\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 43, pp. 1--10.\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 - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/43\hfil Multiplicity of solutions] {Multiplicity of solutions to the sum of polyharmonic equations with \\ critical Sobolev exponents} \author[W. Liu, G. Jia, L.-Q. Guo \hfil EJDE-2015/43\hfilneg] {Wei Liu, Gao Jia, Lu-Qian Guo} \address{Wei Liu \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{liuweimaths@hotmail.com} \address{Gao Jia (corresponding author) \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{gaojia89@163.com} \address{Lu-Quian Guo \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{xiaoshudian.2008@163.com} \thanks{Submitted November 21, 2014. Published February 17, 2015.} \subjclass[2000]{35J30, 35J60} \keywords{Polyharmonic equation; multiple solutions; critical Sobolev exponent} \begin{abstract} In this article, we prove multiplicity of solutions for the sum of polyharmonic equation with critical Sobolev exponent. The proof is based upon the methods of weakly lower semi-continuous of the functionals and the Mountain Pass Lemma without (PS) conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we discuss the multiplicity of solutions for the sum polyharmonic equation \begin{equation} \begin{gathered} \sum_{i = 0}^k {{{( - \Delta )}^i}} u = \lambda {| u |^{q - 2}}u + {| u |^{N - 2}}u + \mu f(x),\quad\text{in }\Omega,\\ u \in H_0^k(\Omega ), \end{gathered} \label{eP} \end{equation} where $\Omega \subset {\mathbb{R}^{n}}$ is a bounded smooth domain, $k$ is positive integer, $q$ is a real number with $22k$ and $f(x)$ be continuous and not identical to 0 in $\Omega$. Then there exist $\lambda_0 >0$ and $\mu_0 >0$, such that for any $\lambda>\lambda_0$ and $0<\mu<\mu_0$, problem \eqref{eP} admits at least two distinct weak solutions $u_1$ with positive energy and $u_2$ with negative energy. \end{theorem} \begin{remark} \label{rmk1.2} \rm For the highest order term ${( - \Delta )^k}u$ of problem \eqref{eP}, we need to discuss that $k$ is odd or even. In fact, no matter $k$ is odd or even, we obtain the similar result of Theorem \ref{thm1.1}. For the sake of simplicity, in the following discussion, we let $k$ be an even, that is $k=2m$ and $m$ is positive integer. \end{remark} Higher-order elliptic boundary problems have abundant applications in physics and engineering \cite{m1} and have also been studied in many areas of mathematics, including conformal geometry \cite{c3}, some geometry invariants \cite{b3} and non-linear elasticity \cite{c4}. The existence of the solutions of the Brezis-Nirenberg problem \cite{b7} for the higher-order equations has been studied in many papers \cite{a1,b1,b5,c2,e1,w1}. Grunau \cite{g1} considered the existence of positive solution for semilinear polyharmonic Dirichlet problem with critical Sobolev exponent \begin{equation}\label{e4} \begin{gathered} (- \Delta )^ku = \lambda u + {{| u |}^{s - 2}}u\quad\text{in }B,\\ {{D^\alpha }u = 0,| \alpha | \le k - 1\quad\text{on }\partial B,} \end{gathered} \end{equation} where $k \in \mathbb{N},~B$ is the unit ball centered at the origin, $\lambda \in \mathbb{R}$, $n >2k$, $s = 2n/(n - 2k)$ is the critical Sobolev exponent. He proved the existence of a positive radial solution for: $\lambda \in (0,{\lambda _1})$, if $n\geq 4k$; $\lambda \in (\overline \lambda ,{\lambda _1})$ for some $\overline \lambda = \overline \lambda (n,k) \in (0,{\lambda _1})$, if $2k + 1 \le n \le 4k - 1$, where $\lambda _1$ is the first eigenvalue of $(- \Delta)^k$ with homogeneous Dirichlet boundary conditions. Recently, Benalili and Tahri \cite{b4} considered the multiplicity of solutions considered for the equation \begin{equation} \label{e1.2} \Delta ^2 u - {\nabla ^i}(a{\rho ^{ - \sigma }}{\Delta _i}u) + b{\rho ^{ - \mu }}u = \lambda {| u |^{q - 2}}u + f(x){| u |^{s - 2}}u \end{equation} where the function $a(x)$ and $b(x)$ are smooth on $M$ and $10$ such that if $\lambda\in (0,\lambda_\ast)$, the equation \eqref{e1.2}~possesses at least two distinct nontrivial solutions in the distribution sense. The multiplicity of solutions for higher-order equations can be founded in \cite{b2} and the references therein. Here, our motivation comes from the recent papers \cite{b4,g1}. We consider the situation of the multiplicity of the higher-order equation with critical Sobolev exponent when $k\geq 1$ and $q>2$. The paper is organized as follows. In Section 2, we will introduce the Sobolev spaces and the embedding theorem which is applicable to problem \eqref{eP}. In Section 3, since a lack of compactness, we use analytic techniques and variational arguments to overcome the difficulty and establish some basic lemmas. In Section 4, we give the proof of two distinct weak solutions of Theorem \ref{thm1.1}. Our methods are mainly based on the weakly lower semi-continuous of the functional and the Mountain Pass Lemma without (PS) condition. \section{Preliminaries} Suppose $\Omega \subset \mathbb{R}^{n}$ is a bounded smooth open domain. We let $H_0^{2m}(\Omega )$ be the Sobolev space which is the completion of the space $C_0^\infty (\Omega )$ with respect to the norm \begin{equation} \label{e2.1} \| u \|_{{H_{2m}}} = (\| {{\Delta ^m}u} \|_2^2 + \| {\nabla {\Delta ^{m - 1}}u} \|_2^2 + \dots + \| {\nabla u} \|_2^2 + \| u \|_2^2)^{1/2}. \end{equation} It is well known that a weak solution of the equation \eqref{eP} is a critical point of the following functional \begin{equation} \label{e2.2} \begin{split} {I_{\lambda ,\mu }}(u) &= \frac{1}{2}\int_\Omega {({{({\Delta ^m}u)}^2} + {{| {\nabla {\Delta ^{m - 1}}u} |}^2} + \dots + {{(\Delta u)}^2} + {{| {\nabla u} |}^2} + {u^2})} \\ &- \frac{1}{q}\lambda \int_\Omega {{{| u |}^q}} - \frac{1}{N}\int_\Omega {{{| u |}^N}} - \mu \int_\Omega {f(x)u}. \end{split} \end{equation} Under the above assumptions, it is easy to