\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 43, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2011/43\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for divergence type elliptic equations} \author[L. Zhao, P. Zhao, X. Xie \hfil EJDE-2011/43\hfilneg] {Lin Zhao, Peihao Zhao, Xiaoxia Xie} \address{Lin Zhao \newline School of Mathematics and Statistics, Lanzhou University\\ Lanzhou, Gansu 730000, China} \email{zhjz9332003@gmail.com} \address{Peihao Zhao \newline School of Mathematics and Statistics, Lanzhou University\\ Lanzhou, Gansu 730000, China} \email{zhaoph@lzu.edu.cn} \address{Xiaoxia Xie \newline School of Mathematics and Statistics, Lanzhou University\\ Lanzhou, Gansu 730000, China} \email{xiexx06@lzu.cn} \thanks{Submitted January 1, 2011. Published March 31, 2011.} \subjclass[2000]{35A15, 35J20, 35J62} \keywords{Nonlinear elliptic equations; uniformly convex; mountain pass lemma; \hfill\break\indent three critical points theorem} \begin{abstract} We establish the existence and multiplicity of weak solutions of a problem involving a uniformly convex elliptic operator in divergence form. We find one nontrivial solution by the mountain pass lemma, when the nonlinearity has a $(p-1)$-superlinear growth at infinity, and two nontrivial solutions by minimization and mountain pass when the nonlinear term has a $(p-1)$-sublinear growth at infinity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article we study the boundary-value problem \begin{gather} -\operatorname{div}(a(x,\nabla u))+|u|^{p-2}u=\lambda f(x,u), \quad x\in\Omega, \label{e1.1} \\ u(x)=\text{constant}, \quad x\in\partial\Omega, \label{e1.2} \\ \int_{\partial\Omega}a(x,\nabla u)\cdot n\,ds=0, \label{e1.3} \end{gather} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, with smooth boundary. We obtain the existence and multiplicity for the equation $$-\operatorname{div}(a(x,\nabla u)) =f(x,u). \label{eP}$$ Such operators arise, for example, from the expression of the $p$-Laplacian in curvilinear coordinates. We refer to the books \cite{k2,s1,z1} for the foundation of the variational methods and refer to the overview papers \cite{b1,d1,k1,l1,l2,n1,p1,r1,y1} for the advances and references of this area. Recently, the Dirichlet problem \eqref{eP} was studied and obtained one weak solution by the mountain pass lemma in \cite{n1}, when the potential satisfies a set of assumptions and $f$ is $(p-1)$-superlinear at infinity. Duc and Vu \cite{d1} extended the result of \cite{n1}, considering the Dirichlet problem \eqref{eP} in the nonuniform case. Krist\'aly, Lisei and Varga \cite{k1} study the Dirichlet problem \eqref{eP}, and obtain three solutions when $f$ is $(p-1)$-sublinear at infinity. Yang, Geng and Yan \cite{y1} deal with the singular $p$-Laplacian type equation and get three solutions with $f$ having $(p-1)$-sublinear growth at infinity. Papageorgiou, Rocha and Staicu \cite{p1} consider the