\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil Two functionals \hfil EJDE--2002/09} {EJDE--2002/09\hfil Yanming Li \& Benjin Xuan \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 09, pp. 1--18. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Two functionals for which $C_0^1$ minimizers are also $W_0^{1,p}$ minimizers % \thanks{ {\em Mathematics Subject Classifications:} 35J60. \hfil\break\indent {\em Key words:} $W_0^{1,p}$ minimizer, $C_0^1$ minimizer, divergence elliptic equation, p-Laplacian. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted November 24, 2001. Published January 24, 2002. \hfill\break\indent Supported by Grants 10071080 and 10101024 from the NNSF of China. } } \date{} % \author{Yanming Li \& Benjin Xuan} \maketitle \begin{abstract} Brezis and Niremberg \cite{BN} showed that for a certain functional the $C_0^1$ minimizer is also the $H_0^1$ minimizer. In this paper, we present two functionals for which a local minimizer in the $C_0^1$ topology is also a local minimizer in the $W_0^{1,p}$ topology. As an application, we show some existence results involving the sub and super solution method for elliptic equations. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}{Remark}[section] \numberwithin{equation}{section} \section{Introduction} It is well known that for a domain $\Omega$ with smooth boundary in $\mathbb{R}^n$, the $W_0^{1,p}$ topology is much weaker than the $C_0^1$ topology. Therefore, a $W^{1,p}_0(\Omega)$ neighborhood of function $u$ contains much more elements than the corresponding $C_0^1(\Omega)$ neighborhood. As an example, let $ B_1 (0)$ be the unit ball in $\mathbb{R}^n$, $$ f(x)=\frac{1}{|x|^{\alpha}} - 1\,, $$ and $\alpha$, $p$ satisfy $0<\alpha0$ as $n\to\infty$. Moreover, $$u'_{n}= \begin{cases} 1, & x\in [a,\delta^1 _{n})\cup (\delta^2_{n},b], \\ -\frac{1}{n}, & [\delta^1 _{n},\delta^2_{n}],\\ 0,& (0,1)\backslash [a,b]\,. \end{cases} $$ Then $\Phi(u_{n}) =\limsup_{x\in \Gamma}\{u'(x)\}= -\frac{1}{n}$ which is less than $\Phi(u_0)=0$. In some special cases, the minimizer in the $C_0^1$ topology of is also the minimizer in the $ W^{1,p}_0$ topology. Brezis and Nirenberg \cite{BN} showed that for the functional $$ \Psi(u) = \int_{\Omega}\frac{1}{2}|\nabla u|^2 - \int_{\Omega}F(x,u), $$ the $C_0^1$ minimizer is also a minimizer in $H^1 _0$, under certain conditions on $F(x,u)$. In this paper, we present another two of these kinds of functionals: The first functional is $$ \Phi(u)=\frac{1}{2} \int_{\Omega}(\sum_{i,j=1}^n a^{ij}u_{x_{i}}u_{x_{j}} + cu^2)- \int_{\Omega}F(x,u), $$ whose $C_0^1$ minimizer is also a minimizer in $H^1 _0$ for coefficient functions $a^{i,j},c$ $(i,j=1,\cdots,n)$ satisfying an ellipticity condition. The second functional is $$ \widetilde{\Phi}(u)=\frac{1}{p} \int_{\Omega}|\nabla u |^{p}- \int_{\Omega}F(x,u). $$ Its $C_0^1$ minimizer is also a minimizer in $ W^{1,p}_0$, $p\leq n$, under certain conditions on $F(x,u)$. We will give examples of functionals for which the $ W^{1,p}_0$ (or the $H^1 _0$) minimizer is not easy to find, but we can find $C_0^1$ minimizers instead. Then we show that it is also a minimizer in the $W_0^{1,p}$ (or the $H^1 _0$) topology. In some cases, it is easier to find minimizers of functional truncated by a constant, and then show that the minimizer of the truncated functional is also the local minimizer of the original functional in the $C_0^1$ topology. Next, we introduce some Lemmas to be used later. \begin{lemma} Let $\Omega$ be a domain in $\mathbb{R}^n $ and $g: \Omega\times\mathbb{R}^n \to \mathbb{R}^n $ be a Carath\'eodory function such that for almost every $x \in\Omega$, $$ |g(x,u)| \leq a(x)(1+|u|) $$ with $a \in L^{n/2}_{{\rm loc}}(\Omega)$. Let $u\in W^{1, 2}_{{\rm loc}}(\Omega)$ be a weak solution of $-\Delta{u} = g(\cdot, u)$ in $\Omega$, then $u\in L^{q}_{{\rm loc}}(\Omega)$, for any $q<\infty$. If $u\in W^{1,2}_0(\Omega)$, and $a \in L^{n/2}(\Omega)$, then $u\in L^{q}(\Omega)$, for any $q<\infty$. \end{lemma} The proof of this lemma is given in the appendix; see also \cite[p.\ 244]{Struwe}. The conclusion can also be obtained for divergence elliptic equations, as stated in the lemma below. \begin{lemma} Suppose $u\in H^1 _0(\Omega)$ is a weak solution of $L u = g(\cdot,u)$ where $$ L u = -\sum^n _{i,j=1}(a^{ij}(x) u_{x_{i}})_{x_{j}} $$ with bounded coefficients $a^{ij}=a^{ji}$ satisfying the uniformly ellipticity condition $$ \sum^n _{i,j=1}a^{ij}\xi_{i}\xi_{j}\geq\theta|\xi|^2,\quad \theta >0,\mbox{ for any } \xi\in \mathbb{R}^n , $$ and $|g(x,u)| \leq \widetilde{a}(x)(1+|u|)$ with $\widetilde{a}(x)\in L^{n/2}(\Omega)$. Then $u\in L^{q}(\Omega)$, for any $q<\infty$. \end{lemma} \noindent\textbf{Proof.