\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 289, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/289\hfil Multiple solutions]
{Multiple solutions for biharmonic  elliptic problems
 with the second Hessian}

\author[F. Fang, C. Ji, B. Zhang \hfil EJDE-2016/289\hfilneg]
{Fei Fang, Chao Ji, Binlin Zhang}

\address{Fei Fang \newline
Department of Mathematics,
Beijing Technology and Business University, \newline
Beijing 100048, China}
\email{fangfei68@163.com}

\address{Chao Ji (corresponding author) \newline
Department of Mathematics,
East China University of Science and Technology, \newline
Shanghai 200237, China}
\email{jichao@ecust.edu.cn}

\address{Binlin Zhang \newline
Department of Mathematics,
Heilongjiang Institute of Technology, \newline
Harbin 150050,  China}
\email{zhangbinlin2012@163.com}

\thanks{Submitted August 18, 2016. Published October 26, 2016.}
\subjclass[2010]{35J50, 35J60, 35J62, 35J96}
\keywords{Biharmonic  elliptic problem; second Hessian; variational methods;
\hfill\break\indent  Nehari manifold}

\begin{abstract}
 In this article, we study the biharmonic elliptic problem with
 the secondnd Hessian
 \begin{gather*}
 \Delta^2u =S_2(D^2u)+\lambda f(x) |u|^{p-1}u,\quad \text{in }
 \Omega \subset \mathbb{R}^3,    \\
  u =\frac{\partial u}{\partial n}=0,   \quad \text{on } \partial\Omega,
 \end{gather*}
 where $f(x)\in C(\bar{\Omega})$ is  a sign-changing weight function.
 By using variational methods and  some properties  of the Nehari manifold,
 we prove that the biharmonic  elliptic problem has at least two nontrivial
 solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$, $0<p<1$.
In this work, we consider the  problem
\begin{equation}\label{e0.1}
\begin{gathered}
\Delta^2u=S_2(D^2u)+\lambda f(x) |u|^{p-1}u,\quad  \text{in } \Omega,    \\
      u=\frac{\partial u}{\partial n}=0,   \quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $f(x)\in C(\bar{\Omega})$ is a sign-changing weight function,
\[
S_2(D^2u)(x)=\sum_{1\leq i<j\leq N} \lambda_i(x)\lambda_j(x), 
\]
$\lambda_i$,
($i=1,\cdots,N$) are the solutions of the equation
$$
\det  (\lambda I-D^2u(x))=0,
$$
and $\Delta^2$ the bi-Laplacian operator.

The case $N = 2$ appears as the stationary part of a model of epitaxial
growth of crystals (see \cite{r1,r9}) initially studied in \cite{r2}.
In dimension $N = 3$ the model can be seen as the
stationary part of a 3-dimensional growth problem driven by the scalar
curvature.

For the case $n=2$, the equation is expressed by the  formula
\begin{equation}\label{e0.2001}
\Delta^2u=\det (D^2u)+\lambda f(x) u,\quad  \text{in }  \Omega \subset \mathbb{R}^2.
\end{equation}
In this case, \eqref{e0.2001} was studied by Escudero and Peral \cite{r2}.
For a Dirichlet boundary condition,
they used  variational methods to prove that
\eqref{e0.2001} has at least two solutions.
However under the Navier boundary condition,  \eqref{e0.2001}
does not have a variational characteristic, so the authors
used fixed point arguments to obtain existence of solutions.

For the evolution formula of  problem \eqref{e0.2001},
Escudero, Gazzola, and Peral  \cite{r6}  proved  existence of local
solutions for arbitrary data and  existence of global solutions for small data.
 Moreover,  by exploiting the boundary conditions and the variational structure
of the equation, according to the size of the data the authors  proved
finite time blow-up of the solution and (or) convergence to a stationary
solution for global solutions.

 For problem \eqref{e0.1},   Ferrari, Medina and  Peral \cite{r4} obtained
the following results for  $f(x)\equiv 1$:
\begin{itemize}
\item[(1)] If $p < 1$ there exists a $\lambda_0 > 0$ such that if
$0 < \lambda < \lambda_0$, problem \eqref{e0.1} has at least two
nontrivial solutions.

\item[(2)]  If $p > 1$  problem \eqref{e0.1}  has at least one nontrivial
solution for every $\lambda\geq 0$.

\item[(3)] If $p = 1$  problem \eqref{e0.1}  has at least one nontrivial
solution whenever $0 < \lambda < \lambda_1$,
where $\lambda_1$ denotes the first eigenvalue of $\Delta^2$ in $\Omega$
with Dirichlet boundary conditions.
\end{itemize}

In the high dimensional case,  Escudero and  Torres \cite{r8}  proved the
existence of radial solutions for the  problem
$$
\Delta^2u=(-1)^kS_k[u]+\lambda f(x),\quad  \text{in } B_1(0)\subset \mathbb{R}^N,
$$
provided either with Dirichlet boundary conditions or Navier boundary conditions,
where the $k$-Hessian $S_k[u]$ is the $k$-th
elementary symmetric polynomial of eigenvalues of the Hessian matrix.

We can state now the following result.

\begin{theorem}\label{thm1}
Let $0<p<1$.  There exists $\lambda_0 > 0$ such that for each
$\lambda \in (0, \lambda_0)$, problem \eqref{e0.1} has at least two nontrivial
solutions.
\end{theorem}

As in \cite{r4}, we will use variational  methods and some properties of
the Nehari manifold to obtain  two nontrivial solutions.
For a study on variational methods and their applications, we refer the reader
 to  \cite{r21, r19, r20,  r22, r23}.
The Nehari manifold was  introduced by Nehari in \cite{r10}
and  has been widely used; see \cite{r11,r12,r13, r26,  r27, r18, r14,r15,r16,r17}.

The main idea for the proof or theorem  \ref{thm1} is dividing the Nehari
manifold into two parts and then considering the  minimum of the functional
on each part.
This article is organized as follows.
In Section 2, we give some preliminary lemmas.
In Section 3, we present the proof of Theorem \ref{thm1}.

