\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 90, pp. 1--14.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/90\hfil Infinitely many weak solutions] {Infinitely many weak solutions for a $p$-Laplacian equation with nonlinear boundary conditions} \author[J.-H. Zhao, P.-H. Zhao\hfil EJDE-2007/90\hfilneg] {Ji-Hong Zhao, Pei-Hao Zhao} % in alphabetical order \address{Department of Mathematics, Lanzhou University\\ Lanzhou, 730000, China} \email[Ji-Hong Zhao]{zhaojihong2007@yahoo.com.cn} \email[Pei-Hao Zhao]{zhaoph@lzu.edu.cn} \thanks{Submitted March 26, 2007. Published June 15, 2007.} \thanks{This work was partly supported by the Fundamental Research Fund for Physics and \hfill\break\indent Mathematics of Lanzhou University} \subjclass[2000]{35J20, 35J25} \keywords{$p$-Laplacian; nonlinear boundary conditions; weak solutions; \hfill\break\indent critical exponent; variational principle} \begin{abstract} We study the following quasilinear problem with nonlinear boundary conditions \begin{gather*} -\Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) \quad \text{in }\Omega, \\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u) \quad \text{on } \partial\Omega, \end{gather*} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary and $\frac{\partial}{\partial \nu}$ is the outer normal derivative, $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian with $10. The study of nonlinear elliptic boundary value problem about$p$-Laplacian of the form \eqref{e1} is an interesting topic in recent years. Many results have been obtained on this kind of problem, for example see \cite{P,CR,BR1,K,MR1,MR2,B} and the references therein. Such problem appear naturally in the study of optimal constants for Sobolev trace embedding and it arises in various applications, e.g. non-Newtonian fluids, reaction-diffusion problems, glaciology, biology etc(see \cite{D,AD,BR2,BMR,B}). The first paper that analyzed \eqref{e1} is \cite{BR1}. In that paper, the authors systematically studied the existence of nontrivial solutions of \eqref{e1} under$f(u)=|u|^{p-2}u$and$g$are subcritical, critical with a subcritical perturbation and supercritical with respect to$u$. Using the ideas from \cite{GP}, they established the existence results, nonexistence result, especially the result of nonlinear eigenvalue problem. In \cite{B}, the author proved the existence of at least three nontrivial solutions for \eqref{e1} under adequate assumptions on the source terms$f$and$g$. On the other hand, when$\Omega$is unbounded, we can see \cite{P,CR,K} for some existence and multiplicity results of solutions to problem \eqref{e1} in some weighted Sobolev spaces. Our aim in this paper is to prove that the infinitely many solutions results for the problem \eqref{e1} under various assumptions on nonlinear terms$f$and$g$. If$f$and$g$are both superlinear and subcritical with respect to$u$, then we prove the existence of infinitely many solutions of problem \eqref{e1} by using fountain theorem" and dual fountain theorem" respectively. In the case, where$g$is superlinear but subcritical and$f$is critical with a subcritical perturbation, namely$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$, we show that there exists at least a nontrivial solution when$p0$such that $$|f(x,u)|\leq C_{1}(1+|u|^{q-1}) \quad \text{for all }x\in \Omega,\; u\in\mathbb{R}.$$ \item[(G1)]$g(x,u)$is a Carath\'{e}odory function and for some$p0$such that $$|g(x,u)|\leq C_{2}(1+|u|^{z-1})\quad \text{for all } x\in \partial\Omega,\; u\in\mathbb{R}.$$ \item[(F2)] There exists$\alpha_{1}>p$and$R>0$such that $$|u|\geq R\Longrightarrow 0<\alpha_{1} F(x,u)\leq uf(x,u)\quad \text{for all } x\in \Omega,$$ where$F(x,u)=\int_{0}^{u}f(x,t)dt$is the primitive function of$f(x,u)$. \item[(G2)] There exists$\alpha_{2}>p$and$R>0$such that $$|u|\geq R\Longrightarrow 0<\alpha_{2} G(x,u)\leq ug(x,u)\quad \text{for all } x\in \partial\Omega,$$ where$G(x,u)=\int_{0}^{u}g(x,t)dt$is the primitive function of$g(x,u)$. \item[(F3)]$f(x,u)$is an odd function with respect to$u$, that is, $$f(x,-u)=-f(x,u) \quad \text{for all } x\in \Omega.