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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 99, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/99\hfil 
 Biharmonic equations with sign-changing coefficients]
{Existence of solutions to biharmonic equations with sign-changing coefficients}

\author[S. Saiedinezhad \hfil EJDE-2018/99\hfilneg]
{Somayeh Saiedinezhad}

\address{Somayeh Saiedinezhad \newline
School of Mathematics,
Iran University of Science and Technology,
Narmak, Tehran, Iran}
\email{ssaiedinezhad@iust.ac.ir}

\dedicatory{Communicated by  Vicentiu D. Radulescu.}

\thanks{Submitted July 17, 2017. Published Aapril 28, 2018.}
\subjclass[2010]{35A01, 35J35, 35D30, 35J91}
\keywords{Bi-Laplacian operator; weak solution; Nehari manifold; fibering map}

\begin{abstract}
 In this article, we study the existence of solutions for the semi-linear 
 elliptic  equation
 $$
 \Delta^2 u-a(x)\Delta u=b(x)| u|^{p-2}u
 $$
 with Navier boundary condition $u=\Delta u=0$ on $\partial\Omega$,
 where $\Omega$ is a bounded domain with smooth boundary and $2<p<2^*$.
 We consider two different assumptions on the potentials $a$ and $b$,
 including the case of sign-changing weights.
 The approach is based on the Nehari manifold with variational arguments
 about the corresponding fibering map, which ensures the multiple results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction and preliminary results}

The literature concerning the existence of solution of the elliptic PDEs 
is very extensive, (for instance see 
\cite{Boccardo-Orsina-2017, Cortazer-etc-2017, Ho-Sim-2017, 
Papageorgiou-Radulescu-2017} and the references therein).
Since  fourth-order PDEs have been appeared in  various models such as 
micro-electro-mechanical systems, phase field models of multiphase
systems (see \cite{Danet-2014, Ferrero- Warnault- 2009, Myers- 1998}), 
a number of articles have been devoted to
the fourth-order elliptic PDEs; we refer the interested readers 
to \cite{Omrane-Khenissy-2014, Boureanu-Radulescu-2016, Kefi-Radulescu-2017, 
Kong-2016, 1998-micheletti, 2014-xu-oregan, 2003-xu-zhang, 2013- pu- wu-tang}.

In particular, the biharmonic equation $\Delta^2 u+c\Delta u=d[(u+1)^+-1]$,
in which $u^+=\max\{u,0\}$, have attracted the attention of the mathematicians.
 This type of elliptic equation furnishes a model to study the traveling
 waves in suspension bridges, which  is first developed by Lazer and 
Mckenna \cite{Lazer}.
For $u=u(x_1,\dots,x_N)$ the bi-Laplacian operator is defined by
$$
\Delta^2 u=\sum_{i=1}^N \frac{\partial ^4 u}{\partial x_i^4}+\sum_{i,j=1;
i\neq j}^N \frac{\partial ^4 u}{\partial x_i^2\partial x_j^2}.
$$

The fourth-order equations,  which are  studied in the most papers, has  the form
$\Delta^2u+c\Delta u=f(x,u)$, in which $f$ satisfied certain conditions, 
 $c<\mu_1$ and sometimes $c>\mu_1$;  where $\mu_1$ is the first eigenvalue 
of $-\Delta u=\lambda u$ with Dirichlet boundary condition.

Micheletti and Pistoia \cite{1998-micheletti}
provided a geometrical structure
of the equation $\Delta^2u+c\Delta u=bg(x,u)$
similar to the linking theorem, by supposing $2G(x,s)\leq s^2$,  
$\limsup_{s\to -\infty}G(x,s)/s^2\leq 0$ and
$\liminf_{s\to 0}G(x,s)/s^2=l(x)$; where 
$G(x,u)=\int_0^ug(x,s)ds$, and consequently they derived the multiplicity 
existence results.