\section{Preliminaries}

To use variational methods and some properties of the Nehari manifold,
we firstly define the corresponding functional and Nehari manifold  with
respect to problem \eqref{e0.1}.
The energy functional for problem \eqref{e0.1} is
\begin{equation}\label{e0}
  I(u)=\frac{1}{2} \int_{\Omega} |\Delta u|^2dx
-\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx
-\frac{\lambda}{p+1}\int_{\Omega} f(x)|u|^{p+1}dx,
\end{equation}
$u\in W_0^{2,2}(\Omega)$.
From \cite{r4}, we know that
\begin{gather*}
\begin{aligned}
(I'(u),v)
&=\int_{\Omega}\Delta u\Delta v\,dx
 -\int_{\Omega} \sum_{1\leq i<j\leq N}\left(\partial_iu\partial_ju\partial_{ij}v
+\partial_{j}u\partial_{ij}u\partial_{i}v
+\partial_iu\partial_{ji}u\partial_{j}v\right)dx \\
&\quad - \int_{\Omega}\lambda f(x)|u|^p v
\end{aligned} \\
J(u)=(I'(u),u)=\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
- \int_{\Omega}\lambda f(x)|u|^{p+1}dx,
\\
(J'(u),u)=2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu\partial_ju\partial_{ij}u\,dx
 - (p+1)\int_{\Omega}\lambda  f(x)|u|^{p+1}dx.
\end{gather*}
As the energy functional $I$ is not bounded on $W_0^{2,2}(\Omega)$,
it is useful to consider the functional on the Nehari manifold
$$
\mathcal{N}=\{u:(I'(u),u)=0\}.
$$
Furthermore, we consider the minimization problem: for $\lambda>0$
$$
\alpha=\inf\{I(u): u\in \mathcal{N}\}.
$$
The Nehari manifold $\mathcal{N}$  can be split three parts:
\[
\mathcal{N}^{+}=\{u:\left(J'(u),u\right)>0\},\quad
\mathcal{N}^{0}=\{u:\left(J'(u),u\right)=0\} \\
\mathcal{N}^{-}=\{u:\left(J'(u),u\right)<0\}.
\]

\begin{lemma}\label{lem1}
   There exists $\lambda_1>0$ such that for each $\lambda\in(0,\lambda_1)$,
$\mathcal{N}^{0}=\emptyset$
 \end{lemma}

\begin{proof}
We consider the following two cases.
\smallskip

\noindent\textbf{Case 1.}
 Assume that $u\in \mathcal{N}$ and   $\int_{\Omega}\lambda f(x)|u|^{p+1}dx=0$.
This implies
$$
(I'(u),u)=\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx=0.
$$
Hence,
$$
(J'(u),u)=-\int_{\Omega}|\Delta u|^2dx<0
$$
and so $u\not\in\mathcal{N}^{0}$
\smallskip

\noindent\textbf{Case 2.}   $u\in \mathcal{N}$ and
 $\int_{\Omega}\lambda f(x)|u|^{p+1}dx\neq 0$.
Assume that $\mathcal{N}^{0}\neq \emptyset$ for all $\lambda>0$.
If $u\in \mathcal{N}^{0}$, then
\begin{equation} \label{e1}
\begin{aligned}
 0=(J'(u),u)
&=2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx \\
&\quad - (p+1)\int_{\Omega}\lambda  f(x)|u|^{p+1}dx \\
 &=(1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)
 \int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx.
\end{aligned}
\end{equation}
Therefore,
\begin{gather}\label{e1.1}
 \int_{\Omega}|\Delta u|^2dx
=\frac{(6-3p)}{(1-p)}\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu\partial_ju\partial_{ij}u\,dx, \\
\label{e2} \begin{aligned}
  \int_{\Omega}\lambda f(x)|u|^{p+1}dx
&=\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx  \\
& =\frac{3}{1-p} \int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx.
\end{aligned}
\end{gather}
Moreover, using H\"{o}lder's inequality, one has
\begin{equation} \label{e3}
\begin{aligned}
  \frac{1}{(2-p)}\int_{\Omega}|\Delta u|^2dx
&=\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx  \\
& =\lambda\int_{\Omega} f(x)|u|^{p+1}dx
  \leq \lambda \|f\|_{L^{m}}\|u\|_{1+q}^{p+1}  \\
&\leq  \lambda \|f\|_{L^{m}}S^{p+1}\Big(\int_{\Omega} |\Delta u|^2dx
\Big)^{\frac{p+1}{2}},
\end{aligned}
\end{equation}
where $m=\frac{1+q}{q-p}$ (so the  conjugate index $m'=\frac{1+q}{p+1}$),
$q+1<\frac{2N}{N-4}$.
By \eqref{e3}, we have
\begin{equation}\label{e4}
  \Big(\int_{\Omega} |\Delta u|^2dx\Big)^{\frac{1-p}{2}}
\leq \lambda(2-p)\|f\|_{L^m}S^{p+1}.
\end{equation}
or
\begin{equation*}
  \Big(\int_{\Omega} |\Delta u|^2dx\Big)
\leq \left(\lambda(2-p)\|f\|_{L^m}S^{p+1}\right)^{\frac{2}{p-1}}.
\end{equation*}
Define the following functional on $W_0^{2,2}(\Omega)$,
$$
A(u)=K(p,q)\Big[\frac{(\int_{\Omega} |\Delta u|^2dx)^{q}}
{\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij}
u\partial_i u\partial_ju\,dx}\Big]^{\frac{1}{q-1}}
-\int_{\Omega} \lambda f(x) |u|^{p+1}dx,
$$
where
\[
K(p,q)=\frac{3}{1-p}\Big(\frac{1-p}{6-3p}\Big)^{\frac{q}{q-1}}.
\]
Then by \eqref{e1.1} and \eqref{e2}, we have $A(u)=0$.

On the other hand, for $u\in W_0^{2,2}(\Omega)$, we have
\begin{equation}\label{e4.1}
\begin{aligned}
\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx
&\leq C\Big(\int_{\Omega} |\Delta u|^2dx\Big)^{1/2}
\Big(\int_{\Omega} |\nabla u|^4dx\Big)^{1/2}\\
&\leq C\Big(\int_{\Omega} |\Delta u|^2dx\Big)^{3/2}
\end{aligned}
\end{equation}
Then using \eqref{e3}, \eqref{e4}, the Holder inequality and Sobolev inequality,
for $u\in \mathcal{N}^0$, we deduce
 %\label{e5}
\begin{align*}
  A(u) & \geq  K(p,q)\Big[\frac{\big(\int_{\Omega} |\Delta u|^2dx\big)^{q}}
{\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}
 \Big]^{\frac{1}{q-1}}-\lambda\|f\|_{L^{m}}\|u\|_{1+q}^{p+1} \\
& \geq K(p,q)\Big[\frac{\big(\int_{\Omega} |\Delta u|^2dx\big)^{q}}
 {C\big(\int_{\Omega} |\Delta u|^2dx\big)^{3/2}}\Big]^{\frac{1}{q-1}}
-C\lambda\|f\|_{L^{m}} \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}} \\
&\geq  \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}}
\Big[ K(p,q)\Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{q-2-pq+p}{2(q-1)}}
 -C\lambda\|f\|_{L^{m}}\Big] \\
&\geq  \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}}
\Big[ K(p,q)\Big((\lambda(2-p)\|f\|_{L^m}S^{p+1})^{\frac{2}{p-1}}
 \Big)^{\frac{q(1-p)+p-2}{2(q-1)}} \\
&\quad  -C\lambda\|f\|_{L^{m}}\Big].
\end{align*}
Since
\[
\frac{q(1-p)+p-2}{2(q-1)}\cdot \frac{2}{p-1}<0,
\]
for $\lambda$ sufficiently small, we have $A(u)>0$. This contradicts $A(u)=0$. Hence
we can conclude that there exits $\lambda_1>0$ such that for
$\lambda\in (0,\lambda_1)$, $\mathcal{N}^0=\emptyset$.
\end{proof}

\begin{lemma}\label{lem2}
  If $u\in \mathcal{N}^{+}$, then $\int_{\Omega}\lambda  f(x)|u|^{p+1}dx>0$.
\end{lemma}

\begin{proof}
For $u\in \mathcal{N}^{+}$, we have
\begin{gather*}
\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
- \int_{\Omega}\lambda f(x)|u|^{p+1}dx=0,\\
2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
- (p+1)\int_{\Omega}\lambda  f(x)|u|^{p+1}dx>0.
\end{gather*}
Combining the above two formulas, we have
$$
(2-p)\int_{\Omega}\lambda f(x)|u|^{p+1}dx>\int_{\Omega}|\Delta u|^2dx>0.
$$
This completes the proof.
\end{proof}