$$ \item[(G3)]$g(x,u)$is an odd function with respect to$u$, that is, $$g(x,-u)=-g(x,u)\quad \text{for all } x\in \Omega.$$ \item[(G4)]$\lim_{u\to0} \frac{g(x,u)}{|u|^{p-1}}=0$. \end{itemize} Define$W^{1,p}(\Omega)=\{u\in L^{p}(\Omega): \int_{\Omega}|\nabla u|^{p}\mathrm{d}x<\infty\}$with the norm $$\label{e3} \| u \|_{1,p}:=(\int_{\Omega}(|\nabla u|^{p}+a(x)|u|^{p})\mathrm{d}x)^{\frac{1}{p}}.$$ Then$W^{1,p}(\Omega)$is a Banach space. For a variational approach, the functional associated to the problem \eqref{e1} is $$\label{e4} \varphi(u)=\frac{1}{p}\int_{\Omega}(|\nabla u|^{p}+a(x)|u|^{p})\mathrm{d}x- \int_{\Omega}F(x,u)\mathrm{d}x-\int_{\partial\Omega}G(x,u)\mathrm{d}S,$$ where$u \in W^{1,p}(\Omega)$and$\mathrm{d}S$is the measure on the boundary. Since (F1) and (G1) we can easily to obtain$\varphi\in C^{1}(W^{1,p}(\Omega),\mathbb{R})and \begin{align*} \langle\varphi'(u);v\rangle &=\int_{\Omega}(|\nabla u|^{p-2}\nabla u\nabla v +a(x)|u|^{p-2}uv)\mathrm{d}x\\ &\quad -\int_{\Omega}f(x,u)v\mathrm{d}x - \int_{\partial\Omega}g(x,u)v\mathrm{d}S \end{align*} for allu,v \in W^{1,p}(\Omega)$. We say that$u$is a weak solution of the problem \eqref{e1} if$u$is the critical point of the functional$\varphi$on$W^{1,p}(\Omega)$. \begin{remark} \label{rmk1.1}\rm According to the regularity theorem of \cite{L}, if$\partial\Omega$is of class$C^{1,\alpha}(0<\alpha\leq1)$and$g$satisfies \begin{equation*} |g(x,u)-g(y,v)|\leq C(|x-y|^{\alpha}+|u-v|^{\alpha}),\quad |g(x,u)|\leq C \end{equation*} for all$x,y\in\Omega, u,v\in\mathbb{R}$, then the regularity up to the boundary of \cite[Theorem 2]{L} shows that every weak solution of \eqref{e1} belongs to$C_{\rm loc}^{1,\beta}(\overline{\Omega})$for some$0<\beta\leq1$. \end{remark} \begin{remark} \label{rmk1.2} \rm Under the assumption \eqref{e2} it is easy to check that the norm \eqref{e3} is equivalent to the usual one, that is the norm with$a(x)\equiv 1$in \eqref{e3}. \end{remark} Our main results are as follows. \begin{theorem}\label{thm1.1} Under the assumptions (F1)--(F3) and (G1)--(G3), problem \eqref{e1} has a sequence of solutions${u_{k}}\in W^{1,p}(\Omega)$such that$\varphi{(u_{k})}\to \infty $as$k\to \infty$. \end{theorem} For a special$f$, we obtain a sequence of weak solutions with negative energy. \begin{theorem}\label{thm1.2} Let$f(x,u)=\mu|u|^{r-2}u+\lambda|u|^{s-2}u$, where$10$,$\mu\in \mathbb{R}$, problem \eqref{e1} has a sequence of solutions${u_{k}}\in W^{1,p}(\Omega)$such that$\varphi(u_{k})\to \infty $as$k\to \infty$, \item for every$\mu>0$,$\lambda\geq 0$, problem \eqref{e1} has a sequence of solutions${v_{k}}\in W^{1,p}(\Omega)$such that$\varphi(v_{k})<0$,$\varphi(v_{k})\to 0 $as$k\to \infty$. \end{enumerate} \end{theorem} Next we consider the critical growth on$f$. In this case, the compactness of the embedding$W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$fails, so to recover some sort of compactness, in spirit of \cite{BN}, we consider a perturbation of the critical power, that is,$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$. We also need much more assumptions on$g$around about the origin. \begin{itemize} \item[(G2')] there exists$\alpha_{2}>p$such that $$0<\alpha_{2}G(x,u)\leq ug(x,u)\quad \text{for all } x\in \partial\Omega, \; u\in \mathbb{R}\setminus\{0\}.