In \cite{2003-xu-zhang}, based on the mountain pass theorem, 
the existence of positive solutions for  the problem 
$\Delta^2 u+c\Delta u=f(x,u)$ is studied in
which $f$ satisfies the local  superlinearity or sublinearity conditions and 
$c<\mu_1$.  The similar problem in \cite{2014-Gu-An} is studied  under 
the conditions  
$\liminf_{| u|\to \infty } G(x,u)/| u|^2=\infty$ and 
$ug(x,u)-2G(x,u)\geq d| u|^\sigma$ where $\sigma>\frac{2N}{N+4}$ and 
by using the variant fountain theorem the existence of multiple solutions is derived.
In \cite{2013- pu- wu-tang} by 	using  the least action principle, 
the Ekeland variational principle and  the mountain pass theorem, 
the multiplicity of solutions for the problem 
$\Delta^2 u+c\Delta u=a(x) | u|^{s-2} u+f(x,u)$ with the combined nonlinearity
 on $f$ is studied.

In \cite{2014-xu-oregan}  the equation $\Delta^2 u+c \Delta u=\lambda u+f(u)$ 
was studied in which $f$ has subcritical growth condition, i.e.,
$| f(s)|\leq d_1| s|+d_2| s|^{p-1}$ for some $p\in [2,2^*) $ and 
$d_1,d_2>0$, under Navier boundary condition by applying the topological 
degree theory.

In this paper, we consider the problem
\begin{equation} \label{eP}
\begin{gathered}
\Delta^2 u-a(x)\Delta u=b(x)| u|^{p-2}u, \quad   x\in\Omega,\\
u=\Delta u=0, \quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded subset of $\mathbb{R}^N$ with $N>4$ and
$2<p<2^*=\frac{2N}{N-2}$. Moreover, one of the following assumptions is satisfied.
\begin{itemize}
\item[(A1)] $a,b\in L^\infty(\Omega)$ and $a(x), b(x)\geq 0$ a.e. in $\Omega$,
or
\item[(A2)] $a,b\in L^\infty(\Omega)$ and $a,b$ may change sign.
\end{itemize}
 The main  results of the article are in
 two subsections. In the first one, we consider problem \eqref{eP}
by assuming condition $(A1)$  and so we seek the solutions through
providing a minimizer sequence.

In the second subsection, where condition (A2) is satisfied,
we study the existence results due to the  behavior of the corresponding 
fibering map, while $a^+<\mu_1$ or $\mu_1<a^-<a^+<\mu_1+\sigma$ for some 
appropriate $\sigma$ which is introduced later,  
$a^+=\operatorname{ess\,sup}\{a(x), x\in \Omega\}$ and
 $a^-=\operatorname{ess\,inf}\{a(x), x\in \Omega\}$.

It is known that if $I(u)$ denotes the energy functional corresponding to an 
equation, all of the critical points of $I$ must lie on the manifold 
$\{u; \langle I'(u),u\rangle=0\}$, which is known as the Nehari manifold 
(see \cite{Nehari, Willem}). Moreover, the fibering map 
$(\varphi_u:t\to I(tu))$ which is closely linked to the Nehari manifold 
is an interesting approach for describing of the  energy functional's 
behavior on the Nehari manifold 
(see \cite{2003-Brown-zhang, 1997- Derabek-pohozaev}).

Consider the Sobolev space
$$
H^1(\Omega):=\{u\in L^2(\Omega): u_{x_i}\in L^2(\Omega), 1\leq i\leq N\}.
$$
It is known that $H^1(\Omega)$ with  the inner product 
$\langle u,v\rangle:=\int_\Omega| \nabla u\nabla v| dx$ is a Hilbert space.
 Moreover, let 
\begin{gather*}
H^1_0(\Omega):=\{u\in H^1(\Omega): u|_{\partial\Omega}=0\},\\
H^2(\Omega):=\{u\in L^2(\Omega): u_{x_i}, u_{x_ix_j}\in L^2(\Omega),\; 
1\leq i,j\leq N\}.
\end{gather*}
We recall that $H^2(\Omega)$ with  the inner products
 $\langle u,v\rangle=\int_\Omega| \triangle u\triangle v| dx$ or 
$$
\langle u,v\rangle=\int_\Omega| \triangle u\triangle v| dx
-c\int_\Omega | \nabla u\nabla v| dx,
$$ 
with $c<\mu_1$ and
 $\mu_1=\inf\{\frac{\int_\Omega |\nabla u|^2 dx}{\int_\Omega | u|^2 dx}:
 0\neq u\in H^1(\Omega)\}$ is a Hilbert space.  We remark that all of 
the derivatives in the above definitions are in the weak sense; 
for more details see \cite{Attouch et al-2014}.

