\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 280, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/280\hfil Kirchhoff problems] {Existence of positive solutions for \\ Kirchhoff problems} \author[J.-F. Liao, P. Zhang, X.-P. Wu \hfil EJDE-2015/280\hfilneg] {Jia-Feng Liao, Peng Zhang, Xing-Ping Wu} \address{Jia-Feng Liao \newline School of Mathematics and Statistics, Southwest University, Chongqing 400715, China. \newline School of Mathematics and Computational Science, Zunyi Normal College, Zunyi, China} \email{liaojiafeng@163.com} \address{Peng Zhang \newline School of Mathematics and Computational Science, Zunyi Normal College, Zunyi, China} \email{gzzypd@sina.com} \address{Xing-Ping Wu (corresponding author)\newline School of Mathematics and Statistics, Southwest University, Chongqing 400715, China} \email{wuxp@swu.edu.cn} \thanks{Submitted June 25, 2015. Published November 10, 2015.} \subjclass[2010]{35B09, 35B33, 35J20} \keywords{Kirchhoff type equation; resonance; positive solution; \hfill\break\indent mountain pass lemma} \begin{abstract} We study problems for the Kirchhoff equation \begin{gather*} -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =\nu u^3+ Q(x)u^{q},\quad \text{in }\Omega, \\ u=0, \quad \text{on }\partial\Omega, \end{gather*} where $\Omega\subset \mathbb{R}^3$ is a bounded domain, $a,b\geq0$ and $a+b>0$, $\nu>0$, $30$ in $\Omega$. By the mountain pass lemma, the existence of positive solutions is obtained. Particularly, we give a condition of $Q$ to ensure the existence of solutions for the case of $q=5$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main results} In this article, we consider the Kirchhoff type problem \begin{equation}\label{1.1} \begin{gathered} -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =\nu u^3+ Q(x)u^{q},\quad \text{in }\Omega, \\ u=0, \quad \text{on }\partial\Omega, \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^3$ is a bounded domain, $a,b\geq0$ and $a+b>0$, $\nu>0,30$, problem \eqref{1.1} is called degenerate, and the case of $a,b>0$ is called non-degenerate. When $a\geq0$ and $b>0$, problem \eqref{1.1} is called the Kirchhoff type problem. Kirchhoff type problems are often referred to as being nonlocal because of the presence of the term $(\int_{\Omega}|\nabla u|^2dx)\Delta u$ which implies that the equation in \eqref{1.1} is no longer a pointwise equation. The existence and multiplicity of solutions for the problem \begin{equation}\label{1.01} \begin{gathered} -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u=f(x,u),\quad \text{in }\Omega, \\ u=0, \quad \text{on }\partial\Omega, \end{gathered} \end{equation} on a smooth bounded domain $\Omega\subset \mathbb{R}^3$ and $f:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ a continuous function, has been extensively studied (see \cite{ACF,GA},\cite{CKW}-\cite{ZP1}, \cite{ST1}-\cite{WTXZ},\cite{XWT,ZP}). Particularly, Sun and Tang \cite{ST} considered the problem \begin{equation}\label{1.02} \begin{gathered} -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =\lambda u^3+ g(u)-h(x),\quad \text{in }\Omega, \\ u=0, \quad \text{on }\partial\Omega, \end{gathered} \end{equation} where $h\in L^2(\Omega)$ and $g\in C(\mathbb{R},\mathbb{R})$ satisfies \begin{equation}\label{1.4} \lim_{|t|\to\infty}\frac{g(t)}{t^3}=0. \end{equation} Under a Landesman-Lazer type condition, by the minimax methods, they obtained the existence of solutions for problem \eqref{1.02}. When $a=1$ and $b=0$, problem \eqref{1.1} reduces to the semilinear elliptic problem \begin{equation}\label{1.2} \begin{gathered} -\Delta u=\nu u^3+\lambda u^{q},\quad \text{in }\Omega, \\ u=0, \quad \text{on }\partial\Omega. \end{gathered} \end{equation} Obviously when $30$. While for $q=5,\lambda=1$, Br\'ezis and Nirenberg \cite{HL} studied problem \eqref{1.2}. By the variant of the mountain pass theorem of Ambrosetti and Rabinowitz without the (PS) condition, they obtained that there exists $\nu_0>0$ such that problem \eqref{1.2} has a positive solution for each $\nu\geq\nu_0$. For $u\in H_0^1(\Omega)$, we define \begin{equation*} I(u)=\frac{a}{2}\int_{\Omega}|\nabla u|^2dx+\frac{b}{4} \Big(\int_{\Omega}|\nabla u|^2dx\Big)^2-\frac{\nu}{4}\int_{\Omega}|u|^4dx -\frac{1}{q+1}\int_{\Omega}Q(x)|u|^{q+1}dx, \end{equation*} where $H_0^1(\Omega)$ is a Sobolev space equipped with the norm $\|u\|=\big(\int_{\Omega}|\nabla u|^2dx\big)^{1/2}$. Note that a function $u$ is called a weak solution of \eqref{1.1} if $u\in H_0^1(\Omega)$ such that \begin{equation}\label{1.00} \Big(a+b\int_{\Omega}|\nabla u|^2dx\Big) \int_{\Omega}(\nabla u,\nabla\varphi)dx -\nu\int_{\Omega}u^3\varphi dx-\int_{\Omega}Q(x)u^{q}\varphi dx=0, \end{equation} for all $\varphi\in H_0^1(\Omega)$. We denote by $\nu_{1}$ is the first eigenfunction of the eigenvalue problem \begin{gather*} -\Big(\int_{\Omega}|\nabla u|^2dx\Big)\Delta u=\nu u^3, \quad x\in\Omega, \\ u=0, \quad x\in\partial\Omega. \end{gather*} From \cite{ZP1}, we know that $\nu_{1}>0$. Let $S$ be the best Sobolev constant, namely \begin{equation}\label{1.3} S:=\inf_{u\in H_0^1(\Omega)\backslash\{0\}} \frac{\int_{\Omega}|\nabla u|^2dx}{\left(\int_{\Omega}|u|^{6}dx\right)^{1/3}}. \end{equation} As well known, the function \begin{equation}\label{1.5} U(x)=\frac{(3\varepsilon^2)^{1/4}}{\left(\varepsilon^2+|x|^2\right)^{1/2}}, \quad x\in \mathbb{R}^3, \end{equation} is an extremal function for the minimum problem \eqref{1.3}; that is, it is a positive solution of the problem \begin{equation*} -\Delta u=u^{5},\quad \forall x\in \mathbb{R}^{N}. \end{equation*} Problem \eqref{1.1} with $30$, $30$, and $I(u^{*})>0$. \end{theorem} \begin{remark} \label{rmk1}\rm Obviously, Theorem \ref{thm1.1} does not apply to \eqref{1.4}. For all $\nu>0$, we obtain the existence of positive solutions for problem \eqref{1.1}. For the degenerate case, that is $a=0$, $b>0$, we can also obtain that problem \eqref{1.1} possesses a positive solution for all $0<\nu0$, $q=5$, $Q\in C(\overline{\Omega})$ is a positive function and satisfies the assumption \begin{itemize} \item[(A1)] There exists $x_0\in\Omega$ such that $Q(x_0)=Q_{M}=\max_{x\in\overline{\Omega}}Q(x)$ and $$ Q(x)-Q(x_0)=o(|x-x_0|^{}),\quad\text{as }x\to x_0. $$ \end{itemize} Then there exists $\nu^{*}>0$ such that \eqref{1.1} possesses a positive solution $u^{*}$ for all $\nu>\nu^{*}$, and $I(u^{*})>0$. \end{theorem} \begin{remark} \label{rmk2} \rm This case is the critical exponent problem, and Theorem \ref{thm1.2} does not apply to \eqref{1.4}. When $Q(x)\equiv1$, the Kirchhoff type problems with critical exponent have been considered by several papers, such as \cite{ACF,HZ,LLT} \cite{ND1}-\cite{DN3}, \cite{SL2,WTXZ,XWT}. Particularly, problem \eqref{1.1} with $Q(x)\equiv1$ was been considered in \cite{DN2}. However, there exists a flaw in the proof of \cite[Theorem 1.3]{DN2} with the case $\theta=4$. \end{remark} To our best knowledge, problem \eqref{1.1} with $Q(x)$ not constant has not been considered yet. When $Q(x)$ is not constant, the analysis of the compactness becomes complicated, which results in much difficulty. It is worth pointing out that (A1) ensures the existence of solutions. Obviously, Theorem \ref{thm1.2} extends the corresponding result of \cite{DN2}. This article is organized as follows. In Section 2, we consider the case of $30$, $30$. \end{lemma} \begin{proof} Suppose