$$ \end{itemize} Here we use the concentration-compactness principle" introduced in \cite{L1,L2}. We prove the following two theorems. \begin{theorem}\label{thm1.3} Let$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$with$p0$depending on$p$,$r$,$N$and$|\Omega|$such that if$\lambda>\lambda_{0}$, problem \eqref{e1} has at least a nontrivial solution in$W^{1,p}(\Omega)$. \end{theorem} \begin{theorem}\label{thm1.4} Let$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$with$10$depending on$p$,$r$,$N$and$|\Omega|$such that if$0<\lambda< \widetilde{\lambda}$, problem \eqref{e1} has infinitely many nontrivial solutions${u_{k}}\in W^{1,p}(\Omega)$such that$\varphi(u_{k})<0$,$\varphi(u_{k})\to 0 $as$k\to \infty$. \end{theorem} This paper is organized as follows. In the second section, we recall some definitions and preliminary theorems, including the well-known fountain theorem" and dual fountain theorem". The$(PS)_{c}$condition and$(PS)_{c}^{*}$condition are also introduced. In the third section, we consider the subcritical case and give the proof of Theorem \ref{thm1.1} and Theorem \ref{thm1.2}. In the last section. We consider the critical case and give the proof of Theorems \ref{thm1.3} and \ref{thm1.4}. \section{Preliminaries} First we introduced some notations:$X$denotes Banach space with the norm$\|\cdot\|_{X}$,$X^{*}$denotes the conjugate space with$X$,$L^{p}(\Omega)$denotes Lebesgue space with the usual norm$|\cdot|_{p}$,$W^{1,p}(\Omega)$denotes Sobolev space with the norm$\|\cdot\|_{1,p}$defined by \eqref{e3},$\langle\cdot;\cdot\rangle$is the dual paring of the space$X^{*}$and$X$,$|\Omega|$denotes the Lebesgue measure of the set$\Omega\subset \mathbb{R}^{N}$,$C_{1},C_{2},\dots$, denote (possibly different) positive constants. One important aspect of applying the standard methods of variational theory is to show that the functional$\varphi$satisfies the$(PS)_{c}$or$(PS)_{c}^{*}$condition which is introduced the following definition. \begin{definition}\label{def2.1} \rm Let$\varphi\in C^{1}(X,\mathbb{R})$and$c\in \mathbb{R}$. The function$\varphi$satisfies the$(PS)_{c}$condition if any sequence$\{u_{n}\}\subset X$such that $$\varphi(u_{n})\to c,\quad \varphi'(u_{n})\to 0\quad \text{in } X^{*}\text{ as } n\to\infty$$ contains a subsequence converging to a critical point of$\varphi$. \end{definition} Let$X$be a reflexive and separable Banach space, then there are${e_{j}}\in X$and${e_{j}^{*}}\in X^{*}$such that \begin{gather*} X=\overline{\mathop{\rm span}\{e_{j}| j=1,2,\dots}\},\quad X^{*}=\overline{\mathop{\rm span}\{{e_{j}^{*}}| j=1,2,\dots}\}, \\ \langle e_{i}^{*};e_{j}\rangle = \begin{cases} 1, \quad i=j,\\ 0, \quad i\neq j. \end{cases} \end{gather*} For convenience, we write$X_{j}:= \mathop{\rm span} \{e_{j}\}$,$Y_{k}:=\oplus_{j=1}^{k}X_{j}$,$Z_{k}:=\overline {\oplus_{j=k}^{\infty}X_{j}}$. And let$B_{k}:=\{u\in Y_{k}:\|u\|_{X}\leq \rho_{k}\}$,$N_{k}:=\{u\in Z_{k}:\|u\|_{X}=\gamma_{k}\}$, where$\rho_{k}>\gamma_{k}>0$. \begin{definition}\label{def2.2} \rm Let$\varphi\in C^{1}(X,\mathbb{R})$and$c\in \mathbb{R}$. The function$\varphi$satisfies the$(PS)_{c}^{*}$condition (with respect to$(Y_{n})$) if any sequence$\{u_{n_{j}}\}\subset Y_{n_{j}}$such that $$\varphi(u_{n_{j}})\to c,\quad \varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \quad \text{in } X^{*} \text{ as } n_{j}\to \infty$$ contains a subsequence converging to a critical point of$\varphi$. \end{definition} \begin{theorem}[{Fountain theorem, \cite[Thm. 3.6]{W}}]\label{thm2.1} Let$\varphi\in C^{1}(X,\mathbb{R})$be an even functional. If , for every$k\in \mathbb{N}$, there exists$\rho_{k}>\gamma_{k}>0$such that \begin{itemize} \item[(A1)]$a_{k}:=\sup_{u\in Y_{k},\,\|u\|_{X}=\rho_{k}} \varphi(u)\leq 0$, \item[(A2)]$b_{k}:=\inf_{u\in Z_{k},\, \|u\|_{X}=\gamma_{k}} \varphi(u)\to \infty$as$k\to \infty$, \item[(A3)]$\varphi$satisfies the$(PS)_{c}$condition for every$c>0$. \end{itemize} Then$\varphi$has an unbounded sequence of critical values. \end{theorem} \begin{theorem}[{Dual fountain theorem, \cite[Theorem 3.18]{W}}]\label{thm2.2} Let$\varphi\in C^{1}(X,\mathbb{R})$be an even functional. If , for every$k\geq k_{0}$, there exists$\rho_{k}>\gamma_{k}>0$such that \begin{itemize} \item[(B1)]$a_{k}:=\inf_{u\in Z_{k},\, \|u\|_{X}=\rho_{k}} \varphi(u)\geq 0$, \item[(B2)]$b_{k}:=\sup_{u\in Y_{k},\, \|u\|_{X}=\gamma_{k}} \varphi(u)< 0$, \item[(B3)]$d_{k}:=\inf_{u\in