The compact embedding $H^1(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)$  
is known, thus  there exists a positive constant $e$ such that
$\| u\|_2\leq e \| \nabla u\|_2$; in which $\| \cdot\|_2$ is the usual norm 
on $L^2(\Omega)$. Indeed, the sharp constant $e$  is equal  to 
$\frac{1}{\sqrt{\mu_1}}$. Hence
\begin{equation}\label{10}
\| u\|_2\leq \frac{1}{\sqrt{\mu_1}} \| \nabla u\|_2;  \quad 
\forall u\in H^1(\Omega).
\end{equation}
Also we have, $H^2(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)$. Let
\begin{equation}\label{11}
\mu_1^2=\inf\{\frac{\int_\Omega |\Delta u|^2 dx}{\int_\Omega | u|^2 dx}:
 0\neq u\in H^2(\Omega)\}.
\end{equation}
By the natural continuous map,  $H^2(\Omega)$ is embedded into $H^1(\Omega)$, 
so for some positive constant $k$, we insert that 
$\| \nabla u\|_2\leq k \|\Delta u\|_2$. By considering  \eqref{10} and \eqref{11},  
the sharp constant $k$ would be $\frac{1}{\sqrt{\mu_1}}$, i.e.,
\begin{equation}\label{mu1}
\mu_1=\inf\{\frac{\int_\Omega |\Delta u|^2 dx}{\int_\Omega | \nabla u|^2 dx}:
 0\neq u\in H^2(\Omega)\}.
\end{equation}
We assume throughout this paper, $\varphi_1$ as a  unit vector in $H^2(\Omega)$, 
which $\mu_1=\frac{\int_\Omega |\Delta \varphi_1|^2 dx}
{\int_\Omega | \nabla \varphi_1|^2 dx}$ and let 
$X=H^2(\Omega)\cap H^1_0(\Omega)$, which is a Hilbert space equipped under the 
inner product
$$
\langle u,v\rangle=\int_\Omega (\triangle u\triangle v+ a(x)\nabla u\nabla v)dx.
$$

\section{Main results}

From the basic variational arguments we insert that the weak solutions of 
\eqref{eP} are corresponded to the local minimizer of
$$
I(u)=\frac{1}{2}\int_\Omega (| \triangle u|^2+a(x)| \nabla u|^2)dx
-\frac{1}{p}\int_\Omega b(x)| u|^pdx.
$$ 
Since $p>2$, for every $u\neq0$, $I(tu)$ tends to $-\infty$ as $t$ tends 
to $+\infty$.
Thus, $I$ is not bounded below and so the minimizing approach in $X$ may fail.

\subsection{Case of nonnegative coefficients}

For every $\alpha\in\mathbb{R}$, let
 $$
S_\alpha:=\{u\in X: \int_\Omega b(x)| u|^p=\alpha\}.
$$ 
Then for every $u\in S_\alpha$,  $I(u)=\frac{1}{2}\| u\|^2-\frac{1}{2}\alpha$. 
Thus $I|_{S_\alpha}$  is certainly bounded below and the process of  minimizing $I$ 
on $S_\alpha$  is equivalent to the process of  minimizing $\| u\|$ or 
$\| u\| ^2$ on $S_\alpha $. Set $\inf_{u\in S_\alpha}\| u\|^2=:m_\alpha$, 
we will show that $m_\alpha$ is achieved by a function, and a multiple of this 
function is a minimizer of $I$ on $X$ and so a weak solution of \eqref{eP}.

\begin{lemma} \label{lem2.1}
 For every $\alpha>0$, there exists a nonnegative function $u_\alpha\in S_\alpha$ 
such that $\| u_\alpha\|^2=m_\alpha$.
 \end{lemma}