know that ${I_{\lambda ,\mu }}(u) \in {C^1}(H_0^{2m}(\Omega ), \mathbb{R})$ and with the G\^ateaux derivative \begin{equation} \label{e2.3} \begin{split} \langle {\nabla {I_{\lambda ,\mu }}(u),v} \rangle &= \int_\Omega {({\Delta ^m}u {\Delta ^m}v) + (\nabla {\Delta ^{m - 1}}u \cdot \nabla {\Delta ^{m - 1}}v) + \dots + \nabla u \cdot \nabla v + uv)} \\ &\quad - \lambda \int_\Omega {{{| u |}^{q - 2}}uv} - \int_\Omega {{{| u |}^{N - 2}}uv} - \mu \int_\Omega {f(x)v} \end{split} \end{equation} for every $v \in H_0^{2m}(\Omega )$ (see \cite{g1}). \begin{lemma}[Mountain Pass Theorem \cite{a2}] \label{lem2.1} Let $E$ be a real Banach space and let $I(u) \in {C^1}(E,\mathbb{R}).$ Suppose $I(0) = 0$ and \begin{itemize} \item[(I1)] there is a constant $\rho >0$ such that ${I|_{\partial {B_\rho }(0)}} >0$, \item[(I2)] there is an $e \in E\backslash \overline {{B_\rho }(0)}$ such that $I(e)\leq 0$. \end{itemize} \end{lemma} Set \begin{equation} \label{e2.4} C = \inf_{\gamma \in \Gamma } \sup_{t \in [0,1]} I(\gamma (t)) > 0 \end{equation} where $\Gamma $ denotes the class of paths joining 0 to $e$. Conclusion: there is a sequence $\{u_k\}$ in $E$, such that $$ I({u_k}) \to C \quad \text{and}\quad \nabla I({u_k}) \to 0\quad \text{in a dual space } E'. $$ \begin{lemma}[Sobolev-Rellich-Knodrakov Theorem \cite{x1}] \label{lem2.2} Assume that $\Omega \subset {\mathbb{R}^{n}}$ is a bounded domain with Lipschitz boundary, $k$ is positive integer and $1 \le p <\infty $. Then the following hold: \begin{itemize} \item if $n>kp$, then $W^{k,p}(\Omega ) \hookrightarrow {L^s}(\Omega )$, for $1 \le s \le p^\ast= \frac{np}{{n - kp}}$; \item the embedding is compact, for $s < \frac{np}{n - kp}$. \end{itemize} \end{lemma} \section{Basic Lemmas} To complete the proof of Theorem \ref{thm1.1}, the following lemmas are our main tools. \begin{lemma} \label{lem3.1} For each fixed $\lambda > 0$, there exist $\delta>0$, $ \mu_0 > 0$ and $ \eta > 0$, such that for all $u \in {H_0^{2m}(\Omega )}$ with ${\| u \|_{{H_{2m}}}} = \delta$ and any $0<\mu<{\mu _0}$, it holds ${I_{\lambda ,\mu }}(u) > {\eta}>0$. \end{lemma} \begin{proof} From \eqref{e2.1}, \eqref{e2.2} and the H\"{o}lder inequality, we deduce that \begin{equation} \label{e3.1} \begin{split} {I_{\lambda ,\mu }}(u) & = \frac{1}{2}\int_\Omega {({{({\Delta ^m}u)}^2}} + {(\nabla {\Delta ^{m - 1}}u)^2} + \dots + {(\Delta u)^2} + {| {\nabla u} |^2} + u^2) \\ &\quad - \frac{\lambda }{q}\int_\Omega {{{| u |}^q}} - \frac{1}{N}\int_\Omega {{{| u |}^N}} - \mu \int_\Omega {f(x)u}\\ &\ge \frac{1}{2}\| u \|_{{H_{2m}}}^2 - \frac{\lambda}{q} {| \Omega |^{1 - \frac{q}{N}}}\| u \|_N^q - \frac{1}{N}\| u \|_N^N - \mu\max_{x \in \Omega } f(x){| \Omega |^{1 - \frac{1}{N}}}{\| u \|_N}. \end{split} \end{equation} By \eqref{e3.1} and Lemma \ref{lem2.2}, we infer that \begin{equation*} \begin{split} {I_{\lambda ,\mu }}(u){\rm{ }} &\ge \frac{1}{2}\| u \|_{{H_{2m}}}^2 - \frac{\lambda}{q} {({C })^q}{| \Omega |^{1 - \frac{q}{N}}}\| u \|_{{H_{2m}}}^q - \frac{1}{N}{({C })^N}\| u \|_{{H_{2m}}}^N\\ &\quad - \mu\max_{x \in \Omega } f(x){| \Omega |^{1 - \frac{1}{N}}}{C }{\| u \|_{{H_{2m}}}}\\ &= \Big( {\big(\frac{1}{2} - \lambda {C_1}\| u \|_{{H_{2m}}}^{q - 2} - {C_2}\| u \|_{{H_{2m}}}^{N - 2}\big)\cdot{{\| u \|}_{{H_{2m}}}} - \mu {C_3}} \Big) \| u \|_{H_{2m}}, \end{split} \end{equation*} with some positive constants $C_1, C_2,C_3 ~\text{and} ~20$, there exist $\delta=\delta(\lambda )>0$, sufficiently small ${{\mu _0}={\mu _0}(\delta) > 0}$, and ${\eta _0}={\eta _0}(\mu _0) > 0$, such that for all $ u \in H_0^{2m}(\Omega )$ with ${\| u \|_{{H_{2m}}}}={\delta}$ and for any $0<\mu<{\mu _0}$, it holds ${I_{\lambda ,\mu }}(u) > {\eta _0}$ \end{proof} \begin{lemma} \label{lem3.2} Suppose $f(x)$ is continuous and not identical to 0 in $ \Omega $. For any ${\mu _0} > 0$, there exist ${\lambda _0} > 0$ and $v_0 \in H_0^{2m}(\Omega )$, such that for any $\lambda \ge \lambda_0$, we have \begin{equation} \label{e3.2} 0 < \sup_{t \ge 0} {I_{\lambda ,\mu }}(tv_0) < \frac{{2m}}{n}{({C^*})^{ -n/(2m)}}, \end{equation} where ${C^ * }$ is the best Sobolev constant of $H_0^{2m}(\Omega )\hookrightarrow {L^N}(\Omega)$, $N = 2n/(n - 4m)$. \end{lemma} \begin{proof} By the conditions of $f(x)$, we can choose $v_0 \in H_0^{2m}(\Omega )$, such that $$ \int_\Omega {f(x)v_0 > 0}\quad\text{and}\quad \int_\Omega {{{| v_0 |}^N} = 1.} $$ Thus from \eqref{e2.2}, we obtain \begin{equation} \label{e3.3} {I_{\lambda ,\mu }}(tv_0 ) = \frac{{{t^2}}}{2}\| v_0 \|_{{H_{2m}}}^2 - {t^q}\frac{\lambda }{q}\int_\Omega {{{| v_0 |}^q}} - \frac{1}{N}{t^N} - t\mu \int_\Omega {f(x)v_0 } . \end{equation} For any $\lambda ,\mu > 0$, we have \begin{equation} \label{e3.4} \lim _{t \to +\infty } {I_{\lambda ,\mu }}(tv_0 ) = - \infty. \end{equation} Using Lemma \ref{lem3.1} and \eqref{e3.4}, there exists $t_{\lambda ,\mu } > 0$, such that \begin{equation} \label{e3.5} {I_{\lambda ,\mu }}(t_{\lambda ,\mu } v_0 ) = \sup_{t \ge 0} {I_{\lambda ,\mu }}(tv_0 ) > 0. \end{equation} By \eqref{e3.3} and \eqref{e3.5}, one gets \begin{equation} \label{e3.6} \begin{split} \frac{1}{2}t_{\lambda ,\mu }^2\| {{v_0}} \|_{{H_{2m}}}^2 - (\frac{\lambda }{q}t_{\lambda ,\mu }^q\| {{v_0}} \|_q^q + \frac{1}{N}t_{\lambda ,\mu }^N) - t_{\lambda ,\mu }^{}\mu \int_\Omega {f(x){v_0}} > 0. \end{split} \end{equation} That is, \[ t_{\lambda ,\mu }^{q - 1}\left( {\frac{\lambda }{q} \| v_0 \|_q^q + \frac{1}{N}t_{\lambda ,\mu }^{N - q}} \right) < \frac{1}{2}\| v_0 \|_{{H_{2m}}}^2 - \mu \int_\Omega {f(x)v_0 }. \] By simple analysis, we obtain $$ \lim _{\lambda \to +\infty } (\frac{\lambda }{q}\| v_0 \|_q^q + \frac{1}{N}t_{\lambda ,\mu }^{N - q}) = + \infty, $$ and \begin{equation} \label{e3.7} \lim _{\lambda\to +\infty } {t_{\lambda,\mu }} = 0. \end{equation} From \eqref{e3.3} and \eqref{e3.7}, we obtain \begin{equation} \label{e3.8} \lim _{\lambda \to + \infty } \lambda t_{\lambda ,\mu }^{q - 1} \le 