According to Lemma \ref{lem2},  for $\lambda\in (0,\lambda_1)$,  we can write
$\mathcal{N}=\mathcal{N}^{+}\cup \mathcal{N}^{-}$ and define
$$
\alpha^{+}=\inf_{u\in \mathcal{N}^{+}}I(u),\quad
\alpha^{-}=\inf_{u\in \mathcal{N}^{-}}I(u).
$$
Next we show that the minimizers on $\mathcal{N}$ are the critical points for $I$.
We denote the dual space of $W_{0}^{2,2}(\Omega)$ by
$\big(W_{0}^{2,2}(\Omega)\big)^{\ast}$.

\begin{lemma}\label{lem3}
  For $\lambda\in (0,\lambda_1)$, if $u_0$ is a local minimizer for $I(u)$
on $\mathcal{N}$, then $I'(u_0)=0$ in $\big((W_{0}^{2,2}(\Omega)\big)^{\ast}$.
\end{lemma}

\begin{proof}
If $u_0$ is a local minimizer for $I(u)$ on $\mathcal{N}$, then $u_0$
is a solution of the optimization problem
$$
\text{minimize $I(u)$ subject to $J(u) = 0$}.
$$
Hence, by the theory of Lagrange multipliers, there exists
$\theta\in  \mathbb{R}$ such that
$$
I'(u_0) = \theta J'(u_0)\quad \text{in }  \big(W_{0}^{2,2}(\Omega)\big)^{\ast}
$$
Thus,
\begin{equation}\label{e6}
 (I'(u_0),u_0) = \theta (J'(u_0),u_0).
\end{equation}
Since $u_0 \in \mathcal{N}$,  $(I'(u_0),u_0) = 0$.  Moreover, since
$N=\emptyset$,  $(J'(u_0),u_0) \neq  0$ and by \eqref{e6}, $ \theta= 0$.
This completes the proof.
\end{proof}

For $u\in W_{0}^{2,2}(\Omega)$, we write
$$
t_{\rm max}=\frac{(1-p)\int_{\Omega} |\Delta u|^2dx}{(6-3p)\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}.
$$

\begin{lemma}\label{lem4}
\begin{itemize}
\item [(1)] If $\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij}
 u\partial_i u\partial_ju\,dx<0$  $(\geq 0)$, then there exits a unique
 $t^{-}>0$ $(t^{+}>0)$ such that $t^{-}u\in \mathcal{N}^{+}$
$(t^{+}u\in \mathcal{N}^{-})$ and $I(t^-u)=\min_{t> 0} I(tu)$
$(I(t^-u)=\max_{t> 0} I(tu))$;

\item[(2)] $t^-(u)$ is a continuous function for nonzero $u$;

\item[(3)]
\[
\mathcal{N}^{+}= \big\{u\in W_0^{2,2}(\Omega)\setminus\{0\}:
t^{-}\Big(\frac{u}{\|u\|}\Big)\frac{1}{\|u\|}=1\big\}.
\]
\end{itemize}
\end{lemma}

\begin{proof}
(1) We firstly define
\begin{equation}\label{e6.1}
\begin{aligned}
 i(t):= I(tu)
&=\frac{t^2}{2} \int_{\Omega} |\Delta u|^2dx-t^3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx \\
&\quad -t^{p+1}\int_{\Omega} \frac{\lambda f(x)}{p+1} |u|^{p+1}dx\,.
\end{aligned}
\end{equation}
We easily compute
\begin{equation}\label{e6.2}
\begin{aligned}
 i'(t):= I'(tu)
&=t \int_{\Omega} |\Delta u|^2dx-3t^2\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx \\
&\quad -t^{p}\int_{\Omega} \lambda f(x) |u|^{p+1}dx
\end{aligned}
\end{equation}
and
\begin{equation} \label{e7}
\begin{aligned}
&(I'(tu),tu) \\
&=t^2\int_{\Omega}|\Delta u|^2dx-3t^3\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu\partial_ju\partial_{ij}u\,dx
- t^{p+1}\int_{\Omega}\lambda f(x)|u|^{p+1}dx  \\
 &=ti'(t)
\end{aligned}
\end{equation}
We distinguish the following two cases.
\smallskip

\noindent\textbf{Case i.}
$\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx<0$.
In this case, $i(t)$ is convex and achieves its minimum  at $t^{-}$ and
$t^{-}\neq 0$. Thus, using \eqref{e6.1} and \eqref{e7}, we obtain
$t^{-}u\in \mathcal{N}^{+}$ and
$$
I''(t)>0\quad \text{for } t=t^{-}.
$$
\smallskip

\noindent\textbf{Case ii.}
$\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx>0$.
Let
$$
s(t)=t^{1-p}\int_{\Omega}|\Delta u|^2dx-3t^{2-p}\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx.
$$
It is easy to show that $s(0)=0$, $s(t)\to  -\infty$ as
$t\to +\infty$ is convex and achieves its maximum at
$$
t_{\rm max}=\frac{(1-p)\int_{\Omega} |\Delta u|^2dx}
{(6-3p)\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij}
 u\partial_i u\partial_ju\,dx}.
$$
Then, using \eqref{e4.1} we obtain
 %\label{e6.3}
\begin{align*}
&s(t_{\rm max}) \\
&= s(t)\\
&=\Big(\frac{(1-p)\int_{\Omega} |\Delta u|^2dx}{(6-3p)
 \int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}
 \Big)^{1-p}\int_{\Omega}|\Delta u|^2dx \\
 &\quad -3\Big(\frac{(1-p)\int_{\Omega} |\Delta u|^2dx}{(6-3p)\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}\Big)^{2-p}
 \int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx. \\
&=(3p-2)\int_{\Omega}|\Delta u|^2dx
\Big(\frac{(1-p)\int_{\Omega} |\Delta u|^2dx}{(6-3p)\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}\Big)^{1-p} \\
&\geq C_1\Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{1+p}{2}}.
\end{align*}
From the above inequality,  there exists a $\lambda_0$ such that  for $\lambda\in (0, \lambda_0)$ small,
\begin{equation} \label{e7.1}
\begin{aligned}
s(0)
&=0<\lambda \int_{\Omega}  f(x) |u|^{p+1}dx \\
& =\lambda\int_{\Omega} f(x)|u|^{p+1}dx
 \leq \lambda \|f\|_{L^{m}}\|u\|_{1+q}^{p+1}  \\
&\leq  \lambda \|f\|_{L^{m}}S^{p+1}\Big(\int_{\Omega} |\Delta u|^2dx
\Big)^{\frac{p+1}{2}}
 \leq s(t_{\rm max}).
\end{aligned}
\end{equation}
where $m=\frac{1+q}{q-p}$  (so the  conjugate index $m'=\frac{1+q}{p+1}$),
 $q+1<\frac{2N}{N-4}$.