} Let $u \in H^1 _0$ be a weak solution of $ Lu=g(\cdot,u)$, in the sense that \begin{equation} \label{eqn:l1} \int_{\Omega}\sum^n _{i,j=1}a^{ij}u_{x_{i}}\cdot\varphi_{x_{j}}= \int_{\Omega}g\cdot \varphi,\mbox{\ \ \ for any } \varphi \in H^1 _0(\Omega). \end{equation} We choose $s\geq0$, $M\geq0$. Let $\varphi =u\min\{|u|^{2s},M^2\}\in H^1 _0(\Omega)$, then $$ \varphi_{x_{j}}= \begin{cases} u_{x_{j}}\,\min\{|u|^{2s},M^2\}+ 2s|u|^{2s}u_{x_{j}}, & |u(x)|^s \leq M, \\[3pt] u_{x_{j}}\,\min\{|u|^{2s},M^2\}, & |u(x)|^s > M. \end{cases} $$ Multiplying (\ref{eqn:l1}) by a test function $\varphi$, we have \begin{eqnarray}\label{eqn:l2} \int_{\Omega}\sum^n _{i,j=1}a^{ij}u_{x_{i}}\cdot\varphi_{x_{j}} &=& \int_{\Omega}\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}\,\min\{|u|^{2s},M^2\} \nonumber\\ &&+ 2s \int_{\{x\in\Omega;\,|u(x)|^s \leq M\}}|u|^{2s}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}) \nonumber\\ &\geq& \theta \int_{\Omega}|\nabla u|^2\,\min\{|u|^{2s},M^2\} \\ &&+2s\theta \int_{\{x\in \Omega;\,|u(x)|^s \leq M\}}|u|^{2s}|\nabla u|^2. \nonumber \end{eqnarray} From the proof of Lemma 1.1, and (\ref{eqn:l2}), we have \begin{eqnarray*} \int_{\Omega}|\nabla (u\,\min\{|u|^s ,M\})|^2 &\leq& C \int_{\Omega}|\nabla u|^2\,\min\{|u|^{2s},M^2\}\\ &\leq& C \int_{\Omega}\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}\,\min\{|u|^{2s},M^2\} \\ & & + \ 2Cs \int_{\{x\in\Omega;\,|u(x)|^s \leq M\}}|u|^{2s}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}})\\ &=& C \int_{\Omega}\sum^n _{i,j=1}a^{ij}u_{x_{i}}\cdot\varphi_{x_{j}} \leq C \int_{\Omega}|g|\cdot|\varphi|\\ &\leq& C \int_{\Omega}|\widetilde{a}||u|^2\,\min\{|u|^{2s},M^2\}. \end{eqnarray*} Here and hereafter we denote all the constants with the same symbol $C$. Then as in the proof of Lemma 1.1, see \cite[p.244]{Struwe}, we obtain $ u\in L^q(\Omega)$, for any $q <\infty$. \hfill$\Box$ \begin{remark} \rm If $ L u = -\sum^n _{i,j=1}(a^{ij}(x) u_{x_{i}})_{x_{j}}+ cu$, with bounded coefficient functions $\textrm{ }a^{ij}$ and $c$ sufficiently smooth, and $a^{ij}=a^{ji}$ satisfying the uniformly ellipticity condition, then with the condition in Lemma 1.2, the conclusion is also true. \end{remark} For the operator $\Delta_p$, $\Delta_{p}u= - \nabla \cdot(|\nabla u|^{p-2}\nabla u)$, we have the following statement. \begin{lemma} Let $1 \leq p\leq n $, $f\in L^s (\Omega)$ for some $s>n/p$, and $u\in W^{1,p}_0(\Omega)$ be a weak solution of \begin{gather*} \Delta_{p}u= f, \quad \mbox{in } \Omega, \\ u= 0, \quad \mbox{on } \partial\Omega. \end{gather*} Then $ u\in L^{\infty}(\Omega)$ and there exists $c=c(n,p,|\Omega|)$ such that $$ \|u\|_{ L^{\infty}(\Omega)}\leq c\,\|f\|^{1/(p-1)}_{ L^s (\Omega)}. $$ \end{lemma} The proof of this lemma is a straightforward application of Moser's iterative scheme(cf. \cite{Morser,ML}). \section{Divergence elliptic differential operator} In this section, we are concerned with the relation between the $H^1 _0$ and $C_0^1$ minimizers of the functional $$ \Phi(u)= \int_{\Omega}[\frac{1}{2}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}+ c u^2)-F(x,u)]dx, $$ where $$ F(x,u) = \int^{u}_0f(x,s)ds\,. $$ Here $f(x,u)$ is a Carath\'eodory function, satisfying the natural growth condition \begin{equation}\label{eqn:l3} |f(x,u)| \leq K(1+|u|^{p}) \end{equation} where $K$ is a constant and $p\leq\frac{(n+2)}{(n-2)}$ for $n>2$. We call $u \in H^1 _0(\Omega)$ a local minimizer of $\Phi$, if $u$ is a weak solution of \begin{equation}\label{eqn:l11} \begin{gathered} L u = f, \quad\mbox{in }\Omega, \\ \ \ u = 0, \quad \mbox{on }\partial\Omega. \end{gathered} \end{equation} where $ L u = -\sum^n _{i,j=1}(a^{ij}(x) u_{x_{i}})_{x_{j}}+ c(x)u$ satisfies the hypotheses in Remark~1.1. Our main theorem is as follows. \begin{theorem} Assume $u_0\in H^1 _0(\Omega)$ is a local minimizer of $\Phi$ in the $C_0^1$ topology: this means that there exits some $r>0$, such that \begin{equation}\label{eqn:2.3} \Phi(u_0)\leq\Phi(u_0+v),\textrm{ }\ \ \ \forall \,v\in {C_0^1(\Omega)}\ \ \mbox{with}\ \ \|v\|_{C_0^1}\leq r. \end{equation} Then $u_0$ is a local minimizer of $\Phi$ in the $ H^1 _0$ topology, i.e. there exists $\epsilon_0>0$ such that \begin{equation}\label{eqn:l4} \Phi(u_0)\leq\Phi(u_0+v),\quad \forall\,v\in { H^1 _0(\Omega)}\ \ \mbox{with}\quad \|v\|_{H^1 _0}\leq\epsilon_0. \end{equation} \end{theorem} \noindent\textbf{Proof.