nonsmooth $p$-Laplacian problem, and obtain at least two solutions. In \cite{l2}, the sub-supersolution method has been applied to find one solution to the problem \eqref{eP} with the boundary condition \eqref{e1.2} and \eqref{e1.3} where the nonlinearity $f$ satisfies the condition: $|f(x,u)|\leq a_3(x)$, with $a_3 \in L^{p'}(\Omega)$, $\frac{1}{p}+\frac{1}{p'}=1$. The first result of this paper is about the existence of solution of \eqref{e1.1}-\eqref{e1.3}. We assume that the nonlinear term $f:\Omega\times \mathbb{R}\to \mathbb{R}$ satisfies the Ambrosetti-Rabinowitz type condition and obtain one weak solution by the mountain pass lemma in Theorem \ref{thm3.1}. The second result of this paper is about the existence and multiplicity of solutions for the problem \eqref{e1.1}-\eqref{e1.3}. Under the growth on $f$, saying, $f$ is $(p-1)$-sublinear at infinity, we obtain two nontrivial solutions by minimization and mountain pass lemma in \cite{b1,k1,p1}, where they do the same thing under different assumptions on $f$. We remark that in \cite{d1,k1,n1}, the function $A$, with $\nabla_{\xi}A=a(x,\xi)$, satisfies the $p$-uniformly convex condition: there exists a constant $k>0$ such that $A(x,\frac{\xi+\eta}{2})\leq\frac{1}{2}A(x,\xi) +\frac{1}{2}A(x,\eta)-k|\xi-\eta|^{p}, \quad x\in \Omega,\; \xi,\eta\in\mathbb{R}^{N}.$ However, for the case $A(\xi)=|\xi|^{p}$, the $p$-uniform convexity condition is satisfied only for $p\in [2,+\infty)$. We assume the function $A$ satisfies the condition (UC) in this paper, while the condition (UC) is satisfied for $A(\xi)=|\xi|^{p}$ for all $p\in(1,+\infty)$ (see \cite{f1}). \section{Preliminaries} Let $X$ be a Banach space and $X^{\ast}$ is its topological dual. We denote the duality brackets for the pair $(X^{\ast},X)$ by $\langle\cdot,\cdot\rangle$ and $W^{1,p}(\Omega)$ $(p>1)$ is the usual Sobolev space, equipped with the norm $$\|u\|=\|u\|_{W^{1,p}(\Omega)}=\Big(\int_{\Omega}|\nabla u|^{p}+|u|^{p}dx\Big)^{1/p}. \label{e2.1}$$ Let $V=\{u\in W^{1,p}(\Omega):u|_{\partial\Omega}=\text{constant}\}.$ We next claim that $V$ is a closed subspace of $W^{1,p}(\Omega)$ and thus a reflexive Banach space with the restricted norm of \eqref{e2.1}. \begin{lemma}[\cite{l2}] \label{lem2.1} $V$ is a Banach space equipped with the norm of \eqref{e2.1}. \end{lemma} \begin{proof} From the definition of $V$, we set $V=\{u+c: u\in W_0^{1,p}(\Omega), c\in\mathbb{R}\}$. We assume that $v_n\in V$, then $v_n=u_n+c_n$, with $u_n\in W_0^{1,p}(\Omega)$. If $\{v_n\}$ is Cauchy sequence in $W^{1,p}(\Omega)$, then for all $\varepsilon>0$, we have \begin{align*} \varepsilon>\|v_n-v_m\|_{W^{1,p}} &= \|u_n+c_n-(u_m+c_m)\|_{W^{1,p}} \\ &= \|\nabla (u_n-u_m)\|_{L^{p}}+\|u_{n}-u_{m}+c_{n}-c_{m}\|_{L^{p}} \\ &\geq \|\nabla (u_n-u_m)\|_{L^{p}} . \end{align*} We obtain that $\{u_n\}$ is Cauchy sequence in $W_0^{1,p}(\Omega)$, so there exists $\tilde{u}\in W_0^{1,p}(\Omega)$ such that $u_n\to \tilde{u} \quad \text{in} W_0^{1,p}(\Omega).