Using \eqref{e7.1}, we easily deduce that there are unique values
$t^{+}$ and $t^{-}$ such that $0<t^{+}<t_{\rm max}<t^{-}$,
\begin{gather*}
s(t^{+})=\lambda \int_{\Omega} f(x) |u|^{p+1}dx= s(t^{-}), \\
s'(t^{+})>0>s'(t^{-}).
\end{gather*}
We have $t^{+}u\in \mathcal{N}^{+}$, $t^{-}u\in \mathcal{N}^{-}$, and
$I(t^{-}u)\geq I(tu)\geq I(t^{+}u)$ for each $t\in [t^{+}, t^{-}]$ and
$I(t^{+}u)\leq I(tu)$ for each $t\in [0, t^{+}]$. Thus
$$
I(t^{-}u)=\max_{t\geq t_{\rm max}}I(tu),\quad
 I(t^{+}u)=\min_{0\leq t\leq  t^{-}}I(tu).
$$
In this case, $i(t)$ is concave and achieves its maximum  at $t^{+}$
and $t^{+}\neq 0$. Thus, using \eqref{e6.1} and \eqref{e7}, we obtain
$t^{+}u\in \mathcal{N}^{-}$ and
$$
I''(t)<0\quad \text{for } t=t^{+}.
$$

(2) By the uniqueness of $t^{-}(u)$ and the external property of $t^{-}(u)$,
we have that $t^{-}(u)$ is a continuous function of $u\neq 0$.

(3) For $u\in \mathcal{N}^{+}$, let $v=\frac{u}{\|u\|}$. Using the discussion (1),
there exists an $t^{-}>0$ such that
$t^{-}v\in \mathcal{N}^{+}$, that is
$t^{-}\Big(\frac{u}{\|u\|}\Big)\frac{u}{\|u\|}\in \mathcal{N}^{+}$.
Since $u \in \mathcal{N}^{+}$, we obtain
$t^{-}\big(\frac{u}{\|u\|}\big)\frac{1}{\|u\|}=1$. This shows that
$$
\mathcal{N}^{+}\subset \big\{u\in W_0^{2,2}(\Omega)\setminus\{0\}:
t^{-}\Big(\frac{u}{\|u\|}\Big)\frac{1}{\|u\|}=1\big\}.
$$
Conversely, let $u\in W_0^{2,2}(\Omega)\setminus\{0\}$ such that
$t^{-}\Big(\frac{u}{\|u\|}\Big)\frac{1}{\|u\|}=1$, then
$$
t^{-}\Big(\frac{u}{\|u\|}\Big)\frac{u}{\|u\|}\in \mathcal{N}^{+}.
$$
Hence,
$$
\mathcal{N}^{+}= \big\{u\in W_0^{2,2}(\Omega)\setminus\{0\}: t^{-}
\Big(\frac{u}{\|u\|}\Big)\frac{1}{\|u\|}=1\big\}.
$$
\end{proof}

Now we consider the  degenerate equation
 \begin{equation}\label{e8}
\begin{gathered}
\Delta^2u=S_2(D^2u),\quad  \text{in }  \Omega,    \\
      u=\frac{\partial u}{\partial n}=0,   \quad  \text{on } \partial\Omega.
\end{gathered}
\end{equation}
The  functional corresponding to \eqref{e8} is
$$
H(u)=\frac{1}{2} \int_{\Omega} |\Delta u|^2dx
-\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx\,.
$$
We consider the minimization problem
$$
\beta=\inf\{H(u):u\in N\},
$$
where $N=\{u:u\in W_0^{2,2}(\Omega)\setminus\{0\}:(H'(u),u)=0$\}.
Next we show that problem \eqref{e8} has a nontrivial solution
$\omega_0$ such that $H(\omega_0)=\beta>0$.

\begin{lemma}\label{lem5}
  For any $u\in W_0^{2,2}(\Omega)\setminus\{0\}$, there exits an unique $t(u)>0$
such $t(u)u\in N$.   The maximum of $H(tu)$ for $t\geq 0$ is achieved at $t=t(u)$.
The function
  $$
W_0^{2,2}(\Omega)\backslash\{0\}\to (0,+\infty):\ u\to  t(u)
$$
  is continuous and  defines a homeomorphism of the unit sphere of
  $W_0^{2,2}(\Omega)$ with $N$.
\end{lemma}

\begin{proof}
  Let $u\in W_0^{2,2}(\Omega)\backslash\{0\}$ be fixed and define the function $g(t):=H(tu)$ on $[0,\infty)$. Obviously, we obtain
  \begin{align}\label{e9}
    g'(t)=0 & \Leftrightarrow tu\in N  \\
     & \Leftrightarrow \int_{\Omega} |\Delta u|^2dx=3t\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx=0.
  \end{align}

If for all $u\in W_0^{2,2}(\Omega)$, it holds
$\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx\leq 0$,
 then $0$ is an unique critical   point of $H(u)$.
 And if $\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i
u\partial_ju\,dx> 0$, using the mountain pass theorem, we can show that
$H(u)$ has a  nontrivial critical point. So for each
$u\in W_0^{2,2}(\Omega)\backslash\{0\}$, it is easy to verify that
$g(0) = 0$ and $g(t) > 0$ for $t > 0$ small and $g(t) < 0$ for $t > 0$ large.
Therefore $\max_{[0,\infty)} g(t)$ is achieved
at an unique $t = t(u)$ such that $g'(t(u)) = 0$ and $t(u)u\in N$.
To prove the continuity of $t(u)$, assume that $u_n \to  u$ in
$W_0^{2,2}(\Omega)\backslash\{0\}$. It is easy to verify that
$\{t(u_n)\}$ is bounded. If a subsequence of $\{t(u_n)\}$ converges to $t_0$,
it follows from \eqref{e9} that $t_0 = t(u)$, but then $t(u_n) \to  t(u)$.
 Finally the continuous map from the unit sphere of
$W_0^{2,2}(\Omega)\backslash\{0\} \to  N$, $u \to  t(u)u$, is inverse
of the retraction $u \to  \frac{u}{\|u\|_a}$.
\end{proof}

