\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 47, 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} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/47\hfil Multiplicity of positive solutions] {Multiplicity of positive solutions for a Navier boundary-value problem involving the $p$-biharmonic with critical exponent} \author[Y. Shen, J. Zhang\hfil EJDE-2011/47\hfilneg] {Ying Shen, Jihui Zhang} % in alphabetical order \address{Ying Shen \newline Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210046, Jiangsu, China} \email{shenying99@126.com} \address{Jihui Zhang \newline Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210046, Jiangsu, China} \email{jihuiz@jlonline.com} \thanks{Submitted December 18, 2010. Published April 6, 2011.} \subjclass[2000]{35J40, 35J67} \keywords{$p$-biharmonic system; Navier condition; Nehari manifold; \hfill\break\indent critical exponent} \begin{abstract} By using the Nehari manifold and variational methods, we prove that a $p$-biharmonic system has at least two positive solutions when the pair the parameters satisfy certain inequality. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we consider the multiplicity results of positive solutions of the semilinear p-biharmonic system $$\label{e1.1} \begin{gathered} \Delta(|\Delta u|^{p-2}\Delta u)=\frac{1}{p^{**}} \frac{\partial F(x,u,v)}{\partial u}+\lambda |u|^{q-2}u \quad\text{in } \Omega, \\ \Delta(|\Delta v|^{p-2}\Delta v) =\frac{1}{p^{**}}\frac{\partial F(x,u,v)}{\partial v} +\mu |v|^{q-2}v \quad\text{in } \Omega, \\ u>0,\quad v>0 \quad\text{in } \Omega, \\ u=v=\Delta u=\Delta v=0 \quad \text{on }\partial \Omega, \end{gathered}$$ where $x_0\in \Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $F\in C^{1}(\overline{\Omega}\times (\mathbb{R}^+)^2, \mathbb{R}^+)$ is positively homogeneous of degree $p^{**}=\frac{pN}{N-2p}$ which is the Sobolev critical exponent; that is, $F(x,tu,tv)=t^{p^{**}}F(x,u,v)$ $(t>0)$ holds for all $(x,u,v)\in \overline{\Omega}\times (\mathbb{R}^+)^2$, $(\frac{\partial F(x,u,v)}{\partial u}, \frac{\partial F(x,u,v)}{\partial v})=\nabla F$. We assume that $10$, $\mu>0$. In recent years, there have been many article concerned with the existence and multiplicity of positive solutions for $p$-biharmonic elliptic problems. Results relating to these problems can be found in \cite{c1,d1,e2,l1,t1,t2,w1,w2} and the references therein. Brown and Wu \cite{b2} considered the semilinear elliptic system $$\label{e1.2} \begin{gathered} -\Delta u+u=\frac{\alpha}{\alpha+\beta}f(x)|u|^{\alpha-2}u|v|^{\beta} \quad\text{in } \Omega, \\ -\Delta v+v=\frac{\beta}{\alpha+\beta}f(x)|u|^{\alpha}|v|^{\beta-2}v \quad\text{in } \Omega, \\ \frac{\partial u}{\partial n}=\lambda g(x)|u|^{q-2}u,\quad \frac{\partial v}{\partial n}=\mu h(x)|v|^{q-2}v\quad \text{on }\partial \Omega. \end{gathered}$$ where $\alpha>1$, $\beta>1$ satisfying $2<\alpha+\beta<2^{*}$ and the weight functions $f,g,h$ are satisfying the following conditions: \begin{itemize} \item[(A)] $f\in C(\overline{\Omega})$ with $\|f\|_{\infty}=1$ and $f^{+}=\max\{f,0\}\not \equiv 0$; \item[(B)] $g,h\in C(\partial \Omega)$ with $\|g\|_{\infty}=\|h\|_{\infty}=1,\ g^{\pm}=\max\{\pm g,0\}\not \equiv 0$ and $h^{\pm}=\max\{\pm h,0\}\not \equiv 0$. \end{itemize} They showed that \eqref{e1.2} has at least two negative solutions if the pair of the parameters $(\lambda, \mu)$ belongs to a certain subset of $\mathbb{R}^2$. Recently, Hsu \cite{h1} considered the case $F(x,u,v)=2|u|^{\alpha}|v|^{\beta},\alpha>1,\beta>1$ satisfying $\alpha+\beta=p^{*}$; i.e., the elliptic system: $$\label{e1.3} \begin{gathered} -\Delta_{p} u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta} +\lambda |u|^{q-2}u \quad\text{in } \Omega, \\ -\Delta_{p} v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v +\mu |v|^{q-2}v \quad\text{in } \Omega, \\ u=v=0\quad \text{on }\partial \Omega. \end{gathered}$$ By variational methods, he proved that \eqref{e1.2} has at least two positive solutions if the pair of the parameters $(\lambda,\mu)$ belongs to a certain subset of $\mathbb{R}^2$. In this article, we give a simple variational method which is similar to the fibering method'' of Pohozaev's ( see \cite{d2,b4}) to prove the existence of at least two positive solutions of problem \eqref{e1.1}. Throughout this paper, we let $S$ be the best Sobolev embedding constant defined by $S=\inf_{u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\Delta u|^pdx}{(\int_{\Omega}|u|^{p^{**}}dx)^{\frac{p}{p^{**}}}},$ and let \begin{gather*} C(p,q,N,K,S,|\Omega|)=(\frac{p-q}{K(p^{**}-q)})^{\frac{p}{p^{**}-q}} (\frac{p^{**}-q}{p^{**}-p}|\Omega|^{\frac{p^{**}-q}{p^{**}}}) ^{-\frac{p}{p-q}}S^{\frac{N}{2p}+\frac{q}{p-q}},\\ C_0=(\frac{q}{p})^{\frac{p}{p-q}}C(p,q,N,K,S,|\Omega|). \end{gather*} For our results, we need the following assumptions: \begin{itemize} \item[(F1)] $F:\overline{\Omega}\times \mathbb{R}^+\times \mathbb{R}^+ \to\mathbb{R}^+$ is a $C^{1}$ function and $F(x,tu,tv)=t^{p^{**}}F(x,u,v)$ for all $t>0$ and $x\in \overline{\Omega}$, $(u,v)\in (\mathbb{R}^+)^2$; \item[(F2)] $F(x,u,0)=F(x,0,v)=\frac{\partial F}{\partial u}(x,u,0)=\frac{\partial F}{\partial v}(x,0,v)=0$, where $u,v\in \mathbb{R}^+$; \item[(F3)] $\frac{\partial F(x,u,v)}{\partial u}, \frac{\partial F(x,u,v)}{\partial v}$ are strictly increasing functions about $u$ and $v$ for all $u>0$, $v>0$. \end{itemize} From assumption (F1), we have the so-called Euler identity $$\label{e1.4} (u,v)\cdot\nabla F(x,u,v)=p^{**} F(x,u,v)$$ and, for a positive constant $K$, $$\label{e1.5} F(x,u,v)\leq K(|u|^p+|v|^p)^{\frac{p^{**}}{p}}.