\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 76, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/76\hfil Elliptic equations with measure data] {Elliptic equations with measure data in Orlicz spaces} \author[G. Dong\hfil EJDE-2008/76\hfilneg] {Ge Dong} \address{Ge Dong \newline 1. Department of Mathematics, Shanghai University \\ No. 99, Shangda Rd., Shanghai, China\newline 2. Department of Basic, Jianqiao College\\ No. 1500, Kangqiao Rd., Shanghai, China} \email{dongge97@sina.com} \thanks{Submitted October 8, 2006. Published May 27, 2008.} \subjclass[2000]{35J15, 35J20, 35J60} \keywords{Orlicz-Sobolev spaces; elliptic equation; nonlinear; measure data} \begin{abstract} This article shows the existence of solutions to the nonlinear elliptic problem $A(u)=f$ in Orlicz-Sobolev spaces with a measure valued right-hand side, where $A(u)=-\mathop{\rm div}a(x,u,\nabla u)$ is a Leray-Lions operator defined on a subset of $W_{0}^{1}L_{M}(\Omega)$, with general $M$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $M:\mathbb{R}\to \mathbb{R}$ be an $N$-function; i.e. $M$ is continuous, convex, with $M(u)>0$ for $u>0$, $M(t)/t\to 0$ as $t\to 0$, and $M(t)/t\to \infty$ as $t\to \infty$. Equivalently, $M$ admits the representation $M(u)=\int_{0}^{u}\phi(t)dt$, where $\phi$ is the derivative of $M$, with $\phi$ non-decreasing, right continuous, $\phi(0)=0$, $\phi(t)>0$ for $t>0$, and $\phi(t)\to \infty$ as $t\to \infty$. The $N$-function $\bar{M}$ conjugate to $M$ is defined by $\bar{M}(v)=\int_{0}^{t}\psi(s)ds$, where $\psi$ is given by $\psi(s)=\sup\{t:\phi(t)\leq s\}$. The $N$-function $M$ is said to satisfy the $\Delta_{2}$ condition, if for some $k>0$ and $u_{0}>0$, $$ M(2u)\leq kM(u), \quad \forall u\geq u_{0}. $$ Let $P,Q$ be two $N$-functions, $P\ll Q$ means that $P$ grows essentially less rapidly than $Q$; i.e. for each $\varepsilon>0$, $P(t)/Q(\varepsilon t)\to 0$ as $t\to \infty$. This is the case if and only if $\lim_{t\to \infty} Q^{-1}(t)/P^{-1}(t)=0$. Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with the segment property. The class $W^{1}L_{M}(\Omega)$ (resp., $W^{1}E_{M}(\Omega)$) consists of all functions $u$ such that $u$ and its distributional derivatives up to order 1 lie in $L_{M}(\Omega)$ (resp., $E_{M}(\Omega)$). Orlicz spaces $L_{M}(\Omega)$ are endowed with the Luxemburg norm $$ \|u\|_{(M)}=\inf\big\{ \lambda>0: \int_{\Omega}M\big(\frac{|u(x)|} {\lambda}\big)dx\leq1\big\}. $$ The classes $W^{1}L_{M}(\Omega)$ and $W^{1}E_{M}(\Omega)$ of such functions may be given the norm $$ \|u\|_{\Omega,M}=\sum_{|\alpha|\leq1}\|D^{\alpha}u\|_{(M)}. $$ These classes will be Banach spaces under this norm. I refer to spaces of the forms $W^{1}L_{M}(\Omega)$ and $W^{1}E_{M}(\Omega)$ as Orlicz-Sobolev spaces. Thus $W^{1}L_{M}(\Omega)$ and $W^{1}E_{M}(\Omega)$ can be identified with subspaces of the product of $N+1$ copies of $L_{M}(\Omega)$. Denoting this product by $\Pi L_{M}$, we will use the weak topologies $\sigma(\Pi L_{M},\Pi E_{\bar{M}})$ and $\sigma(\Pi L_{M},\Pi L_{\bar{M}})$. If $M$ satisfies $\Delta_{2}$ condition, then $L_{M}(\Omega)=E_{M}(\Omega)$ and $W^{1}L_{M}(\Omega)=W^{1}E_{M}(\Omega)$. The space $W_{0}^{1}E_{M}(\Omega)$ is defined as the (norm) closure of $C_{0}^{\infty}(\Omega)$ in $W^{1}E_{M}(\Omega)$ and the space $W^{1}_{0}L_{M}(\Omega)$ as the $\sigma(\Pi L_{M},\Pi E_{\bar{M}})$ closure of $C_{0}^{\infty}(\Omega)$ in $W^{1}L_{M}(\Omega)$. Let $W^{-1}L_{\bar{M}}(\Omega)$ (resp. $W^{-1}E_{\bar{M}}(\Omega)$) denote the space of distributions on which can be written as sums of derivatives of order $\leq1$ of functions in $L_{\bar{M}}(\Omega)$ (resp. $E_{\bar{M}}(\Omega)$). It is a Banach space under the usual quotient norm (see \cite{g1}). If the open set $\Omega$ has the segment property, then the space $C_{0}^{\infty}(\Omega)$ is dense in $W^{1}_{0}L_{M}(\Omega)$ for the modular convergence and thus for the topology $\sigma(\Pi L_{M},\Pi L_{\bar{M}})$ (cf. \cite{g1,g2}). Let $A(u)=-\mathop{\rm div} a(x,u,\nabla u)$ be a Leray-Lions operator defined on $W^{1,p}(\Omega)$, $11$ and $\mu$ is a given Radon measure on $\Omega$. For some classical and recent results on elliptic and parabolic problems in Orlicz spaces, I refer the reader to \cite{a2,b1,b4,e3,f1,g1,g3,m1,v1}. \section{Preliminaries} We define a subset of $N$-functions as follows. \begin{align*} \mathcal{P}_{M}=\Big\{&B:\mathbb{R}^{+}\to \mathbb{R}^{+} \text{ is an $N$-function, } B''/B'\leq M''/M'\\ &\text{and } \int_{0}^{1}B\circ H^{-1}(1/t^{1-1/N})dt<\infty\Big\} \end{align*} where $H(r)=M(r)/r$. Assume that \begin{equation} \mathcal{P}_{M}\neq\emptyset \label{e2.1} \end{equation} Let $\Omega\subset \mathbb{R}^{N}$ be a bounded domain with the segment property, $M,P$ be two $N$-functions such that $P\ll M$, $\bar{M},\bar{P}$ be the complementary functions of $M,P$, respectively, $A:D(A)\subset W_{0}^{1}L_{M}(\Omega)\to W^{-1}L_{\bar{M}}(\Omega)$ be a mapping given by $A(u)=-\mathop{\rm div} a(x,u,\nabla u)$ where $a:\Omega\times \mathbb{R}\times \mathbb{R}^{N}\to \mathbb{R}^{N}$ be a Caratheodory function satisfying for a.e. $x\in \Omega$ and all $s\in \mathbb{R}$, $\xi,\eta\in \mathbb{R}^{N}$ with $\xi\neq\eta$: \begin{gather} |a(x,s,\xi)|\leq \beta M(|\xi|)/|\xi| \label{e2.2}\\ [a(x,s,\xi)-a(x,s,\eta)][\xi-\eta]>0 \label{e2.3}\\ a(x,s,\xi)\xi\geq \alpha M(|\xi|) \label{e2.4} \end{gather} where $\alpha,\beta,\gamma>0$. Furthermore, assume that there exists $D\in \mathcal{P}_{M}$ such that \begin{equation} D\circ H^{-1} \text{ is an $N$-function}. \label{e2.5} \end{equation} Set $T_{k}(s)=\max(-k,\min(k,s))$, $\forall s\in \mathbb{R}$, for all $ k\geq0$. Define by $\mathcal{M}_{b}(\Omega)$ as the set of all bounded Radon measure defined on $\Omega$ and by $T_{0}^{1,M}(\Omega)$ as the set of measurable functions $\Omega\to \mathbb{R}$ such that $T_{k}(u)\in W_{0}^{1}L_{M}(\Omega)\cap D(A)$. Assume that $f\in \mathcal{M}_{b}(\Omega)$, and consider the