 \begin{proof}
By the  coercivity of $I$ on $S_\alpha$
 (i.e., $\lim_{\| u\|\to\infty, u\in S_\alpha} I(u)=\infty$), there exists a 
bounded minimizer sequence $\{u_n^{(\alpha)}\}$ for $f(u):=\| u\|^2$ on $S_\alpha$. 
Obviously, since $\{| u_n^{(\alpha)}|\}$ is still  a minimizer sequence in 
$S_\alpha$, we can suppose  that $u_n^{(\alpha)}(x)\geq0$ a.e. in $\Omega$. 
By reflexivity of $X$, there exists a subsequence of $u_n^{(\alpha)}$ 
(still  denote it by $u_n^{(\alpha)}$), which  is weakly convergent to 
$u_\alpha\in X$ ($u_n^{(\alpha)}\rightharpoonup u_\alpha$) and therefore the
 Sobolev compact embedding ensures that
$u_n^{(\alpha)}$ is  strongly convergent in $L^p(\Omega)$. Hence 
$$
lim_{n\to\infty}\int_\Omega b(x)| u_n^{(\alpha)}|^p dx
=\int_\Omega b(x)| u_\alpha|^p,
$$ 
which means $u_\alpha\in S_\alpha$. If $u_n^{(\alpha)}\not\to u_\alpha$ 
in $X$, we have  that $\| u_\alpha\|^2<\liminf \| u_n^{(\alpha)}\|^2=m_\alpha$, 
which is a contradiction, since $u_\alpha\in S_\alpha$. Hence $u_n\to u_\alpha$ in 
$X$ and since $u_\alpha\in S_\alpha$, $u$ does not  vanish identically.
 \end{proof}

 \begin{theorem} \label{thm2.2}
Suppose that $a,b$ satisfy  condition {\rm (A1)}, then problem \eqref{eP} 
admits at least one weak solution in $X$.
 \end{theorem}

\begin{proof}
Let  $g(u):=\int_\Omega b(x)| u|^p dx$ and $f(u):=\| u\|^2$. 
Relying on the Lagrange multiplier theorem, if $u_\alpha$ is a minimizer 
of $f$ under  the condition $g(u)=\alpha$, then  there exists 
$\lambda\in\mathbb{R}$ such that $f'(u_\alpha)=\lambda g'(u_\alpha)$; that is
\begin{equation}\label{1}
\langle u_\alpha,v\rangle
=\frac{p\lambda}{2}\int_\Omega b(x)|\nabla u_\alpha|^{p-2}
 \nabla u_\alpha\nabla v dx,
\end{equation}
for every $v\in X$.
By taking $u_\alpha=C w_\alpha$ for an appropriate constant $C$, which will 
be introduced in the sequel, it yields
$$
C\langle w_\alpha,v\rangle=\frac{p\lambda}{2} C^{p-1}
\int_\Omega b(x)|\nabla w_\alpha|^{p-2} \nabla w_\alpha\nabla v dx.
$$
Now, by considering $C=(\frac{2}{p\lambda})^{\frac{1}{p-2}}$ we have
 $\langle w_\alpha,v\rangle=\int_\Omega b(x)|\nabla w_\alpha|^{p-2} 
\nabla w_\alpha\nabla v dx$, namely $w_\alpha$ is a weak solution of \eqref{eP}.
\end{proof}

\begin{remark} \label{rmk2.3} \rm
For $\alpha\neq \beta$ the minimizers of $f$ on $S_\alpha$ and $S_\beta$ 
give the same weak solution of \eqref{eP}.
\end{remark}

\begin{proof}
For $\alpha\neq \beta$, one can readily check that 
$m_\alpha=\big(\frac{\alpha}{\beta}\big)^{2/p}m_\beta$. Indeed,
\[
S_\alpha=\big\{u\in X: \int_\Omega b(x) | u| ^p=\alpha\big\}
=\big\{\big(\frac{\alpha}{\beta}\big)^{1/p}v: v\in X, \int_\Omega b(x)
| v| ^p=\beta  \big\}.
\]
Thus
\begin{align}\label{2}
m_\alpha=\inf_{u\in S_\alpha} \| u\|^2
=\big(\frac{\alpha}{\beta}\big)^{2/p}m_\beta.
\end{align}
So $u_\alpha$ minimizes $\| u\|^2$ on $S_\alpha$ if and only if 
$(\frac{\beta}{\alpha})^{1/p} u_\alpha$ minimizes  $\| u\|^2$ on $S_\beta$. Moreover, it is easy to see that $\lambda_\alpha=\frac{2m_\alpha}{p_\alpha}$
and $C_\alpha=(\frac{\alpha}{m_\alpha})^{\frac{1}{p-2}}$; indeed,
it is sufficient to rewrite \eqref{1}  by  substituting $v=u_\alpha$.
Therefore
\begin{align*}
w_\alpha
&=\frac{1}{C_\alpha} u_\alpha=(\frac{m_\alpha}{\alpha})^{\frac{1}{p-2}}
\big(\frac{\alpha}{\beta}\big)^{1/p}u_\beta\\
&= (\frac{m_\beta}{\beta})^{\frac{1}{p-2}}u_\beta=\frac{u_\beta}{c_\beta}=w_\beta.
\end{align*}
\end{proof}