$$ \begin{theorem} \label{thm1.1} If $\lambda,\mu$ satisfy $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}0\};\\ N_{\lambda,\mu}^{0}=\{(u,v)\in N_{\lambda,\mu}|\langle \Phi'_{\lambda,\mu}(u,v),(u,v)\rangle=0\};\\ N_{\lambda,\mu}^{-}=\{(u,v)\in N_{\lambda,\mu}|\langle \Phi'_{\lambda,\mu}(u,v),(u,v)\rangle<0\}. \end{gather*} Then, we have the following results. \begin{lemma} \label{lem2.5} Suppose that$(u_0,v_0)$is a local minimizer for$J_{\lambda,\mu}$on$N_{\lambda,\mu}$and that$(u_0,v_0)\not\in N_{\lambda,\mu}^{0}$. Then$J'_{\lambda,\mu}(u_0,v_0)=0$in$E^{-1}$(the dual space of the Sobolev space$E$). \end{lemma} \begin{proof} If$(u_0,v_0)$is a local minimizer for$J_{\lambda,\mu}$on$N_{\lambda,\mu}$, then$(u_0,v_0)$is a solution of the optimization problem minimize$J_{\lambda,\mu}(u,v)$subject to$\Phi_{\lambda,\mu}(u,v)=0$. Hence, by the theory of Lagrange multiplies, there exists$\theta\in R $, such that $J'_{\lambda,\mu}(u_0,v_0)=\theta \; \Phi'_{\lambda,\mu}(u_0,v_0)\quad \text{in } E^{-1}(\Omega),$ Thus, $$\label{e2.9} \langle J'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}=\theta \langle \Phi'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}.$$ Since$(u_0,v_0)\in N_{\lambda,\mu}$, we have$\langle J'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}=0$. Moreover,\\$\langle\Phi'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}\neq0$, by \eqref{e2.9},$\theta=0$. Thus,$J'_{\lambda,\mu}(u_0,v_0)=0$in$ E^{-1}$(the dual space of the Sobolev space$E$). \end{proof} \begin{lemma} \label{lem2.6} If $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}0,\mu>0 with \[ 0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}d_0 for some constant \[ d_0=d_0(p,q,N,K,S,|\Omega|,\lambda,\mu)>0.$ \end{itemize} \end{lemma} \begin{proof} (i) Let$(u,v)\in N_{\lambda,\mu}^{+}. By \eqref{e2.7}, $\frac{p-q}{p^{**}-q}\|(u,v)\|^p>\int_{\Omega}F(x,u,v)dx$ and so \begin{align*} J_{\lambda,\mu}(u,v) &= (\frac{1}{p}-\frac{1}{q})\|(u,v)\|^p +(\frac{1}{q}-\frac{1}{p^{**}})\int_{\Omega}F(x,u,v)dx\\ &< [(\frac{1}{p}-\frac{1}{q}) +(\frac{1}{q}-\frac{1}{p^{**}})\frac{p-q}{p^{**}-q}]\|(u,v)\|^p\\ &= -\frac{2(p-q)}{qN}\|(u,v)\|^p<0. \end{align*} Thus, from the definition of\theta_{\lambda,\mu}$and$\theta_{\lambda,\mu}^{+}$, we can deduce that$\theta_{\lambda,\mu}\leq\theta_{\lambda,\mu}^{+}<0$. (ii) Let$(u,v)\in N_{\lambda,\mu}^{-}. By \eqref{e2.7}, $\frac{p-q}{p^{**}-q}\|(u,v)\|^p<\int_{\Omega}F(x,u,v)dx.$ Moreover, by the Minkowski inequality, the Sobolev imbedding theorem, and \eqref{e1.5}, $$\label{e2.10} \int_{\Omega}F(x,u,v)dx\leq KS^{-\frac{p^{**}}{p}}\|(u,v)\|^{p^{**}}.$$ This implies $$\label{e2.11} \|(u,v)\|>(\frac{p-q}{K(p^{**}-q)})^{\frac{1}{p^{**}-p}} S^{\frac{N}{2p^2}} \quad \text{for all } (u,v)\in N_{\lambda,\mu}^{-}.