Define
$$
c_1:=\inf_{u\in W_0^{2,2}(\Omega)\backslash\{0\}}\max_{t\geq 0}H(tu), \quad
c:=\inf_{r\in \Gamma}\max_{t\in[0,1]}H(\gamma(tu)),
$$
where
\[
\Gamma:=\big\{\gamma\in C[0,1], W_0^{2,2}(\Omega):
 \gamma(0)=0, H(\gamma(1))<0 \big\}.
\]


\begin{lemma}\label{lem6}
  $c_1=c=\beta>0$ and $c$ is a critical value of $H(u)$.
\end{lemma}

\begin{proof}
From Lemma \ref{lem5}, we easily know that  $\beta = c_1$.
Since $H(tu) < 0$ for $ u\in W_0^{2,2}(\Omega)\backslash\{0\}$
and $t$ large, we have $c \leq c_1$. The manifold $\mathcal{N}$ separates
$W_0^{2,2}(\Omega)$ into two components. The component containing the
origin also contains a small ball around the origin.
 Moreover $H(u) \geq 0$ for all $u$ in this component, because
$(H(tu), u) \geq 0$ for all $0 \geq t \geq t(u)$. 
Thus every $\gamma\in \Gamma$ has to cross $N$ and $\beta\leq c$. 
Since the embedding $W_0^{2,2}(\Omega) \hookrightarrow L^{m}(\Omega)\ (m<2^{\ast})$ 
is compact, it is easy to prove that $c > 0$ is
a critical value of $H(u)$ and $\omega_0$ a nontrivial solution corresponding to $c$.
\end{proof}

\begin{lemma}\label{lem7}
{\rm (1)} There exist $\hat{t}>0$ such that
$$
\alpha\leq \alpha^{+}<\frac{p-1}{6p+6} \hat{t}^2\beta<0.
$$
{\rm (2)}  $I(u)$ is coercive and bounded below on $\mathcal{N}$ for 
$\lambda$ sufficiently small.
\end{lemma}

\begin{proof}
(1) Let $\omega_0$ be a nontrivial solution of problem \eqref{e8}) such that 
$H(\omega_0) = \beta> 0$. Then
$$
\int_{\Omega}|\Delta \omega_0|^2dx-3\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_i\omega_0\partial_j\omega_0\partial_{ij}\omega_0dx=0.
$$
 Set $\hat{t}=t^{+}(\Omega)$ as defined by Lemma \ref{lem4}. Hence 
$\hat{t}\omega_0\in \mathcal{N}^{+}$  and
\begin{equation} \label{e10}
\begin{aligned}
   I(\hat{t}\omega_0)
&=\frac{\hat{t}^2}{2} \int_{\Omega} |\Delta \omega_0|^2dx
 -\hat{t}^3\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij}
 \omega_0\partial_i \omega_0\partial_j\omega_0dx \\
&\quad -\hat{t}^{p+1}\int_{\Omega} \frac{\lambda f(x)}{p+1} |\omega_0|^{p+1}dx   \\
&= \big(\frac{1}{2}-\frac{1}{p+1} \big) \hat{t}^2\int_{\Omega}
|\Delta \omega_0|^2dx \\
&\quad +\big(\frac{3}{p+1} -1\big)\hat{t}^3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_{ij} \omega_0\partial_i \omega_0\partial_j\omega_0dx
\\
&< \frac{p-1}{6p+6} \hat{t}^2\beta.
 \end{aligned}
\end{equation}
This yields
$$
\alpha\leq \alpha^{+}<\frac{p-1}{6p+6} \hat{t}^2\beta<0.
$$

(2) For $u\in \mathcal{N}$, we have
\begin{align*}
J(u)&=(I'(u),u) \\
&=\int_{\Omega}|\Delta u|^2dx-3\int_{\Omega} 
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
- \lambda\int_{\Omega} f(x)|u|^{p+1}dx=0.
\end{align*}
Then by H\"{o}lder and Young inequalities,
\begin{equation} \label{e11}
\begin{aligned}
  I(u)
&=\frac{1}{2} \int_{\Omega} |\Delta u|^2dx-\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx
  -\frac{\lambda}{p+1}\int_{\Omega} f(x) |u|^{p+1}dx \\
&=\frac{1}{6} \int_{\Omega} |\Delta u|^2dx
  - \big(\frac{\lambda }{p+1}-\frac{\lambda}{3}\big)\int_{\Omega} f(x)|u|^{p+1}dx \\
&\geq  \frac{1}{6} \int_{\Omega} |\Delta u|^2dx
 -\big(\frac{\lambda }{p+1}-\frac{\lambda}{3}\big) \|f\|_{L^{m}}S^{p+1}
 \Big(\int_{\Omega} |\Delta u|^2dx\Big)^{\frac{p+1}{2}} \\
&\geq \Big(\frac{1}{6}-\frac{2\lambda(2-p)}{3(p+1)^2}\Big)
 \int_{\Omega} |\Delta u|^2dx-\frac{\lambda(2-p)}{3(p+1)}
 \big(\|f\|_{L^{m}}S^{p+1}\big)^{\frac{2}{1-p}}.
\end{aligned}
\end{equation}
In  \eqref{e11},  since $p<1$, for $\lambda$ small, we have  $I(u)>0$
on $\mathcal{N}$. So we easily know that
 $I(u)$ is coercive and bounded below on $\mathcal{N}$ for  $\lambda$
 sufficiently small.
\end{proof}

\section{Proof of Theorem \ref{thm1}}
We need the following lemmas.

\begin{lemma}\label{lem8}
  For each $u\in \mathcal{N}$, there exist
 $\varepsilon > 0$ and a differentiable function
$\xi :B(0,\varepsilon)\subset W^{2,2}_0(\Omega)\to  \mathbb{R}^{+}$ 
such that $\xi(0)=1$,   the function $\xi(v)(u-v)\in \mathcal{N}$ and
\begin{align*}
&(\xi'(0),v) \\
&=\frac{2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega} 
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
   - \lambda(p+1)\int_{\Omega} f(x)|u|^{p+1}dx}{(1-p)
 \int_{\Omega}|\Delta u|^2dx-(6-3p)\int_{\Omega} 
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx},
\end{align*}
for all $v\in W_0^{2,2}(\Omega)$
\end{lemma}