\begin{corollary} \label{coro2.4}
Let $a\in L^\infty(\Omega)$ which is a.e. nonnegative. 
 Every $\mu>0$ is an eigenvalue of problem $\eqref{ePm}$ where
\begin{equation} \label{ePm}
\begin{gathered}
\Delta^2 u-a(x)\Delta u=\mu| u|^{p-2}u, quad   x\in\Omega,\\
u=\Delta u=0, \quad x\in\partial\Omega;
\end{gathered}
\end{equation}
\end{corollary}

\subsection{Case of sign-changing coefficients}

Now we consider problem \eqref{eP} in which $a,b$ meet the condition (A2). 
The fibering map corresponding to the Euler-Lagrange functional of problem
 \eqref{eP} is defined as a map $\varphi:[0,\infty)\to \mathbb{R}$ with 
$\varphi_u(t)=I(tu)$. Hence,
\begin{gather*}
\varphi_u(t)=\frac{t^2}{2}\int_\Omega (| \triangle u|^2-a(x)| \nabla u|^2)dx
-\frac{t^p}{p}\int_\Omega b(x)| u|^pdx,\\
\varphi'_u(t)=t\int_\Omega (| \triangle u|^2-a(x)| \nabla u|^2)dx
 -t^{p-1}\int_\Omega b(x)| u|^pdx.
\end{gather*}
Obviously, $\varphi_u'(1)=0$ if and only if 
$u\in N:=\{u\in X; \langle I'(u),u\rangle=0$.
It is natural to divide the critical points of $\varphi'_u(t)$ 
into three subsets containing local minimuma, local maximuma and inflection
 points and so we define
$N^+:=\{u\in N, \varphi''_u(1)>0\}$, 
$N^-:=\{u\in N, \varphi''_u(1)<0\}$ and 
$N^0:=\{u\in N, \varphi''_u(1)=0\}$.

In this section, we consider $X$ with the norm
$\| u\|=(\int_\Omega | \triangle u|^2 dx)^{1/2}$; 
moreover $A(u):=\int_\Omega (| \triangle u|^2-a(x)| \nabla u|^2)dx$ and 
$B(u):=\int_\Omega b(x)| u|^pdx$. 
Hence for each $u\in X$ we have $\varphi'_u(t)=0$ if and only if 
$A(u)=t^{p-2}B(u) $. Moreover, if $A(u)B(u)>0$ then there exists $t_0>0$ 
such that $\varphi_u(t_0)=0$, i.e. $t_0u\in N$ and otherwise no multiple of
 $u$ belongs to $N$.
Finally, if $t_0u\in N$, then
$$
\varphi''_{t_0u}(1)=(2-p)A(t_0u)=(2-p)t_0^2A(u).
$$
Hence, for $p>2$, if $A(u)>0$ we derive $t_0u\in N^-$ and if $A(u)<0$ we 
conclude $t_0u\in N^+$.

\begin{lemma} \label{lem4}
If $a^+<\mu_1$, then there exists  $\delta>0$ such that for every $u\in X$, 
$A(u)\geq \delta\| u\|^2$.
\end{lemma}

\begin{proof}
If $\int_\Omega a(x)| \nabla u|^2dx \leq0$ then the  assertion is obvious. 
Let us suppose that $\int_\Omega a(x)| \nabla u|^2dx>0$ and argue by contradiction. 
If for each $\delta>0$ there exists $u\in X$ such that $A(u)<\delta\| u\|^2$,
 we derive that
\begin{align}\label{3}
\| u\|^2<\frac{\int_\Omega a(x)| \nabla u|^2dx}{1-\delta}
<\frac{a^+\int_\Omega | \nabla u|^2dx}{1-\delta}.
\end{align}
Now, by considering $\delta<1-\frac{a^+}{\mu_1}$ we have 
$\frac{a^+}{1-\delta}<\mu_1$
and thus \eqref{3} leads to a  contradiction with \eqref{mu1}.
\end{proof}