} \noindent\textbf{1.}\ We claim that $u_0\in C^{1,\alpha}_0(\Omega)$, for any $0<\alpha<1$. In the case $p<(n+2)/(n-2)$ we can prove the regularity of $u_0$ by a bootstrap argument \cite{H.B}. For $p = (n+2)/(n-2)$, the standard bootstrap procedure does not work. We now define $$ \widetilde{a}(x)= \frac{f(x,u_0)}{1+|u_0|}\,, $$ then by (\ref{eqn:l3}), $$ |\widetilde{a}(x)|\leq C\,|u_0(x)|^{p-1}\leq C\,|u_0(x)|^{\frac{4}{n-2}}. $$ Note that $u_0(x)\in H^1 _0(\Omega)\hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$, so we have $\widetilde{a}(x) \in L^{n/2}(\Omega)$. Then we can deduce from Lemma 1.2 that $ u_0\in L^{q}(\Omega)$, for any $q<\infty$, furthermore, since $|f(x,u_0)| \leq K(1+|u_0|^p)$, then $f(x,u_0)\in L^{q}(\Omega)$, for any $q<\infty$. From (\ref{eqn:l11}) we deduce that $u_0\in W^{2,q}_0(\Omega)$, for any $q<\infty$. By a Sobolev embedding with $q$ large enough, $ W^{2,q}_0\hookrightarrow C^{1,\alpha}_0(\Omega)$; therefore $u_0\in C^{1,\alpha}_0(\Omega) $, for any $0<\alpha<1$. Without loss of generality we may now assume that $u_0= 0$. \noindent\textbf{2.}\ Now we prove Theorem 2.1 in the subcritical case $p<(n+2)/(n-2)$. Suppose the conclusion (\ref{eqn:l4}) does not hold. Then \begin{equation}\label{eqn:l5} \forall\, \epsilon >0,\ \exists \,v_{\epsilon}\in B_{\epsilon}\quad\mbox{such that}\quad \Phi(v_{\epsilon})<\Phi(0), \end{equation} where $ B_{\epsilon} = \{u\in H^1 _0(\Omega): \|u\|_{H^1 _0}\leq\epsilon\}$, For each $j$ consider the truncation map $$ T_{j}(r)= \begin{cases} -j, & \mbox{if } r \leq -j, \\ r, & \mbox{if } -j \leq r \leq j,\\ j, & \mbox{if } r \geq j. \end{cases} $$ Set \begin{gather*} f_{j}(x,s)= f(x,T_{j}(s)) ,\quad F_{j}(x,u)= \int^{u}_0f_{j}(x,s)ds, \\ \Phi_{j}(u)= \int_{\Omega}[\frac{1}{2}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}+ c\,u^2)- F_{j}(x,u)]dx. \end{gather*} Then, for each $u\in H^1 _0(\Omega)$, $\Phi_{j}(u)\to \Phi(u)$ as $j\to\infty$. Hence, from (\ref{eqn:l5}) we know for each $\epsilon > 0$ there is some $j=j(\epsilon)$ s.t. $ \Phi_{j}(v_{\epsilon})<\Phi(0)$. Now we point out $\Phi_{j}$ is coercive and weakly lower semi-continuous. $\Phi_{j}$ is coercive, because \begin{eqnarray*} \Phi_{j}(u)&\geq& \theta \int_{\Omega}(|\nabla u|^2+|c| |u|^2)dx- \int_{\Omega} \int^{u}_0(1+|T_{j}(s)|^{p})dsdx\\ &\geq& \theta\int_{\Omega}(|\nabla u|^2+ c|u|^2)dx- \int_{\Omega}(1+j^{p})|u|dx)\\ &\geq& \theta\int_{\Omega}(|\nabla u|^2+c|u|^2)dx-C\epsilon \int_{\Omega}|u|^2dx-C(\epsilon)\\ &\geq& \theta\int_{\Omega}(|\nabla u|^2+C'|u|^2)dx-C(\epsilon)\\ &\geq& C\,\|u\|^2_{H^{1}_{0}}-C(\epsilon). \end{eqnarray*} Note that $\int_{\Omega}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}}+cu^2)$ is equivalent to the norm $\|u\|_{H^1 _0}$ and is weakly lower semi-continuous. Using Lemma \ref{eqn:l6} below, we can deduce that $$ E_j(u)\doteq \int_{\Omega}F_{j}(x,u) dx= \int_{\Omega}(F_{j}(x,u)+0\cdot \nabla u )dx $$ is weakly lower semi-continuous, so $\Phi_{j}$ is weakly lower semi-continuous. \begin{lemma} \label{eqn:l6} Assume that $ F:\Omega \times \mathbb{R}^{N}\times\mathbb{R}^{nN}\to \mathbb{R}$ is a Carath\'eodory function satisfying the conditions \begin{enumerate} \item $F(x,u,p)\geq \phi(x)$ for almost every $x, u, p$, where $\phi \in L^1 (\Omega)$. \item $F(x,u,\cdot)$ is convex in $p$ for almost every $x, u$. \end{enumerate} Then, if $u_{m}$, $u\in W^{1,1}_{{\rm loc}}(\Omega)$ and $u_{m}\to u $ in $ L^1 (\Omega')$, $\nabla u_{m}\rightharpoondown \nabla u$ weakly in $ L^1 (\Omega')$ for all bounded $\Omega'\subset\subset\Omega$, it follows that $$ E(u)\leq \liminf_{m\to\infty}E(u_{m}) $$ where $ E(u) = \int_{\Omega}F(x,u,\nabla u )dx$. \end{lemma} The proof of this lemma can be found in \cite[p.