Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Abstract

The purpose of this article is to deal with the following biharmonic critical Choquard equation\begin{align*}\begin{cases}\Delta^{2}u = \lambda f(x) |u|^{q-2}u+ g(x)\bigg(\displaystyle \int_{\Omega}\frac{g(y)|u(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}dy\bigg)|u|^{2_\alpha^*-2}u & \text{in } \Omega,\\u,\ \nabla u = 0 & \text{on } \partial\Omega,\end{cases}\end{align*}where $\Omega$ is a bounded domain in $\mathbb R^N$ with smooth boundary $\partial \Omega$, $N\geq 5$, $1< q < 2$, $0< \alpha < N$, $2_\alpha^*=({2N-\alpha})/({N-4})$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\lambda > 0$ is a parameter. The functions $f, g\colon \overline {\Omega}\rightarrow \mathbb R$ are continuous sign-changing weight functions. Using the Nehari manifold and fibering map analysis, we prove the existence of two nontrivial solutions of the problem with respect to parameter $\lambda$.