\begin{proof}
For $u\in \mathcal{N}$, define  a function by 
$F:\mathbb{R}\times W_0^{2,2}(\Omega)\to  \mathbb{R}$ by
\begin{equation} \label{e12}
\begin{aligned}
&F_u(\xi,\omega)\\
&=(I(\xi(u-\omega), \xi(u-\omega))) \\
  &=\xi^2\int_{\Omega}|\Delta (u-\omega)|^2dx-3\xi^3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_i(u-\omega)\partial_j(u-\omega)
 \partial_{ij}(u-\omega)dx \\
&\quad - \lambda\xi^{p+1}\int_{\Omega} f(x)|(u-\omega)|^{p+1}dx.
\end{aligned}
\end{equation}
Then $F_u(1,0)=(I'(u),u)=0$ and
\begin{equation} \label{e13}
\begin{aligned}
  \frac{d}{dt}F_u(1,0)
&=2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx \\
&\quad - \lambda(p+1)\int_{\Omega} f(x)|u|^{p+1}dx. \\
&=(1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx\neq 0.
\end{aligned}
\end{equation}
According to the implicit function theorem, there exist $\varepsilon>0$
and a differentiable function
$\xi :B(0,\varepsilon)\subset W^{2,2}_0(\Omega)\to  \mathbb{R}^{+}$
such that $\xi(0) = 1$ and
\begin{align*}
&(\xi'(0),v)\\
&=\frac{2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
   - \lambda(p+1)\int_{\Omega} f(x)|u|^{p+1}dx}{(1-p)
 \int_{\Omega}|\Delta u|^2dx-(6-3p)\int_{\Omega}
 \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx}
\end{align*}
and
$$
F_u(\xi(v),v)=0\quad \text{for all } v\in B(0,\varepsilon);
$$
that is, $\xi(v)(u-v)\in \mathcal{N}$.
\end{proof}

Similarity, we have the following result.

\begin{lemma}\label{lem9}
  For each $u\in \mathcal{N}^{-}$, there exist
 $\varepsilon > 0$ and a differentiable function
$\xi^{-} :B(0,\varepsilon)\subset W^{2,2}_0(\Omega)\to  \mathbb{R}^{+}$ 
such that $\xi^{-}(0)=1$,   the function 
$\xi^{-}(v)(u-v)\in \mathcal{N^{-}}$ and
\begin{align*}
&(\xi'(0),v)\\
&=\frac{2\int_{\Omega}|\Delta u|^2dx-9\int_{\Omega} 
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
 - \lambda (p+1)\int_{\Omega} f(x)|u|^{p+1}dx}
 {(1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)\int_{\Omega} 
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx},
\end{align*}
for all $v\in W_0^{2,2}(\Omega)$.
\end{lemma}

\begin{proof}
As in the proof in Lemma \ref{lem8}, there exist $\varepsilon>0$ and a 
differentiable function 
$\xi^{-}: B(0,\varepsilon)\subset W_0^{2,2}(\Omega)\to  \mathbb{R}^{+}$ 
such that $\xi^{0}=1$ and $\xi^{-}(v)(u-v)\in \mathcal{N}$ for all 
$v\in B(0,\varepsilon)$. Since
\[
(J'(u),u)=(1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)
\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx<0.
\]
Thus, by the continuity of the function $J'(u)$ and $\xi^{-}$, we have
\begin{equation} \label{e14}
\begin{aligned}
&(J'(\xi^{-}(v)(u-v)),\xi^{-}(v)(u-v)) \\
&=(1-p)\int_{\Omega}|\Delta \left(\xi^{-}(v)(u-v)\right)|^2dx \\
&\quad -(6-3p)\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_i
\left(\xi^{-}(v)(u-v)\right)\partial_j\left(\xi^{-}(v)(u-v)\right) \\
&\quad
\times \partial_{ij}\left(\xi^{-}(v)(u-v)\right)dx<0.
\end{aligned}
\end{equation}
For $\varepsilon$ sufficiently small, this implies
 $\xi^{-}(v)(u-v)\in \mathcal{N}^{-}$.
\end{proof}


\begin{lemma}\label{lem10}
Let $\lambda_0=\inf\{\lambda_1, \lambda_2\}$.
\begin{itemize}
\item[(1)] There exists a minimizing sequences 
$\{u_n\}\subset \mathcal{N}$  such that
\begin{equation} \label{e15}
  I(u_n)=\alpha+o(1), \quad
  I'(u_n) =o(1)\quad \text{for } \big(W_0^{2,2}(\Omega)\big)^{\ast}.
\end{equation}