$ As $\|u_n-u_m\|_{L^{p}}\leq c_{p}\|\nabla(u_n-u_m)\|_{L^{p}}\leq c_{p}\varepsilon,$ we have \begin{align*} \|c_n-c_m\|_{L^p} &= \|u_n+c_n-(u_m+c_m)-u_n+u_m\|_{L^p} \\ &\leq \|u_n+c_n-(u_m+c_m)\|_{L^p}+\|u_n-u_m\|_{L^p} \\ &\leq \|v_n-v_m\|_{L^p}+c_{p}\|u_n-u_m\|_{L^p} \\ &\leq \varepsilon+c_{p}\varepsilon . \end{align*} We conclude that $\{c_n\}$ is a Cauchy sequence in $L^p(\Omega)$, and so is in $\mathbb{R}$. We conclude that there exists $\tilde{c}\in \mathbb{R}$, such that \begin{equation*} u_n+c_n\to \tilde{u}+\tilde{c} \quad \text{in $V$ as $c_n\to \tilde{c}$ in $\mathbb{R}$}. \end{equation*} \end{proof} \begin{definition} \label{def2.2}\rm We say that $u\in V$ is a weak solution of the boundary-value problem \eqref{e1.1}-\eqref{e1.3}if $$\int_{\Omega}a(x,\nabla u)\cdot\nabla v\,dx+\int_{\Omega}|u|^{p-2}uv\,dx -\lambda\int_{\Omega}f(x,u)v\,dx=0, \quad \forall v \in V.$$ \end{definition} \begin{definition}[\cite{f1}] \label{def2.3}\rm Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}, A=A(x,\xi)$ be a continuous function in $\Omega\times\mathbb{R}^N$ with continuous derivative with respect to $\xi$, $a(x,\xi)=\nabla_{\xi} A(x,\xi)=A'$. Define $A^{|\vee|}:\Omega\times\mathbb{R}\to\mathbb{R}$ as follows, $A^{|\vee|}(x,t)=\sup_{|\xi|=t}A(x,\xi), \quad \forall x\in\Omega.$ For every $\varepsilon, b\in (0,1)$ and $x\in\Omega$, define \begin{align*} E_{\varepsilon,b}(x) =\Big\{&(\xi,\eta)\in\mathbb{R}^{N}\times\mathbb{R}^{N}: A(x,\frac{\xi-\eta}{2})\geq\frac{1}{2}\max\{A(x,\varepsilon\xi), A(x,\varepsilon\eta)\}, \\ &A(x,\frac{\xi+\eta}{2})>(1-b)\frac{A(x,\xi)+A(x,\eta)}{2}\Big\}, \end{align*} and $q_{\varepsilon,b}(x)=\sup\{\frac{|\xi-\eta|}{2}: (\xi,\eta)\in E_{\varepsilon,b}(x)\}.$ We say that $A$ satisfies condition (UC) if $\lim_{b\to0}\int_{\Omega}A^{|\vee|}(x,q_{\varepsilon,b}(x))dx=0 \quad \text{for every } \varepsilon\in(0,1).$ So a function $A$ is said to be uniformly convex if $A$ satisfies condition (UC). \end{definition} As in \cite{f1}, we remark that for $A(\xi)=|\xi|^{p}$, the $p$-uniform convexity condition $A(x,\frac{\xi+\eta}{2})\leq\frac{1}{2}A(x,\xi) +\frac{1}{2}A(x,\eta)-k|\xi-\eta|^{p}, \quad \forall x\in\Omega,\; \xi,\eta\in \mathbb{R}^{N},$ where $k$ is a positive constant, is satisfied only if $p\in[2,+\infty)$, but (UC) is satisfied for all $p\in(1,+\infty)$. \begin{lemma}[\cite{k2,s1,z1}] \label{lem2.4} Let $X$ be a Banach space and $I\in C^{1}(X;\mathbb{R})$ satisfy the Palais-Smale condition. Suppose \begin{itemize} \item[(i)] $I(0)=0$; \item[(ii)] there exists constants $r>0, a>0$ such that $I(u)\geq a$ if $\|u\|=r$; \item[(iii)] there exists $u_1\in X$ such that $\|u_1\|\geq r$ and $I(u_1)1$, $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}$, $a(x,\xi)$ be derivative of $A(x,\xi)$ with respect to $\xi$, and we assume that the following conditions hold \begin{itemize} \item[(A1)] $A(x,0)=0$ for all $x\in\Omega$; \item[(A2)] $a$ satisfies the growth condition $|a(x,\xi)|\leq c_{2}(1+|\xi|^{p-1})$ for all $x\in\Omega$, $\xi\in \mathbb{R}^{N}$, for some constant $c_{2}>0$; \item[(A3)] $A$ is uniformly convex; \item[(A4)] $A$ is $p$-subhomogeneous, $0\leq a(x,\xi)\xi\leq p A(x,\xi)$ for all $x\in\Omega$, $\xi\in \mathbb{R}^{N}$. \item[(A5)] $A$ satisfies $A(x,\xi)\geq \Lambda |\xi|^{p}$ for all $x\in\Omega$, $\xi\in \mathbb{R}^{N}$, where $\Lambda>0$ is a constant. \end{itemize} Let $f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous function satisfying the following conditions: \begin{itemize} \item[(F1)] The subcritical growth condition $|f(x,s)|\leq c_{3}(1+|s|^{q-1}), \quad\forall x\in\Omega,s\in \mathbb{R},$ where $pN$; \item[(F2)] (The Ambrosetti-Rabinowitz condition) $F(x,s)=\int_0^{s}f(x,t)dt$ is $\theta$-super\-homogeneous at infinity; i.e., there exists $s_0>0$ such that $0<\theta F(x,s)\leq f(x,s)s, \quad \text{for } |s|\geq s_0,\, x\in\Omega,$ where $\theta>p$; \item[(F3)] $\lim_{|s|\to0}\frac{f(x,s)}{|s|^{p-1}}=0$; \item[(F4)] $\lim_{|s|\to\infty}\frac{f(x,s)}{|s|^{p-1}}=0$; \item[(F5)] There exists $s^{\ast}>0, s^{\ast}\in \mathbb{R}$ such that $F(x,s^{\ast})>0$, $\forall x\in \Omega$. \end{itemize} Our main result is as follows. \begin{theorem} \label{thm3.1} Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}^{N}$ be a potential which satisfies {\rm (A1)--(A5)}, and let $f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous function. If $f$ satisfies {\rm (F1)--(F3)}, then \eqref{e1.1}-\eqref{e1.3} has at least one nontrivial weak solution in $V$, for every $\lambda\in \mathbb{R}$. \end{theorem} \begin{theorem} \label{thm3.2} Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}^{N}$ be a potential which satisfies {\rm (A1)--(A5)}, and let $f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous function. If $f$ satisfies {\rm (F3)--(F5)}, then there exists a constant $\mu>0$, such that for $\lambda\in(\mu,+\infty)$, problem \eqref{e1.1}-\eqref{e1.3} has at least two nontrivial weak solutions in $V$. \end{theorem} \subsection{Proof of Theorem \ref{thm3.1}} Under the assumptions of Theorem \ref{thm3.1} we define the functional $J(u)=\int_{\Omega}A(x,\nabla u)\,dx +\frac{1}{p}\int_{\Omega}|u|^{p}dx-\lambda\int_{\Omega}F(x,u)\,dx.$ It is easy to see that $J:V\to \mathbb{R}$ is well defined and $J\in C^{1}(V;\mathbb{R})$. Its derivative is given by $\langle J'(u),\varphi\rangle=\int_{\Omega}a(x,\nabla u)\cdot\nabla\varphi\, dx+\int_{\Omega}|u|^{p-2}u\varphi\, dx -\lambda\int_{\Omega}f(x,u)\varphi\,dx,$ for all $u,\varphi\in V$. Thus the weak solution of \eqref{e1.1}--\eqref{e1.3} corresponds to the critical point of the functional $J$ on $V$. To prove Theorem \ref{thm3.1}, we apply the mountain pass lemma to this functional. We will show $J$ satisfies the Palais-Smale condition in the first. Let $\{u_{n}\}\subset V$ be a Palais-Smale sequence; i.e., $J'(u_{n})\to 