\begin{theorem} \label{thm2.6}
If $a^+<\mu_1$, then $I$ admits a minimizer on $N$.
\end{theorem}

\begin{proof}
Since $a^+<\mu_1$, we deduce that $N^+=\emptyset$; thus 
$\inf_{u\in N} I(u)=\inf_{u\in N^-} I(u)$.
We will show that $\inf_{u\in N^-} I(u)>0$.
For $u\in N$, $A(u)=B(u)$ and hence $\| u\|^2=(\frac{A(v)}{B(v)})^{\frac{2}{p-2}}$ 
where $v=\frac{u}{\| u\|}$.
Consequently, for $u\in N$ we have
$$
I(u)=(\frac{1}{2}-\frac{1}{p})A(u)= (\frac{1}{2}-\frac{1}{p})\| u\|^2 A(v)
= (\frac{1}{2}-\frac{1}{p})\frac{A(v)^{\frac{p}{p-2}}}{B(v)^{\frac{2}{p-2}}}.
$$
Lemma \ref{lem4} ensures that  $A(v)\geq \delta$ for some $\delta>0$. 
Moreover, by Sobolev embedding $X\hookrightarrow L^p(\Omega)$,  
for a positive constant $C$ we have, $\int_\Omega| v|^pdx<C$. Hence
$$
I(u)\geq (\frac{1}{2}-\frac{1}{p})
\frac{\delta^{\frac{p}{p-2}}}{(b^+C)^{\frac{2}{p-2}}};
$$ 
and  thus  $\inf_{u\in N^-} I(u)>0$.
Set $m:=\inf_{u\in N^-}I(u)$ and let us  consider $\{u_n\}\subset N^-$, 
 which $\lim_{n\to\infty}I(u_n)=m$. In this cae, the coercivity  of $I$ on $N^-$, 
$\{u_n\}$ would be bounded and so by reflexivity of $X$, up to subsequence, 
there exists $u_0\in X$ such that $u_n$ is weakly convergent to $u_0$, 
$(u_n\rightharpoonup u_0)$. Since $u_n\to u_0$ in $L^p(\Omega)$ and $u_n\in N$, 
then
$$
m=\lim_{n\to\infty}I(u_n)=(\frac{1}{2}-\frac{1}{p})\lim_{n\to\infty}B(u_n)
=(\frac{1}{2}-\frac{1}{p}) B(u_0).
$$
Thus $B(u_0)>0$ and hence $u_0\neq 0$. Moreover,
since $a^+<\mu_1$ we have $A(u_0)>0$. Therefore, a  multiple of $u_0$ 
($t_0u_0$; $t_0^{p-2}=\frac{A(u_0)}{B(u_0)}$) belongs to $N^-$. 
If $u_n\not\to u_0$ in $X$ then
$\| u_0\|<\liminf _{n\to \infty} \| u_n\|$ and so 
$$
A(u_0)-B(u_0)<\liminf _{n\to \infty} (A(u_n)-B(u_n))=0.
$$
Consequently,  $t_0<1$ and $\varphi'_{u_0}(1)<0$. Therefore
$$
I(t_0u_0)<\liminf _{n\to \infty} I(t_0u_n)
=\liminf _{n\to \infty}\varphi_{u_n}(t_0)
<\liminf _{n\to \infty}\varphi_{u_n}(1)=\liminf _{n\to \infty}I(u_n)=m,
$$
which is in contrast with $t_0u_0\in N^-$. Hence, $u_n\to u_0$ in $X$ 
and  $u_0\in N$, since $A(u_0)=B(u_0)$.		
\end{proof}

\begin{lemma} \label{lem5}
There exists $\sigma>0$ in a way that
for every $\mu\in(\mu_1,\mu_1+\sigma)$ if 
$\int_\Omega(| \triangle u|^2-\mu|\nabla u|^2)dx\leq 0$ then $u=k\varphi_1$ 
for some $k\in \mathbb{R}$.
\end{lemma}