Z_{k},\, \|u\|_{X}\leq\rho_{k}} \varphi(u)\to 0$as$k\to \infty$, \item[(B4)]$\varphi$satisfies the$(PS)_{c}^{*}$condition for every$c\in[d_{k_{0}},0[$. \end{itemize} Then$\varphi$has a sequence of negative critical values converging to$0$. \end{theorem} \section{Proof of Theorem \ref{thm1.1}} \subsection*{Proof of the$(PS)_{c}$condition} Let us introduce the following lemmas which will be helpful in the proof. \begin{lemma}[{\cite[Lemma 2.1]{MR2}}] \label{lem3.1} Let$A: W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$be the function given by$\langle A(u);v\rangle:=\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\mathrm{d}x + \int_{\Omega}a(x)|u|^{p-2}uv\mathrm{d}x$. Then$A$is continuous, odd,$(p-1)$-homogeneous, and continuously invertible. \end{lemma} \begin{lemma}[{\cite[Lemma 2.2]{MR2}}] \label{lem3.2} Let$B:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$be the function given by$\langle B(u);v\rangle:=\int_{\partial\Omega}g(x,u)v\mathrm{d}S$, where$g(x,u)$be a Carath\'{e}odory function with subcritical growth. Then$B$is continuous and compact. \end{lemma} \begin{lemma}[{\cite[Lemma 2.3]{MR2}}] \label{lem3.3} Let$C:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$be the function given by$\langle C(u);v\rangle=\int_{\Omega}f(x,u)v\mathrm{d}x$, where$f(x,u)$is a Carath\'{e}odory function with subcritical growth. Then$C$is continuous and compact. \end{lemma} \begin{lemma}\label{lem3.4} Under the hypotheses of Theorem \ref{thm1.1},$\varphi$satisfies the$(PS)_{c}$condition with$c>0$. \end{lemma} \begin{proof} Suppose that$\{u_{n}\}\subset W^{1,p}(\Omega)$, for every$c>0$, $$\varphi(u_{n})\to c,\text{ } \varphi'(u_{n})\to 0 \text{ in } W^{1,p}(\Omega)^{*} \text{ as }n\to\infty.$$ First we prove the boundness of$\{u_{n}\}$. After integrating, we obtain from the assumptions (F2) and (G2) that there exist$C_{1}$,$C_{2}>0$such that \begin{gather} C_{1}(|u|^{\alpha_{1}}-1)\leq F(x,u)\quad \text{for all }x\in\Omega,\; u\in\mathbb{R}, \label{e5}\\ C_{2}(|u|^{\alpha_{2}}-1)\leq G(x,u)\quad \text{for all }x\in\partial\Omega,\; u\in\mathbb{R}. \label{e6} \end{gather} Set$\alpha=\min\{\alpha_{1},\alpha_{2}\}$and choose$\frac{1}{\beta}\in(\frac{1}{\alpha},\frac{1}{p})$, and from \eqref{e5} and \eqref{e6}, we obtain for$nsufficiently large, \begin{align*} & c + 1 + \|u_{n}\|_{1,p}\\ &\geq \varphi(u_{n})-\frac{1}{\beta}\langle\varphi'(u_{n}),u_{n}\rangle\\ &\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n}\|_{1,p}^{p}+(\frac{\alpha_{1}}{\beta}-1)\int_{\Omega}F(x,u_{n})\mathrm{d}x -(\frac{\alpha_{2}}{\beta}-1)\int_{\partial\Omega}G(x,u_{n})\mathrm{d}S\\ &\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n}\|_{1,p}^{p}+C_{1}(\frac{\alpha_{1}}{\beta}-1)|u_{n}|_{\alpha_{1}}^{\alpha_{1}}+C_{2}(\frac{\alpha_{2}}{\beta}-1)|u_{n}|_{_{L^{\alpha_{2}}(\partial\Omega)}}^{\alpha_{2}}-C_{3}. \end{align*} Note that\frac{\alpha_{i}}{\beta}-1>0 (i=1,2)$, then$\{u_{n}\}$is bounded in$W^{1,p}(\Omega)$. Next we show that the strongly convergence of$\{u_{n}\}$in$W^{1,p}(\Omega)$. Since$\{u_{n}\}$is bounded, up to a subsequence (which we still denote by$\{u_{n}\}$), we may assume that there exists$u\in W^{1,p}(\Omega)$such that$u_{n}\rightharpoonup u$weakly in$W^{1,p}(\Omega)$as$n\to \infty$. Note that$\varphi'(u_{n})=A(u_{n})-B(u_{n})-C(u_{n})\to 0$. By the compactness of$B$,$C$and the continuity of$A^{-1}$, we have $$u_{n}\to A^{-1}(a(x)B(u)-C(u)) \text{ in } W^{1,p}(\Omega)\text{ as } n\to \infty.$$ Thus$u_{n}\to u$in$W^{1,p}(\Omega)$. \end{proof} To prove Theorem \ref{thm1.1} we also need the following two lemmas. \begin{lemma}[\cite{FH}] \label{lem3.5} If$1\leq q0$. So from Theorem \ref{thm2.1}, we need only to verify$\phi$satisfying the condition (A1) and (A2). As for \eqref{e5} and \eqref{e6} in Lemma \ref{lem3.4}, we have $$\varphi(u)\leq\frac{\|u_{n}\|_{1,p}^{p}}{p}-C_{1}| u|_{\alpha_{1}}^{\alpha_{1}}-C_{2}|u|_{L^{\alpha_{2}} (\partial\Omega)}^{\alpha_{2}}-C_{1}|\Omega|-C_{2}|\partial\Omega|.