0$ in $X^{\ast}$ and $J(u_{n})\to l$, where $l$ is a constant. We first show that $\{u_{n}\}$ is bounded in $V$, \begin{align*} J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle &= \int_{\Omega}[A(x,\nabla u_n)-\frac{1}{\theta}a(x,\nabla u_n)\cdot\nabla u_{n}]dx\\ &\quad + (\frac{1}{p}-\frac{1}{\theta}) \int_{\Omega}|u_n|^{p}dx +\lambda\int_{\Omega}[\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx, \end{align*} where $\theta>p$. From condition (A4), we have \begin{align*} J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle &\geq (1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla u_{n})dx+(\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx \\ &\quad + \lambda\int_{\Omega}[\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx, \end{align*} then \begin{align*} &(1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla u_n)dx + (\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx\\ &\leq J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle -\lambda\int_{\{x: |u_{n}(x)|>s_0\}} [\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx+Mm(\Omega), \end{align*} where $M=\sup\{| \frac{1}{\theta}f(x,s)s-F(x,s)|:x\in\Omega, |s|\leq s_0\}$, and $m(\Omega)$ denotes the Lebesgue measure of $\Omega$. By (F2) (the Ambrosetti-Rabinowitz condition), we have $(1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla u_n)dx +(\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx\leq J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle+Mm(\Omega).$ By (A5), $%(3.1) (1-\frac{p}{\theta})\min\{\Lambda,\frac{1}{p}\} (\int_{\Omega}|\nabla u_{n}|^{p}+|u_n|^{p})dx\leq J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle+Mm(\Omega),$ where $\min\{\Lambda,\frac{1}{p}\}$ denotes the minimum of $\Lambda$ and $\frac{1}{p}$. As $\|u_n\|=\Big(\int_{\Omega}|\nabla u_n|^{p}+|u_n|^{p}dx\Big)^{1/p},$ we conclude that $\{u_{n}\}$ is bounded in $V$. Since $V$ is a closed subspace of $W^{1,p}(\Omega)$ and the reflexivity of $W^{1,p}(\Omega)$, we may extract a weakly convergent subsequence that we call $\{u_{n}\}$ for simplicity. So we may assume that $u_{n}\rightharpoonup u$ weakly in $W^{1,p}(\Omega)$. Next, we will prove that ${u_{n}}$ converges strongly to $u\in V$. From the derivative of $J$ we obtain $$\label{e3.2} \begin{split} &\int_{\Omega}a(x,\nabla u_n)\cdot\nabla(u_n-u)dx+ \int_{\Omega}|u_n|^{p-2}u_n(u_n-u)dx \\ &= \langle J'(u_n),u_n-u\rangle-\lambda\int_{\Omega}f(x,u_n)(u_n-u)\, dx. \end{split}$$ Since $\|J'(u_n)\|_{W^{-1,p'}}\to0$ and $\{u_n-u\}$ is bounded in $V\subset W^{1,p}(\Omega)$, by the $|\langle J'(u_n),u_n-u\rangle|\leq\|J'(u_n)\|_{W^{-1,p'}}\|u_n-u\|$ it follows that $\langle J'(u_n),u_n-u\rangle\to0.