\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 131, pp. 1--15.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/131\hfil A semilinear elliptic problem] {A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential} \author[T.-F. Wu\hfil EJDE-2006/131\hfilneg] {Tsung-Fang Wu} \address{Tsung-Fang Wu \newline Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan} \email{tfwu@nuk.edu.tw} \date{} \thanks{Submitted July 6, 2006. Published October 17, 2006} \thanks{Partially supported by the National Science Council of Taiwan (R.O.C.)} \subjclass[2000]{35J65, 35J50, 35J55} \keywords{Semilinear elliptic equations; Nehari manifold; \hfill\break\indent Nonlinear boundary condition} \begin{abstract} In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation: \begin{gather*} -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\ \frac{\partial u}{\partial \nu }=g(x)|u| ^{p-2}u \quad \text{on }\partial \Omega , \end{gather*} has at least two nontrivial nonnegative solutions for $\lambda$ is sufficiently small. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} In this paper, we consider the multiplicity of nontrivial nonnegative solutions for the following semilinear elliptic equation $$\begin{gathered} -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\ \frac{\partial u}{\partial \nu }=g(x)|u| ^{p-2}u \quad \text{on }\partial \Omega , \end{gathered} \label{Efg}$$ where $10$, $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary, $\frac{\partial}{\partial \nu }$ is the outer normal derivative and $f,g:\overline{\Omega }\to \mathbb{R}$ are continuous functions which change sign in $\overline{\Omega }$. Associated with \eqref{Efg}, we consider the energy functional $J_{\lambda }$ in $H^{1}(\Omega )$, $J_{\lambda }(u)=\frac{1}{2}\| u\|_{H^{1}}^{2}- \frac{\lambda }{q}\int_{\Omega }f|u|^{q}dx-\frac{1}{p} \int_{\partial \Omega }g|u|^{p}ds.$ where $ds$ is the measure on the boundary and $\| u\|_{H^{1}}^{2}=\int_{\Omega }|\nabla u|^{2}+u^{2}dx$. It is well known that $J_{\lambda }$ is of $C^{1}$ in $H^{1}(\Omega)$ and the solutions of equation \eqref{Efg} are the critical points of the energy functional $J_{\lambda }$. The fact that the number of solutions of equation \eqref{Efg} is affected by the nonlinear boundary conditions has been the focus of a great deal of research in recent years. Garcia-Azorero, Peral and Rossi \cite{GPR} have investigated \eqref{Efg} when $f\equiv g\equiv 1$. They found that there exist positive numbers $\Lambda _{1},\Lambda _{2}$ with $\Lambda _{1}\leq \Lambda _{2}$ such that equation \eqref{Efg} admits at least two positive solutions for $\lambda \in (0,\Lambda _{1})$ and no positive solution exists for $\lambda >\Lambda _{2}$. Also see Chipot-Chlebik-Fila-Shafrir \cite{CCF}, Chipot-Shafrir-Fila \cite{CSF}, Flores-del Pino \cite{FDP}, Hu \cite{H}, Pierrotti-Terracini \cite{PT} and Terraccini \cite{Te} where problems similar to equation \eqref{Efg} have been studied. The purpose of this paper is to consider the multiplicity of nontrivial nonnegative solutions of equation \eqref{Efg} with sign-changing potential. We prove that equation \eqref{Efg} has at least two nontrivial nonnegative solutions for $\lambda$ is sufficiently small. \begin{theorem}\label{t1} There exists $\lambda _{0}>0$ such that for $\lambda \in (0,\lambda _{0})$, equation \eqref{Efg} has at least two nontrivial nonnegative solutions. \end{theorem} Among the other interesting problems which are similar of equation \eqref{Efg}, Ambro\-setti-Brezis-Cerami \cite{ABC} have investigated the equation $$\begin{gathered} -\Delta u=\lambda |u|^{q-2}u+|u| ^{p-2}u \quad \text{in }\Omega , \\ u=0 \quad \text{on }\partial \Omega , \end{gathered} \label{Elambda}$$ where $10$ such that \eqref{Elambda} admits at least two positive solutions for $\lambda \in (0,\lambda _{0})$, has a positive solution for $\lambda =\lambda _{0}$, and no positive solution for $\lambda >\lambda _{0}$. Actually, Adimurthi-Pacella-Yadava \cite{APY}, Damascelli-Grossi-Pacella \cite{DGP}, Ouyang-Shi \cite{OS} and Tang \cite{Ta} proved that there exists $\lambda _{0}>0$ such that equation \eqref{Elambda} in the unit ball $B^{N}(0;1)$ has exactly two positive solutions for $\lambda \in (0,\lambda _{0})$, has exactly one positive solution for $\lambda =\lambda _{0}$ and no positive solution exists for $\lambda >\lambda _{0}$. Generalizations of the result of equation \eqref{Elambda} were done by Ambrosetti-Azorero-Peral \cite{AAP}, de Figueiredo-Gossez-Ubilla \cite{FGU} and Wu \cite{Wu}. This paper is organized as follows. In section 2, we give some notation and preliminaries. In section 3, we prove that \eqref{Efg} has at least two nontrivial nonnegative solutions for $\lambda$ is sufficiently small. \section{Notation and Preliminaries} Throughout this section, we denote by $S_{p}$, $C_{p}$ the best Sobolev embedding and trace constant for the operators $H^{1}(\Omega ) \hookrightarrow L^{p}(\Omega )$, $H^{1}(\Omega ) \hookrightarrow L^{p}(\partial \Omega )$, respectively. Now, we consider the Nehari minimization problem: For $\lambda >0$, $\alpha _{\lambda }=\inf \{J_{\lambda }(u) : u\in \mathbf{M}_{\lambda }\},$ where $\mathbf{M}_{\lambda }=\{u\in H^{1}(\Omega ) \backslash \{0\}: \langle J_{\lambda }'( u),u\rangle =0\}$. Define $\psi _{\lambda }(u)=\langle J_{\lambda }'( u),u\rangle =\| u\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u|^{q}dx-\int_{\partial \Omega }g|u|^{p}ds.$ Then for $u\in \mathbf{M}_{\lambda }$, $\langle \psi _{\lambda }'(u),u\rangle =2\| u\|_{H^{1}}^{2}-\lambda q\int_{\Omega }f| u|^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds.$ Similarly to the method used in Tarantello \cite{T}, we split $\mathbf{M} _{\lambda }$ into three parts: \begin{gather*} \mathbf{M}_{\lambda }^{+} =\{u\in \mathbf{M}_{\lambda }: \langle \psi _{\lambda }'(u) ,u\rangle >0\}, \\ \mathbf{M}_{\lambda }^{0} =\{u\in \mathbf{M}_{\lambda }: \langle \psi _{\lambda }'(u) ,u\rangle =0\}, \\ \mathbf{M}_{\lambda }^{-} =\{u\in \mathbf{M}_{\lambda }: \langle \psi _{\lambda }'(u) ,u\rangle <0\}. \end{gather*} Then, we have the following results. \begin{lemma}\label{g3} There exists $\lambda _{1}>0$ such that for each $\lambda \in (0,\lambda _{1})$ we have $\mathbf{M}_{\lambda }^{0}=\phi$. \end{lemma} \begin{proof} We consider the following two cases. \noindent Case (I): $u\in \mathbf{M}_{\lambda }$ and $\int_{\partial \Omega }g|u|^{p}ds\leq 0$. We have $\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u|^{p}ds.$ Thus, \begin{align*} \langle \psi _{\lambda }'(u),u\rangle &=2\| u\|_{H^{1}}^{2}-\lambda q\int_{\Omega }f| u|^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds \\ &=(2-q)\| u\|_{H^{1}}^{2}+(q-p) \int_{\partial \Omega }g|u|^{p}ds>0 \end{align*} and so $u\in \mathbf{M}_{\lambda }^{+}$. \noindent Case (II): $u\in \mathbf{M}_{\lambda }$ and $\int_{\partial \Omega }g|u|^{p}ds>0$. Suppose that $\mathbf{M}_{\lambda }^{0}\neq \phi$ for all $\lambda >0$. If $u\in \mathbf{M}_{\lambda }^{0}$, then we have \begin{align*} 0 &=\langle \psi _{\lambda }'(u),u\rangle =2\| u\|_{H^{1}}^{2}-\lambda q\int_{\Omega }f|u|^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds \\ &=(2-q)\| u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|u|^{p}ds. \end{align*} Thus, $$\| u\|_{H^{1}}^{2}=\frac{p-q}{2-q}\int_{\partial \Omega }g|u|^{p}ds \label{1}$$ and $$\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u| ^{p}ds=\frac{p-2}{2-q}\int_{\partial \Omega }g|u|^{p}ds. \label{2}$$ Moreover, \begin{align*} (\frac{p-2}{p-q})\| u\|_{H^{1}}^{2} &= \| u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u|^{p}ds\\ &=\lambda \int_{\Omega }f|u|^{q}dx \\ &\leq \lambda \| f\|_{L^{p^{\ast }}}\| u\|_{L^{p}}^{q} \\ &\leq \lambda \| f\|_{L^{p^{\ast }}}S_{p}^{q}\| u\|_{H^{1}}^{q}, \end{align*} where $p^{\ast }=\frac{p}{p-q}$. This implies $$\| u\|_{H^{1}}\leq \Big[ \lambda (\frac{p-q}{p-2} )\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{1/(2-q)}. \label{4}$$ Let $I_{\lambda }:\mathbf{M}_{\lambda }\to \mathbb{R}$ be given by $I_{\lambda }(u)=K(p,q)\Big(\frac{\| u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega }g|u|^{p}ds}\Big)^{1/(p-2)}-\lambda \int_{\Omega }f|u|^{q}dx,$ where $K(p,q)=(\frac{2-q}{p-q})^{(p-1)/(p-2)} (\frac{p-2}{2-q})$. Then $I_{\lambda }(u)=0$ for all $u\in \mathbf{M}_{\lambda }^{0}$. Indeed, from $(\ref{1})$ and $(\ref{2})$ it follows that for $u\in \mathbf{M}_{\lambda }^{0}$ we have \begin{aligned} I_{\lambda }(u) &= K(p,q)\Big(\frac{\| u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega }g|u|^{p}ds}\Big)^{1/(p-1)}-\lambda \int_{\Omega }f|u|^{q}dx \\ &= \big(\frac{2-q}{p-q}\big)^{\frac{p}{p-1}}(\frac{p-2}{2-q}) \Big(\frac{(\frac{p-q}{2-q})^{p-1} \big(\int_{\partial \Omega }g|u|^{p}ds\big)^{p-1}}{ \int_{\partial \Omega }g|u|^{p}ds}\Big)^{\frac{1}{p-2}}\\ &\quad -\frac{p-2}{2-q}\int_{\partial \Omega }g|u|^{p}ds = 0. \end{aligned} \label{5} However, by \eqref{4}, the H\"{o}lder and Sobolev trace inequality, for $u\in \mathbf{M}_{\lambda }^{0}$ \begin{align*} I_{\lambda }(u) &\geq K(p,q)\Big(\frac{ \| u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega }g|u|^{p}ds}\Big)^{1/(p-2)}-\lambda S_{p}^{q}\| f\|_{L^{p^{\ast }}}\| u\|_{H^{1}}^{q} \\ &\geq \| u\|_{H^{1}}^{q}\Big(K(p,q)\Big( \frac{\| u\|_{H^{1}}^{2(p-1)}}{ C_{p}^{p}\| g\|_{\infty }\| u\| _{H^{1}}^{p+q(p-2)}}\Big)^{1/(p-2)}-\lambda S_{p}^{q}\| f\|_{L^{p^{\ast }}}\Big)\\ &\geq \| u\|_{H^{1}}^{q}\big\{K(p,q)C_{p}^{ \frac{p}{2-p}}\lambda ^{\frac{1-q}{2-q}}\Big[ \big(\frac{p-q}{p-2}\big) \| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{\frac{1-q}{2-q}} -\lambda S_{p}^{q}\| f\|_{L^{p^{\ast }}}\big\}. \end{align*} This implies that for $\lambda$ sufficiently small we have $I_{\lambda}(u)>0$ for all $u\in \mathbf{M}_{\lambda }^{0}$, this contradicts \eqref{5}. Thus, we can conclude that there exists $\lambda _{1}>0$ such that for $\lambda \in (0,\lambda _{1})$, we have $\mathbf{M}_{\lambda }^{0}=\phi$. \end{proof} By Lemma \ref{g3}, for $\lambda \in (0,\lambda _{1})$ we write $\mathbf{M}_{\lambda }=\mathbf{M}_{\lambda }^{+}\cup \mathbf{M}_{\lambda }^{-}$ and define $\alpha _{\lambda }^{+}=\inf_{u\in \mathbf{M}_{\lambda }^{+}}J_{\lambda }(u);\quad\text{}\alpha _{\lambda }^{-}(\Omega ) =\inf_{u\in \mathbf{M}_{\lambda }^{-}}J_{\lambda }(u).$ The following lemma shows that the minimizers on $\mathbf{M}_{\lambda }$ are usually'' critical points for $J_{\lambda }$. \begin{lemma} \label{g2} For $\lambda \in (0,\lambda _{1})$. If $u_{0}$ is a local minimizer for $J_{\lambda }$ on $\mathbf{M}_{\lambda }$, then $J_{\lambda }'(u_{0})=0$ in $H^{\ast }(\Omega )$. \end{lemma} \begin{proof} If $u_{0}$ is a local minimizer for $J_{\lambda }$ on $\mathbf{M}_{\lambda}$, then $u_{0}$ is a solution of the optimization problem $\text{minimize }J_{\lambda }(u)\quad\text{subject to }\psi_{\lambda }(u)=0.$ Hence, by the theory of Lagrange multipliers, there exists $\theta \in \mathbb{R}$ such that $J_{\lambda }'(u_{0})=\theta \psi _{\lambda }'(u_{0})\quad \text{in }H^{\ast }(\Omega ).$ Thus, $$\langle J_{\lambda }'(u_{0}),u_{0}\rangle _{H^{1}}=\theta \langle \psi _{\lambda }'(u_{0}) ,u_{0}\rangle _{H^{1}}. \label{6}$$ By Lemma \ref{g3}, $u_{0}\in \mathbf{M}_{\lambda }^{+}\cup \mathbf{M} _{\lambda }^{-}$, we have $\langle \psi _{\lambda }'(u_{0}),u_{0}\rangle _{H^{1}}\neq 0$ and so by \eqref{6} $\theta =0$. This completes the proof. \end{proof} \begin{lemma} \label{g1} \begin{itemize} \item[(i)] If $u\in \mathbf{M}_{\lambda }^{+}$, then $\int_{\Omega }f|u|^{q}dx>0$; \item[(ii)] If $u\in \mathbf{M}_{\lambda }^{-}$, then $\int_{\partial \Omega }g|u|^{p}ds>0$. \end{itemize} \end{lemma} \begin{proof} (i) Case (I): $\int_{\partial \Omega}g|u|^{p}ds\leq 0$. We have $\lambda \int_{\Omega }f|u|^{q}dx =\|u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u|^{p}ds>0.$ Case (II): $\int_{\partial \Omega }g|u|^{p}ds>0$. We have $\| u\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u|^{q}dx-\int_{\partial \Omega }g|u|^{p}ds=0$ and $\| u\|_{H^{1}}^{2}>\frac{p-q}{2-q}\int_{\partial \Omega }g|u|^{p}ds.$ Thus, $\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u| ^{p}ds>\frac{p-2}{2-q}\int_{\partial \Omega }g|u| ^{p}ds>0.$ (ii) Since $(2-q)\| u\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega }g|u|^{p}ds=\langle \psi _{\lambda }'(u) ,u\rangle <0.$ It follows that $\int_{\partial \Omega }g|u| ^{p}ds>0$. This completes the proof. \end{proof} For each $u\in \mathbf{M}_{\lambda }^{-}$, we write $t_{\rm max}=\Big(\frac{(2-q)\| u\|_{H^{1}}^{2} }{(p-q)\int_{\partial \Omega }g|u|^{p}ds} \Big)^{1/(p-2)}<1.