\ 9]{Struwe}. Then $\Phi_{j}$ is bounded from below and attains its infimum in $ B_{\epsilon}$ (which is closed and convex, so it is a weakly closed subset), suppose $$ \Phi_{j_{\epsilon}}(w_{\epsilon})= \min_{u\in B_{\epsilon}}\Phi_{j_{\epsilon}}(u), $$ we have \begin{equation}\label{eqn:l7} \Phi_{j_{\epsilon}}(w_{\epsilon})\leq \Phi_{j_{\epsilon}}(v_{\epsilon})\leq \Phi(0). \end{equation} The corresponding Euler equation for $w_{\epsilon}$ involves a Lagrange multiplier $\mu_{\epsilon}\leq 0$ (cf. Generalized Kuhn-Tucker Theory in \cite{Eber}), namely, $w_{\epsilon}$ satisfies $$ \langle\,\Phi_{j}'(w_{\epsilon}),\, \zeta\,\rangle_{H^{-1},H^1 _0}= \mu_{\epsilon}(w_{\epsilon}\cdot\zeta)_{H^1 _0},\quad \forall\,\zeta \in H^1 _0(\Omega), $$ i.e. $$ \int_{\Omega}[\sum^n _{k,l=1}a^{kl}(w_{\epsilon})_{x_{k}}\zeta_{x_{l}}+ c w_{\epsilon} \zeta - f_{j}(x,w_{\epsilon}) \zeta] = \mu_{\epsilon} \int_{\Omega}\nabla w_{\epsilon} \cdot \nabla\zeta, \quad \forall \zeta\in H^1 _0(\Omega). $$ This implies $ (L w_{\epsilon} - \mu_{\epsilon}\Delta w_{\epsilon}) = f_{j}(x,w_{\epsilon})$. Let $ L'w_{\epsilon} = (L w_{\epsilon} - \mu_{\epsilon}\Delta w_{\epsilon})$, then $$ L'w_{\epsilon} = \sum^n _{k,l=1}[\hat{a}^{\textrm{ }kl}\cdot (w_{\epsilon})_{x_{k}}]_{x_{l}} + cw_{\epsilon},\quad \mbox{where}\quad \hat{a}^{kl} = \begin{cases} a^{kl}-\mu_{\epsilon}, & k=l, \\ a^{kl}, & k\neq l. \end{cases} $$ Note that $\mu_{\epsilon}\leq 0$. It is easy to check that $\hat{a}^{kl}$ still satisfy the uniformly ellipticity condition. Since $L'w_{\epsilon} = f_{j}(x,w_{\epsilon})$ and $p < (n-2)/(n+2)$, using a bootstrap procedure, we can derive from $\|w_{\epsilon}\|_{H^1 _0}\leq C$, that $\|w_{\epsilon}\|_{C^{1,\alpha}_0}\leq C$, where $C$ is a constant independent of $\epsilon$. Then $$ \sup_{x,y \in \Omega}\frac{|w_{\epsilon}(x)-w_{\epsilon}(y)|}{|x-y|^{\alpha}}< C< \infty, \quad \forall \epsilon > 0. $$ This implies that the $w_{\epsilon}$'s are equicontinuous and $|w_{\epsilon}|< 2C \mathop{\rm diam}(\Omega)$, for all $x\in \Omega$, which means $w_{\epsilon}$ are uniformly bounded. Then by Ascoli Theorem, $\{w_{\epsilon}\}$ has a subsequence converging in $C_0^1(\Omega)$, still denoted by $\{w_{\epsilon}\}$. Then since $\|w_{\epsilon}\|_{H^1 _0}\to 0$ as $\epsilon\to 0 $, we can derive $\|w_{\epsilon}\|_{C_0^1}\to 0$ as $\epsilon\to 0 $, i.e. $w_{\epsilon}\to 0$ in $ C_0^1(\Omega) $, as $\epsilon\to 0$. Then for $\epsilon$ small enough, we have $$ \Phi(w_{\epsilon})= \Phi_{j(\epsilon)}(w_{\epsilon})< \Phi(0). $$ This contradicts (\ref{eqn:2.3}); therefore, (\ref{eqn:l5}) can not hold. Thus Theorem 2.1 is proved for the subcritical case. \noindent\textbf{3.}\ In the critical case $p=(n+2)/(n-2)$, since $H^1 _0(\Omega)\hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ is not compact, a bootstrap argument does not work. But we still have $$ L'w_{\epsilon} = (L w_{\epsilon} - \mu_{\epsilon}\Delta w_{\epsilon}) = f_{j}(x,w_{\epsilon}) $$ and \begin{equation}\label{eqn:l8} f_{j}(x,w_{\epsilon})= f(x,T_{j}(w_{\epsilon}))\leq K\, (1+|T_{j}(w_{\epsilon})|^{p})\leq K\,(1+|w_{\epsilon}|^{p}). \end{equation} Let $$ \widetilde{a}(x)= \frac{f_{j}(x,w_{\epsilon})}{1+|w_{\epsilon}|}, $$ then $$ |\widetilde{a}(x)|\leq C\,|w_{\epsilon}|^{p-1}\leq C\,|w_{\epsilon}|^{\frac{4}{n-2}}\in L^{n/2}(\Omega). $$ By Remark 1.1, $w_{\epsilon} \in L^{q}(\Omega)$, for any $q < \infty$. From (\ref{eqn:l8}) $f_{j}(x,w_{\epsilon})\in L^{q}(\Omega)$, for any $q < \infty$. Then $ w_{\epsilon}\in W^{2,q}_0(\Omega)$, for any $q < \infty$, then $w_{\epsilon}\in C^{1,\alpha}_0(\Omega)$, for any $q < \infty$. Consequently, $w_{\epsilon}\to 0\ in\ C_0^1$ since $w_{\epsilon}\to 0\ in\ H^1 _0$. Thus Theorem 2.1 is proved. \hfill$\Box$\smallskip As an application of Theeorem 2.1, we will obtain an existence result of divergence elliptic equation involving the sub and super solution method. First, we give a lemma: \begin{lemma} Let $\Omega$ be a bounded domain in $\mathbb{R}^n $ with smooth boundary $\partial\Omega$. Let $u\in L^1 _{{\rm loc}}(\Omega)$ and assume that, for some constant $k\geq 0$, $u$ satisfies \begin{gather*} Lu + ku \geq 0, \quad \mbox{in } \Omega,\\ u\geq 0 \quad \mbox{in } \Omega. \end{gather*} Then either $u\equiv 0$, or there exists $\epsilon > 0$ such that $$ u(x) \geq \epsilon\mathop{\rm dist}(x, \partial\Omega),\quad\mbox{in } \Omega. $$ \end{lemma} \textbf{Proof.} Let $\mu = L u + k u$, then $\mu \geq 0$, we may assume $u \not\equiv 0$.