that $\{u_{n}\}$ is a (PS)c sequence of $I$, that is, \begin{equation}\label{2.0} I(u_{n})\to c,\quad I'(u_{n})\to0, \end{equation} as $n\to+\infty$. We claim that $\{u_{n}\}$ is bounded in $H_0^1(\Omega)$. In fact, from \eqref{2.0} one has \begin{align*} 1+c+o(1)\|u_{n}\| &\geq I(u_{n})-\frac{1}{4}\langle I'(u_{n}),u_{n}\rangle\\ &= \frac{a}{4}\|u_{n}\|^2+\big(\frac{1}{4}-\frac{1}{q+1}\big) \int_{\Omega}Q(x)|u_{n}|^{q+1}dx\\ &\geq \frac{a}{4}\|u_{n}\|^2. \end{align*} Hence, we conclude that $\{u_{n}\}$ is bounded in $H_0^1(\Omega)$. Going if necessary to a subsequence, still denoted by $\{u_n\}$, there exists $u\in H_0^1(\Omega)$ such that \begin{equation}\label{2.1} \begin{gathered} u_n\rightharpoonup u,\quad \text{weakly in } H_0^1(\Omega),\\ u_n\to u,\quad \text{strongly in } L^{s}(\Omega),\; 1\leq s<6,\\ u_n(x)\to u(x),\quad \text{a.e. in } \Omega, \end{gathered} \end{equation} as $n\to\infty$. Now, we only need to prove that $u_n\to u$ as $n\to\infty$ in $H_0^1(\Omega)$. As usually, letting $w_{n}=u_{n}-u$, we need prove that $\|w_{n}\|\to0$ as $n\to\infty$. By the Vitali theorem (see \cite[p.133]{RW}), we claim that \begin{equation}\label{2.2} \lim_{n\to\infty}\int_{\Omega}Q(x)|u_n|^{q+1}dx =\int_{\Omega}Q(x)|u|^{q+1}dx. \end{equation} Indeed, we only need to prove that $\{\int_{\Omega}Q(x)|u_n|^{p+1}dx,n\in N\}$ is equi-absolutely-continuous. Note that $\{u_{n}\}$ is bounded in $H_0^1(\Omega)$, by the Sobolev embedding theorem, then exists a constant $C>0$ such that $|u_{n}|_{6}\leq C<\infty$. From the H$\ddot{\mathrm{o}}$lder inequality, for every $\varepsilon>0$, setting $\delta>0$, when $E\subset\Omega$ with $\operatorname{meas}E<\delta$, we have \begin{equation*} \int_{E} Q(x)|u_{n}|^{q+1}dx \leq |u_n|_{6}^{q+1}\Big(\int_{E}Q^{\frac{6}{5-q}}(x)dx\Big)^{\frac{5-q}{6}} <\varepsilon, \end{equation*} where the last inequality is from the absolutely-continuity of $\int_{\Omega}Q^{\frac{6}{5-q}}(x)dx$. Thus, our claim is proved. Moreover, one also has \begin{gather}\label{2.3} \int_{\Omega}|\nabla u_{n}|^2dx =\int_{\Omega}|\nabla w_{n}|^2dx+\int_{\Omega}|\nabla u|^2dx+o(1), \\ \label{2.4} \Big(\int_{\Omega}|\nabla u_n|^2dx\Big)^2 =\|w_n\|^4+\|u\|^4+2\|w_n\|^2\|u\|^2+o(1). \end{gather} Since $I'(u_{n})\to0$, one obtains $$ a\|u_{n}\|^2+b\|u_{n}\|^4-\nu\int_{\Omega}|u_n|^4dx -\int_{\Omega}Q(x)|u_n|^{q+1}dx=o(1), $$ consequently, from \eqref{2.1}-\eqref{2.4}, we deduce that \begin{equation}\label{2.5} a\|w_{n}\|^2+a\|u\|^2+b\|w_{n}\|^4+2b\|w_{n}\|^2\|u\|^2+b\|u\|^4 -\nu|u|_{4}^4-\int_{\Omega}Q(x)|u|^{q+1}dx=o(1). \end{equation} From \eqref{2.0} it follows that \begin{equation}\label{2.6} \lim_{n\to\infty}\langle I'(u_{n}), u\rangle= a\|u\|^2+bl^2\|u\|^2+b\|u\|^4 -\nu|u|_{4}^4-\int_{\Omega}Q(x)|u|^{q+1}dx=0, \end{equation} where $l=\lim_{n\to\infty}\|w_{n}\|$. According to \eqref{2.5} and \eqref{2.6}, we have $$ a\|w_{n}\|^2+b\|w_{n}\|^4+b\|w_{n}\|^2\|u\|^2=o(1), $$ consequently, one has $al^2+bl^4+bl^2\|u\|^2=0$. Thus $l=0$; that is, $u_n\to u$ as $n\to\infty$ in $H_0^1(\Omega)$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] The main idea is to construct a suitable geometry of mountain pass lemma (see\cite{AR}). Then obtain a critical point of $I$ in $H_0^1(\Omega)$. We claim that $I$ has the geometry of mountain pass lemma in $H_0^1(\Omega)$. Indeed, since \begin{equation*} I(u)=\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4 -\frac{\nu}{4}\int_{\Omega}|u|^4dx-\frac{1}{q+1}\int_{\Omega}Q(x)|u|^{q+1}dx, \end{equation*} then $I(0)=0$, and for every $u\in H_0^1(\Omega)\backslash\{0\}$ one has $$ \lim_{t\to0+}\frac{I(tu)}{t^2}=\frac{a}{2}\|u\|^2,\quad \lim_{t\to+\infty}\frac{I(tu)}{t^{q+1}}=-\frac{1}{q+1}\int_{\Omega}Q(x)|u|^{q+1}dx. $$ Since $a>0$ and $\int_{\Omega}Q(x)|u|^{q+1}dx>0$, then there exist $R,\alpha>0$ and $e\in H_0^1(\Omega)$ with $\|e\|>R$ such that $I|_{\partial B_{R}}\geq \alpha$ and $I(e)<0$, where $\partial B_{R}=\{u\in H_0^1(\Omega)~|~\|u\|=R\}$. Thus, $I$ satisfies the geometry of the mountain-pass lemma. Let \begin{equation*} c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t)), \end{equation*} where $\Gamma=\{\gamma\in C([0,1], H_0^1(\Omega)): \gamma(0)=0, \gamma(1)=e\}$. Then $c\geq\alpha$. According to Lemma \ref{lem2.1}, $I$ satisfies the conditions of the mountain pass lemma. Applying the mountain-pass lema, there exists a sequence $\{u_n\}\subset H_0^1(\Omega)$, such that $I(u_{n})\to c$ and $I'(u_{n})\to0$ as $n\to\infty$. Then $c$ is a critical value of $I$ and $c>\alpha>0$. Moreover, $\{u_n\}\subset H_0^1(\Omega)$ has a convergent subsequence, still denoted by $\{u_n\}$, we may assume that $u_n\to u^{*}$ in $H_0^1(\Omega)$ as $n\to\infty$. Thus $I(u^{*})=c>0$ and $u^{*}$ is a nonzero solution of \eqref{1.1}. Since $I(|u|)=I(u)$, by a result due to Br\'ezis and Nirenberg \cite[Theorem 10]{BCN}, we conclude that $u^{*}\geq0$. By the strong maximum principle, one has $u^{*}>0$ in $\Omega$. Therefore, $u^{*}$ is a positive solution of problem \eqref{1.1} with $I(u^{*})>0$. This completes the proof. \end{proof} \section{The case q=5} In this part, assume that $Q\in C(\overline{\Omega})$ is a positive function and satisfies (A1). We study the case of $q=5$. This case is more delicate, because of the Sobolev embedding $H_0^1(\Omega)\hookrightarrow L^{6}(\Omega)$ is not compact. Thus the functional $I$ does not satisfy the $(PS)_{c}$ condition. When $Q(x)$ is not constant, the analysis of $(PS)$ sequences becomes complicated, which results in much difficulty. We will complete the proof of Theorem \ref{thm1.2} by the mountain pass lemma. Now, we prove that $I$ satisfies the local $(PS)_{c}$ condition. \begin{lemma} \label{lem3.1} Assume $a,b>0$ and the positive function $Q\in C(\overline{\Omega})$ satisfies (A1), then $I$ satisfies the $(PS)_{c}$ condition, where $c\in(0,\Lambda)$ with $$ \Lambda=\frac{abS^3}{4Q_{M}}+\frac{b^3S^{6}}{24Q_{M}^2} +\frac{aS\sqrt{b^2S^4+4aSQ_{M}}}{6Q_{M}} +\frac{b^2S^4\sqrt{b^2S^4+4aSQ_{M}}}{24Q_{M}^2}. $$ \end{lemma} \begin{proof} Suppose that $\{u_{n}\}$ is a $(PS)_{c}$ sequence for $c\in (0,\Lambda)$; that is, \begin{equation}\label{3.1} I(u_{n})\to c,\quad I'(u_{n})\to0, \end{equation} as $n\to+\infty$. According to Lemma \ref{lem2.1}, we can easy obtain that $\{u_{n}\}$ is bounded in $H_0^1(\Omega)$. Going if necessary to a subsequence, there exists $u\in H_0^1(\Omega)$ such that \eqref{2.1} holds. As usually, letting $w_{n}=u_{n}-u$, we need prove that $\|w_{n}\|\to0$ as $n\to\infty$. We denote $\lim_{n\to\infty}\|w_{n}\|=l$. As in Lemma \ref{lem2.1}, we have \eqref{2.3} and \eqref{2.4}. By Br\'ezis-Lieb's Lemma \cite{BL}, one has \begin{equation}\label{3.2} \int_{\Omega}Q(x)|u_{n}|^{6}dx =\int_{\Omega}Q(x)|w_{n}|^{6}dx+\int_{\Omega}Q(x)|u|^{6}dx+o(1). \end{equation} From \eqref{3.1} and \eqref{2.1}, one obtains \begin{equation*} a\|u_{n}\|^2+b\|u_{n}\|^4-\nu\int_{\Omega}|u|^4dx -\int_{\Omega}Q(x)|u_{n}|^{6}dx=o(1), \end{equation*} consequently, from \eqref{2.3}-\eqref{2.4} and \eqref{3.2} it follows that \begin{equation}\label{3.3} \begin{aligned} & a\|u\|^2+a\|w_n\|^2+b\|u\|^4+b\|w_n\|^4+2b\|w_n\|^2\|u\|^2\\ &-\int_{\Omega}Q(x)|w_n|^6dx-\int_{\Omega}Q(x)|u|^6dx -\nu\int_{\Omega}|u|^4dx=o(1). \end{aligned} \end{equation} From \eqref{3.1} it follows that \begin{equation}\label{3.4} \lim_{n\to\infty}\langle I'(u_{n}),u\rangle=a\|u\|^2+b\|u\|^4+bl^2\|u\|^2 -\int_{\Omega}Q(x)|u|^6dx-\nu\int_{\Omega}|u|^4dx=0. \end{equation} On the one hand, from \eqref{3.4}, we have \begin{equation}\label{3.0} \begin{aligned} I(u) &= \frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\frac{\nu}{4}\int_{\Omega}|u|^4dx -\frac{1}{6}\int_{\Omega}Q(x)|u|^{6}dx\\ &= \frac{a}{4}\|u\|^2+\frac{1}{12}\int_{\Omega}Q(x)|u|^{6}dx -\frac{bl^2}{4}\|u\|^2\\ &\geq -\frac{bl^2}{4}\|u\|^2. \end{aligned} \end{equation} On the other hand, from \eqref{3.3} and \eqref{3.4} it follows that \begin{equation}\label{3.5} a\|w_n\|^2+b\|w_n\|^4+b\|w_n\|^2\|u\|^2-\int_{\Omega}Q(x)|w_n|^{6}dx=o(1), \end{equation} and \begin{equation}\label{3.6} I(u_{n})=I(u)+\frac{a}{2}\|w_n\|^2+\frac{b}{4}\|w_n\|^4 +\frac{b}{2}\|w_n\|^2\|u\|^2 -\frac{1}{6}\int_{\Omega}Q(x)|w_n|^{6}dx+o(1). \end{equation} From (A1) and \eqref{1.3}, one has $$ \int_{\Omega}Q(x)|w_n|^{6}dx\leq Q_{M}\int_{\Omega}|w_n|^{6}dx \leq Q_{M}\frac{\|w_{n}\|^{6}}{S^3}, $$ consequently, it follows from \eqref{3.5} that $$ al^2+bl^4+bl^2\|u\|^2\leq Q_{M}\frac{l^{6}}{S^3}, $$ which implies that \begin{equation}\label{3.7} l^2\geq\frac{1}{2}\Big[\frac{bS^3}{Q_{M}}+\frac{\sqrt{b^2S^{6} +4S^3Q_{M}(a+b\|u\|^2)}}{Q_{M}}\Big]. \end{equation} Thus, from \eqref{3.5}-\eqref{3.7}, we obtain \begin{align*} I(u) &= \lim_{n\to\infty}\Big[I(u_{n})-\frac{a}{2}\|w_n\|^2-\frac{b}{4}\|w_n\|^4 -\frac{b}{2}\|w_n\|^2\|u\|^2 +\frac{1}{6}\int_{\Omega}Q(x)|w_n|^{6}dx\Big]\\ &= c-\Big(\frac{a}{3}l^2+\frac{b}{12}l^4+\frac{b}{3}l^2\|u\|^2\Big)\\ &\leq c-\Big[\frac{a}{6}\Big(\frac{bS^3}{Q_{M}}+\frac{\sqrt{b^2S^{6} +4S^3Q_{M}(a+b\|u\|^2)}}{Q_{M}}\Big)\\ &\quad +\frac{b}{48}\Big(\frac{bS^3}{Q_{M}}+\frac{\sqrt{b^2S^{6} +4S^3Q_{M}(a+b\|u\|^2)}}{Q_{M}}\Big)^2\\ &\quad +\frac{b\|u\|^2}{24}\Big(\frac{bS^3}{Q_{M}}+\frac{\sqrt{b^2S^{6} +4S^3Q_{M}(a+b\|u\|^2)}}{Q_{M}}\Big)\Big] -\frac{bl^2}{4}\|u\|^2\\ &\leq c-\Big(\frac{abS^3}{4Q_{M}}+\frac{b^3S^{6}}{24Q_{M}^2} +\frac{aS\sqrt{b^2S^4+4aSQ_{M}}}{6Q_{M}} +\frac{b^2S^4\sqrt{b^2S^4+4aSQ_{M}}}{24Q_{M}^2}\Big)\\ &\quad -\frac{bl^2}{4}\|u\|^2\\ &<-\frac{bl^2}{4}\|u\|^2, \end{align*} which contradicts \eqref{3.0}. Hence, $l\equiv0$; that is, $u_{n}\to u$ in $H_0^1(\Omega)$ as $n\to\infty$. Therefore, $I$ satisfies the $(PS)_{c}$ condition for all $c<\Lambda$. This completes the proof. \end{proof} Next, we estimate the level value of functional $I$ and obtain the following lemma. \begin{lemma} \label{lem3.2} Assume that $a,b>0$ and the positive function $Q\in C(\overline{\Omega})$ satisfies {\rm (A1)}. Then there exists $u_0\in H_0^1(\Omega)$, such that $\sup_{t\geq0}I(tu_0)<\Lambda$ for all $\nu>\nu^{*}$, where $\Lambda$ is defined by Lemma \ref{lem3.1} and $\nu^{*}$ independent of $u_0$ is a positive constant. \end{lemma} \begin{proof} Define a cut-off function $\eta\in C_0^{\infty}(\Omega)$ such that $0\leq\eta\leq1$, $|\nabla \eta|\leq C_{1}$. For some $\tilde{\delta}>0$, we define \begin{equation*} \eta(x)=\begin{cases} 1, &|x-x_0|\leq\frac{\tilde{\delta}}{2},\\ 0, &|x-x_0|\geq \tilde{\delta}, \end{cases} \end{equation*} where $x_0$ is defined by (A1). Set $u_{\varepsilon}=\eta(x)U(x-x_0)$. As well known(see \cite{HL,MW}), one has \begin{gather}\label{3.8} \|u_{\varepsilon}\|^2=\|U_{\varepsilon}\|^2 +O(\varepsilon)=S^{3/2}+O(\varepsilon), \\ \label{3.9} |u_{\varepsilon}|_{6}^{6}=|U_{\varepsilon}|_{6}^{6}+O(\varepsilon^3) =S^{3/2}+O(\varepsilon^3), \end{gather} and \begin{equation}\label{4.9} \begin{gathered} C_{2}\varepsilon^{\frac{s}{2}}\leq\int_\Omega u_\varepsilon^sdx \leq C_{3}\varepsilon^{\frac{s}{2}}, \quad 1\leq s<3,\\ C_{4}\varepsilon^{\frac{s}{2}}|\ln\varepsilon| \leq\int_\Omega u_\varepsilon^sdx \leq C_{5}\varepsilon^{\frac{s}{2}}|\ln\varepsilon|, \quad