$ Then we have the following lemma. \begin{lemma} \label{g4} Let $p^{\ast }=\frac{p}{p-q}$ and $\lambda _{2}=(\frac{p-2}{p-q})(\frac{2-q}{p-q})^{\frac{2-q}{p-2}} C_{p}^{\frac{p(2-q)}{2-p}}S_{p}^{-q}\| f\|_{L^{p^{\ast}}}^{-1}$. Then for each $u\in \mathbf{M}_{\lambda }^{-}$ and $\lambda \in(0,\lambda _{2})$, we have \begin{itemize} \item[(i)] if $\int_{\Omega }f|u|^{q}dx\leq 0$, then $J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(tu)>0;$ \item[(ii)] if $\int_{\Omega }f|u|^{q}dx>0$, then there is a unique $0t_{\rm max}$ such that $h(t^{-})=\lambda\int_{\Omega }f|u|^{q}dx$ and $h'(t^{-})<0$. Now, \begin{align*} &(2-q)\| t^{-}u\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega }|t^{-}u|^{p}ds \\ &= (t^{-})^{1+q}\Big[ (2-q)(t^{-}) ^{1-q}\| u\|_{H^{1}}^{2}-(p-q)( t^{-})^{p-q-1}\int_{\partial \Omega }g|u|^{p}ds\Big] \\ &= (t^{-})^{1+q}h'(t^{-})<0, \end{align*} and \begin{align*} &\langle J_{\lambda }'(t^{-}u),t^{-}u\rangle \\ &= (t^{-})^{2}\| u\|_{H^{1}}^{2}-( t^{-})^{q}\lambda \int_{\Omega }f|u| ^{q}dx-(t^{-})^{p}\int_{\partial \Omega }g|u|^{p}ds \\ &= (t^{-})^{q}\Big[ h(t^{-})-\lambda \int_{\Omega }f|u|^{q}dx\Big] =0. \end{align*} Thus, $t^{-}u\in \mathbf{M}_{\lambda }^{-}$ or $t^{-}=1$. Since for $t>t_{\rm max}$, we have \begin{gather*} (2-q)\| tu\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|tu|^{p}ds < 0, \\ \frac{d^{2}}{dt^{2}}J_{\lambda }(tu) < 0, \\ \frac{d}{dt}J_{\lambda }(tu)=t\| u\| _{H^{1}}^{2}-\lambda t^{q-1}\int_{\Omega }f|u| ^{q}dx-t^{p-1}\int_{\partial \Omega }g|u|^{p}ds=0 \quad \text{for }t=t^{-}. \end{gather*} Thus, $J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }( tu)$. Moreover, $J_{\lambda }(u)\geq J_{\lambda }(tu)\geq \frac{ t^{2}}{2}\| u\|_{H^{1}}^{2}-\frac{t^{p}}{p}\int_{\partial \Omega }g|u|^{p}ds\quad\text{for all }t\geq 0.$ By routine computations, $g(t)=\frac{t^{2}}{2}\| u\|_{H^{1}}^{2}-\frac{t^{p}}{p}\int_{\partial \Omega }g|u|^{p}ds$ achieves its maximum at $t_{0}=(\| u\|_{H^{1}}^{2}/\int_{\partial \Omega }g|u| ^{p}ds)^{1/(p-2)}$. Thus, $J_{\lambda }(u)\geq \frac{p-2}{2p}\Big(\frac{\| u\|_{H^{1}}^{p}}{\int_{\partial \Omega }g|u| ^{p}ds}\Big)^{\frac{2}{p-2}}>0.$ (ii): $\int_{\Omega }f|u|^{q}dx>0$. By (\ref{12}) and \begin{align*} h(0)&= 0<\lambda \int_{\Omega }f|u| ^{q}dx\leq \lambda \| f\|_{L^{p^{\ast }}}S_{p}^{q}\| u\|_{H^{1}}^{q} \\ &< \| u\|_{H^{1}}^{q}(\frac{p-2}{p-q})( \frac{2-q}{p-q})^{\frac{2-q}{p-2}}C_{p}^{\frac{p(2-q)}{ 2-p}}\\ &\leq h(t_{\rm max})\quad \text{for }\lambda \in (0,\lambda _{2}), \end{align*} there are unique $t^{+}$ and $t^{-}$ such that $00>h'(t^{-}). \end{gather*} We have$t^{+}u\in \mathbf{M}_{\lambda }^{+}$,$t^{-}u\in \mathbf{M}_{\lambda }^{-}$, and$J_{\lambda }(t^{-}u)\geq J_{\lambda }(tu)\geq J_{\lambda }(t^{+}u)$for each$t\in [ t^{+},t^{-}] $and$J_{\lambda }(t^{+}u)\leq J_{\lambda }(tu)$for each$t\in [ 0,t^{+}]$. Thus,$t^{-}=1$and $J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(tu) ,J_{\lambda }(t^{+}u)=\inf_{0\leq t\leq t_{\rm max}}J_{\lambda }(tu).$ This completes the proof.\end{proof} Next, we establish the existence of nontrivial nonnegative solutions for the equation $$\begin{gathered} -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\ u=0 \quad \text{on }\partial \Omega , \end{gathered} \label{13}$$ Associated with equation \eqref{13}, we consider the energy functional $K_{\lambda }(u)=\frac{1}{2}\| u\|_{H^{1}}^{2}- \frac{\lambda }{q}\int_{\Omega }f|u|^{q}dx$ and the minimization problem $\beta _{\lambda }=\inf \{K_{\lambda }(u) : u\in \mathbf{N}_{\lambda }\},$ where$\mathbf{N}_{\lambda }=\{u\in H_{0}^{1}(\Omega ) \backslash \{0\}: \langle K_{\lambda }'(u),u\rangle =0\}$. Then we have the following result. \begin{theorem} \label{m4} Suppose that$\lambda >0$. Then equation \eqref{13} has a nontrivial nonnegative solution$v_{\lambda }$with$K_{\lambda }(v_{\lambda })=\beta _{\lambda }<0$. \end{theorem} \begin{proof} First, we need to show that$K_{\lambda }$is bounded below on$\mathbf{N}_{\lambda }$and$\beta _{\lambda }<0$. Then for$u\in \mathbf{N}_{\lambda}$, $\| u\|_{H^{1}}^{2}=\lambda \int_{\Omega }f|u|^{q}dx\leq \lambda \| f\|_{L^{q^{\ast }}}S_{p}^{-\frac{q}{2}}\| u\|_{H^{1}}^{q}.$ where$p^{\ast }=\frac{p}{p-q}. This implies $$\| u\|_{H^{1}}\leq (\lambda \| f\| _{L^{p^{\ast }}}S_{p}^{-\frac{q}{2}})^{\frac{1}{2-q}}. \label{11}$$ Hence, \begin{align*} K_{\lambda }(u)&= \frac{1}{2}\| u\|_{H^{1}}- \frac{\lambda }{q}\int_{\Omega }f|u|^{q}dx \\ &= \big(\frac{1}{2}-\frac{1}{q}\big)\| u\|_{H^{1}}^{2} \\ &\leq \big(\frac{1}{2}-\frac{1}{q}\big)\big(\lambda \| f\|_{L^{p^{\ast }}}S_{p}^{-\frac{q}{2}}\big)^{\frac{1}{2-q}} \end{align*} for allu\in \mathbf{N}_{\lambda }$and$\beta _{\lambda }<0$. Let$\{ v_{n}\}$be a minimizing sequence for$K_{\lambda }$on$\mathbf{N} _{\lambda }$. Then by$(\ref{11})$and the compact imbedding theorem, there exist a subsequence$\{v_{n}\}$and$v_{\lambda } $in$H_{0}^{1}(\Omega )$such that $v_{n}\rightharpoonup