\begin{proof}
Suppose the sequences $\{\mu_n\}$ and $\{u_n\}$ are such that 
$\mu_n\to \mu_1^+$ (i.e., $\mu_n\to \mu_1$ and $\mu_n>\mu_1$ ) and
$\int_\Omega(| \triangle u_n|^2-\mu_n|\nabla u_n|^2)dx\leq 0$. 
Without loss of generality, let $\| u_n\|=1$. Since
$\{u_n\}$ is bounded, there exists $u_0\in X$ such that  
$u_n\rightharpoonup u_0$. If this convergence is not strong in $X$ then
$$
\int_\Omega(| \triangle u_0|^2-\mu_1|\nabla u_0|^2)dx
<\liminf\int_\Omega(| \triangle u_n|^2-\mu_n|\nabla u_n|^2)dx\leq 0
$$ 
which is impossible. Hence $u_n\to u_0$ and so $\| u_0\|=1$.
Moreover, we deduce that
$\int_\Omega(| \triangle u_0|^2-\mu_1|\nabla u_0|^2)dx\leq 0$ which  
holds if and only if  $u_0=k\varphi_1$, for some constant $k$.
\end{proof}

\begin{theorem} \label{thm2}
Suppose that $B(\varphi_1)\neq 0$ and let $\sigma>0$ as introduced in lemma 
\ref{lem5}. If   $\mu_1<a^-\leq  a^+<\mu_1+\sigma$ then $I$ admits 
a minimizer on $N^+$.
\end{theorem}

\begin{proof}
Firstly, we show that $N^+$ is bounded. Let us  argue by contradiction, 
so assume that  there exists an unbounded sequence $\{u_n\}\subseteq N^+$, 
which $\| u_n\|\to\infty$. Let $v_n:=\frac{u_n}{\| u_n\|}$, thus  
by boundedness of $v_n$, up to a subsequence, it would be weakly convergent 
to some $v_0\in X$.
We have $A(u_n)=B(u_n)$ then
\begin{equation} \label{6}
A(v_n)=\| u_n\|^{p-2} B(v_n).
\end{equation}
  Moreover, $| A(v_n)|\leq 1+|\int_\Omega a(x)| \nabla v_n|^2 dx|<1+C^2 a^+$, so
$A(v_n)$ is uniformly bounded   and this in conjunction with
 \eqref{6} ensures that  $\lim_{n\to\infty}B(v_n)=0$ and since
$v_n\to v_0$ in $L^p(\Omega)$, we get  $B(v_0)=0$.
 If $v_n\not\to v_0$ in $X$ we have
 \begin{align}\label{7}
  A(v_0)<\liminf A(v_n)\leq 0;
\end{align}
therefore by regarding to the  lemma \ref{lem5} we deduce $v_0=k\varphi_1$.
Since  $B(v_0)=0$,  while $B(\varphi_1)\neq 0$, we insert that $k=0$,
which contradicts \eqref{7}. Hence
  $v_n\to v_0$ in $X$ and so $\| v_0\|=1$ and further
$A(v_0)=\liminf A(v_n)\leq 0$. Due to the lemma \ref{lem5} and since
$B(v_0)=0$, we get $v_0=0$, which contradicts $\| v_0\|=1$,
hence $N^+$ is bounded.

Hence, let us suppose  $\{u_n\}$ as a bounded minimizer  sequence of $I$ on 
$N^+$ and set 
$$
m:=\inf I(u)_{u\in N^+}=\lim_{n\to\infty} I(u_n).
$$ 
Then, up to a subsequence, there exists $u_0\in X$ in a way that 
$u_n\rightharpoonup u_0$ in $X$ and  $u_n\to u_0$ in $L^p(\Omega)$. Hence
\begin{gather*}
B(u_0)=\lim_{n\to\infty}B(u_n)=(\frac{2p}{p-2})m<0, \\
A(u_0)\leq \liminf A(u_n)=(\frac{2p}{p-2})m<0.
\end{gather*}
Consequently,  a multiple of $u_0$ ($t_0u_0$; $t_0^{p-2}=\frac{A(u_0)}{B(u_0)}$) 
belongs to $N$ and since 
$$
\varphi_{t_0u_0}''(1)=(2-p)t_0^2 A(u_0)>0,
$$ 
then $t_0u_0\in N^+$. If $u_0\not\to u_0$ in $X$, we have
  $$
A(u_0)<\liminf A(u_n)=\liminf B(u_n)=B(u_0)
$$ 
and thus $t_0<1$.  Therefore
  $$
I(t_0u_0)=(\frac{1}{2}-\frac{1}{p}) t_0^2A( u_0)<(\frac{1}{2}-\frac{1}{p}) A(u_0)
<(\frac{1}{2}-\frac{1}{p})\liminf A(u_n)=(\frac{1}{2}-\frac{1}{p}) m<m;
$$
 which contradicts $t_0u_0\in N^+$. Hence, we deduce that $u_n$ converge 
strongly to $u_0$ in $X$  and $A(u_0)=B(u_0)$, i.e., $u_0\in N$ and since 
$B(u_0)<0$ we derive that $u_0\in N^+$.
\end{proof}