\\ Case 1: $\mu \equiv 0$. In this case $u \in C^{\infty}(\Omega)$, $$ L u + k u = 0, \quad u\geq 0,\quad \mbox{in } \Omega. $$ Since $u\not\equiv 0$, $u\geq\delta\geq 0 $ in some closed ball $B$ in $\Omega$. Suppose $h$ solves \begin{gather*} (L+k) h = 0 \quad \mbox{in } \Omega\backslash{B}, \\ h=\delta \quad \mbox{on } \partial{B},\\ h=0 \quad \mbox{on } \partial\Omega. \end{gather*} Using the maximum principle, we have $u - h \geq 0$ in $\Omega\backslash{B}$. Using the Hopf Lemma, we have $$ \frac{|h(x)- 0|}{\mathop{\rm dist}(x, \partial\Omega)}\geq\epsilon > 0,\quad \mbox{in }\Omega\backslash{B} $$ for some $\epsilon > 0$; that is $ h(x) \geq\epsilon\mathop{\rm dist}(x,\partial\Omega)$ in $\Omega\backslash{B}$. Then $$ u(x) \geq\epsilon \mathop{\rm dist}(x, \partial\Omega),\quad \mbox{in }\Omega\backslash{B}. $$ Since $\overline{\Omega}$ is compact, there can be found $\epsilon$, such that $u(x) \geq\epsilon\mathop{\rm dist}(x,\partial\Omega)$ in $\Omega$. \\ Case 2: $\mu\not\equiv 0$. Let $\zeta \in C^{\infty}_0(\Omega)$ be a cutoff function, $0 \leq \zeta \leq 1$, such that $\zeta u \not\equiv 0$. Let $v$ be the solution of \begin{gather*} (L+k) v = \zeta u \quad \mbox{in } \Omega, \\ v= 0, \quad\mbox{on }\partial\Omega. \end{gather*} Since $\zeta u \geq 0$, using Hopf Lemma, we have $v(x) \geq\epsilon\mathop{\rm dist}(x,\partial\Omega)$ in $\Omega$. Now we claim $u \geq v$ in $\Omega$. Given any $\alpha>0$, we will prove that $\overline{u}= u+\alpha \geq v$ in $\Omega$. Let $w = \overline{u}-v$, then we have \begin{eqnarray*} (L+k)w &=& (L+k)(u+\alpha-v) = (L+k)u - \zeta\mu + (L+k)\alpha\\ &=& (L+k)u - \zeta\mu + (c(x)+k)\alpha\\ &=& (1-\zeta)\mu + (c(x)+k)\alpha. \end{eqnarray*} Let $k$ be large enough, so that $c(x)+ k \geq 0$ for all $x\in\Omega$, then $w$ satisfies \begin{equation}\label{eqn:l10} (L+k)w = (1-\zeta)\mu + (c(x)+k)\alpha \geq 0 ,\ \ \ \ \quad\mbox{in }\Omega \end{equation} and \begin{equation}\label{eqn:l9} w \geq 0,\quad \mbox{in } N_{\eta}\{x \in \Omega;\mathop{\rm dist}(x,\partial \Omega)< \eta\}, \end{equation} provided $\eta$ is sufficiently small (depend on $\alpha$). The last property (\ref{eqn:l9}) follows from the fact $v$ is smooth near $\partial\Omega$ and $v=0$ on $\partial\Omega$. Let $\{\rho_{j}\}$ be a sequence of mollifiers with $ \mbox{supp}\,\rho_{j} \subset \Omega/N_{1/j}$. Set $w_{j}(x)= \int_{\Omega}\rho_{j}(x-y)w(y)$.\\ Clearly $w_{j}$ is smooth,and by (\ref{eqn:l10}) we have $$ (L + k)w_{j} \geq 0, \quad\mbox{in } \Omega/\overline{N}_{1/j}. $$ On the other hand, from (\ref{eqn:l9}) we deduce $w_{j} \geq 0$ in $N_{(\eta-1/j)}$. Provided that $j$ is large enough, $2/j < \eta$, then $w_{j}\geq 0\ in\ N_{(\frac{1}{j}+\epsilon)}$, thus $w_{j} \geq 0\ $ on $\partial(\Omega/\overline{N}_{1/j})$, using the maximum principle, we have $$ w_{j} \geq 0, \quad\mbox{in }\Omega/\overline{N}_{1/j}. $$ Passing to the limit as $j \to \infty$ we see that $w \geq 0$ in $\Omega$, which is the desired conclusion. \hfill$\Box$\smallskip \begin{theorem} Assume that $\underline{u}$ and $\overline{u}$ are sub and super solutions in $C^1(\overline{\Omega})$ in the weak sense: \begin{gather*} L\underline{u}- f(x,\underline{u}) \leq 0 \leq L\overline{u}- f(x,\overline{u}) \quad\mbox{in }\Omega, \\ \underline{u} \leq 0 \leq \overline{u} \quad\mbox{on } \partial\Omega. \end{gather*} Moreover, assume that neither $\underline{u}$ nor $\overline{u}$ is a solution of (\ref{eqn:l11}). Then there is a solution $u_0$ of (\ref{eqn:l11}), $\underline{u} \le u_0 \le \overline{u}$, such that, in addition, $u_0$ is a local minimum of $\Phi$ in $ H_0^1 (\Omega)$. \end{theorem} \noindent\textbf{Proof.} \noindent\textbf{1.}\ We introduce an auxiliary function $$ \widetilde{f}(x, s) = \begin{cases} f(x,\underline{u}(x)), & \mbox{if } s < \underline{u}(x), \\ f(x,s), & \mbox{if } \underline{u}(x) \leq s \leq \overline{u}(x), \\ f(x,\overline{u}(x)), & \mbox{if } s > \overline{u}(x). \end{cases} $$ which is continuous in $s$. Also set \begin{gather*} \widetilde{F}(x,u) = \int^{u}_0 \widetilde{f}(x, s)ds \\ \widetilde{\Phi}(u) = \int_{\Omega}[\frac{1}{2}(\sum^n _{i,j=1}a^{ij}u_{x_{i}}u_{x_{j}} + c(x)u^2) - \widetilde{F}(x,u)]dx. \end{gather*} Let $u_0$ be a minimizer of $\widetilde{\Phi}$ on $H^1 _0(\Omega)$; as before, we can say that the minimizer is achieved and satisfies $$ Lu_0 = \widetilde{f}(x, u_0) \quad\mbox{in }\Omega. $$ Thus $u_0 \in W^{2.p}_0(\Omega)$, $\forall p < \infty$. \noindent\textbf{2.}\ We claim that $\underline{u} \leq u_0 \leq \overline{u}$; we will just prove the first inequality. Indeed we have \begin{equation}\label{eqn:l12} L(\underline{u}-u_0) \leq f(x, \underline{u}) - \widetilde{f}(x, u_0), \end{equation} and in particular $$ L(\underline{u} - u_0) \leq 0, \quad\mbox{in }A=\{ x \in \Omega; u_0(x) < \underline{u}(x)\}. $$ Since $\underline{u} - u_0 \leq 0$ on $\partial A$, it follows from the maximum principle that $\underline{u} - u_0 \leq 0$ in $A$. Therefore $A = \emptyset $ and the claim is proved. \noindent\textbf{3.