v_{\lambda }\quad\text{weakly in }H_{0}^{1}(\Omega )$ and $$v_{n}\to v_{\lambda }\quad\text{strongly in }L^{q}(\Omega ). \label{10}$$ First, we claim that$\int_{\Omega }f|v_{\lambda }| ^{q}dx>0$. If not, $K_{\lambda }(v_{n})\geq \frac{1}{2}\| v_{\lambda }\|_{H^{1}}^{2}-\frac{\lambda }{q}\int_{\Omega }f|v_{\lambda }|^{q}dx+o(1) \geq \frac{1}{2}\| v_{\lambda }\|_{H^{1}}^{2}+o(1),$ this contradicts$K_{\lambda }(v_{n})\to \beta_{\lambda }(\Omega )<0$as$n\to \infty $. Thus,$\int_{\Omega }f|v_{\lambda }|^{q}dx>0$. In particular,$v_{\lambda }\not\equiv 0$. Now, we prove that$v_{n}\to v_{\lambda }$strongly in$H_{0}^{1}(\Omega )$. Suppose otherwise, then$\|v_{\lambda }\|_{H^{1}}<\liminf_{n\to \infty }\| v_{n}\|_{H^{1}}$and so $\| v_{\lambda }\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|v_{\lambda }|^{q}dx<\liminf_{n\to \infty } \Big(\| v_{n}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|v_{n}|^{q}dx\Big)=0.$ Since$\int_{\Omega }f|v_{\lambda }|^{q}dx>0$, there is a unique$t_{0}\neq 1$such that$t_{0}v_{\lambda }\in \mathbf{N}_{\lambda }. Thus, $t_{0}v_{n}\rightharpoonup t_{0}v_{\lambda }\quad\text{weakly in } H_{0}^{1}(\Omega ).$ Moreover, K_{\lambda }(t_{0}v_{\lambda })0 and a differentiable function \xi :B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R}^{+} such that \xi (0)=1, the function \xi (v)(u-v)\in \mathbf{M}_{\lambda } and $$\langle \xi '(0),v\rangle =\frac{ 2\int_{\Omega }\nabla u\nabla vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial \Omega }g|u| ^{p-2}uvds}{(2-q)\| u\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega }g|u|^{p}ds} \label{21}$$ for all v\in H^{1}(\Omega ). \end{lemma} \begin{proof} For u\in \mathbf{M}_{\lambda }, define a function F:\mathbb{R}\times H^{1}(\Omega )\to \mathbb{R} by \begin{align*} F_{u}(\xi ,w) &= \langle J_{\lambda }'(\xi (u-w)),\xi (u-w)\rangle \\ &= \xi ^{2}\int_{\Omega }|\nabla (u-w)|^{2}+(u-w)^{2}dx -\xi ^{q}\lambda \int_{\Omega }f|u-w|^{q}dx \\ &\quad -\xi ^{p}\int_{\partial \Omega }g|u-w|^{p}ds. \end{align*} Then F_{u}(1,0)=\langle J_{\lambda }'(u),u\rangle =0 and \begin{align*} \frac{d}{d\xi }F_{u}(1,0) &= 2\| u\|_{H^{1}}^{2}-\lambda q\int_{\partial \Omega }f|u| ^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds \\ &= (2-q)\| u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|u|^{p}ds\neq 0. \end{align*} According to the implicit function theorem, there exist \epsilon >0 and a differentiable function \xi :B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R} such that \xi (0)=1, \[ \langle \xi '(0),v\rangle =\frac{ 2\int_{\Omega }\nabla u\nabla vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial \Omega }g|u| ^{p-2}uvds}{(2-q)\| u\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega }g|u|^{p}ds} and $F_{u}(\xi (v),v)=0\quad\text{for all }v\in B( 0;\epsilon )$ which is equivalent to $\langle J_{\lambda }'(\xi (v)(u-v)),\xi (v)(u-v)\rangle =0 \quad \quad\text{for all }v\in B(0;\epsilon ),$ that is\xi (v)(u-v)\in \mathbf{M}_{\lambda }$. \end{proof} \begin{lemma} \label{m7} For each$u\in \mathbf{M}_{\lambda }^{-}$, there exist$\epsilon>0 $and a differentiable function$\xi ^{-}:B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R}^{+}$such that$\xi ^{-}(0)=1$, the function$\xi ^{-}(v)(u-v)\in \mathbf{M}_{\lambda }^{-}$and $$\langle (\xi ^{-})'(0) ,v\rangle =\frac{2\int_{\Omega }\nabla u\nabla vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial \Omega }g|u|^{p-2}uvds}{(2-q)\| u\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega }g|u|^{p}ds} \label{22}$$ for all$v\in H^{1}(\Omega )$. \end{lemma} \begin{proof} Similar to the argument in Lemma \ref{m6}, there exist$\epsilon >0$and a differentiable function$\xi ^{-}:B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R}$such that$\xi ^{-}( 0)=1$and$\xi ^{-}(v)(u-v)\in \mathbf{M} _{\lambda }$for all$v\in B(0;\epsilon )$. Since $\langle \psi _{\lambda }'(u),u\rangle =(2-q)\| u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|u|^{p}ds<0.