$$ Since on the finite-dimensional space$Y_{k}$all norms are equivalent, so$\alpha_{i}>p$($i=1,2$) implies that (A1) is satisfied for$\rho_{k}>0$large enough. After integrating, we obtain from the assumptions (F1) and (G1) that there exist constants$C_{1}, C_{2}>0$such that $$F(x,u)\leq C_{1}(1+|u|^{q}),\quad G(x,u)\leq C_{2}(1+|u|^{z}).$$ Let us define $$\beta_{k}:=\sup_{u\in Z_{k},\, |u\|_{1,p}=1} |u|_{q},\quad\,d\sigma_{k} :=\sup_{u\in Z_{k},\, \|u\|_{1,p}=1} |u|_{L^{z}(\partial\Omega)}.$$ On$Z_{k}, we have \begin{align*} \varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\int_{\Omega}F(x,u)\mathrm{d}x -\int_{\partial\Omega}G(x,u)\mathrm{d}S\\ &\geq \frac{1}{p}\|u\|_{1,p}^{p}-C_{1}| u|_{q}^{q}-C_{1}|\Omega|-C_{2}|u|_{L^{z}(\partial\Omega)}^{z}-C_{2}|\partial\Omega|\\ &\geq \frac{1}{p}\|u\|_{1,p}^{p}-C_{1}\beta_{k}^{q}\|u\|_{1,p}^{q}-C_{2}\sigma_{k}^{z}\|u\|_{1,p}^{z} -C_{3}. \end{align*} Let \begin{gather*} \frac{1}{4p}\rho^{p}-C_{1}\beta_{k}^{q}\rho^{q}=0, \\ \frac{1}{4p}\rho^{p}-C_{2}\sigma_{k}^{z}\rho^{z}=0. % 7,8 \end{gather*} From these two equations, we have\rho_{k}:=(4pC_{1}\beta_{k}^{q})^{\frac{1}{p-q}}$,$\rho_{k}':=(4pC_{2}\sigma_{k}^{z})^{\frac{1}{p-z}}$. From Lemmas \ref{lem3.5} and \ref{lem3.6} we know that$\beta_{k}\to 0$,$\sigma_{k}\to 0$as$k\to \infty$. So, we know $$\label{e9} \rho_{k}\to \infty \text{ as } k\to \infty,\quad \rho_{k}'\to \infty \text{ as } k\to \infty.$$ Let $$\label{e10} \gamma_{k}=\min\{\rho_{k},\rho_{k}'\},$$ we obtain, if$u\in Z_{k}$and$\|u\|_{1,p}=\gamma_{k}$, then$\varphi(u)\geq \frac{1}{2p}\gamma_{k}^{p}-C_{3}$. From \eqref{e9} and \eqref{e10}, so (A2) is proved. It suffices then to use the fountain theorem to complete the proof. Here, we show two examples for readers for special cases of$f$to understand our theorem. \begin{example} \label{exa3.1} \rm Let$p0$,$\mu \in \mathbb{R}$, this problem has a sequence of solutions$\{u_{k}\}$such that$\varphi(u_{k})\to \infty$as$k\to \infty$. \end{example} \subsection*{Proof of Theorem \ref{thm1.2}} The first conclusion of Theorem \ref{thm1.2} is just example \ref{exa3.2}. We shall prove Theorem \ref{thm1.2} by using Theorem \ref{thm2.2}, so we need to verify the condition (B1)--(B4). Now we assume that$\mu>0$. To verify (B1), we define$\beta_{k}:=\underset{\substack {u\in Z_{k}\\ \|u\|_{1,p}=1}}{\sup}| u|_{r}$. From the assumptions (G1) and (G4), we have$G(x,u)\leq \varepsilon|u|^{p}+C|u|^{z}$, where$\varepsilon\to 0$as$|u|\to 0. So from the Sobolev trace embedding, we have \begin{align*} \varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r}-\frac{\lambda}{s}|u|_{s}^{s} -\int_{\partial\Omega}G(x,u)\mathrm{d}S\\ &\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s} -\varepsilon|u|_{L^{p}(\partial\Omega)}^{p}-C_{2}|u|_{L^{z}(\partial\Omega)}^{z}\\ &\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s} -\varepsilon C_{3}\|u\|_{1,p}^{p}-C_{4}\|u\|_{1,p}^{z}.\\ &\geq (\frac{1}{p}-\varepsilon C_{3})\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s} -C_{4}\|u\|_{1,p}^{z}. \end{align*} Sincep0$such that$\|u\|_{1,p}\leq R$. We have \begin{gather*} \frac{1}{4p}\|u\|_{1,p}^{p}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s}\geq 0,\\ \frac{1}{4p}\|u\|_{1,p}^{p}-C_{4}\|u\|_{1,p}^{z}\geq 0. \end{gather*} From these two inequalities, it follows that $$\label{e15} \varphi(u)\geq(\frac{1}{2p}-\varepsilon C_{3})\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r} .