}\ Returning to (\ref{eqn:l12}), we have $$ L(\underline{u} - u_0) + K( \underline{u} - u_0) \leq (f(x, \underline{u}) + k \underline{u})-(f(x, u_0) + ku_0) \leq 0, $$ here we let k be large enough, so that $f(x, u) + ku$ is nondecreasing in $u$, for a.e. $x \in \Omega$. Since $ \underline{u}$ is not a solution, it follows from Lemma 2.3 that there is some $\epsilon > 0$ such that $$ \underline{u}(x) - u_0(x) \leq -\epsilon\mathop{\rm dist}(x, \partial\Omega),\quad\forall\,x\in\Omega. $$ Similar inequality is obtained for for $ \overline{u}$. therefore, $$ \underline{u}(x) + \epsilon\mathop{\rm dist}(x, \partial\Omega) \leq u_0(x) \leq \overline{u}(x) - \epsilon\mathop{\rm dist}(x, \partial\Omega), \quad\forall\,x\in\Omega. $$ It follows that if $ u \in C_0^1(\overline{\Omega})$ and $\|u - u_0\|_{C_0^1} \leq \epsilon$ then $$ \underline{u} \leq u \leq \overline{u} \quad\mbox{in }\Omega. $$ Next, we use the fact that $\widetilde{F}(x, u) - F(x, u)$ is a function of $x$ alone for $u \in [ \underline{u}(x), \overline{u}(x)]$. In particular, $\Phi(u) - \widetilde{\Phi}(u)$ is constant for $\|u - u_0\|_{C_0^1} \leq \epsilon$. Hence, $u_0$ is a local minimum of $\Phi$ in $C_0^1$ topology (since it is a global minimum for $\widehat{\Phi}$). Now, from Theorem 2.1, we claim that $u_0$ is also a local minimum of $\Phi$ in $H^1 _0$ topology. \section{The $\Delta_{p}$ operator} Let $u\in W^{1, p}_0(\Omega)$ be a weak solution of \begin{equation}\label{eqn:l13} \begin{gathered} \Delta_{p}u = - \nabla \cdot(|\nabla u|^{p-2}\nabla u)= f(x,u) \quad\mbox{in }\Omega, \\ u = 0 \quad\mbox{on }\partial\Omega. \end{gathered} \end{equation} in the sense that $u$ satisfy the equation $$ \int_{\Omega}(\nabla u |\nabla u |^{p-2}\nabla\varphi - f\varphi)dx = 0,\quad\forall\,\varphi\in W^{1, p}_0(\Omega). $$ We consider the functional \begin{equation}\label{eqn:l14} \Phi(u) = \frac{1}{p} \int_{\Omega}[|\nabla u|^{p} - F(x, u)]dx, \end{equation} where $F(x,u) = \int^{u}_0f(x,s)ds$, with $f(x,r)$ continuous in $\overline{\Omega}\times \mathbb{R}^n $, and $$ |f(x,r)| \leq K(1+|r|^{\gamma}) \quad\mbox{with} \ \ \gamma\leq p^{*}-1. $$ Here $p^{*}=\frac{np}{n-p}$ is the critical Sobolev exponent corresponding to the noncompact embedding of $W^{1,p}_0(\Omega)$ into $L^{p^{*}(\Omega)}$. Note that $u$ is a weak solution of (\ref{eqn:l13}) if $u$ is a minimizer of (\ref{eqn:l14}). For the operator $\Delta_{p}$, the conclusion of Theorem 2.1 is still true. In fact, we have the following statement. \begin{theorem} Assume $\Phi$ is as in (\ref{eqn:l14}), $u_0\in W^{1,p}_0(\Omega)$ is a local minimizer of $\Phi$ in the $C_0^1$ topology. Then $u_0$ is a local minimizer of $\Phi$ in the $ W^{1,p}_0$ topology. \end{theorem} To prove this theorem, we need the following Lemma, whose proof relies partially on Lemma 1.3. The rest of the proof is almost the same as that of Theorem 2.1. \begin{lemma} Assume $1 \overline{u}(x); \end{cases}\\ \widetilde{F}(x,u) = \int^{u}_0 \widetilde{f}(x, s)ds,\quad \widetilde{\Phi}(u) = \frac{1}{p} \int_{\Omega}[|\nabla u|^{p} - \widetilde{F}(x,u)]dx. \end{gather*} Since $|f'_{u}|< C$, we can fix a number $\lambda > 0$ large enough so that the mapping \begin{equation}\label{eqn:l16} z\mapsto \widetilde{f}(\cdot,z)+ \lambda z \mbox{ is nondecreasing.} \end{equation} Now write $u_0 = \underline{u}$, and define $u_{k}$ ($k =0,1,2,\cdots$) inductively: $u_{k+1}\in W^{1,p}_0(\Omega)$ is a nonzero weak solution of \begin{equation}\label{eqn:l15} \begin{gathered} - \nabla \cdot(|\nabla u_{k+1}|^{p-2}\nabla u_{k+1})+ \lambda u_{k+1}= \widetilde{f}(x,u_{k})+ \lambda u_{k} \quad\mbox{in }\Omega, \\ u_{k+1}=0 \quad\mbox{on }\partial\Omega. \end{gathered} \end{equation} The nonzero weak solution exists. In fact, $u_{k+1}$ is a weak solution of (\ref{eqn:l15}), in the sense that $u_{k+1}$ is a local minimum of the functional $$ \hat{\Phi}(u) = \int_{\Omega}[\frac{1}{p}|\nabla u|^{p}+ \frac{\lambda}{2}u^2 - (\widetilde{f}(x,u_{k})+\lambda u_{k})u]dx. $$ It is easy to check that $\hat{\Phi}(u)$ is coercive and weakly lower semi-continuous, hence it attains a local minimum in $ W^{1,p}_0(\Omega)$. \noindent\textbf{2.