$ Thus, by the continuity of the function$\xi ^{-}, we have \begin{align*} &\langle \psi _{\lambda }'(\xi ^{-}(v) (u-v)),\xi ^{-}(v)(u-v) \rangle \\ &= (2-q)\| \xi ^{-}(v)(u-v) \|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega }g|\xi ^{-}(v)(u-v)|^{p}ds <0 \end{align*} if\epsilon $sufficiently small, this implies that$\xi ^{-}(v)(u-v)\in \mathbf{M}_{\lambda }^{-}$. \end{proof} \begin{proposition} \label{m8} Let$\lambda _{0}=\min \{\lambda _{1},\lambda _{2},\frac{p-1 }{p-q}\}$, Then for$\lambda \in (0,\lambda _{0})$: \begin{itemize} \item[(i)] There exists a minimizing sequence$\{u_{n}\} \subset \mathbf{M}_{\lambda }such that \begin{align*} J_{\lambda }(u_{n})&= \alpha _{\lambda }+o(1), \\ J_{\lambda }'(u_{n})&= o(1)\quad \text{in } H^{\ast }(\Omega ); \end{align*} \item[(ii)] there exists a minimizing sequence\{u_{n}\}\subset \mathbf{M}_{\lambda }^{-}such that \begin{align*} J_{\lambda }(u_{n})&= \alpha _{\lambda }^{-}+o(1), \\ J_{\lambda }'(u_{n})&= o(1)\quad \text{in }H^{\ast }(\Omega ). \end{align*} \end{itemize} \end{proposition} \begin{proof} (i) By Lemma \ref{g5} (ii) and the Ekeland variational principle \cite{E}, there exists a minimizing sequence\{u_{n}\}\subset \mathbf{M}_{\lambda }$such that \begin{gather} J_{\lambda }(u_{n})<\alpha _{\lambda }+\frac{1}{n}, \label{16} \\ J_{\lambda }(u_{n})\frac{-pq }{2\lambda (p-q)}\beta _{\lambda }>0. \label{24} Consequently,$u_{n}\neq 0$and putting together \eqref{23}, \eqref{24} and the H\"{o}lder inequality, we obtain \begin{gather} \| u_{n}\|_{H^{1}}>\Big[ \frac{-pq}{2\lambda ( p-q)}\beta _{\lambda }S_{p}^{-q}\| f\|_{L^{p^{\ast }}}^{-1}\Big] ^{1/q} \label{25-1} \\ \| u_{n}\|_{H^{1}}<\Big[ \frac{2(p-q)}{ (p-2)q}\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{1/(2-q)} \label{25-2} \end{gather} Now, we show that $\| J_{\lambda }'(u_{n})\| _{H^{-1}}\to 0\quad \text{as }n\to \infty .$ Applying Lemma \ref{m6} with$u_{n}$to obtain the functions$\xi_{n}:B(0;\epsilon _{n})\to \mathbb{R}^{+}$for some$\epsilon _{n}>0$, such that$\xi _{n}(w)(u_{n}-w) \in \mathbf{M}_{\lambda }$. Choose$0<\rho <\epsilon _{n}$. Let$u\in H^{1}(\Omega )$with$u\not\equiv 0$and let$w_{\rho }=\frac{\rho u}{\| u\|_{H^{1}}}$. We set$\eta _{\rho }=\xi_{n}(w_{\rho })(u_{n}-w_{\rho })$. Since$\eta_{\rho }\in \mathbf{M}_{\lambda }, we deduce from \eqref{17} that $J_{\lambda }(\eta _{\rho })-J_{\lambda }(u_{n}) \geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\|_{H^{1}}$ and by the mean value theorem, we have $\langle J_{\lambda }'(u_{n}),\eta _{\rho}-u_{n}\rangle +o(\| \eta _{\rho }-u_{n}\|_{H^{1}})\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\|_{H^{1}}.$ Thus, \label{18} \begin{aligned} &\langle J_{\lambda }'(u_{n}),-w_{\rho }\rangle +(\xi _{n}(w_{\rho })-1) \langle J_{\lambda }'(u_{n}),(u_{n}-w_{\rho})\rangle \\ &\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\| _{H^{1}}+o(\| \eta _{\rho }-u_{n}\|_{H^{1}}). \end{aligned} Since\xi _{n}(w_{\rho })(u_{n}-w_{\rho })\in \mathbf{M}_{\lambda }$and$(\ref{18})it follows that \begin{align*} &-\rho \langle J_{\lambda }'(u_{n}),\frac{u}{ \| u\|_{H^{1}}}\rangle +(\xi _{n}( w_{\rho })-1)\langle J_{\lambda }'( u_{n})-J_{\lambda }'(\eta _{\rho }),( u_{n}-w_{\rho })\rangle \\ &\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\| _{H^{1}}+o(\| \eta _{\rho }-u_{n}\|_{H^{1}}). \end{align*} Thus, \label{19} \begin{aligned} \langle J_{\lambda }'(u_{n}),\frac{u}{\| u\|_{H^{1}}}\rangle &\leq \frac{\| \eta _{\rho}-u_{n}\|_{H^{1}}}{n\rho } +\frac{o(\| \eta _{\rho}-u_{n}\|_{H^{1}})}{\rho } \\ &\quad +\frac{(\xi _{n}(w_{\rho })-1)}{\rho } \langle J_{\lambda }'(u_{n})-J_{\lambda }'(\eta _{\rho }), (u_{n}-w_{\rho })\rangle . \end{aligned} Since\| \eta _{\rho }-u_{n}\|_{H^{1}}\leq \rho \|\xi _{n}(w_{\rho })\|+\|\xi _{n}(w_{\rho})-1\|\| u_{n}\|_{H^{1}}$and $\lim_{\rho \to 0}\frac{\|\xi _{n}(w_{\rho }) -1\|}{\rho }\leq \| \xi _{n}'(0) \|,$ if we let$\rho \to 0$in$(\ref{19})$for a fixed$n$, then by$(\ref{25-2})$we can find a constant$C>0$, independent of$\rho $, such that $\langle J_{\lambda }'(u_{n}),\frac{u}{\| u\|_{H^{1}}}\rangle \leq \frac{C}{n}(1+\| \xi _{n}'(0)\|).$ The proof will be complete once we show that$\| \xi _{n}'(0)\|$is uniformly bounded in$n$. By \eqref{21}, \eqref{25-2} and the H\"{o}lder inequality, we have $\langle \xi _{n}'(0),v\rangle \leq \frac{ b\| v\|_{H^{1}}}{|(2-q)\| u_{n}\|_{H^{1}}-(p-q)\int_{\partial \Omega }g|u_{n}|^{p}ds|}\quad \text{for some }b>0.$ We only need to show that $$|(2-q)\| u_{n}\|_{H^{1}}-( p-q)\int_{\partial \Omega }g|u_{n}| ^{p}ds|>c \label{26}$$ for some$c>0$and$n$large enough. We argue by contradiction. Assume that there exists a subsequence$\{u_{n}\}$, we have $$(2-q)\| u_{n}\|_{H^{1}}-(p-q) \int_{\partial \Omega }g|u_{n}|^{p}ds=o(1). \label{27}$$ Combining$(\ref{27})$with$(\ref{25-1})$, we can find a suitable constant$d>0$such that $$\int_{\partial \Omega }g|u_{n}|^{p}ds\geq d\quad \text{for n sufficiently large.