\Omega
)}<\infty .
\end{equation}
Indeed, from the growth condition (2.4) imposed to $g$, we get
\begin{align*}
\int_{\Omega }\big|\frac{\partial g}{\partial
x_{i}}(v^{\varepsilon })\big|^{\overline{q}}dx
&\leq C\int_{\Omega }\big( 1+|v^{\varepsilon }|
^{q\overline{q}}\big) |\frac{\partial v^{\varepsilon
}}{\partial x_{i}}|^{\overline{q}}dx\\
&\leq C( 1+( \int_{\Omega }|v^{\varepsilon }|^{q\overline{q}\gamma }dx)
^{1/\gamma }) ( \int_{\Omega }|\nabla
v^{\varepsilon }|^{\overline{q}\delta }dx) ^{1/\delta },
\end{align*}
where we took $\gamma $ and $\delta $ such that $\overline{q}\delta =2$, $%
1/\gamma +1/\delta =1$ and $q\overline{q}\gamma =2n/(n-2)$. Note that from
here we get $\overline{q}={\frac{2n}{q(n-2)+n}}$. Also, since
$0\leq q< n/(n-2)$, we have $\overline{q}> 1$. Now, since
\[
\sup \| v^{\varepsilon }\|_{L^{\frac{2n}{n-2}}(\Omega )}<\infty,
\]
we get immediately (2.12). Hence, to get (2.11), it remains only to prove
that
\begin{equation}
g(v^{\varepsilon })\rightarrow g(v)\quad \text{strongly in }L^{\overline{q}%
}(\Omega ).
\end{equation}
But this is just a consequence of the following well-known result
(see \cite{DalMaso} and \cite{Lions}).
\begin{theorem} \label{thm2.4}
Let $G:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ be a Carath\'{e}odory
function, i.e.
\begin{itemize}
\item[(a)] For every $v$ the function $G(\cdot ,v)$ is measurable with respect to
$x\in \Omega $.
\item[(b)] For every (a.e.) $x\in \Omega $, the function $G(x,\cdot )$ is continuous
with respect to $v$.
\end{itemize}
Moreover, if we assume that there exists a positive constant $C$
such that
\[
|G(x,v)|\leq C\big( 1+|v|^{r/t}\big) ,
\]
with $r\geq 1$ and $t<\infty $, then the map $v\in L^{r}(\Omega )\mapsto
G(x,v(x))\in L^{t}(\Omega )$ is continuous in the strong topologies.
\end{theorem}
Indeed, since
\[
|g(v)|\leq C(1+|v|^{q+1}),
\]
applying the above theorem for $G(x,v)=g(v)$, $t=\overline{q}$ and $%
r=(2n/(n-2))-r'$, with $r'>0$ such that $q+1