$ From (F1), we have \begin{align*} &\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx \\ &\leq c_3\int_{\Omega}|u_n(x)-u(x)|dx + c_3\int_{\Omega}|u_n(x)|^{q-1}|u_n(x)-u(x)|dx \\ &\leq c_3((m(\Omega))^{1/q'}+\|u_n\|_{L^{q}}^{q-1}) \|u_n-u\|_{L^q}, \end{align*} where $\frac{1}{q}+\frac{1}{q'}=1$. Since the embedding $W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ is compact, with $q<\frac{Np}{N-p}$, we obtain $u_n\to u$ strongly in $L^q(\Omega)$. So we obtain $\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx\to0.$ Considering the inequality \begin{align*} \int_{\Omega}||u_n(x)|^{p-2}u_n(x)(u_n(x)-u(x))|dx &= \int_{\Omega}|u_n(x)|^{p-1}|u_n(x)-u(x)|dx \\ &\leq \|u_n\|_{L^p}^{p-1}\|u_n-u\|_{L^p}, \end{align*} and $u_n\to u$ strongly in $L^p(\Omega)$, we have $\int_{\Omega}||u_n(x)|^{p-2}u_n(x)(u_n(x)-u(x))|dx .$ From \eqref{e3.2}, we may conclude $%3.3 \limsup_{n\to\infty}\langle a(x,u_{n}),u_{n}-u\rangle =\limsup_{n\to\infty}\int_{\Omega}a(x,\nabla u_n) \cdot\nabla(u_n-u)dx\leq0,$ where $\langle a(x,u_{n}),u_{n}-u\rangle$ denotes $\int_{\Omega}a(x,\nabla u_n)\cdot\nabla(u_n-u)dx$. Therefore, from condition (A3), $A$ is uniformly convex, and the operator $a(x,\xi)=D_{\xi}A(x,\xi)$ satisfies the $(S_+)$ property. From the $(S_+)$ condition in \cite[Proposition 2.1]{n1}, so we have $u_{n}\to u$ strongly in $W^{1,p}(\Omega)$. Since $\{u_{n}\}\subset V$, $V$ is a closed subspace of $W^{1,p}(\Omega)$, and we have $u\in V$. So $u_{n}\to u$ strongly in $V$. Next, we show that $J$ satisfies the geometry condition of the mountain pass lemma; i.e., \begin{itemize} \item[(1)] There exists $r>0$, such that $\inf_{\| u\|=r}J(u)=b>0$. \item[(2)] There exists $u_0\in V$ such that $J(tu_0)\to -\infty$, as $t\to+\infty$. \end{itemize} \noindent\textbf{Step 1.} Fix $\lambda\in \mathbb{R}$, we choose $\varepsilon>0$ small enough satisfying $\Lambda>\frac{\lambda\varepsilon}{pc_{p}}$. Then by (F3), there exists $\delta>0$ such that $|f(x,s)|\leq\varepsilon|s|^{p-1}$ for $|s|\leq\delta$, for all $x\in\Omega$. Integrating the above inequality, we deduce that $F(x,s)\leq\frac{\varepsilon}{p}|s|^{p},\quad \text{for } |s|\leq\delta.$ Consequently, using (F1) and the Sobolev embedding, we have \begin{align*} J(u) &\geq \int_{\Omega}A(x,\nabla u)dx+\frac{1}{p}\int_{\Omega}|u|^{p}dx -\lambda\int_{\{x\in\Omega:|u(x)|\leq\delta\}} \frac{\varepsilon}{p}|u|^{p}dx\\ &\quad -\lambda\int_{\{x\in\Omega:|u|>\delta\}}c_{4} |u|^{q}dx\\ &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^{p}-\frac{\lambda\varepsilon}{p}c_{p}\|u\|^{p}-\lambda c_{4}\|u\|^{q} \\ &\geq \Big(\min\{\Lambda,\frac{1}{p}\} -\frac{\lambda\varepsilon}{p}c_{p}\Big)\|u\|^{p} -\lambda c_{4}\|u\|^{q}=\Phi(r), \end{align*} where $r=\|u\|^{p}$, $\min\{\Lambda,\frac{1}{p}\}>\frac{\lambda\varepsilon}{p}c_p$, as $\varepsilon$ is small enough. Moreover, $\Phi(r)>0$ for $r>0$ small enough, since $q>p$. \noindent\textbf{Step 2.} Since $A$ is $p$-subhomogeneous, can be restated as a differential inequality for the function $F$ in the form $s|s|^{\theta}\frac{d}{ds}(|s|^{-\theta}F(x,s))\geq0, \quad \text{for } |s|\geq s_0.$ We infer that for $|s|\geq s_0$, we have $F(x,s)\geq\gamma_0(x)|s|^{\theta}$, where $\gamma_0=s_0^{-\theta}\min\{F(x,s_0),F(x,-s_0)\}>0.