} \label{28}$$ In addition$(\ref{27})$, and the fact that$u_{n}\in \mathbf{M}_{\lambda }$also give $\lambda \int_{\Omega }f|u_{n}|^{q}dx=\| u_{n}\|_{H^{1}}^{2}-\int_{\partial \Omega }g| u_{n}|^{p}ds=\frac{p-2}{2-q}\int_{\partial \Omega }g| u_{n}|^{p}ds+o(1)$ and $$\| u_{n}\|_{H^{1}}\leq \Big[ \lambda (\frac{p-q}{p-2 })\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{\frac{1}{ 2-q}}+o(1). \label{37}$$ This implies $$I_{\lambda }(u_{n})=K(p,q) \Big(\frac{\| u_{n}\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega }g|u_{n}|^{p}ds}\Big)^{1/(p-2)}-\lambda \int_{\Omega }f|u_{n}|^{q}dx=o(1). \label{29}$$ However, by \eqref{28}, \eqref{37} and$\lambda \in (0,\lambda _{0}), \begin{align*} I_{\lambda }(u_{n}) &\geq K(p,q)\Big(\frac{\| u_{n}\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega }g|u_{n}|^{p}ds}\Big)^{1/(p-2)}-\lambda S_{p}^{q}\| f\|_{L^{p^{\ast }}}\| u_{n}\| _{H^{1}}^{q} \\ &\geq \| u_{n}\|_{H^{1}}^{q}\Big(K(p,q) \Big(\frac{\| u_{n}\|_{H^{1}}^{2(p-1)}}{ C_{p}^{p}\| u_{n}\|_{H^{1}}^{p+q(p-2)}} \Big)^{1/(p-2)}-\lambda S_{p}^{q}\| f\| _{L^{p^{\ast }}}\Big)\\ &\geq \| u_{n}\|_{H^{1}}^{q}\Big\{K(p,q) C_{p}^{\frac{p}{2-p}}\lambda ^{\frac{1-q}{2-q}}\big[ (\frac{p-q}{p-2} )\| f\|_{L^{p^{\ast }}}S_{p}^{q}\big] ^{\frac{1-q}{2-q}} -\lambda \| f\|_{L^{p^{\ast }}}\Big\}. \end{align*} this contradicts \eqref{29}. We get $\langle J_{\lambda }'(u_{n}),\frac{u}{\| u\|_{H^{1}}}\rangle \leq \frac{C}{n}.$ This completes the proof of(i)$. \noindent (ii) Similarly, by using Lemma \ref{m7}, we can prove (ii). We will omit detailed proof here. \end{proof} Now, we establish the existence of a local minimum for$J_{\lambda }$on$\mathbf{M}_{\lambda }^{+}$. \begin{theorem} \label{t2} Let$\lambda _{0}>0$as in Proposition \ref{m8}, then for$\lambda \in (0,\lambda _{0})$the functional$J_{\lambda }$has a minimizer$u_{0}^{+}$in$\mathbf{M}_{\lambda }^{+}$and it satisfies \begin{itemize} \item[(i)]$J_{\lambda }(u_{0}^{+})=\alpha _{\lambda }=\alpha _{\lambda }^{+}$; \item[(ii)]$u_{0}^{+}$is a nontrivial nonnegative solution of equation \eqref{Efg}; \item[(iii)]$J_{\lambda }(u_{0}^{+})\to 0$as$\lambda \to 0$. \end{itemize} \end{theorem} \begin{proof} Let$\{u_{n}\}\subset \mathbf{M}_{\lambda }$be a minimizing sequence for$J_{\lambda }$on$\mathbf{M}_{\lambda }$such that $J_{\lambda }(u_{n})=\alpha _{\lambda }+o(1)\quad \text{and}\quad J_{\lambda }'(u_{n})=o(1)\quad \text{in }H^{\ast }(\Omega ).$ Then by Lemma \ref{g5} and the compact imbedding theorem, there exist a subsequence$\{u_{n}\}$and$u_{0}^{+}\in H^{1}(\Omega )$such that $u_{n}\rightharpoonup u_{0}^{+}\quad\text{weakly in }H^{1}(\Omega ), u_{n}\to u_{0}^{+}\quad\text{strongly in }L^{p}(\partial \Omega )$ and $$u_{n}\to u_{0}^{+}\quad\text{strongly in }L^{q}(\Omega ). \label{7}$$ First, we claim that$\int_{\Omega }f(x)\|u_{0}^{+}\|^{q}dx\neq 0. Suppose otherwise, by \eqref{7} we can conclude that $\int_{\Omega }f|u_{n}|^{q}dx\to \int_{\Omega }f|u_{0}^{+}|^{q}dx=0\quad \text{as }n\to \infty$ and so $\| u_{n}\|_{H^{1}}^{2}=\int_{\partial \Omega }g| u_{n}|^{p}ds+o(1).$ Thus, \begin{align*} J_{\lambda }(u_{n}) &= \frac{1}{2}\| u_{n}\|_{H^{1}}^{2}-\frac{\lambda }{q} \int_{\Omega }f|u_{n}|^{q}dx-\frac{1}{p}\int_{\partial \Omega } g|u_{n}|^{p}ds \\ &= (\frac{1}{2}-\frac{1}{p})\int_{\partial \Omega }g| u_{n}|^{p}ds+o(1)\\ &= (\frac{1}{2}-\frac{1}{p})\int_{\partial \Omega }g| u_{0}^{+}|^{p}ds\quad\text{as }n\to \infty , \end{align*} this contradictsJ_{\lambda }(u_{n})\to \alpha _{\lambda }<0$as$n\to \infty $. Moreover, $o(1)=\langle J_{\lambda }'(u_{n}) ,\phi \rangle =\langle J_{\lambda }'(u_{0}) ,\phi \rangle +o(1)\quad\text{for all }\phi \in H^{1}( \Omega ).$ Thus,$u_{0}^{+}\in \mathbf{M}_{\lambda }$is a nonzero solution of equation \eqref{Efg} and$J_{\lambda }(u_{0}^{+})\geq \alpha _{\lambda }$. Now we prove that$u_{n}\to u_{0}^{+}$strongly in$H^{1}(\Omega )$. Suppose otherwise, then$\| u_{0}^{+}\|_{H^{1}}<\liminf_{n\to \infty } \| u_{n}\|_{H^{1}}and so \begin{align*} &\| u_{0}^{+}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u_{0}^{+}|^{q}dx-\int_{\partial \Omega }g| u_{0}^{+}|^{p}ds \\ &< \liminf_{n\to \infty } \Big(\| u_{n}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f| u_{n}|^{q}dx-\int_{\partial \Omega }g|u_{n}| ^{p}ds\Big)=0, \end{align*} this contradictsu_{0}^{+}\in \mathbf{M}_{\lambda }$. Hence$u_{n}\to u_{0}^{+}$strongly in$H^{1}(\Omega )$and $J_{\lambda }(u_{n})\to J_{\lambda }( u_{0}^{+})=\alpha _{\lambda }\quad\text{as }n\to \infty .