$$ By \eqref{e2.4} in the proof of Lemma \ref{lem2.4} \begin{align*} J_{\lambda,\mu}(u,v) &\geq \|(u,v)\|^{q}[\frac{p^{**}-p}{p^{**}p}\|(u,v)\|^{p-q} -\frac{p^{**}-q}{p^{**}q}S^{-\frac{q}{p}} |\Omega|^{\frac{p^{**}-q}{p^{**}}}(\lambda^{\frac{p}{p-q}} +\mu^{\frac{p}{p-q}})^{\frac{p-q}{p}}]\\ &>(\frac{p-q}{K(p^{**}-q)})^{\frac{q}{p^{**}-p}} S^{\frac{qN}{2p^2}}[\frac{p^{**}-p}{p^{**}p} S^{\frac{(p-q)N}{2p^2}}(\frac{p-q}{K(p^{**}-q)}) ^{\frac{p-q}{p^{*}-p}}\\ &\quad -\frac{p^{**}-q}{p^{**}q} S^{-\frac{q}{p}}|\Omega|^{\frac{p^{**}-q}{p^{**}}} (\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})^{\frac{p-q}{p}}]. \end{align*} Thus, if0<|\lambda|^{\frac{p}{p-q}}+|\mu|^{\frac{p}{p-q}}d_0 \quad \text{for all } (u,v)\in N_{\lambda,\mu}^{-}, \] for some $d_0=d_0(p,q,N,K,S,|\Omega|,\lambda,\mu)>0$. This completes the proof. \end{proof} For each $(u,v)\in E$ with $\int_{\Omega}F(x,u,v)dx>0$, set $t_{\rm max}=(\frac{(p-q)\|(u,v)\|^p}{(p**-q) \int_{\Omega}F(x,u,v)dx})^{\frac{1}{p**-p}}>0.$ Then the following lemma holds, which is similar to the one in Brown and Wu \cite[Lemma 2.6]{b2}. \begin{lemma} \label{lem2.8} For each $(u,v)\in E$ with $\int_{\Omega}F(x,u,v)dx>0$, there are unique $00$. Thus, $(u_0^{+},v_0^{+})$ is a nontrivial solution of \eqref{e1.1}. Now it follows that $u_{n}\to u_0^{+}$ strongly in $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$, $v_{n}\to v_0^{+}$ strongly in $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ and $J_{\lambda,\mu}(u_0^{+},v_0^{+})=\theta_{\lambda,\mu}$. By $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}$ and applying Fatou's lemma, we obtain \begin{align*} \theta_{\lambda,\mu} &\leq J_{\lambda,\mu}(u_0^{+},v_0^{+})\\ &= \frac{2}{N}\|(u_0^{+},v_0^{+})\|^p -\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_0^{+},v_0^{+})\\ &\leq \liminf_{n\to\infty}(\frac{2}{N}\|(u_{n},v_{n})\|^p -\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_{n},v_{n}))\\ &\leq \liminf_{n\to\infty}J_{\lambda,\mu}(u_{n},v_{n}) =\theta_{\lambda,\mu}. \end{align*} This implies $J_{\lambda,\mu}(u_0^{+},v_0^{+})=\theta_{\lambda,\mu},\quad \lim_{n\to\infty}\|(u_{n},v_{n})\|^p=\|(u_0^{+},v_0^{+})\|^p.$ Let $(\widetilde{u}_{n},\widetilde{v}_{n}) =(u_{n},v_{n})-(u_0^{+},v_0^{+})$, then by Br\'ezis-Lieb lemma \cite{b1}, $\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p=\|(u_{n},v_{n})\|^p-\|(u_0^{+},v_0^{+})\|^p.$ Therefore, $u_{n}\to u_0^{+}$ strongly in $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$, $v_{n}\to v_0^{+}$ strongly in $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$. Moreover, we have $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}^{+}$. In fact, if $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}^{-}$, by Lemma \ref{lem2.8}, there are unique $t_0^{+}$ and $t_0^{-}$ such that $(t_0^{+}u_0^{+},t_0^{+}v_0^{+})\in N_{\lambda,\mu}^{+}$ and $(t_0^{-}u_0^{+},t_0^{-}v_0^{+})\in N_{\lambda,\mu}^{-}$. In particular, we have $t_0^{+}0, \] there exists$t_0^{+}<\overline{t}\leq t_0^{-}$such that$J_{\lambda,\mu}(t_0^{+}u_0^{+},t_0^{+}v_0^{+}) 0\}. \] We need also the following version of Br\'ezis-Lieb lemma \cite{b1}. \begin{lemma} \label{lem3.4} Consider $F\in C^{1}(\overline{\Omega},(\mathbb{R}^+)^2)$ with $F(x,0,0)=0$ and $|\frac{\partial F(x,u,v)}{\partial u}|,|\frac{\partial F(x,u,v)}{\partial v}|\leq C_{1}(|u|^{p-1}+|v|^{p-1})$ for some $1\leq p<\infty, C_{1}>0$. Let $(u_{k},v_{k})$ be a bounded sequence in $L^p(\overline{\Omega},(\mathbb{R}^+)^2)$, and such that $(u_{k},v_{k})\rightharpoonup(u,v)$ weakly in $E$. Then as $k\to\infty$, $\int_{\Omega}F(x,u_{k},v_{k})dx\to \int_{\Omega}F(x,u_{k}-u,v_{k}-v)dx+\int_{\Omega}F(x,u,v)dx.$ \end{lemma} \begin{lemma} \label{lem3.5} $J_{\lambda,\mu}$ satisfies the $(PS)_{c}$ condition with $c$ satisfying $-\infty0, then from \eqref{e3.7}, we obtain \[ S_{F}l^{\frac{p}{p^{**}}}= S_{F}\lim_{n\to\infty} (\int_{\Omega}F(x,\widetilde{u}_{n},\widetilde{v}_{n})dx) ^{p/p^{**}} \leq \lim_{n\to\infty}\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p=l,$ which implies $l\geq S_{F}^{N/(2p)}$. In addition, from Lemma \ref{lem3.2}, \eqref{e3.6} and \eqref{e3.7}, we obtain $c= (\frac{1}{p}-\frac{1}{p^{**}})l+J_{\lambda,\mu}(u,v) \geq \frac{2}{N}S_{F}^{N/(2p)} -\Lambda(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}),$ which contradicts $c<\frac{2}{N}S_{F}^{N/(2p)}-\Lambda(\lambda^{\frac{p}{p-q}} +\mu^{\frac{p}{p-q}})$. \end{proof} \begin{lemma} \label{lem3.6} There exist a nonnegative function $(u,v)\in E\backslash\{(0,0)\}$ and $C^{*}>0$ such that for $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}0$ such that $B^{N}(x_0;2\rho_0)\subset\Omega$. Now, we consider the functional $I:E\to R$ defined by $I(u,v)=\frac{1}{p}\|(u,v)\|^p-\frac{1}{p^{**}} \int_{\Omega}F(x,u,v)dx$ and define a cut-off function $\eta(x)\in C_0^{\infty}(\Omega)$ such that $\eta(x)=1$ for $|x-x_0|<\rho_0,\eta(x)=0$ for $|x-x_0|>2\rho_0, 0\leq\eta\leq1$ and $|\nabla \eta|\leq C$. For $\varepsilon>0$, let $u_{\varepsilon}(x)=\eta(x)U(\frac{x}{\varepsilon}),$ where $U(\cdot)$ is a radially symmetric minimizer of $\{\frac{\|\Delta u\|^p_{L^p}}{\|u\|^p_{L^{p^{**}}}}\}_{u\in W^{2,p}(\mathbb{R}^N)\backslash\{0\}}$. Similar to the work of Brown and Wu \cite{b3}, we have the following estimates: $$\label{e3.8} \begin{gathered} \Big(\int_{\Omega}|u_{\varepsilon}|^{p^{**}}dx\Big)^{\frac{p}{p^{**}}}= \varepsilon^{-\frac{N-2p}{p}}\|U\|_{L^{p^{**}}(\mathbb{R}^N)}^p +O(\varepsilon),\\ \int_{\Omega}|\Delta u_{\varepsilon}|^pdx=\varepsilon^{-\frac{N-2p}{p}}\|\Delta U\|_{L^p(\mathbb{R}^N)}^p+O(1), \\ \frac{\int_{\Omega}|\Delta u_{\varepsilon}|^pdx}{(\int_{\Omega}|u_{\varepsilon}|^{p^{**}}dx)^{\frac{p}{p^{**}}}}= S+O(\varepsilon^{\frac{N-2p}{p}}), \end{gathered}$$ Thus, we obtain $\frac{\|\Delta U\|^p_{L^p(\mathbb{R}^N)}}{\|U\|^p_{L^{p^{**}}(\mathbb{R}^N)}}=S=\inf_{u\in W^{2,p}(\mathbb{R}^N)\backslash\{0\}}\frac{\|\Delta u\|^p_{L^p(\mathbb{R}^N)}}{\|u\|^p_{L^{p^{**}}(\mathbb{R}^N)}}.