$ Considering condition (A4), we obtain that for some constant $k(u)>0$ there holds \begin{align*} J(tu_0) &= \int_{\Omega}A(x,t\nabla u_0)dx+\frac{1}{p}\int_{\Omega}|tu_0|^{p}dx -\lambda\int_{\Omega}F(x,tu_0)dx\\ &\leq t^{p}\int_{\Omega}A(x,\nabla u_0)dx+\frac{1}{p}t^p\int_{\Omega}|u_0|^{p}dx-k(u)|\lambda| t^{\theta}+|\lambda|M_{1}m(\Omega) , \end{align*} where $M_{1}=\sup\{|F(x,s)|:x\in\Omega,|s|\leq s_0\}$. Since $\theta>p$, we choose $u_0$ such that $m\{x\in\Omega:u_0(x)\geq s_0\}>0$. We deduce that $J(tu_0)\to-\infty$, as $t\to +\infty$. For fixed $u_0\neq0$ and sufficiently large $t>0$, we let $u_{1}=tu_0$. By Lemma \ref{lem2.4} (mountain pass lemma), we obtain the existence of a non-trivial solution $u$ to \eqref{e1.1}-\eqref{e1.3}. The proof is completed. \subsection{Proof of Theorem \ref{thm3.2}} We denote $\mathcal{A}(u)=\int_{\Omega}A(x,\nabla u)dx+\frac{1}{p}\int_{\Omega}|u|^pdx$ and $\mathcal{F}(u)=\int_{\Omega}F(x,u)dx$, then the functional $J$ is given by $J(u)=\mathcal{A}(u)-\lambda\mathcal{F}(u)$. \begin{lemma}[\cite{k1}] \label{lem3.3} For every $\lambda\in \mathbb{R}$, the functional $J:V\to \mathbb{R}$ is sequentially weakly lower semicontinuous. \end{lemma} \begin{proof} The functional $\mathcal{A}$ being locally uniformly convex is weakly lower semicontionous. From the condition $(F_4)$, we have $|f(x,s)|\leq c_5(1+|s|^{p-1})$ for every $s\in \mathbb{R}$. Since the embedding $V\subset W^{1,p}(\Omega) \hookrightarrow L^{p}(\Omega)$ is compact, we obtain that $\mathcal{F}$ is sequentially weakly lower semicontinuous in the standard method. \end{proof} \begin{lemma} \label{lem3.4} For every $\lambda\in \mathbb{R}$, the functional $J$ is coercive and satisfies the Palais-Smale condition. \end{lemma} \begin{proof} By (F4), for $\varepsilon>0$ small enough, there exists $\delta$ such that $|f(x,s)|\leq\varepsilon|s|^{p-1}$ for every $|s|\geq\delta$. Integrating this inequality, we have $|F(x,s)|\leq\frac{\varepsilon}{p}|s|^{p} +\max_{|t|\leq\delta}|f(x,t)||s|,\quad \forall s\in \mathbb{R}.$ Thus, for every $u\in V$, we obtain \begin{align*} J(u) &\geq \mathcal{A}(u)-|\lambda||\mathcal{F}(u)|\\ &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^p -|\lambda|\frac{\varepsilon}{p}\int_{\Omega}|u|^{p}dx -|\lambda|\max_{|t|\leq\delta}|f(x,t)|\int_{\Omega}|u|dx\\ &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^p -\frac{\varepsilon|\lambda|}{p}\int_{\Omega}|u|^{p}dx -|\lambda|m(\Omega)^{1/p'}\max_{|t|\leq\delta}|f(x,t)| \Big(\int_{\Omega}|u|^{p}dx\Big)^{1/p}\\ &\geq \Big(\min\{\Lambda,\frac{1}{p}\}-\frac{\varepsilon| \lambda|c_{p}}{p}\Big)\|u\|^p-c_{p}^{1/p}| \lambda|m(\Omega)^{1/p'}\max_{|t|\leq\delta}|f(x,t)| \|u\|. \end{align*} Since $\varepsilon$ is small enough, $\min\{\Lambda,\frac{1}{p}\}>\frac{\varepsilon|\lambda|c_{p}}{p}$, so we have $J(u)\to+\infty$, whenever $\|u\|\to+\infty$. Hence $J$ is