$ Moreover, we have$u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$. If not, then$u_{0}^{+}\in \mathbf{M}_{\lambda }^{-}$and by Lemma \ref{g4}, there are unique$t_{0}^{+}$and$t_{0}^{-}$such that$t_{0}^{+}u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$and$t_{0}^{-}u_{0}^{+}\in \mathbf{M}_{\lambda }^{-}$. In particular, we have$t_{0}^{+}0, \] there exists $t_{0}^{+}<\bar{t}\leq t_{0}^{-}$ such that $J_{\lambda }(t_{0}^{+}u_{0}^{+})J_{\lambda }(u_{0}^{+})\geq -\lambda \Big(\frac{( p-q)(2-q)}{2pq}\Big)(\| f\|_{L^{p^{\ast }}}S_{p}^{q})^{\frac{2}{2-q}} \] and so$J_{\lambda }(u_{0}^{+})\to 0$as$\lambda \to 0$. \end{proof} Next, we establish the existence of a local minimum for$J_{\lambda }$on$\mathbf{M}_{\lambda }^{-}$. \begin{theorem} \label{t3} Let$\lambda _{0}>0$as in Proposition \ref{m8}. Then for$\lambda \in (0,\lambda _{0})$the functional$J_{\lambda }$has a minimizer$u_{0}^{-}$in$\mathbf{M}_{\lambda }^{-}$and satisfies \begin{itemize} \item[(i)]$J_{\lambda }(u_{0}^{-})=\alpha _{\lambda }^{-}$; \item[(ii)]$u_{0}^{-}$is a nontrivial nonnegative solution of equation \eqref{Efg}. \end{itemize} \end{theorem} \begin{proof} By Proposition \ref{m8} (ii), there exists a minimizing sequence$\{u_{n}\}$for$J_{\lambda }$on$\mathbf{M}_{\lambda }^{-}$such that $J_{\lambda }(u_{n})=\alpha _{\lambda }^{-}+o(1) \quad\text{and}\quad J_{\lambda }'(u_{n})=o(1) \quad\text{in }H^{\ast }(\Omega ).$ By Lemma \ref{g5} and the compact imbedding theorem, there exist a subsequence$\{u_{n}\}$and$u_{0}^{-}\in H^{1}(\Omega)$such that \begin{gather*} u_{n}\rightharpoonup u_{0}^{-}\quad \text{weakly in }H^{1}(\Omega ),\\ u_{n}\to u_{0}^{-}\quad \text{strongly in }L^{p}(\partial \Omega ),\\ u_{n}\to u_{0}^{-}\quad\text{strongly in }L^{q}(\Omega ). \end{gather*} Since$(2-q)\| u_{n}\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega }g|u_{n}|^{p}ds<0$, by the Sobolev trace inequality there exists$C>0$such that$\int_{\partial \Omega }g|u_{n}|^{p}ds>C. Moreover, $o(1)=\langle J_{\lambda }'(u_{n}) ,\phi \rangle =\langle J_{\lambda }'(u_{0}) ,\phi \rangle +o(1)\quad\text{for all }\phi \in H^{1}( \Omega )$ and \begin{align*} &(2-q)\| u_{0}\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega }g|u_{0}|^{p}ds \\ &\leq \liminf_{n\to \infty } \Big((2-q) \| u_{n}\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega }g|u_{n}|^{p}ds\Big) \leq 0. \end{align*} Thus,u_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$is a nonzero solution of equation \eqref{Efg}. Now we prove that$u_{n}\to u_{0}^{-}$strongly in$H^{1}(\Omega )$. Suppose otherwise, then$\| u_{0}^{-}\|_{H^{1}}<\liminf_{n\to \infty } \| u_{n}\|_{H^{1}}and so \begin{align*} &\| u_{0}^{-}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u_{0}^{-}|^{q}dx-\int_{\partial \Omega }g| u_{0}^{-}|^{p}ds \\ &< \liminf_{n\to \infty } \Big(\| u_{n}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f| u_{n}|^{q}dx-\int_{\partial \Omega }g|u_{n}| ^{p}ds\Big)=0, \end{align*} this contradictsu_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$. Hence$u_{n}\to u_{0}^{-}$strongly in$H^{1}(\Omega )$. This implies $J_{\lambda }(u_{n})\to J_{\lambda }( u_{0}^{-})=\alpha _{\lambda }^{-}\quad\text{as }n\to \infty .$ Since$J_{\lambda }(u_{0}^{-})=J_{\lambda }(|u_{0}^{-}|)$and$|u_{0}^{-}|\in \mathbf{M}_{\lambda }^{-}$, by Lemma \ref{g2} we may assume that$u_{0}^{-}$is a nontrivial nonnegative solution of equation \eqref{Efg}. \end{proof} Now, we complete the proof of Theorem \ref{t1}. By Theorems \ref{t2}, \ref{t3}, we obtain equation \eqref{Efg} has two nontrivial nonnegative solutions$u_{0}^{+}$and$u_{0}^{-}$such that$u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$and$u_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$. Since$\mathbf{M}_{\lambda }^{+}\cap \mathbf{M}_{\lambda }^{-}=\phi $, this implies that$u_{0}^{+}$and$u_{0}^{-}$are different. \begin{thebibliography}{00} \bibitem{APY} Adimurthi, F. Pacella, and L. Yadava; \emph{On the number of positive solutions of some semilinear Dirichlet problems in a ball}, Diff. Int. 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