following nonlinear elliptic problem with Dirichlet boundary \begin{equation} A(u)=f\quad \text{in } \Omega. \label{e2.6} \end{equation} The following lemmas can be found in \cite{b2}. \begin{lemma} \label{lem2.1} Let $F:\mathbb{R}\to \mathbb{R}$ be uniformly Lipschitzian, with$F(0)= 0$. Let $M$ be an $N$-function, $u\in W^{1}L_{M}(\Omega)$ (resp. $W^{1}E_{M}(\Omega)$). Then $F(u)\in W^{1}L_{M}(\Omega)$ (resp. $W^{1}E_{M}(\Omega)${\rm). Moreover, if the set $D$ of discontinuity points of} $F'$ is finite, then $$ \frac{\partial(F\circ u)}{\partial x_i} =\begin{cases} F'(u)\frac{\partial u}{\partial x_i} &\mbox{a.e. in } \{x\in \Omega: u(x)\not\in D\}\\ 0 &\mbox{a.e. in } \{x\in \Omega: u(x)\in D\}. \end{cases} $$ \end{lemma} \begin{lemma} \label{lem2.2} Let $F:\mathbb{R}\to \mathbb{R}$ be uniformly Lipschitzian, with $F(0)= 0$. I suppose that the set of discontinuity points of $F'$ is finite. Let $M$ be an $N$-function, then the mapping $F:W^{1}L_{M}(\Omega) \to W^{1}L_{M}(\Omega)$ is sequentially continuous with respect to the weak$\ast$ topology $\sigma(\Pi L_{M},\Pi E_{\bar{M}})$. \end{lemma} \section{Existence theorem} \begin{theorem} \label{thm3.1} Assume that \eqref{e2.1}-\eqref{e2.5} hold and $f\in \mathcal{M}_{b}(\Omega) $. Then there exists at least one weak solution of the problem \begin{gather*} u\in T_{0}^{1,M}(\Omega)\cap W_{0}^{1}L_{B}(\Omega), \quad \forall B\in \mathcal{P}_{M} \\ \int_{\Omega}a(x,u,\nabla u)\nabla \phi dx =\langle f,\phi\rangle, \quad \forall\phi\in\mathcal{D}(\Omega) \end{gather*} \end{theorem} \begin{proof} Denote $V=W_{0}^{1}L_{M}(\Omega)$. (1) Consider the approximate equations \begin{equation} \begin{gathered} u_{n}\in V \\ -\mathop{\rm div}a(x,u_{n},\nabla u_{n})=f_{n} \end{gathered} \label{e3.1} \end{equation} where $f_{n}$ is a smooth function which converges to $f$ in the distributional sense that such that $\|f_{n}\|_{L^{1}(\Omega)}\leq\|f\|_{\mathcal{M}_{b}(\Omega)}$. By \cite[Theorem 3.1]{b2} or \cite{e1}, there exists at least one solution $\{u_{n}\}$ to \eqref{e3.1}. For $k>0$, by taking $T_{k}(u_{n})$ as test function in \eqref{e3.1}, one has $$ \int_{\Omega}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\nabla T_{k}(u_{n})dx \leq Ck. $$ In view of \eqref{e2.4}, we get \begin{equation} \int_{\Omega}M(|\nabla T_{k}(u_{n})|)dx\leq Ck. \label{e3.2} \end{equation} Hence $\nabla T_{k}(u_{n})$ is bounded in $(L_{M}(\Omega))^{N}$. By \cite{e2} there exists $u$ such that $u_{n}\to u$ almost everywhere in $\Omega$ and \begin{equation} T_{k}(u_{n})\rightharpoonup T_{k}(u)\quad \text{weakly in $W_{0}^{1}L_{M}(\Omega)$ for $\sigma\left(\Pi L_{M},\Pi E_{\bar{M}}\right)$.