\begin{theorem} \label{thm2.9}
Suppose that  $B(\varphi_1)< 0$ and  let $\sigma>0$ as introduced 
in lemma \ref{lem5}. If $\mu_1<a^-\leq  a^+<\mu_1+\sigma$ then $I$  
admits a minimizer on $N^-$.
\end{theorem}

 \begin{proof}
In the first step, by an argument similar  to the proof of theorem \ref{thm2}, 
we deduce that every minimizer sequence of $I$ on $N^-$ is bounded.
In what follows,  we will show that $\inf_{u\in N^-}I(u)\neq 0$.
 Let us  argue by contradiction. Suppose that, for a bounded minimizer 
sequence $\{u_n\}\subset N^-$, $A(u_n)=B(u_n)\to 0$ as $n\to\infty$.

Let $v_n=\frac{u_n}{\| u_n\|}$, then $A(v_n)\to 0$	as $n\to\infty$, 
and for some $v_0\in X$, up to a subsequence, $v_n\rightharpoonup v_0$ in $X$. 
If  $v_n\not\to v_0$ then
 \begin{align}\label{8}
 	A(v_0)<\liminf A(v_n)=0.									
 \end{align}
Thus, by lemma \ref{lem5}, $v_0$ is a multiple of $\varphi_1$ such as 
$v_0=k\varphi_1$.

Further, $B(v_n)\to B(v_0)$ which  $B(v_n)=\| u_n\|^{-p} B(u_n)>0$, 
thus $B(v_0)\geq 0$.
But  since $v_0=k\varphi_1$ and $B(\varphi_1)<0$ we derive that $k=0$ and 
so $v_0=0$, which gives a contradiction with \eqref{8},
 hence $v_n\to v_0$ in $X$ and $\| v_0\|=1$.

In addition, if $A(v_0)\leq 0$, by applying lemma \ref{lem5} we deduce $v_0=0$, 
which contradicts  $\| v_0\|=1$, hence $\inf_{u\in N^-} I(u)>0$.
In the sequel we will show that, $I$ achieves its minimum on $N^-$.
We insert that $u_n\rightharpoonup u_0$ for some $u_0\in X$.
 One can derive  that $A(u_0)\leq 0$; indeed, if $A(u_0)>0$ by lemma \ref{lem5}, 
$u_0=k\varphi_1$ and yields
$$
| k|^p B(\varphi_1) =B(u_0)=(\frac{2p}{p-2})\inf_{u\in N^-}I(u)>0,
$$ 
which is not compatible with the assumption $B(\varphi_1)<0$.

Hence $A(u_0)>0$ and so a multiple of $u_0$ 
($t_0u_0;  t_0^{p-2}=\dfrac{A(u_0)}{B(u_0)}$) belongs to  $N^-$. 
If $u_n\not\to u_0$ then $A(u_0)<B(u_0)$ and thus $t_0<1$, which leads to
\begin{align*}
I(t_0u_0)&<\liminf I(t_0u_n)=\liminf\varphi_{u_n}(t_0) \\
&\leq \liminf\varphi_{u_n}(1)=\liminf E(u_n) 
=\inf_{u\in N^-} I(u).
\end{align*}
This is in contrast with $t_0u_0\in N^-$, hence $u_0$ is a nontrivial weak 
solution of the problem, which belongs to $N^-$.
\end{proof}

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\end{document}