0. \end{equation} Taking into account of \eqref{e3.5}, \eqref{e3.7} and \eqref{e3.8}, we obtain \begin{equation} \label{e3.9} \lim _{\lambda\to +\infty } \sup_{t \ge 0} {I_{\lambda ,\mu }} ({t_{\lambda ,\mu }}v_0 ) = 0. \end{equation} Then there exist ${\lambda _0}$ such that for any $\lambda > {\lambda _0}$, we have $$ 0 < \sup_{t \ge 0} {I_{\lambda ,\mu }}(tv_0 ) < \frac{{2m}}{n}{({C^ * })^{-n/(2m)}}. $$ The proof is complete. \end{proof} \begin{lemma} \label{lem3.3} For any $\lambda > 0,~\text{there exists sufficiently small} ~{\mu _0}>0$, such that for any $0 < \mu < {\mu _0}$, the $I_{\lambda ,\mu }(u)$ satisfies the $(PS)_{C_{\lambda ,\mu }}$-condition for all ${C_{\lambda ,\mu }}$ in the interval \begin{equation} \label{e3.10} 0 < {C_{\lambda ,\mu }} < \frac{{2m}}{n}{({C^ * })^{{-n/(2m)}}}. \end{equation} \end{lemma} \begin{proof} First we prove that each $(PS)_{C_{\lambda ,\mu }}$ sequence is bounded in ${H_0^{2m}(\Omega )}$. Let $\{u_k\} \subset {H_0^{2m}(\Omega )}$ be a $(PS)_{C_{\lambda ,\mu }}$ sequence for ${I_{\lambda ,\mu }}(u)$, defined by \eqref{e2.2}, i.e., $$ {I_{\lambda ,\mu }}(u_{k}) \to {C_{\lambda ,\mu }}, \quad\text{and}\quad \nabla {I_{\lambda ,\mu }}({u_k}) \to 0,\quad \text{in }{H_0^{2m}(\Omega )}', \text{ as }k\to \infty. $$ That is, \begin{equation} \label{e3.11} \begin{split} {I_{\lambda ,\mu }}({u_k}) &= \frac{1}{2}\int_\Omega {({{({\Delta ^m}{u_k})}^2}} + {(\nabla {\Delta ^{m - 1}}{u_k})^2} + \dots + {(\Delta {u_k})^2} + {| {\nabla {u_k}} |^2} + u_k^2)\\ &\quad - \frac{\lambda }{q}\int_\Omega {{{| {{u_k}} |}^q}} - \frac{1}{N}\int_\Omega {{{| {{u_k}} |}^N}} - \mu \int_\Omega {f(x){u_k}}\\ &= {C_{\lambda ,\mu }}+ o(1) \end{split} \end{equation} and \begin{equation} \label{e3.12} \begin{split} \langle {\nabla {I_{\lambda ,\mu }}({u_k}),{u_k}} \rangle &=\int_\Omega {({{({\Delta ^m}{u_k})}^2}} + {(\nabla {\Delta ^{m - 1}}{u_k})^2} + \dots + {(\Delta {u_k})^2} + {| {\nabla {u_k}} |^2} + u_k^2)\\ &\quad- \lambda \int_\Omega {{{| {{u_k}} |}^q}} - \int_\Omega {{{| {{u_k}} |}^N}} - \mu \int_\Omega {f(x){u_k}}\\ &=o(1){\| {{u_k}} \|_{{H_{2m}}}}, \end{split} \end{equation} as $k\to \infty$. By \eqref{e3.11}, \eqref{e3.12}, the H\"{o}lder inequality and Lemma \ref{lem2.2}, we obtain \begin{equation} \label{e3.13} \begin{split} &{I_{\lambda ,\mu }}({u_k}) - \frac{1}{q}\langle {\nabla {I_{\lambda ,\mu }} ({u_k}),{u_k}} \rangle \\ &= (\frac{1}{2} - \frac{1}{q})\| {{u_k}} \|_{{H_{2m}}}^2 + (\frac{1}{q} - \frac{1}{N})\int_\Omega {{{| {{u_k}} |}^N}} -(1-\frac{1}{q})\mu \int_\Omega {f(x){u_k}} \\ & \ge (\frac{1}{2} - \frac{1}{q})\| u_k \|_{{H_{2m}}}^2 - (1-\frac{1}{q})\mu \int_\Omega {f(x){u_k}} \\ & \ge (\frac{1}{2} - \frac{1}{q})\| u_k \|_{{H_{2m}}}^2 - (1 - \frac{1}{q})\mu\max_{x \in \Omega } f(x){| \Omega |^{1 - \frac{1}{N}}}(C^\ast){\| u_k \|_{{H_{2m}}}}. \end{split} \end{equation} It follows from \eqref{e3.11}, \eqref{e3.12} and \eqref{e3.13} that \begin{align*} &o(1)+{C_{\lambda ,\mu }} + o(1){\| u_k \|_{{H_{2m}}}} \\ &\ge (\frac{1}{2} - \frac{1}{q})\| u_k \|_{{H_{2m}}}^2 - (1 - \frac{1}{q})\mu\max_{x \in \Omega } f(x){| \Omega |^{1 - \frac{1}{N}}}(C^\ast){\| u_k \|_{{H_{2m}}}}, \end{align*} i.e. \begin{equation} \label{e3.14} \begin{aligned} &(\frac{1}{2} - \frac{1}{q})\| u_k \|_{{H_{2m}}}^2 \\ &\leq o(1)+{C_{\lambda ,\mu }} + \big[ {(1 - \frac{1}{q}) \mu\max_{x \in \Omega } f(x){{| \Omega |}^{1 - \frac{1}{N}}}({C^ * }) + o(1)} \big]{\| u_k \|_{{H_{2m}}}}, \end{aligned} \end{equation} where $q>2$. Hence, for each $\lambda ~,~\mu > 0$, fixed $ {C_{\lambda ,\mu }} \in \mathbb {R}$, we conclude that the sequence $\{u_k\}$ is bounded in $ H_0^{2m}(\Omega )$. Now, we show that the $(PS)_{C_{\lambda ,\mu }}$ sequence contains a strongly convergent subsequence. Since the sequence $\{u_k\}$ is bounded in ${H_0^{2m}(\Omega )}$ and the well-known Sobolev's embedding, there exists a subsequence, still denoted by $\{u_k\}$, and $ u \in {H_0^{2m}(\Omega )}$, such that \begin{gather*} {u_k} \rightharpoonup u\quad \text{weakly in } {H_0^{2m}(\Omega )},\\ {u_k} \to u\quad \text{strongly in ${L^{{i}}}(\Omega )$, for $1 < i$} < N = \frac{{2n}}{{n - 4m }},\\ \nabla {u_k} \to \nabla u\quad \text{strongly in }{L^{{2}}}(\Omega ),\\ \Delta {u_k} \to \Delta u\quad \text{strongly in }{L^{{2}}}(\Omega ),\\ \dots\\ \nabla {\Delta ^{m - 1}}{u_k} \to \nabla {\Delta ^{m - 1}}u,\quad \text{strongly in }{L^{{2}}}(\Omega ),\\ {u_k} \to u\quad\text{a.e. in }\Omega. \end{gather*} Thus \begin{gather*} \int_\Omega {u_k^2} \to \int_\Omega u^2,\\ \int_\Omega {{{| {\nabla {u_k}} |}^2}} \to {\int_\Omega {| {\nabla u} |} ^2},\\ \dots \\ \int_\Omega {{{| {\nabla {\Delta ^{m - 1}}{u_k}} |}^2}} \to {\int_\Omega {| {\nabla {\Delta ^{m - 1}}u} |} ^2},\\ \int_\Omega {{{| {{u_k}} |}^q}} \to \int_\Omega {{{| u |}^q}},~q 0$ and $\mu $ is sufficiently small, we have \[ \frac{\mu}{2}| { \int_\Omega {f{u_k}} } | < \varepsilon. \] Thus for any $\lambda >0$ and $\mu $ is sufficiently small, we obtain \begin{equation} \label{e3.23} (\frac{1}{2} - \frac{1}{N})\| {{u_k}} \|_N^N \leq {C_{\lambda ,\mu }}. \end{equation} By the assumption $0 < {C_{\lambda ,\mu }} < \frac{{2m}}{n}{({C^ * })^{-n/(2m)}}$, we have thus \eqref{e3.21}. \end{proof} \begin{lemma} \label{lem3.4} For all $\lambda > 0$ and $\mu > 0$, the function ${I_{\lambda ,\mu }(u)}$ is weak lower semicontinuous on the set $$ \{ {u \in H_0^{2m}(\Omega ):{{\| u \|}_{{H_{2m}}}} \le {r_0}}\}, $$ where ${r_0} = \big( {\frac{N}{{{2^{2N - 2}}{{({C^*})}^N}}}} \big)^{1/(N - 2)}$. \end{lemma} \begin{proof} Let $\{u_k\}$ be a sequence in ${H_0^{2m}(\Omega )}$, and $0< r < \big( \frac{N}{2^{2N - 3}(C^\ast)^N}\big)^{1/(N - 2)}$, such that $$ {u_k} \rightharpoonup u , \quad\text{in ${H_0^{2m}(\Omega )}$ and } {{{\| {{u_k}} \|}_{{H_{2m}}}} \le r}. $$ Then we have ${\| u \|_{{H_{2m}}}} \le r$. Up to a subsequence, we obtain \begin{gather*} {u_k} \to u,\quad \text{strongly in $L^p(\Omega )$, for all }p < N,\\ {u_k} \to u\quad\text{a.e. in }\Omega. \end{gather*} Thus \begin{gather} \label{e3.24} \int_\Omega {{{| {{u_k}} |}^q}} \to \int_\Omega {{{| u |}^q}},\quad 20$. For ${\mu _0} > 0$ small enough such that for any $0 < \mu < {\mu _0}$, then \eqref{eP} has a solution with negative energy. \end{proposition} \begin{proof} Since $f(x)$ is continuous and not identical to 0 in $\Omega$, then there exists $ \phi \in H_0^{2m}(\Omega )$, such that $\int_\Omega {f(x)\phi} > 0$. For any $t > 0$, we have \begin{equation} \label{e4.1} {I_{\lambda ,\mu }}(t\phi ) = \frac{1}{2}{t^2}\| \phi \|_{{H_{2m}}}^2 - \frac{1}{N}{t^N}\| \phi \|_N^N - \lambda \frac{1}{q}{t^q}\| \phi \|_q^q - \mu t\int_\Omega {f(x)\phi }. \end{equation} Hence, there exists ${t_0}(\lambda,\mu) > 0$, such that $ 00 $ and $v \in H_0^{2m}(\Omega )$ with ${\| v \|_{{H_{2m}}}} \le r_0 $, such that \begin{equation} \label{e4.2} {I_{\lambda ,u}}(v) = \inf _{{{\| u \|}_{{H_{2m}}}} \le r_0 } {I_{\lambda ,\mu }}(u) < 0. \end{equation} Thus $v$ is a weak solution of \eqref{eP} with negative energy. \end{proof} \begin{proposition} \label{prop4.2} Suppose $f(x)$ is continuous and not identical to 0 in $\Omega$. If $\lambda > 0$ is sufficiently large and $\mu >0$ is enough small, then \eqref{eP} has a weak solution with positive energy. \end{proposition} \begin{proof} Lemma \ref{lem3.1} implies that ${I_{\lambda, \mu}}(u)$ satisfies the condition (I1) in Lemma \ref{lem2.1}. On the other hand, from \eqref{e4.1}, we obtain \[ \lim _{t \to +\infty } {I_{\lambda ,\mu}}(t \phi ) = - \infty. \] There exists a constant $T>0$, taking $e=T\phi$ with ${\| e \|_{{H_{2m}}}} > \delta $ such that $$ {I_{\lambda, \mu}}(e) < 0, $$ where $\delta>0$ is the constant in Lemma \ref{lem3.1}. Thus the condition (I2) of Lemma \ref{lem2.1} holds. Denote \begin{equation} \label{e4.3} {C_{\lambda ,\mu }} = \inf _{\gamma \in \Gamma } \sup_{t \in [0,1]} {I_{\lambda ,\mu }}(\gamma (t)), \end{equation} where \[ \Gamma = \{ \gamma \in C([0,1],H_0^{2m}(\Omega )):\gamma (0) = 0,\quad \gamma (1) = e\}. \] From Lemma \ref{lem3.2} and \eqref{e4.3}, it follows that \[ 0 < {C_{\lambda ,\mu }} < \frac{{2m}}{n}{({C^ * })^{-n/(2m)}}. \] Applying Lemma \ref{lem2.1}, there exists a sequence $\{ {u_k}\} \subset H_0^{2m}(\Omega )$, such that $$ {I_{\lambda ,\mu }}(u_{k}) \to {C_{\lambda ,\mu }}, \quad\text{and}\quad \nabla {I_{\lambda ,\mu }}({u_k}) \to 0,\quad\text{as } k\to \infty. $$ By Lemma \ref{lem3.3}, there exists a subsequence of $\{ {u_k}\}$ which strongly converges to $u$ in $H_0^{2m}(\Omega )$. Thus ${I_{\lambda ,\mu }}(u)$ has a critical point $u$ with ${I_{\lambda ,\mu }}({u}) = {C_{\lambda ,\mu }} > 0$. Hence we obtain a weak solution of equation \eqref{eP} with positive energy. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] From Propositions \ref{prop4.1} and \ref{prop4.2}, problem \eqref{eP} has two distinctic solutions $u_1, u_2$ with $I_{\lambda ,\mu }(u_1)<0