\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 $2
\mu_1$; 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 $2
2$, 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