$$ Choose$\varepsilon$so small such that$\frac{1}{2p}-\varepsilon C_{3}>0$and let$\rho_{k}:=[(\frac{1}{2p}-\varepsilon C_{3})^{-1}\frac{\mu}{r}\beta_{k}^{r}]^{\frac{1}{p-r}}$, by Lemma \ref{lem3.5},$\beta_{k}\to 0$as$k\to \infty$, it follows that$\rho_{k}\to 0$as$k\to \infty$. So there exists$k_{0}$such that$\rho_{k}\leq R$when$k\geq k_{0}$. Thus, for$k\geq k_{0}$,$u\in Z_{k}$and$\|u\|_{1,p}=\rho_{k}$, we have$\varphi(u)\geq 0$and (B1) is proved. From (G2) we know there exists$C>0$such that$ C(|u|^{\alpha_{2}}-1)\leq G(x,u). Then, we have \begin{align*} \varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r} -\frac{\lambda}{s}|u|_{s}^{s} -\int_{\partial\Omega}G(x,u)\mathrm{d}S\\ &\leq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r} -\frac{\lambda}{s}|u|_{s}^{s} -C|u|_{L^{\alpha_{2}}(\partial\Omega)}^{\alpha_{2}}-C|\partial\Omega|. \end{align*} Since on the finite dimensional spaceY_{k}$all norms are equivalent, as$r0$then (B2) is satisfied for every$r_{k}>0$small enough. We obtain from \eqref{e15}, for$k\geq k_{0}$,$u\in Z_{k},\|u\|_{1,p}\leq\rho_{k}$,$\varphi(u)\geq -\frac{\mu}{r}\beta_{k}^{r}\rho_{k}^{r}$, since$\beta_{k}\to 0$and$\rho_{k}\to 0$as$k\to \infty$, (B3) is also satisfied. Finally we proved the$(PS)_{c}^{*}$condition. Consider a sequence$u_{n_{j}}\in Y_{n_{j}}$such that $$\varphi(u_{n_{j}})\to c,\text{ } \varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \text{ in } W^{1,p}(\Omega)^{*} \text{ as } n_{j}\to \infty.$$ For$n_{j}$big enough, let$\zeta=\min\{s,\alpha_{2}\}$and choose$\frac{1}{\beta}\in (\frac{1}{\zeta},\frac{1}{p})$. Now as$\lambda\geq0we have \begin{align*} c + 1 + \|u_{n_{j}}\|_{1,p} &\geq \varphi(u_{n_{j}})-\frac{1}{\beta}\langle\varphi'(u_{n_{j}}); u_{n_{j}}\rangle\\ &\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n_{j}}\|_{1,p}^{p} -\mu(\frac{1}{r}-\frac{1}{\beta})|u_{n_{j}}|_{r}^{r}. \end{align*} We can obtain the boundness of(u_{n_{j}})$in$W^{1,p}(\Omega)$since$10$,$\mu\in \mathbb{R}$, problem \eqref{e16} has a sequence of solutions${u_{k}}\in W^{1,p}(\Omega)$such that$\varphi(u_{k})\to \infty $as$k\to \infty$, \item for every$\mu>0$,$\lambda\in\mathbb{R}$, problem \eqref{e16} has a sequence of solutions${v_{k}}\in W^{1,p}(\Omega)$such that$\varphi(v_{k})<0$,$\varphi(v_{k})\to 0 $as$k\to \infty$. \end{enumerate} \end{corollary} \begin{proof} We need only to prove the boundness of$\{u_{n_{j}}\}$in$(PS)_{c}^{*}$sequence. Consider a sequence$u_{n_{j}}\in Y_{n_{j}}$such that $$\varphi(u_{n_{j}})\to c,\text{ } \varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \text{ in } W^{1,p}(\Omega)^{*} \text{ as } n_{j}\to\infty.$$ For$n_{j}big enough, from (G2'') we have \begin{align*} & c + 1 + \|u_{n_{j}}\|_{1,p}\\ &\geq \varphi(u_{n_{j}})-\frac{1}{s}\langle\varphi'(u_{n_{j}});u_{n_{j}} \rangle\\ &= (\frac{1}{p}-\frac{1}{s})\|u_{n_{j}}\|_{1,p}^{p} -\mu(\frac{1}{r}-\frac{1}{s})|u_{n_{j}}|_{r}^{r} +\int_{\partial\Omega}(\frac{1}{s}g(x,u_{n_{j}})u_{n_{j}}-G(x,u_{n_{j}}))\mathrm{d}S\\ &\geq (\frac{1}{p}-\frac{1}{s})\|u_{n_{j}}\|_{1,p}^{p} -\mu(\frac{1}{r}-\frac{1}{s})|u_{n_{j}}|_{r}^{r}. \end{align*} We obtain the boundness of\{u_{n_{j}}\}$in$W^{1,p}(\Omega)$since$10,\\ d\mu\geq|\nabla u|^{p}+\sum_{j=1}^{l}\mu_{j}\delta_{x_{j}},\quad \mu_{j}>0,\\ (\sigma_{j})^{\frac{p}{p^{*}} }\leq \frac{\mu_{j}}{S}. \end{gather*} \end{lemma} Now, we can prove a local $(PS)_{c}$ condition by using Lemma \ref{lem4.1}. \begin{lemma}\label{lem4.2} Let $\{u_{j}\}\subset W^{1,p}(\Omega)$ be a $(PS)_{c}$ sequence for $\varphi$ with energy level $c$. If $c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}$, where $S$ is the best constant in the Sobolev embedding $W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$, then there exists a subsequence that converges strongly in $W^{1,p}(\Omega)$. \end{lemma} \begin{proof} Let $\{u_{j}\}$ be a $(PS)_{c}$ sequence, it follows that $\{u_{j}\}$ is bounded in $W^{1,p}(\Omega)$(see Lemma \ref{lem3.4}). By Lemma \ref{lem4.1}, there exists a subsequence, that we still denote $\{u_{j}\}$ such that \begin{gather} u_{j}\rightharpoonup u \text{ weakly in } W^{1,p}(\Omega), \nonumber \\ u_{j}\to u \text{ strongly in } L^{r}(\Omega),\quad 10, \label{e17}\\ |u_{j}|^{p^{*}}\rightharpoonup \,d\sigma=|u|^{p^{*}}+ \sum_{k=1}^{l}\sigma_{k}\delta_{x_{k}}, \quad \sigma_{k}>0. \label{e18} \end{gather} Choose $\phi\in C_{0}^{\infty}(\mathbb{R^{N}})$ such that $$\phi \equiv 1 \text{ in } B(x_{k},\varepsilon), \quad \phi \equiv 0 \text{ in } B(x_{k},2\varepsilon)^{c},\quad |\nabla\phi|\leq \frac{2}{\varepsilon},$$ where $x_{k}$ belongs to the support of $\,d\sigma$. Considering $\{{u_{j}\phi}\}$, it is easy to see this sequence is bounded in $W^{1,p}(\Omega)$. Since $\varphi'(u_{j})\to 0$ in $W^{1,p}(\Omega)^{*}$ as $j\to \infty$, we obtain that $$\label{e19} \lim_{j\to\infty} \langle\varphi'(u_{j});\phi u_{j}\rangle=0.