\item[(2)]
There exists a minimizing sequences $\{u_n\}\subset \mathcal{N}^{-}$  such that
\begin{equation} \label{e16}
  I(u_n)=\alpha^{-}+o(1) ,\quad
  I'(u_n) =o(1)\quad \text{for }\big(W_0^{2,2}(\Omega)\big)^{\ast}.
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
Using Lemma \ref{lem7} and Ekeland variational principle \cite{r3}, 
there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}$
such that
\begin{gather}\label{e17}
 I(u_n)<\alpha+\frac{1}{n}, \\
\label{e18}
  I(u_n)<I(\omega)+\frac{1}{n}\|\omega-u_n\|\quad \text{for each }
 \omega\in \mathcal{N}.
\end{gather}
By taking $n$ enough large, from Lemma \ref{lem7} (1), we have
\begin{equation} \label{e19}
\begin{aligned}
 I(u_n)
&= \frac{1}{2}\int_{\Omega}|\Delta u_n|^2dx
 -\frac{(2-p)\lambda}{p+1} \int_{\Omega} f(x)|u_n|^{p+1}dx \\
 &<\alpha+\frac{1}{n}<\frac{p-1}{6p+6} \hat{t}^2\beta<0.
\end{aligned}
\end{equation}
This implies
\begin{align}\label{e20}
   \|f\|_{L^{m}}S^{p+1}\Big(\int_{\Omega} |\Delta u_n|^2dx\Big)^{\frac{p+1}{2}}
\geq \int_{\Omega} f(x)|u_n|^{p+1}dx
>\frac{1-p}{6\lambda(2-p)}\hat{t}^2\beta.
\end{align}
Consequently $u_n\neq 0$ and combining the above two estimates and the
Holder inequality, we obtain
\begin{gather}\label{e21}
  \int_{\Omega} |\Delta u_n|^2dx>
\Big[\frac{1-p}{6\lambda(2-p)}\hat{t}^2\beta \|f\|^{-1}_{L^{m}}S^{-p-1}
 \Big]^{\frac{2}{p+1}}, \\
\label{e22}
  \int_{\Omega} |\Delta u_n|^2dx<\Big[\frac{(4-2p)\lambda}{(p+1)}
\|f\|_{L^{m}}S^{p+1}\Big]^{\frac{2}{1-p}}.
\end{gather}
Next we show that
$$
\|I'(u_n)\|_{(W_0^{2,2}(\Omega))^{\ast}}\to  0\quad\text{as } n\to  +\infty.
$$
Applying Lemma \ref{lem8} with $u_n$ to obtain the function
$\xi_n: B(0,\varepsilon_n)\subset W_0^{2,2}(\Omega)\to  \mathbb{R}^{+}$ for some
$\varepsilon_n>0$, such that $\xi_n(\omega)(u_n-\omega)\in \mathcal{N}$.
Choose $0<\rho<\varepsilon_n$. Let $u\in W_0^{2,2}(\Omega)$ with
$u\not \equiv 0$ and let $\omega_{\rho}=\frac{\rho u}{\|u\|}$.
We set $\eta_{\rho}=\xi_n(\xi_{\rho})(u-\omega_{\rho})$.
Since $\eta_{\rho}\in \mathcal{N}$,
we deduce that from \eqref{e18} that
$$
I(\eta_{\rho})-I(u_n)\geq -\frac{1}{n}\|\eta_{\rho}-u_n\|,
$$
and by the mean value theorem, we have
\begin{equation}\label{e23}
  (I'(u_n),\eta_{\rho}-u_n)+o(\|\eta_{\rho}-u_n\|\geq \frac{-1}{n}\|\eta_{\rho}-u_n\|).
\end{equation}
Thus
\begin{equation}\label{e24}
\begin{aligned}
&(I'(u_n),-\omega_{\rho})+(\xi_n(\omega_{\rho})-1)
(I'(u_n),(u_n-\omega_{\rho})) \\
&\geq -\frac{1}{n}\|\eta_{\rho}-u_n\|+o(\|\eta_{\rho}-u_n\|).
\end{aligned}
\end{equation}
It follows from $(\xi_n(\omega_{\rho}))(u_n-\omega_{\rho})\in \mathcal{N}$
and \eqref{e24} that
\begin{equation}\label{e24b}
\begin{aligned}
&-\rho\big(I'(u_n),\frac{u}{\|u\|}\big)
 +(\xi_n(\omega_{\rho})-1)(I'(u_n)-I'(\eta_{\rho}),(u_n-\omega_{\rho})) \\
&\geq -\frac{1}{n}\|\eta_{\rho}-u_n\|+o(\|\eta_{\rho}-u_n\|).
\end{aligned}
\end{equation}
Thus
\begin{equation}\label{e25}
\begin{aligned}
  \big(I'(u_n),\frac{u}{\|u\|}\big)
&\leq \frac{(\xi_n(\omega_{\rho})-1)}{\rho}(I'(u_n)-I'(\eta_{\rho}),
 (u_n-\omega_{\rho})) \\
&\quad + \frac{1}{n\rho}\|\eta_{\rho}-u_n\|+\frac{o(\|\eta_{\rho}-u_n\|)}{\rho}.
\end{aligned}
\end{equation}
Since
$\|\eta_{\rho}-u_n\|\leq |\xi_n(\omega_{\rho}-1)|\|u_n\|+\rho|\xi_n(\omega_{\rho})|$
and
$$
\lim_{\rho\to  0}\frac{|\xi_n(\omega_{\rho}-1)|}{\rho}\leq \|\xi'_n(0)\|.
$$
If we let $\rho\to  0$ in \eqref{e25} for a fixed $n$, then by \eqref{e22}
we can find a constant $C>0$,
independent of $\rho$, such that
\begin{equation}\label{e26}
  \big(I'(u_n),\frac{u}{\|u\|}\big)\leq \frac{C}{n}(1+\|\xi'_n(0)\|).
\end{equation}
We are done once we show that $\|\xi'_n(0)\|$ is uniformly bounded in $n$.
By  \eqref{e22} and Lemma \ref{lem8} and H\"{o}lder inequality, we have
\[
(\xi'(0),v)=\frac{b\|v\|}{\big|(1-p)\int_{\Omega}|\Delta u|^2dx
-(6-3p)\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx
\big|},
\]
for some $b>0$.
We only need to show that
\begin{equation}\label{e27}
  \Big|(1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx\Big|>0,
\end{equation}
for some $c>0$ and $n$ large enough. We argue by contradiction.
Assume that there exists a subsequence $\{u_n\}$ such that
\begin{equation}\label{e28}
  (1-p)\int_{\Omega}|\Delta u|^2dx-(6-3p)
\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu\partial_ju\partial_{ij}u\,dx=o(1).
\end{equation}
Using \eqref{e4.1},  \eqref{e28}  and \eqref{e21}, we can find a constant $d>0$
such that
\begin{equation}\label{e28.5}
\Big|\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_iu_n
 \partial_ju_n\partial_{ij}u_ndx\Big|\geq d
\end{equation}
for $n$ sufficiently large.
In addition \eqref{e28}, and the fact $\{u_n\}\subset {\mathcal{N}}$  also give
\begin{align}\label{e29}
  \lambda\int_{\Omega} f(x)|u_n|^{p+1}dx
&=\int_{\Omega}|\Delta u_n|^2dx-3\int_{\Omega}
\sum_{1\leq i<j\leq N}\partial_iu_n\partial_ju_n\partial_{ij}u_ndx \\
  &=\frac{3}{1-p}\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu_n\partial_ju_n\partial_{ij}u_ndx+o(1)
\end{align}
and
\begin{equation} \label{e30}
  \int_{\Omega} |\Delta u_n|^2dx
<\Big[\frac{(4-2p)\lambda}{(p+1)}\|f\|_{L^{m}}S^{p+1}\big]^{\frac{2}{1-p}}+o(1).
\end{equation}
This implies
\begin{equation} \label{e31}
\begin{aligned}
  A(u)
&=K(p,q)\Big[\frac{\left(\int_{\Omega} |\Delta u|^2dx\right)^{q}}
{\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}
 \Big]^{\frac{1}{q-1}}-\int_{\Omega} \lambda f(x) |u|^{p+1}dx, \\
&\leq \frac{3}{1-p}\big(\frac{1-p}{6-3p}\big)^{\frac{q}{q-1}}
\Big[\frac{\big(\frac{6-3p}{1-p}\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu\partial_ju\partial_{ij}u\,dx\big)^{q}}
{\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i u\partial_ju\,dx}
\Big]^{\frac{1}{q-1}} \\
&\quad -\frac{3}{1-p}\int_{\Omega} \sum_{1\leq i<j\leq N}
\partial_iu_n\partial_ju_n\partial_{ij}u_ndx+o(1)=o(1).
\end{aligned}
\end{equation}
However, from \eqref{e28.5} and \eqref{e30}, for $\lambda$ small,  we have
\begin{equation}\label{e32}
\begin{aligned}
  A(u) & \geq  K(p,q)\Big[\frac{\big(\int_{\Omega} |\Delta u|^2dx\big)^{q}}
{\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u\partial_i 
 u\partial_ju\,dx}\Big]^{\frac{1}{q-1}}-\lambda\|f\|_{L^{m}}\|u\|_{1+q}^{p+1} \\
   & \geq K(p,q)\Big[\frac{\big(\int_{\Omega} |\Delta u|^2dx\big)^{q}}
{C\big(\int_{\Omega} |\Delta u|^2dx\big)^{3/2}}\Big]^{\frac{1}{q-1}}
-C\lambda\|f\|_{L^{m}} \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}} \\
   &\geq  \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}}
\Big[ K(p,q)\Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{q-2-pq+p}{2(q-1)}}
-C\lambda\|f\|_{L^{m}}\Big] \\
 &\geq  \Big(\int_{\Omega}|\Delta u|^2dx\Big)^{\frac{p+1}{2}}
\Big[ K(p,q)\left(\left(\lambda(2-p)\|f\|_{L^m}S^{p+1}\right)^{\frac{2}{p-1}}
\right)^{\frac{q(1-p)+p-2}{2(q-1)}} \\
&\quad -C\lambda\|f\|_{L^{m}}\Big].
\end{aligned}
\end{equation}
This contradicts \eqref{e31}. We deduce that
\begin{equation}\label{e33}
  \Big(I'(u_n),\frac{u}{\|u\|}\Big)\leq \frac{C}{n}.
\end{equation}
The proof is complete.