}\ Now we claim \begin{equation}\label{eqn:l17} \underline{u}=u_0\leq u_1 \leq\cdots\leq u_{k}\leq\cdots\quad a.e \quad\mbox{in }\Omega. \end{equation} Note from (\ref{eqn:l15}) for k=0, we have $$ - \Delta_{p}u_1 + \lambda u_1 = \widetilde{f}(x,u_0)+ \lambda u_0, $$ in the sense that $u_1 $ satisfies \begin{equation}\label{eqn:l18} \int_{\Omega}(\nabla u_1 |\nabla u_1 |^{p-2}\nabla \varphi +\lambda u_1 \varphi)dx = \int_{\Omega}(\widetilde{f}(x,u_0)+\lambda u_0)\varphi dx. \end{equation} for each $\varphi \in H^1 _0(\Omega)$. From its definition, $\underline{u}$ satisfies \begin{equation}\label{eqn:l19} \int_{\Omega}\nabla \underline{u}|\nabla \underline{u}|^{p-2}\nabla \varphi dx \leq \int_{\Omega}\widetilde{f}(x,\underline{u})\varphi dx, \ \ \forall\,\varphi \in W^{1,p}_0(\Omega). \end{equation} Compare (\ref{eqn:l19}) with (\ref{eqn:l18}), note that $u_0=\underline{u}$,. For any $\varphi \in W^{1,p}_0(\Omega)$ we get \begin{equation}\label{eqn:l20} \begin{gathered} \int_{\Omega}[(\nabla u_0|\nabla u_0|^{p-2}-\nabla u_1 |\nabla u_1 |^{p-2})\nabla \varphi + \lambda (u_0-u_1 )\varphi]dx \leq 0, \\ \varphi=(u_0-u_1 )^{+} = \begin{cases} u_0-u_1 , & u_0> u_1 , \\ 0, & u_0\leq u_1 . \end{cases} \end{gathered}\end{equation} Then $\varphi \in W^{1,p}_0$, and $$ D\varphi=D(u_0-u_1 )^{+}= \begin{cases} D(u_0-u_1 ), & \mbox{a.e.\ on}\{u_0> u_1 \}, \\ 0, & \mbox{a.e.\ on}\{u_0\leq u_1 \}. \end{cases} $$ Multiplying (\ref{eqn:l20}) with $\varphi$, we have $$ \int_{\{u_0> u_1 \}}[(\nabla u_0|\nabla u_0|^{p-2}-\nabla u_1 |\nabla u_1 |^{p-2})(\nabla u_0-\nabla u_1 ) + \lambda (u_0-u_1 )^2]dx \leq 0, $$ so that ${\mathcal{L}}\{u_0>u_1 \}=0$ with ${\mathcal{L}}$ for Lebesgue measurement. If $\nabla u_0=\nabla u_1$ a.e. in $\Omega$, then testing (\ref{eqn:l20}) with $\varphi =(u_0-u_1 )^{+}$, we have $$ \int_{\{u_0> u_1 \}}\lambda (u_0-u_1 )^2dx \leq 0, $$ there still has ${\mathcal{L}}\{u_0>u_1 \}=0$, that is, $u_0\leq u_1 \ \mbox{a.e.\ in}\ \Omega$. Now assume inductively $u_{k-1}\leq u_{k}$ a.e. in $\Omega$, from (\ref{eqn:l15}), for any $\varphi \in W^{1,p}_0(\Omega)$, we have \begin{gather}\label{eqn:l22} \int_{\Omega}(\nabla u_{k+1}|\nabla u_{k+1}|^{p-2}\nabla \varphi +\lambda u_{k+1}\varphi)dx = \int_{\Omega}(\widetilde{f}(x,u_{k})+ \lambda u_{k})\varphi dx,\\ \label{eqn:l23} \int_{\Omega}(\nabla u_{k}|\nabla u_{k}|^{p-2}\nabla \varphi +\lambda u_{k}\varphi)dx = \int_{\Omega}(\widetilde{f}(x,u_{k-1})+ \lambda u_{k-1})\varphi dx , \end{gather} Subtract (\ref{eqn:l22}) from (\ref{eqn:l23}) and set $\varphi = (u_{k}- u_{k+1})^{+}$, noting $\widetilde{f}(\cdot,z)+ \lambda z$ is nondecreasing to $z$, we deduce \begin{multline*} \int_{\{u_{k}> u_{k+1}\}}\Big[(\nabla u_{k}|\nabla u_{k}|^{p-2}-\nabla u_{k+1}|\nabla u_{k+1}|^{p-2})(\nabla u_{k}-\nabla u_{k+1}) \\ + \lambda (u_{k}-u_{k+1})^2\Big]dx \\ = \int_{\Omega}[(\widetilde{f}(u_{k-1})+\lambda u_{k-1})-(\widetilde{f}(u_{k})+\lambda u_{k})](u_{k}-u_{k+1})^{+}dx \leq 0, \end{multline*} while ${\mathcal{L}}\{\nabla u_{k}\neq\nabla u_{k+1}\}\neq 0$; and we have \begin{multline*} \int_{\{u_{k}> u_{k+1}\}}\lambda (u_{k}-u_{k+1})^2dx \\ = \int_{\Omega}[(\widetilde{f}(u_{k-1})+\lambda u_{k-1})-(\widetilde{f}(u_{k})+\lambda u_{k})](u_{k}-u_{k+1})^{+}dx \leq 0, \end{multline*} while $\nabla u_{k}= \nabla u _{k+1}$ a.e. in $\Omega$. Hence ${\mathcal{L}}\{ u_{k}> u_{k+1}\}= 0$, i.e. $u_{k}\leq u_{k+1}\ \mbox{a.e.\ in}\ \Omega$, as asserted. \noindent\textbf{3.}\ Next we show that $$ u_{k} \leq \overline{u},\quad\mbox{a.e\ in} \Omega\ (k=0,1,\cdots), $$ while $k=0$, there hold $u_0 =\underline{u}\leq \overline{u}$. Assume inductively $u_{k} \leq \overline{u}$, a.e in $\Omega$. Then from the definition of $ \overline{u}$, there holds $$ \int_{\Omega}\nabla \overline{u}|\nabla \overline{u}|^{p-2}\nabla \varphi dx \geq \int_{\Omega}\widetilde{f}(x,\overline{u})\varphi dx,\quad \forall\,\varphi \in W^{1,p}_0(\Omega), $$ compare with (\ref{eqn:l22}) and setting $\varphi=(u_{k+1}-\overline{u})^{+}$, we find \begin{multline*} \int_{\{u_{k+1}> \overline{u}\}}[(\nabla u_{k+1}|\nabla u_{k+1}|^{p-2}-\nabla \overline{u}|\nabla \overline{u}|^{p-2})(\nabla u_{k+1}-\nabla \overline{u}) + \lambda (u_{k+1}-\overline{u})^2]dx \\ = \int_{\Omega}[(\widetilde{f}(u_{k})+\lambda u_{k})-(\widetilde{f}(\overline{u})+\lambda \overline{u})](u_{k+1}-\overline{u})^{+}dx \leq 0. \end{multline*} As before, we conclude $u_{k+1}\leq \overline{u}$ a.e. in $\Omega$. \noindent\textbf{4.