$$ Then from \eqref{e17} and \eqref{e18}, we obtain \begin{align*} & \lim_{j\to \infty} \int_{\Omega}|\nabla u_{j}|^{p-2}\nabla u_{j}\nabla\phi u_{j}\mathrm{d}x\\ &=\int_{\Omega}\phi \,d\sigma+\lambda\int_{\Omega}|u|^{r}\phi \mathrm{d}x+\int_{\partial\Omega}ug(x,u)\phi \mathrm{d}S-a(x)\int_{\Omega}|u|^{p}\phi \mathrm{d}x-\int_{\Omega}\phi d\mu. \end{align*} Now, by H\"{o}lder inequality and weak convergence, we obtain \begin{align*} 0&\leq\lim_{j\to \infty} |\int_{\Omega}|\nabla u_{j}|^{p-2}\nabla u_{j}\nabla\phi u_{j}\mathrm{d}x|\\ &\leq \lim_{j\to \infty} (\int_{\Omega}|\nabla u_{j}|^{p}\mathrm{d}x)^{\frac{p-1}{p}}(\int_{\Omega}|\nabla\phi|^{p}| u_{j}|^{p}\mathrm{d}x)^{\frac{1}{p}}\\ &\leq C(\int_{B(x_{k},2\varepsilon)\cap \Omega}|\nabla\phi|^{p}| u|^{p}\mathrm{d}x)^{\frac{1}{p}}\\ &\leq C(\int_{B(x_{k},2\varepsilon)\cap \Omega}|\nabla\phi|^{N}\mathrm{d}x)^{\frac{1}{N}}(\int_{B(x_{k},2\varepsilon)\cap \Omega}| u|^{\frac{Np}{N-p}}\mathrm{d}x)^{\frac{N-p}{Np}}\\ &\leq C(\int_{B(x_{k},2\varepsilon)\cap \Omega}| u|^{\frac{Np}{N-p}}\mathrm{d}x)^{\frac{N-p}{Np}}\to 0 \quad \text{as}\quad\varepsilon \to 0. \end{align*} Then from \eqref{e19} we have \begin{align*} %20 &\lim_{\varepsilon\to 0}\big[\int_{\Omega}\phi \,d\sigma+\lambda\int_{\Omega}| u|^{r}\phi \mathrm{d}x+\int_{\partial\Omega}ug(x,u)\phi \mathrm{d}S-a(x)\int_{\Omega}| u|^{p}\phi \mathrm{d}x-\int_{\Omega}\phi d\mu\big]\\ &= \sigma_{k}-\mu_{k}=0. \end{align*} Then from Lemma \ref{lem4.1}, we have that $(\sigma_{k})^{p/p^*}S\leq \mu_{k}$. Therefore by the above equality, \begin{equation*} (\sigma_{k})^{p/p^*}S\leq \sigma_{k}. \end{equation*} Then, either $\sigma_{k}=0$ or \begin{equation*} %21 \sigma_{k}\geq S^{\frac{p*}{p*-p}}. \end{equation*} If this inequality occurs for some $k_{0}$, then, from the fact that $\{u_{j}\}$ is a $(PS)_{c}$ sequence and from (G2) we obtain \begin{align*} c&=\lim_{j\to \infty} \varphi(u_{j}) = \lim_{j\to \infty} \varphi(u_{j})-\frac{1}{p}\langle\varphi'(u_{j});u_{j}\rangle\\ &\geq (\frac{1}{p}-\frac{1}{p^{*}})\int_{\Omega}| u|^{p^{*}}\mathrm{d}x +(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}+\lambda (\frac{1}{p}-\frac{1}{r})\int_{\Omega}| u|^{r}\mathrm{d}x\\ &\geq (\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}, \end{align*} which contradicts our hypothesis. Since $c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}$, it follows that $$\int_{\Omega}|u_{j}|^{p^{*}} \mathrm{d}x\to \int_{\Omega}|u|^{p^{*}}\mathrm{d}x,$$ so we have $u_{j}\to u$ in $L^{p^{*}}(\Omega)$. Now the proof is complete with the continuity of the operator $A^{-1}$. \end{proof} \subsection*{Proof of Theorem \ref{thm1.3}} We want to obtain our result by using mountain pass theorem. First from the assumption (G1) and (G4), we have $$\label{e22} G(x,u)\leq \varepsilon|u|^{p}+C|u|^{z},$$ where $\varepsilon\to 0$ as $|u|\to 0$. From the Sobolev embedding theorem and Sobolev trace inequality, we have \begin{align*} \varphi(u) &\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}| u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}| u|^{r}\mathrm{d}x-\varepsilon\int_{\partial\Omega}|u|^{p}\mathrm{d}S-C_{1}\int_{\partial\Omega}|u|^{z}\mathrm{d}S\\ &\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}| u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}| u|^{r}\mathrm{d}x-\varepsilon C_{2}\|u\|_{1,p}^{p}- C_{3}\|u\|_{1,p}^{z}\\ &\geq (\frac{1}{p}-\varepsilon C_{2})\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}| u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}| u|^{r}\mathrm{d}x- C_{3}\|u\|_{1,p}^{z}\\ &\geq (\frac{1}{p}-\varepsilon C_{2})\|u\|_{1,p}^{p}-\frac{1}{p^{*}}S^{p^{*}}\|u\|_{1,p}^{p^{*}}-\frac{\lambda}{r}C_{4}\|u\|_{1,p}^{r}- C_{3}\|u\|_{1,p}^{z}. \end{align*} Choose $\varepsilon>0$ sufficiently small such that $\frac{1}{p}-\varepsilon C_{2}>0$ and let $$g(t)=(\frac{1}{p}-\varepsilon C_{2})t^{p}-\frac{1}{p^{*}}S^{p^{*}}t^{p^{*}} -\frac{\lambda}{r}C_{4}t^{r}-C_{3}t^{z},$$ it is easy to check that $g(R)>r>0$ for some $R$ sufficiently small since $p<\min\{r, p^{*}, z\}$. On the other hand, since $p<\min\{r, p^{*}, z\}$, so for fixed $\omega\in {W^{1,p}(\Omega)}$ with $\omega|_{\Omega}\neq 0$, we have $\lim_{t\to \infty} \varphi(t\omega)=-\infty$. Then there exists $v_{0}\in W^{1,p}(\Omega)$ such that $\|v_{0}\|_{1,p}>R$ and $\varphi(v_{0})0$ such that $\sup_{t>0}\varphi(t\omega)=h(t_{\lambda})$. Differentiating $h$, we obtain $$\label{e24} 0=h'(t_{\lambda})=t_{\lambda}^{p-1}\|\omega\|_{1,p}^{p}- t_{\lambda}^{p^{*}-1}-t_{\lambda}^{r-1}\lambda|\omega|_{r}^{r}- \int_{\partial\Omega}g(x,t_{\lambda}\omega)\omega \mathrm{d}S.$$ From assumptions (G1) and (G4), we obtain \begin{align*} |\int_{\partial\Omega}g(x,t_{\lambda}\omega)\omega \mathrm{d}S| &\leq \int_{\partial\Omega}|g(x,t_{\lambda}\omega)||\omega|\mathrm{d}S\\ &\leq \varepsilon t_{\lambda}^{p-1}\int_{\partial\Omega}|\omega|^{p} \mathrm{d}S+C_{1}t_{\lambda}^{z-1}\int_{\partial\Omega}|\omega|^{z} \mathrm{d}S\\ &=\varepsilon t_{\lambda}^{p-1}|\omega|_{L^{p}(\partial\Omega)}^{p} +C_{1}t_{\lambda}^{z-1}|\omega|_{L^{z}(\partial\Omega)}^{z}. \end{align*} From \eqref{e24}, $$t_{\lambda}^{p-1}\|\omega\|_{1,p}^{p}- t_{\lambda}^{p^{*}-1}-t_{\lambda}^{r-1}\lambda|\omega|_{r}^{r}-\varepsilon t_{\lambda}^{p-1}|\omega|_{L^{p}(\partial\Omega)}^{p} -C_{1}t_{\lambda}^{z-1}|\omega|_{L^{z}(\partial\Omega)}^{z}\leq 0.$$ Then $$\label{e25} t_{\lambda}^{p^{*}-p}+t_{\lambda}^{r-p}\lambda|\omega|_{r}^{r} +C_{3}t_{\lambda}^{z-p}\|\omega\|_{1,p}^{z}\leq (1-\varepsilon C_{2})\|\omega\|_{1,p}^{p}.$$ Hence, $t_{\lambda}\leq C\|\omega\|_{1,p}^{\frac{p}{p^{*}-p}}$. So from \eqref{e25}, $t_{\lambda}^{p^{*}-r}+\lambda|\omega|_{r}^{r}+C_{3}t_{\lambda}^{z-r}\|\omega\|_{1,p}^{z}\to \infty$ as $\lambda\to \infty$. we obtain $$\label{e26} \underset{\lambda\to \infty}{\lim}t_{\lambda}=0.$$ On the other hand, it is easy to check that if $\lambda>\overline{\lambda}$ we could have $\varphi(t_{\overline{\lambda}}\omega)\geq \varphi(t_{\lambda}\omega)$. So by \eqref{e26}, we get $\lim_{\lambda\to \infty} \varphi(t_{\lambda}\omega)=0$. But this equality means that there exists a constant $\lambda_{0}>0$ such that if $\lambda>\lambda_{0}$, then $\underset{t\geq 0}{\sup}\varphi(t\omega) <(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}$. We choose $v_{0}=t_{0}\omega$ with $t_{0}$ sufficiently large to have $\varphi(t_{0}\omega)<0$. This completes the proof. \subsection{Critical case 2} In this subsection we study $f$ has critical and sublinear terms in problem \eqref{e1}, that is, $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$ with $10$ such that if $0<\lambda<\widetilde{\lambda}$, then $\varphi$ satisfies a local $(PS)_{c}$ condition for $c\leq0$. \end{lemma} \begin{proof} We need only to check the local $(PS)_{c}$ condition. Obviously observe that every $(PS)_{c}$ sequence for $\varphi$ with energy level $c\leq 0$ must be bounded. Therefore by Lemma \ref{lem4.3} if $\lambda$ verifies $$0<\lambda<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}} -K\lambda^{\frac{p^{*}}{p^{*}-r}},$$ then their exists a convergent subsequence. \end{proof} \subsection*{Proof of Theorem \ref{thm1.4}} The proof is analogous to that of Theorem \ref{thm1.2}. Here we use Lemma \ref{lem4.3} and Lemma \ref{lem4.4} respectively to work with the functional $\varphi$ and complete the proof. \begin{thebibliography}{00} \bibitem{AD} D. 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