} \label{e3.3} \end{equation} For $t>0$, by taking $T_{h}(u_{n}-T_{t}(u_{n}))$ as test function, we deduce that $$ \int_{t<|u_{n}|\leq t+h}a(x,u_{n},\nabla u_{n})\nabla u_{n}dx \leq h\|f\|_{M_{b}(\Omega)} $$ which gives $$ \frac{1}{h}\int_{t<|u_{n}|\leq t+h}M(|\nabla u_{n}|)dx\leq \|f\|_{M_{b}(\Omega)} $$ and by letting $h\to 0$, $$ -\frac{d}{dt}\int_{|u_{n}|>t}M(|\nabla u_{n}|)dx\leq \|f\|_{M_{b}(\Omega)}. $$ Let now $B\in \mathcal{P}_{M}$. Following the lines of \cite{t1}, it is easy to deduce that \begin{equation} \int_{\Omega}B(|\nabla u_{n}|)dx\leq C,\quad \forall n. \label{e3.4} \end{equation} We shall show that $a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))$ is bounded in $(L_{\bar{M}}(\Omega))^{N}$. Let $\varphi\in (E_{M}(\Omega))^{N}$ with $\|\varphi\|_{(M)}=1$. By \eqref{e2.2} and Young inequality, one has \begin{align*} \int_{\Omega}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\varphi dx &\leq \beta\int_{\Omega}\bar{M}\Big(\frac{M(|\nabla T_{k}(u_{n})|)} {|\nabla T_{k}(u_{n})|}\Big)dx +\beta \int_{\Omega}M(|\varphi|)dx \\ &\leq \beta\int_{\Omega}M(|\nabla T_{k}(u_{n})|)dx+\beta \end{align*} This last inequality is deduced from $\bar{M}(M(u)/u)\leq M(u)$, for all $u>0$, and $\int_{\Omega}M(|\varphi|)dx\leq1$. In view of \eqref{e3.2}, $$ \int_{\Omega}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\varphi dx\leq Ck+\beta, $$ which implies $\{a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\}_{n}$ being a bounded sequence in $(L_{\bar{M}}(\Omega))^{N}$. (2) For the rest of this article, $\chi_{r}$, $\chi_{s}$ and $\chi_{j,s}$ will denoted respectively the characteristic functions of the sets $\Omega_{r}=\{x\in\Omega;|\nabla T_{k}(u(x))|\leq r\}$, $\Omega_{s}=\{x\in\Omega;|\nabla T_{k}(u(x))|\leq s\}$ and $\Omega_{j,s}=\{x\in\Omega;|\nabla T_{k}(v_{j}(x))|\leq s\}$. For the sake of simplicity, I will write only $\varepsilon(n,j,s)$ to mean all quantities (possibly different) such that $$ \lim_{s\to \infty}\lim_{j\to \infty} \lim_{n\to \infty} \varepsilon(n,j,s)=0. $$ Take a sequence $(v_{j})\subset\mathcal{D}(\Omega)$ which converges to $u$ for the modular convergence in $V$ (cf. \cite{g2}). Taking $T_{\eta}(u_{n}-T_{k}(v_{j}))$ as test function in \eqref{e3.1}, we obtain \begin{equation} \int_{\Omega}a(x,u_{n},\nabla u_{n})\nabla T_{\eta}(u_{n}-T_{k}(v_{j}))dx \leq C\eta \label{e3.5} \end{equation} On the other hand, \begin{align*} & \int_{\Omega}a(x,u_{n},\nabla u_{n}) \nabla T_{\eta}(u_{n}-T_{k}(v_{j}))dx \\ &= \int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|\leq k\}} a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))(\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j}))dx \\ &\quad +\int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,u_{n},\nabla u_{n})(\nabla u_{n}- \nabla T_{k}(v_{j}))dx \\ &= \int_{\{|T_{k}u_{n}-T_{k}(v_{j})|\leq\eta\}} a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))(\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j}))dx \\ &\quad +\int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,u_{n},\nabla u_{n})\nabla u_{n}dx \\ &\quad -\int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,u_{n},\nabla u_{n})\nabla T_{k}(v_{j})dx \end{align*} By \eqref{e2.4} the second term of the right-hand side satisfies $$ \int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,u_{n},\nabla u_{n})\nabla u_{n}dx\geq0. $$ Since $a(x,T_{k+\eta}(u_{n}),\nabla T_{k+\eta}(u_{n}))$ is bounded in $(L_{\bar{M}}(\Omega))^{N}$, there exists some $h_{k+\eta}\in(L_{\bar{M}}(\Omega))^{N}$ such that $$ a(x,T_{k+\eta}(u_{n}),\nabla T_{k+\eta}(u_{n}))\rightharpoonup h_{k+\eta} $$ weakly in $(L_{\bar{M}}(\Omega))^{N}$ for $\sigma\left(\Pi L_{\bar{M}}, \Pi E_{M}\right)$. Consequently the third term of the right-hand side satisfies \begin{align*} & \int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,u_{n},\nabla u_{n})\nabla T_{k}(v_{j})dx \\ &= \int_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}} a(x,T_{k+\eta}(u_{n}),\nabla T_{k+\eta}(u_{n}))\nabla T_{k}(v_{j})dx \\ &= \int_{\{|u-T_{k}(v_{j})|\leq\eta\}\cap\{|u|> k\}} h_{k+\eta} \nabla T_{k}(v_{j})dx +\varepsilon(n) \end{align*} since $$ \nabla T_{k}(v_{j})\chi_{\{|u_{n}-T_{k}(v_{j})|\leq\eta\}\cap\{|u_{n}|> k\}}\to \nabla T_{k}(v_{j}) \chi_{\{|u-T_{k}(v_{j})|\leq\eta\}\cap\{|u|> k\}} $$ strongly in $(E_{M}(\Omega))^{N}$ as $n\to \infty$. Hence \begin{align*} & \int_{\{|T_{k}u_{n}-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))[\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})]dx \\ &\leq C\eta +\varepsilon(n)+\int_{\{|u-T_{k}(v_{j})|\leq\eta\}\cap\{|u|> k\}} h_{k+\eta} \nabla T_{k}(v_{j})dx \end{align*} Let $0<\theta<1$. Define $$ \Phi_{n,k}=[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))][\nabla T_{k}(u_{n})-\nabla T_{k}(u)]. $$ For $r>0$, I have \begin{align*} 0 &\leq \int_{\Omega_{r}}\{[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))][\nabla T_{k}(u_{n})-\nabla T_{k}(u)]\}^{\theta}dx \\ &= \int_{\Omega_{r}}\Phi_{n,k}^{\theta} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}}dx +\int_{\Omega_{r}}\Phi_{n,k}^{\theta} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}dx \end{align*} Using the H\"{o}lder Inequality (with exponents $1/\theta$ and $1/(1-\theta)$), the first term of the right-side hand is less than $$ \Big(\int_{\Omega_{r}}\Phi_{n,k}dx\Big)^{\theta} \Big(\int_{\Omega_{r}}\chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}}dx\Big) ^{1-\theta}. $$ Noting that \begin{align*} &\int_{\Omega_{r}}\Phi_{n,k}dx\\ &=\int_{\Omega_{r}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\nabla T_{k}(u_{n})dx -\int_{\Omega_{r}}a(x,T_{k}(u_{n}),\nabla T_{k}(u))\nabla T_{k}(u_{n})dx \\ &\quad -\int_{\Omega_{r}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\nabla T_{k}(u)dx +\int_{\Omega_{r}}a(x,T_{k}(u_{n}),\nabla T_{k}(u))\nabla T_{k}(u)dx \\ &\leq Ck+\beta\int_{\Omega_{r}}\bar{M} \Big(\frac{M(|\nabla T_{k}(u)|)}{|\nabla T_{k}(u)|}\Big)dx +\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u_{n})|)dx \\ &\quad +\beta\int_{\Omega_{r}}\bar{M} \Big(\frac{M(|\nabla T_{k}(u_{n})|)}{|\nabla T_{k}(u_{n})|}\Big)dx +\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u)|)dx \\ &\quad +\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u)|)dx\\ &\leq Ck+\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u)|)dx +\beta\int_{\Omega}M(|\nabla T_{k}(u_{n})|)dx \\ &\quad +\beta\int_{\Omega}M(|\nabla T_{k}(u_{n})|)dx +\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u)|)dx +\beta\int_{\Omega_{r}}M(|\nabla T_{k}(u)|)dx \\ &\leq (2\beta+1)Ck+3M(r)\mathop{\rm meas}\Omega \end{align*} it follows that $$ \int_{\Omega_{r}}\Phi_{n,k}^{\theta} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}}dx \leq \tilde{C}(\mathop{\rm meas}\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}) ^{1-\theta}, $$ where $\tilde{C}=[(2\beta+1)Ck+3M(r)\mathop{\rm meas}\Omega]^{\theta}$. Using the H\"{o}lder Inequality (with exponents $1/\theta$ and $1/(1-\theta)$), \begin{align*} &\int_{\Omega_{r}}\Phi_{n,k}^{\theta} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}dx \\ &\leq \Big(\int_{\Omega_{r}}\Phi_{n,k} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}dx\Big)^{\theta} \Big(\int_{\Omega_{r}}dx\Big) ^{1-\theta} \\ &\leq\Big(\int_{\Omega_{r}}\Phi_{n,k} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}dx\Big)^{\theta} \big(\mathop{\rm meas}\Omega\big)^{1-\theta} \end{align*} Hence \begin{align*} 0 &\leq \int_{\Omega_{r}}\{[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))][\nabla T_{k}(u_{n})-\nabla T_{k}(u)]\}^{\theta}dx \\ &\leq \tilde{C}\big(\mathop{\rm meas}\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}\big) ^{1-\theta} \\ &\quad +\Big(\int_{\Omega_{r}}\Phi_{n,k} \chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}dx\Big)^{\theta} \big(\mathop{\rm meas}\Omega\big)^{1-\theta} \\ &=\tilde{C}\big(\mathop{\rm meas}\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}\big) ^{1-\theta} \\ &\quad +\Big(\int_{\Omega_{r}\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))\big]\\ &\quad\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big] dx\Big)^{\theta} \big(\mathop{\rm meas}\Omega\big)^{1-\theta} \end{align*} For each $s\geq r$ one has \begin{align*} 0 &\leq \int_{\Omega_{r}\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))\big]\\ &\quad\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big]dx \\ &\leq \int_{\Omega_{s}\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))\big]\\ &\quad\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)]dx \\ &= \int_{\Omega_{s}\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u)\chi_{s})\big] \\ &\quad \times\big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\chi_{s}\big]dx \\ &\leq \int_{\Omega\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u)\chi_{s})\big]\\ &\quad\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\chi_{s}]dx \\ &=\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(v_{j})\chi_{j,s})\big] \\ &\quad \times\big[\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})\chi_{j,s}\big]dx \\ &\quad +\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n})) \big[\nabla T_{k}(v_{j})\chi_{j,s}-\nabla T_{k}(u)\chi_{s}\big]dx \\ &\quad +\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(v_{j})\chi_{j,s})\\ &\quad -a(x,T_{k}(u_{n}), \nabla T_{k}(u)\chi_{s})\big] \nabla T_{k}(u_{n})dx \\ &\quad -\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(v_{j})\chi_{j,s})\nabla T_{k}(v_{j})\chi_{j,s}dx \\ &\quad +\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(u)\chi_{s})\nabla T_{k}(u)\chi_{s}dx\\ &= I_{1}(n,j,s)+I_{2}(n,j,s)+I_{3}(n,j,s)+I_{4}(n,j,s)+I_{5}(n,j,s) \end{align*} On the other hand, \begin{align*} & \int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))[\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})]dx \\ &= \int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(v_{j})\chi_{j,s})\big] \\ &\quad\times\big[\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})\chi_{j,s}\big]dx\\ &\quad +\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(v_{j})\chi_{j,s}) \big[\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})\chi_{j,s}\big]dx \\ &\quad -\int_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\nabla T_{k}(v_{j})\chi_{\{|\nabla T_{k}(v_{j})|> s\}}dx \end{align*} The second term of the right-hand side tends to $$ \int_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}}a(x,T_{k}(u),\nabla T_{k}(u)\chi_{s})[\nabla T_{k}(u)-\nabla T_{k}(v_{j})\chi_{s}]dx $$ since $a(x,T_{k}(u_{n}),\nabla T_{k}(u)\chi_{s})\chi_{\{|T_{k}(u_{n}) -T_{k}(v_{j})|\leq\eta\}}$ tends to $$ a(x,T_{k}(u),\nabla T_{k}(u)\chi_{s})\chi_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}} $$ in $(E_{\bar{M}}(\Omega))^{N}$ while $\nabla T_{k}(u_{n})-\nabla T_{k}(v_{j})\chi_{s}$ tends weakly to $\nabla T_{k}(u)-\nabla T_{k}(v_{j})\chi_{s}$ in $(L_{M}(\Omega))^{N}$ for $\sigma(\Pi L_{M}, \Pi E_{\bar{M}})$. Since $a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))$ is bounded in $(L_{\bar{M}}(\Omega))^{N}$ there exists some $h_{k}\in(L_{\bar{M}}(\Omega))^{N}$ such that (for a subsequence still denoted by $u_{n}$) $$ a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\rightharpoonup h_{k} \quad\text{weakly in $(L_{\bar{M}}(\Omega))^{N}$ for $\sigma(\Pi L_{\bar{M}}, \Pi E_{M})$.} $$ In view of the fact that $\nabla T_{k}(v_{j})\chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \to \nabla T_{k}(v_{j})\chi_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}}$ strongly in $(E_{M}(\Omega))^{N}$ as $n\to \infty$ the third term of the right-hand side tends to $$-\int_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}}h_{k}\nabla T_{k}(v_{j})\chi_{\{|\nabla T_{k}(v_{j})|> s\}}dx.$$ Hence in view of the modular convergence of $(v_{j})$ in $V$, one has \begin{align*} I_{1}(n,j,s) &\leq C\eta+\varepsilon(n) +\int_{\{|u-T_{k}(v_{j})|\leq\eta\}\cap\{|u|> k\}} h_{k+\eta} \nabla T_{k}(v_{j})dx \\ &\quad +\int_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}}h_{k}\nabla T_{k}(v_{j})\chi_{\{|\nabla T_{k}(v_{j})|> s\}}dx \\ &\quad - \int_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}} a(x,T_{k}(u),\nabla T_{k}(u)\chi_{s})[\nabla T_{k}(u)-\nabla T_{k}(v_{j})\chi_{s}]dx \\ &= C\eta+\varepsilon(n)+\varepsilon(j) +\int_{\Omega}h_{k}\nabla T_{k}(u)\chi_{\{|\nabla T_{k}(u)|> s\}}dx \\ &\quad -\int_{\Omega} a(x,T_{k}(u),0)\chi_{\{|\nabla T_{k}(u)|> s\}}dx \end{align*} Therefore, \begin{equation} I_{1}(n,j,s)=C\eta+\varepsilon(n,j,s) \label{e3.6} \end{equation} For what concerns $I_{2}$, by letting $n\to \infty$, one has $$ I_{2}(n,j,s)=\int_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}}h_{k} [\nabla T_{k}(v_{j})\chi_{j,s}-\nabla T_{k}(u)\chi_{s}]dx+\varepsilon(n) $$ since $$ a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))\rightharpoonup h_{k} \quad\text{weakly in $(L_{\bar{M}})^{N}$ for $\sigma(\Pi L_{\bar{M}},\Pi E_{M})$} $$ while $\chi_{\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} [\nabla T_{k}(v_{j})\chi_{j,s}-\nabla T_{k}(u)\chi_{s}]$ approaches $$ \chi_{\{|T_{k}(u)-T_{k}(v_{j})|\leq\eta\}} [\nabla T_{k}(v_{j})\chi_{j,s}-\nabla T_{k}(u)\chi_{s}] $$ strongly in $(E_{M})^{N}$. By letting $j\to \infty$, and using Lebesgue theorem, then \begin{equation} I_{2}(n,j,s)=\varepsilon(n,j).\label{e3.7} \end{equation} Similar tools as above, give \begin{equation} I_{3}(n,j,s)=-\int_{\Omega}a(x,T_{k}(u),\nabla T_{k}(u)\chi_{s})\nabla T_{k}(u)\chi_{s}dx+\varepsilon(n,j) \label{e3.8} \end{equation} Combining \eqref{e3.6}, \eqref{e3.7}, and \eqref{e3.8}, we have \begin{align*} &\int_{\Omega_{r}\cap\{|T_{k}(u_{n})-T_{k}(v_{j})|\leq\eta\}} \big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))\big]\\ &\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big]dx\\ &\leq \varepsilon(n,j,s). \end{align*} Therefore, \begin{align*} 0 &\leq \int_{\Omega_{r}}\{[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))][\nabla T_{k}(u_{n})-\nabla T_{k}(u)]\}^{\theta}dx \\ &\leq \tilde{C}(\mathop{\rm meas}\{|T_{k}(u_{n})-T_{k}(v_{j})|>\eta\}) ^{1-\theta}+(\mathop{\rm meas}\Omega)^{1-\theta} (\varepsilon(n,j,s))^{\theta} \end{align*} Which yields, by passing to the limit superior over $n,j,s$ and $\eta$, \begin{align*} &\lim_{n\to \infty}\int_{\Omega_{r}}\big\{\big[a(x,T_{k}(u_{n}),\nabla T_{k}(u_{n}))-a(x,T_{k}(u_{n}),\nabla T_{k}(u))\big]\\ &\times \big[\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big]\big\}^{\theta}dx=0\,. \end{align*} Thus, passing to a subsequence if necessary, $\nabla u_{n}\to \nabla u$ a.e. in $\Omega_{r}$, and since $r$ is arbitrary, $$ \nabla u_{n}\to \nabla u \quad\text{a.e. in }\Omega. $$ By \eqref{e2.2} and \eqref{e2.5}, $$\int_{\Omega}D\circ H^{-1}\Big(\frac{|a(x,u_{n},\nabla u_{n})|}{\beta}\Big)dx\leq \int_{\Omega}D(|\nabla u_{n}|)dx\leq C $$ Hence $$ a(x,u_{n},\nabla u_{n})\rightharpoonup a(x,u,\nabla u) \quad\text{weakly for $\sigma(\Pi L_{D\circ H^{-1}} \Pi E_{\overline{D\circ H^{-1}}})$.} $$ Going back to approximate equations \eqref{e3.1}, and using $\phi\in\mathcal{ D}(\Omega)$ as the test function, one has $$ \int_{\Omega}a(x,u_{n},\nabla u_{n})\nabla \phi dx=\langle f_{n},\phi\rangle $$ in which I can pass to the limit. 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