$ Set $u_0(x)=e_{1}u_{\varepsilon}(x-x_0), v_0(x)=e_2u_{\varepsilon}(x-x_0)$ and $(u_0,v_0)\in E$, where $x_0\in \Omega$, $(e_{1},e_2)\in (\mathbb{R}^+)^2$, and $e^p_{1}+e^p_2=1$ are such that $F(x_0,e_{1},e_2)=\max_{x\in\overline{\Omega},g_{1}^p +g_2^p=1,g_{1},g_2>0}F(x,g_{1},g_2)=:K .$ Then, by (F1), \eqref{e1.5}, the definition of $S_{F}$ and \eqref{e3.8}, we obtain \label{e3.9} \begin{aligned} \sup_{t\geq0}I(tu_0,tu_0) &\leq \frac{2}{N} (\frac{(e^p_{1}+e^p_2)\int_{\Omega}|\Delta u_{\varepsilon}|^pdx}{(\int_{\Omega}F(x,e_{1}u_{\varepsilon}(x-x_0), e_2u_{\varepsilon}(x-x_0))dx)^{\frac{p}{p^{**}}}})^{N/(2p)}\\ &= \frac{2}{N}(\frac{\int_{\Omega}|\Delta u_{\varepsilon}|^pdx}{\int_{\Omega}(|u_{\varepsilon}(x-x_0)|^{p^{**}} F(x,e_{1},e_2)dx)^{\frac{p}{p^{**}}}})^{N/(2p)}\\ &\leq \frac{2}{N}(\frac{1}{K^{\frac{p}{p^{**}}}})^{N/(2p)}(S+O(\varepsilon^{\frac{N-2p}{p}}))^{N/(2p)}\\ &= \frac{2}{N}(\frac{1}{K^{\frac{p}{p^{**}}}})^{N/(2p)} (S^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}}))\\ &\leq \frac{2}{N}S_{F}^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}}), \end{aligned} where we have used that $\sup_{t\geq0}(\frac{t^p}{p}A-\frac{t^{p^{**}}}{p^{**}}B) =\frac{2}{N}(\frac{A}{B^{\frac{p}{p^{**}}}})^{N/(2p)},\quad A,B>0.$ We can choose $\delta_{1}>0$ such that for all $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<\delta_{1}$, so we have $c_{\infty}=\frac{2}{N}S_{F}^{N/(2p)}-\Lambda(\lambda^{\frac{p}{p-q}} +\mu^{\frac{p}{p-q}})>0.$ Using the definitions of $J_{\lambda,\mu}$ and $(u_0,v_0)$, we obtain $J_{\lambda,\mu}(tu_0,tv_0)\leq\frac{t^p}{p}\|(u_0,v_0)\|^p \quad \text{for all }t\geq0,\; \lambda,\mu>0,$ which implies that there exists $t_0\in (0,1)$ satisfying $\sup_{0\leq t\leq t_0}J_{\lambda,\mu}(t_0u_0,t_0v_0)0 such that for all 0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<\delta_2, we obtain \[ O(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})-\frac{t_0^{q}}{q}mC_2(\lambda+\mu) <-\Lambda(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}).$ If we set $C^{*}=\min\{\delta_{1},\rho_0^{\frac{N-2p}{p-1}},\delta_2\}$ and $\varepsilon=(\lambda^{\frac{p}{p-q}} +\mu^{\frac{p}{p-q}})^{\frac{p}{N-2p}}$ then for $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}0. \] Combining this with Lemma \ref{lem2.8}, from the definition of$\theta^{-}_{\lambda,\mu}$and \eqref{e3.12}, we obtain that there exists$t_0>0$such that$(t_0u_0,t_0v_0)\in N_{\lambda,\mu}^{-}$and \[ \theta^{-}_{\lambda,\mu}\leq J_{\lambda,\mu}(t_0u_0,t_0v_0)\leq\sup_{t\geq 0}J_{\lambda,\mu}(tu_0,tv_0)0$. Since $J_{\lambda,\mu}$ is coercive on $N_{\lambda,\mu}$, we obtain that $(u_{n},v_{n})$ is bounded in $E$. Therefore, there exist a subsequence still denoted by $(u_{n},v_{n})$ and $(u_0^{-},v_0^{-})\in N_{\lambda,\mu}^{-}$ such that $(u_{n},v_{n})\to(u_0^{-},v_0^{-})$ strongly in $E$ and $J_{\lambda,\mu}(u_0^{-},v_0^{-})=\theta^{-}_{\lambda,\mu}>0$ for all \$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}