coercive. The proof of the functional $J$ satisfying the Palais-Smale condition is similar to Theorem \ref{thm3.1} The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.2}] From condition (F5), we have \begin{equation*} \rho:=\sup_{u\in V,u\neq0}\frac{\mathcal{F}(u)}{\mathcal{A}(u)} \geq\frac{\mathcal{F}(s^\ast)}{\mathcal{A}(s^\ast)}>0. \end{equation*} Let $\mu=1/\rho$. Fix $\lambda\in(\mu,+\infty)$. From the definition of $\rho$, there exists some $u^\ast\in V$, with $\min\{\mathcal{A}(u^\ast),\mathcal{F}(u^\ast)\}>0$, such that $\frac{1}{\lambda}<\frac{\mathcal{F}(u^\ast)}{\mathcal{A}(u^\ast)}.$ This implies $J(u^\ast)=\mathcal{A}(u^\ast)-\lambda\mathcal{F}(u^\ast)<0$. By Lemma \ref{lem3.4}, the functional $J$ is bounded from below, coercive and satisfies the (P-S) condition on $V$ for every $\lambda>0$. This implies the functional $J$ has a global minimizer $u_1$; i.e., $J(u_1)\leq J(u) \quad \forall u\in V.$ Let $u=u^\ast$. We have $J(u_1)\leq J(u^\ast)<0.$ By (F3), there exists $\delta>0$ such that $|f(x,s)|\leq\varepsilon|s|^{p-1}$ for $|s|<\delta$, for all $x\in \Omega$. We have $$\label{e3.5} |F(x,s)|\leq\frac{\varepsilon}{p}|u|^p \text{for} |s|\leq\delta.$$ Using (F4), there exists $k(\delta)>0$ such that $|F(x,s)|\leq k(\delta)|s|^p\leq k(\delta)|s|^q$, $p\delta$. Considering this fact and \eqref{e3.5}, for $\lambda\in(\mu,+\infty)$ we have \begin{align*} J(u) &\geq \int_{\Omega}A(x,\nabla u)dx+\frac{1}{p}\int_{\Omega}|u|^{p}dx-\lambda\int_{\{x\in\Omega:|u(x)|\leq\delta\}}\frac{\varepsilon}{p}|u|^{p}dx\\ &\quad -\lambda\int_{\{x\in\Omega:|u|>\delta\}}k(\delta) |u|^{q}dx\\ &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^{p}-\frac{\lambda\varepsilon}{p}c_{p}\|u\|^{p}-\lambda k(\delta)\|u\|^{q} \\ &\geq \Big(\min\{\Lambda,\frac{1}{p}\}-\frac{\lambda\varepsilon}{p}c_{p} \Big)\|u\|^{p}-\lambda k(\delta)\|u\|^{q}=\Phi(r), \end{align*} where $r=\|u\|^{p}$ and $q>p$. We can take $\varepsilon$ small enough, such that $\min\{\Lambda,\frac{1}{p}\}>\frac{\lambda\varepsilon}{p}c_p$. Moreover, $\exists r>0$ small enough and $a>0$, such that $\Phi(r)\geq a>0$.\\ Obviously, $J(0)=0$. If we denote by $\Gamma$ the set of all continuous functions $\gamma:[0,1]\to V$, such that $\gamma(0)=0$ and $\gamma(1)=u_1$. From the mountain pass lemma, there exists $u_2$ such that $J'(u_2)=0$ and $J(u_2)=\beta=\inf_{\gamma\in\Gamma}\sup_{u\in \gamma}J(u)\geq a>0.$ This completes the proof. \end{proof} \subsection*{Acknowledgements} The authors thank Professor V. Radulescu for his valuable comments which helped to improve this article. \begin{thebibliography}{86} \bibitem{b1} G. Bonanno; \emph{Some remarks on a three critical points theorem}, Nonlinear Anal. TMA 54 (2003) 651--665. \bibitem{d1} D. M. Duc, N. T. Vu; \emph{Nonuniformly elliptic equations of $p$-Laplacian type}, Nonlinear Anal. TMA 61 (2005) 1483--1495. \bibitem{f1} X. L. Fan, C. X. 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