}\ From steps 2 and 3, we have $$ \underline{u}\leq\cdots\leq u_{k}\leq u_{k+1}\leq\cdots\leq\overline{u},\quad\mbox{a.e\quad in } \Omega, $$ as $u_{k}$ $(k=1,2\cdots)$ are solutions of (\ref{eqn:l15}), using Moser's iterative scheme which we cited in the proof of Lemma 1.3, we have $u_{k} \in L^{q}$ for all $q<\infty$, $(k=1,2\cdots)$, since $|f'_{z}(\cdot,z)|< C$, then $\widetilde{f}(\cdot,z)\leq c(1+|z|)$, so that $f(\cdot,u_{k})\in L^{q}$, then from (\ref{eqn:l15}), we deduce $u_{k} \in W^{2,q}_0$ for all $q<\infty$, $(k=1,2\cdots)$. We can also deduce from (\ref{eqn:l15}) that \begin{eqnarray*} \|u_{k}\|^{q}_{ W^{2,q}_0}&\leq& C(\|\widetilde{f}(u_{k-1})\|^{q}_{L^{q}}+ \|u_{k-1}\|^{q}_{L^{q}})\\ &\leq& C(1+ \|u_{k-1}\|^{q}_{L^{q}})\\ &\leq& C(1+ \max\{\|\underline{u}\|^{q}_{L^{q}},\, \|\overline{u}\|^{q}_{L^{q}} \}). \end{eqnarray*} So that $u_{k}$ is unified bounded in $ W^{2,q}_0(\Omega)$, hence there exists a subsequence converging in $ W^{2,q}_0(\Omega)$, still denote $u_{k}$, i.e. \begin{equation}\label{eqn:l24} u_{k}\rightharpoonup u_0,\quad\mbox{in } W^{2,q}_0(\Omega), \end{equation} set $q$ be large enough, then $ W^{2,q}_0\hookrightarrow\hookrightarrow C^{1,\alpha}_0$, and $$ u_{k}\to u_0,\quad\mbox{in } C^{1,\alpha}_0(\Omega). $$ From (\ref{eqn:l24}), let $k\to\infty$ in (\ref{eqn:l22}) and cancelling the same item on both sides, we get $$ \int_{\Omega}\nabla u_0|\nabla u_0|^{p-2}\nabla \varphi dx = \int_{\Omega}\widetilde{f}(x,u_0)\varphi dx, \quad \forall\,\varphi\in W^{1,p}_0(\Omega). $$ This means $u_{0}\in C^{1,\alpha}_0$ is a weak solution of \begin{gather*} \Delta_{p}u = \widetilde{f} \quad\mbox{in }\Omega, \\ u = 0 \quad\mbox{on }\partial\Omega. \end{gather*} Hence we have a local minimizer of $\widetilde{\Phi}$. Then given $\overline{u}$ and $\underline{u}$ in $C^1 (\overline{\Omega})$ satisfying the assumption (\ref{eqn:xg}), we can deduce $$ \underline{u} < u_0 < \overline{u}, \quad\mbox{a.e.\quad in } \Omega, $$ Let $\Lambda=\{x\in\Omega : \underline{u}(x) = u_0(x)\}\cup\{x\in\Omega : \overline{u}(x) = u_0(x)\}$, then $\mathcal{L}(\overline{\Lambda})=0$ (if not, $\mathcal{L}(\overline{\Lambda})>0$ , since $u_0$, $\underline{u}$, and $\overline{u}$ are all continuous, then there must be $\underline{u}(x) = u_0(x)$ or $\overline{u}(x) = u_0(x)$ on $\overline{\Lambda}$, so $\Delta_{p}\underline{u}=\Delta_{p}u_0$ or $ \Delta_{p}\overline{u}=\Delta_{p}u_0$ on $\overline{\Lambda}$, and this contradicts (\ref{eqn:xg}).). Thus $\Omega':\,=\Omega\backslash\overline{\Lambda}\subset\subset\Omega$ is still a domain in $\mathbb{R}^n $. and $$ \underline{u}(x) < u_0(x) < \overline{u}(x), \quad\mbox{for any x\ in}\ \Omega', $$ so when set $\epsilon$ be small enough, and $\|u-u_0\|_{C_0^1}\leq \epsilon$ there has $$ \underline{u} \leq u \leq \overline{u}, \quad\mbox{in }\Omega'. $$ Denote $\hat{\Phi}(u) = \frac{1}{p} \int_{\Omega'}[|\nabla u|^{p} - F(x, u)]dx$, and $\widetilde{\hat{\Phi}}(u) = \frac{1}{p} \int_{\Omega'}[|\nabla u|^{p} - \widetilde{F}(x, u)]dx$, then $u_0$ is a $C^1 (\overline{\Omega'})$ local minimizer of $\hat{\Phi}$, then as we do in Lemma 2.4, noted the fact that $\hat{\Phi}(u) - \widetilde{\hat{\Phi}}(u)$ is constant for $\|u - u_0\|_{C_0^1} \leq \epsilon$, we deduce $u_0$ is a local minimizer of $\hat{\Phi}$ in $C^1 $ topology, and since $\mathcal{L}\{\Omega\backslash\Omega'\}=0$, so the integral functional $\Phi$ and $\hat{\Phi}$ share the same minimizers, thus we have $u_0$ is a local minimizer of $\Phi$ in $C_0^1$ topology. Finally, using Theorem 3.1, we deduce that $u_0$ is a local minimizer of (\ref{eqn:l14}) in $ W^{1,p}_0(\Omega)$ topology, $ \underline{u}\leq u_0\leq \overline{u}$. Thus complete the proof of Theorem 3.3. \hfill$\Box$ \section{Appendix} \textbf{Proof of lemma 1.1} 1. Note that $u$ is a weak solution of equation $-\Delta u = g(\cdot,u)$ in $\Omega$, in the sense that $u$ satisfies \begin{equation}\label{eqn:aa} \int_{\Omega}\nabla u \cdot\nabla\varphi = \int_{\Omega} g\cdot\varphi ,\quad \forall\, \varphi\in H^1 _0(\Omega) \end{equation} Then we choose $\eta\in\textbf{C}^{\infty}_0$ and for $s\geq 0, M\geq 0$, let $\varphi = u\min \{ |u|^{2s},{M}^2\}\eta^2\in W^{1,2}_0(\Omega)$, with $supp\,\varphi\subset\subset\Omega$. then we have \begin{equation}\label{1.2} \nabla\varphi= \begin{cases} \nabla u\min \{|u|^{2s},M^2\}\eta^2 + 2s |u|^{2s}\nabla u \eta^2 \\ \quad + 2u \min \{|u|^{2s},M^2\}\eta\nabla\eta, & \text{if } \min\{|u|^{2s}, {M}^2\}= |u|^{2s}, \\[3pt] \nabla u\min \{|u|^{2s},M^2\}\eta^2\\ \quad + 2u\, \min \{|u|^{2s},M^2\}\eta\nabla\eta, &\text{otherwise}. \end{cases} \end{equation} Multiplying (\ref{eqn:aa}) with $\varphi$, we obtain \begin{eqnarray}\label{eqn:bb} \nonumber \int_{\Omega}\nabla u \nabla\varphi &=&\int_{\Omega}|\nabla u|^2\, \min \{|u|^{2s},M^2\}\eta^2+ 2s\int_{\{|u|^s