Infinitely many weak solutions for a fourth-order Kirchhoff type elliptic equation

Abstract

In this article, by using critical point theory, we prove the existence
of infinitely many weak solutions for a fourth-order Kirchhoff type elliptic
equation involving multi-singular inverse square potentials.
Precisely this work is devoted to consider a fourth order elliptic equation
involving multi-singular inverse square potentials on the smooth
bounded domain $\Omega{\subset}\mathbb{R}^N(N\geq 5)$
\begin{equation*}
\left\{
\begin{array}{cc}
\Delta^2{u} &- M \left(\int_{\Omega} \left( \vert \nabla{ u}
\vert^2- \displaystyle \sum_{i=1}^k\frac{\mu_i}{|x-a_i|^2} \vert u
\vert ^2 \right) dx \right) \left( \Delta{ u} - \displaystyle
\sum_{i=1}^k\frac{\mu_i}{|x-a_i|^2}u\right) \textrm{ for } x \in{\Omega{\backslash}}\{a_1,...,a_k\}\\
&= \lambda f \left( x,u \right) + u \vert u \vert ^{2^{**} -2}\\
&u= \Delta u=0 \textrm{ for } x \in \partial \Omega,
\end{array}
\right.
\end{equation*}
where $ \Delta^2 $ is the biharmonic operator and $2^{**}$ is the
critical Sobolev exponent,
$a_i\in{\Omega},i=1,2,...,k,$ for $k\geq1$ are
different points, $0\leq{\mu_i} \in \mathbb{R}$ and $ \sum_{i=1}^k
\mu _ i < \bar{\mu}:=\left(\dfrac{N-2}{2} \right)^2 $ which $
\bar{\mu} $ is the best constant in the Hardy inequality, $ f:
\Omega \times \mathbb{R}\longrightarrow \mathbb{R} $
is $ L^1- $Carath\'{e}odory function and $ M \in C^1 \left( \left[0, + \infty \right[ , \mathbb{R} \right) $.