(2) Similar to the proof of (1), we may prove (2).
\end{proof}

 Now we establish the existence of a local minimum for $I$ on $\mathcal{N}^{+}$.

\begin{lemma}\label{lem11}
  For $\lambda$ small, the functional $I$ has a minimizer 
$u_0^{+}\in \mathcal{N}^{+}$ and it satisfies
  \begin{itemize}
\item[(1)]  $I(u_0^{+})=\alpha=\alpha^{+}$;
\item[(2)]  $u_0^{+}$ is a nontrivial nonnegative solution of problem \eqref{e0.1};
\item[(3)]  $I(u_0^{+})\to  0$ as $\lambda\to  0$.
 \end{itemize}
\end{lemma}

\begin{proof}
Let $\{u_n\}\subset \mathcal{N}$  be a minimizing sequence for $I$ on 
$\mathcal{N}$ such that
\begin{equation}\label{e34}
  I(u_n)=\alpha+o(1),\quad  I'(u_n)=o(1),\quad \text{for }
 \big(W_0^{2,2}(\Omega)\big)^{\ast}.
\end{equation}
Then by Lemma \ref{lem7}  and the compact embedding theorem, there exists 
a subsequence $\{u_n\}$ and $u_0^{+}\in W_0^{2,2}(\Omega)$ such that
\begin{gather*}
u_n\rightharpoonup u_0^{+}\quad \text{in } W_0^{2,2}(\Omega), \\
u_n \to  u_0^{+}\quad \text{in } L^{h}(\Omega),
\end{gather*}
where $1<h<2^{\ast}$. We now show that  $\int_{\Omega} f(x)|u_0|^{p+1}dx\neq 0$. 
If not, by \eqref{e34}, we can conclude that
\begin{gather*}
\int_{\Omega} f(x)|u_n|^{p+1}dx=0, \\
\int_{\Omega} f(x)|u_n|^{p+1}dx\to  0\ \ \text{as}\ \ n\to \infty.
\end{gather*}
Thus,
$$
\int_{\Omega}|\Delta u_n|^2dx=3\int_{\Omega} 
\sum_{1\leq i<j\leq N}\partial_iu_n\partial_ju_n\partial_{ij}u_ndx+o(1),
$$
and
\begin{equation} \label{e35}
\begin{aligned}
  I(u_n)
&=\frac{1}{2} \int_{\Omega} |\Delta u_n|^2dx
 -\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u_n\partial_i u_n\partial_ju_ndx\\
&\quad -\frac{\lambda}{p+1}\int_{\Omega} f(x) |u_n|^{p+1}dx \\
&=\frac{1}{2} \int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u_n\partial_i
  u_n\partial_ju_ndx+o(1) \\
&\to \frac{1}{2} \int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij}
u_0\partial_i u_0\partial_ju_0dx\quad  \text{as }  n\to +\infty,
\end{aligned}
\end{equation}
Similar to  Lemma \ref{lem5}, we can see that
$\int_{\Omega} \sum_{1\leq i<j\leq N}\partial_{ij} u_0\partial_i u_0\partial_ju_0dx>0$.
 So \eqref{e35} contradicts $I(u_n)\to  \alpha<0$ as $n\to +\infty$.
In particular, $u_0^{+} \in \mathcal{N}^{+}$
is a nontrivial solution of problem \eqref{e0.1} and $I(u_0^{+})\geq \alpha$.
Similar to the proof of \cite[Lemma 3.1]{r4},
 we can prove that $u_n\to  u_0^{+}$ strongly in $W_0^{2,2}\Omega)$.
In fact, if $u_0^{+}\in \mathcal{N}^{-}$, by Lemma \ref{lem4},
 there are unique $t_0^{+}$ and $t_0^{-}$
such that $t_0^{+}u_0^{+}\in \mathcal{N}^{+}$ and
 $t_0^{-}u_0^{+}\in \mathcal{N}^{-}$, we have $t_0^{+}<t_0^{-}=1$.
 Since
$$
\frac{d}{dt}I(t_0^{+}u_0^{+})=0\quad \text{and}\quad
\frac{d^2}{dt^2}I(t_0^{+}u_0^{+})>0,
$$
there exists $t_0^{+}<\bar{t}\leq t_0^{-}$ such that
$I(t^{+}_0 u_0^{+})<I(\bar{t}u_0^{+})$. By
$$
I(t^{+}_0u_0^{+})<I(\bar{t}u_0^{+})\leq I(t^{-}_0u_0^{+})=I(u_0^{+}),
$$
which is a contradiction. By Lemma \ref{lem3}, we know that $u^+_0$
is a nontrivial solution.
Moreover, from \eqref{e11}, we know that
$$
0>I(u_0^{+})\geq -\frac{\lambda(2-p)}{3(p+1)}
\left(\|f\|_{L^{m}}S^{p+1}\right)^{\frac{2}{1-p}}.
$$
It is clear that $I(u_0^{+})\to  0$ as $\lambda \to  0$.
\end{proof}

As in the proof of Lemma \ref{lem11}, we establish the existence of a local 
minimum for $I$ on $\mathcal{N}^{-}$.

\begin{lemma}\label{lem12}
  For $\lambda$ small, the functional $I$ has a minimizer 
$u_0^{-}\in \mathcal{N}^{-}$ and it satisfies
  \begin{itemize}
\item[(1)]  $I(u_0^{-})=\alpha^{-}$;
\item[(2)]  $u_0^{-}$ is a nontrivial nonnegative solution of problem \eqref{e0.1}.
 \end{itemize}
\end{lemma}

Combining  Lemma \ref{lem11} and \ref{lem12}, for problem \eqref{e0.1}, 
there exist two nontrivial  solutions $u_0^+$
 and $u^0_-$ such that $u^+_0\in \mathcal{N}^{+}$,  $u^-_0\in \mathcal{N}^{-}$.  
Since $\mathcal{N}^{+} \cap \mathcal{N}^{-}=\emptyset$, this shows that 
$u_0^+$ and $u^0_-$ are different.


\subsection*{Acknowledgements}
The first author is supported by Young Teachers Foundation  of BTBU 
(No. QNJJ2016-15).
The second author is supported by NSFC (No. 11301181) and China
 Postdoctoral Science Foundation.
The third author is supported by Research Foundation of Heilongjiang 
Educational Committee (No. 12541667).


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\end{document}