s=3,\\ C_{6}\varepsilon^{\frac{6-s}{2}} \leq\int_\Omega u_\varepsilon^sdx \leq C_{7}\varepsilon^{\frac{6-s}{2}}, \quad 30$ is a small constant. Thus $\sup_{t\geq0}I(tu_\varepsilon)$ attains for some $t_\varepsilon>0$. Moreover, we can claim that there exist two constants $t_0,T_0>0$, which independent of $\varepsilon$, such that $t_00$, then there exists $t_0>0$ such that $|I(t_0u_\varepsilon)|=|I(t_0u_\varepsilon)-I(0)|<\epsilon$. Then according to the monotonicity of $I(tu_\varepsilon)$ near $t=0$, we have $t_\varepsilon>t_0$. Similarly, we can obtain that $t_\varepsilon0$ for all $0T_{\varepsilon}$, so $I_{\varepsilon,1}(t)$ attains its maximum at $T_{\varepsilon}$. From (A1), let $\varepsilon\to 0^{+}$, we claim that \begin{equation}\label{3.11} \left(\int_\Omega Q(x)u_{\varepsilon}^{6}dx\right)^{1/3} =Q_{M}^{1/3}|u_\varepsilon|_{6}^2+o(\varepsilon). \end{equation} In fact, for all $\varepsilon>0$, it follows that \begin{equation}\label{3.12} \begin{aligned} \left|\int_{\Omega}Q(x)u_{\varepsilon}^{6}dx -\int_{\Omega}Q_{M}u_{\varepsilon}^{6}dx\right| &\leq \int_{\Omega}|Q(x)-Q(x_0)|u_{\varepsilon}^{6}dx\\ &\leq \int_{\{x\in\Omega:~ |x-x_0|\leq \tilde{\delta}\}} |Q(x)-Q(x_0)|u_{\varepsilon}^{6}dx. \end{aligned} \end{equation} From (A1), for all $\eta>0$, there exists $\delta>0$ such that $$ |Q(x)-Q(x_0)|<\eta|x-x_0|,\quad \mathrm{for~all}~0<|x-x_0|<\delta. $$ When $\varepsilon>0$ small enough, for $\delta>\varepsilon^{1/2}$, it follows from \eqref{3.12} and (A1) that \begin{align*} &\Big|\int_{\Omega}Q(x)u_\varepsilon^{6}dx -\int_{\Omega}Q_{M}u_\varepsilon^{6}dx\Big|\\ &\leq \int_{\{x\in\Omega:|x-x_0|\leq \tilde{\delta}\}} |Q(x)-Q(x_0)|u_\varepsilon^{6}dx\\ &< \int_{\{x\in\Omega:|x-x_0|\leq\delta\}}\eta|x-x_0| \frac{(3\varepsilon^2)^{3/2}}{[\varepsilon^2+|x-x_0|^2]^3}dx\\ &\quad +\int_{\{x\in\Omega:~\delta<|x-x_0|\leq \tilde{\delta}\}} \frac{(3\varepsilon^2)^{3/2}}{[\varepsilon^2+|x-x_0|^2]^3}dx\\ &= \sqrt{27}\eta\int_0^{\delta}r^3\frac{\varepsilon^3}{(\varepsilon^2+r^2)^3}dr + \sqrt{27}\int_{\delta}^{\tilde{\delta}}\frac{\varepsilon^3r^2}{(\varepsilon^2+r^2)^3}dr\\ &= \sqrt{27}\eta\varepsilon\int_0^{\frac{\delta}{\varepsilon}} \frac{r^3}{(1+r^2)^3}dr + \sqrt{27}\int_{\frac{\delta}{\varepsilon}}^{\frac{\tilde{\delta}}{\varepsilon}}\frac{r^2}{(1+r^2)^3}dr\\ &\leq C_{8}\eta\varepsilon+C_{9}\varepsilon^3. \end{align*} Consequently, one has $$ \frac{\left|\int_{\Omega}Q(x)u_\varepsilon^{6}dx -\int_{\Omega}Q_{M}u_\varepsilon^{6}dx\right|}{\varepsilon} \leq C_{8}\eta+C_{9}\varepsilon^2, $$ which implies $$ \limsup_{\varepsilon\to 0^{+}}\frac{\big|\int_{\Omega}Q(x)u_\varepsilon^{6}dx -\int_{\Omega}Q_{M}u_\varepsilon^{6}dx\big|}{\varepsilon} \leq C_{8}\eta. $$ Then from the arbitrariness of $\eta$, we obtain \eqref{3.11}. Thus, from \eqref{3.9} and \eqref{3.11}, one gets \begin{equation}\label{3.13} \int_\Omega Q(x)u_{\varepsilon}^{6}dx =Q_{M}|u_\varepsilon|_{6}^{6}+o(\varepsilon)=Q_{M}S^{3/2}+o(\varepsilon). \end{equation} Thus from \eqref{3.8},\eqref{3.10},\eqref{4.10} and \eqref{3.13}, we have \begin{align} I_{\varepsilon,1}(t) &\leq I_{\varepsilon,1}(T_{\varepsilon}) \nonumber\\ &= T_{\varepsilon}^2\Big(\frac{a}{2}\|u_{\varepsilon}\|^2 +\frac{b}{4}T_{\varepsilon}^2\|u_{\varepsilon}\|^4 -\frac{T_{\varepsilon}^4}{6}\int_{\Omega}Q(x)u_{\varepsilon}^{6}dx\Big) \nonumber\\ &= T_{\varepsilon}^2\Big(\frac{a}{3}\|u_{\varepsilon}\|^2 +\frac{b}{12}T_{\varepsilon}^2\|u_{\varepsilon}\|^4\Big) \nonumber\\ &= \frac{ab\|u_\varepsilon\|^6+a\|u_\varepsilon\|^2 \sqrt{b^2\|u_\varepsilon\|^8+4a\|u_\varepsilon\|^2 \int_{\Omega}Q(x)u_\varepsilon^6dx} }{6\int_{\Omega}Q(x)u_\varepsilon^6dx} \nonumber\\ &\quad + \frac{b^3\|u_\varepsilon\|^{12}+2ab\|u_\varepsilon\|^{6} \int_{\Omega}Q(x)u_\varepsilon^6dx} {24 \big(\int_{\Omega}Q(x)u_\varepsilon^6dx\big)^2} \nonumber\\ &\quad + \frac{b^2\|u_\varepsilon\|^4\sqrt{b^2\|u_\varepsilon\|^8 +4a\|u_\varepsilon\|^2\int_{\Omega}Q(x)u_\varepsilon^6dx} } {24\big(\int_{\Omega}Q(x)u_\varepsilon^6dx\big)^2} \nonumber\\ &= \frac{ab\|u_\varepsilon\|^6}{4\int_{\Omega}Q(x)u_\varepsilon^6dx} +\frac{b^3\|u_\varepsilon\|^{12}}{24(\int_{\Omega}Q(x)u_\varepsilon^6dx)^2} \nonumber\\ &\quad +\frac{a\|u_\varepsilon\|^2 \sqrt{b^2\|u_\varepsilon\|^8+4a\|u_\varepsilon\|^2 \int_{\Omega}Q(x)u_\varepsilon^6dx} }{6\int_{\Omega}Q(x)u_\varepsilon^6dx} \nonumber\\ &\quad + \frac{b^2\|u_\varepsilon\|^4\sqrt{b^2\|u_\varepsilon\|^8 +4a\|u_\varepsilon\|^2\int_{\Omega}Q(x)u_\varepsilon^6dx} } {24\big(\int_{\Omega}Q(x)u_\varepsilon^6dx\big)^2} \nonumber\\ &= \frac{ab(S^{\frac{9}{2}}+O(\varepsilon))}{4(Q_{M}S^{3/2}+o(\varepsilon))}+ \frac{b^3(S^9+O(\varepsilon))}{24(Q_{M}S^{3/2}+o(\varepsilon))^2} \nonumber\\ &\quad +\frac{a(S^{3/2}+O(\varepsilon)) \sqrt{b^2S^{6}+4aS^3+O(\varepsilon)}}{6(Q_{M}S^{3/2}+o(\varepsilon))} \nonumber\\ &\quad +\frac{b^2(S^6+O(\varepsilon))\sqrt{b^2S^{6}+4aS^3+O(\varepsilon)}} {24(Q_{M}S^{3/2}+o(\varepsilon))^2} \nonumber\\ &= \frac{abS^3}{4Q_{M}}+\frac{b^3S^6}{24Q_{M}^2} +\frac{aS\sqrt{b^2S^4+4aSQ_{M}}}{6Q_{M}} \nonumber\\ &\quad +\frac{b^2S^4\sqrt{b^2S^4+4aSQ_{M}}}{24Q_{M}^2}+O(\varepsilon) \nonumber \\ &= \Lambda+O(\varepsilon). \label{3.14} \end{align} Second, we estimate the value of $I_{\varepsilon,2}$. From \eqref{4.9}, since $00$ such that $I(tu_\varepsilon)<\Lambda$ for all $\nu>\nu^{*}$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] As in the proof of Theorem \ref{thm1.1}, we can obtain that $I$ has the geometry of mountain pass lemma in $H_0^1(\Omega)$. According to Lemmas \ref{lem3.1} and \ref{lem3.2}, it follows that $I$ satisfies the conditions of the mountain pass lemma. Then as in the proof of Theorem \ref{thm1.1}, we obtain that \eqref{1.1} has a positive solution $u^{*}$ with $I(u^{*})>0$ as long as $\nu>\nu^{*}$. The proof is complete. \end{proof} \subsection*{Acknowledgments} This research was supported by the National Science Foundation of China (No. 11471267), by the Science and Technology Foundation of Guizhou Province (No. LKZS[2014]30, No. LH[2015]7001, No. LH[2015]7049). The authors would like to thank the anonymous referees and Professor G. Molica Bisci for their valuable suggestions. \begin{thebibliography}{99} \bibitem{ACF} C. O. Alves, F. J. S. A. Corr\^ea, G. M. Figueiredo; \emph{On a class of nonlocal elliptic problems with critical growth}, Differ. Equ. Appl. 2 (2010) 409--417. \bibitem{AR} A. Ambrosetti, P. H. Rabinowitz; \emph{Dual variational methods in critical point theory and applications}, J. Funct. Anal., 14 (1973), 349--381. \bibitem{GA} G. Anello; \emph{A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems}, J. Math. Anal. Appl., 373 (2011), 248--251. \bibitem{BCN} H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg; \emph{Variational methods for indefinite superlinear homogeneous elliptic problems}, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553--572. \bibitem{BL} H. Br\'ezis, E. Lieb; \emph{A relation between pointwise convergence of functions and convergence of functionals}, Proc. Amer. Math. Soc. 88 (1983), 486--490. \bibitem{HL} H. Br\'ezis, L. Nirenberg; \emph{Positive solutions of nonlinear elliptic equations involving critical exponents}, Comm. Pure Appl. Math., 36 (1983), 437--477. \bibitem{CKW} C. Y. Chen, Y. C. Kuo, T. F. Wu; \emph{he Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions}, J. Differential Equations, 250 (2011), 1876--1908. \bibitem{FMBR} G. M. Figueiredo, G. Molica Bisci, R. Servadei; \emph{On a fractional Kirchhoff-type equation via Krasnoselskii's genus}, Asymptot. Anal., 94 (2015), 347--361. \bibitem{HZ} X. M. He, W. M. Zou; \emph{Infnitely many solutions for Kirchhoff-type problems}, Nonlinear Anal., 70 (2009), 1407--1414. \bibitem{HZ1}X. M. He, W. M. Zou; \emph{Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$}, J. Differential Equations, 252 (2012), 1813--1834. \bibitem{HZ2} X. M. He, W. M. Zou; \emph{Ground states for nonlinear Kirchhoff equations with critical growth}, Ann. Mat. Pura Appl., 193 (2014), 473--500. \bibitem{LLT} C. Y. Lei, J. F. Liao, C. L. Tang; \emph{Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents}, J. Math. Anal. Appl., 421 (2015), 521--538. \bibitem{LLS} Y. H. Li, F. Y. Li, J. P. Shi; \emph{Existence of a positive solution to Kirchhoff type problems without compactness conditions}, J. Differential Equations, 253 (2012), 2285--2294. \bibitem{SL} X. Liu, Y.J. Sun; \emph{Multiple positive solutions for Kirchhoff type problems with singularity}, Commun. Pure Appl. Anal., 12 (2013), 721--733. \bibitem{AL} A. M. Mao, S. X. Luan; \emph{Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems}, J. Math. Anal. Appl., 383 (2011), 239--243. \bibitem{MZ} A. M. Mao, Z. T. Zhang; \emph{Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition}, Nonlinear Anal., 70 (2009), 1275--1287. \bibitem{MBP} G. Molica Bisci, P. F. Pizzimenti; \emph{Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition}, Proc. Edinb. Math. Soc., 57 (2014), 779--809. \bibitem{MBR} G. Molica Bisci, V. D. R\u{a}dulescu; \emph{Mountain Pass Solutions for Nonlocal Equations}, Ann. Acad. Fenn. Math., 39 (2014), 579--592. \bibitem{MBRD} G. Molica Bisci, D. Repov\v{s}; \emph{On doubly nonlocal fractional elliptic equations}, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 161--176. \bibitem{ND1} D. Naimen; \emph{The critical problem of Kirchhoff type elliptic equations in dimension four}, J. Differential Equations, 257 (2014), 1168--1193. \bibitem{DN2} D. Naimen; \emph{Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent}, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885--914. \bibitem{DN3} D. Naimen; \emph{On the Brezis-Nirenberg problem with a Kirchhoff type perturbation}, Adv. Nonlinear Stud., 15 (2015), 135--156. \bibitem{ZP1} K. Perera, Z. T. Zhang; \emph{Nontrivial solutions of Kirchhoff-type problems via the Yang index}, J. Differential Equations 221 (2006), 246--255. \bibitem{RW} W. Rudin; \emph{Real and complex analysis}, McGraw-Hill, New York, London etc. 1966. \bibitem{ST1} J. J. Sun, C. L. Tang; \emph{Existence and multipicity of solutions for Kirchhoff type equations}, Nonlinear Anal. 74 (2011), 1212--1222. \bibitem{ST} J. J. Sun, C. L. Tang; \emph{Resonance problems for Kirchhoff type equations}, Discrete Contin. Dyn. Syst., 33 (2013), 2139--2154. \bibitem{SL2} Y. J. Sun, X. Liu; \emph{Existence of positive solutions for Kirchhoff type problems with critical exponent}, J. Partial Differ. Equ., 25 (2012), 187--198. \bibitem{WTXZ} J. Wang, L.X. Tian, J. X. Xu, F. B. Zhang; \emph{Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth}, J. Differential Equations, 253 (2012), 2314--2351. \bibitem{MW} M. Willem; \emph{Minimax Theorems}, Birth\"auser, Boston, 1996. \bibitem{XWT} Q. L. Xie, X. P. Wu, C. L. Tang; \emph{Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent}, Commun. Pure Appl. Anal., 12 (2013), 2773--2786. \bibitem{ZP} Z. T. Zhang, K. Perera; \emph{Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow}, J. Math. Anal. Appl., 317 (2006), 456--463. \end{thebibliography} \end{document}