A fourth-order equation with critical growth: the effect of the domain topology

Abstract

In this paper we prove the existence of multiple classical solutions for the fourth-order problem$$\begin{cases}\Delta^2 u = \mu u+ u ^{2_* -1} & \text{in } \Omega,\\u,\quad -\Delta u\gt 0 & \text{in } \Omega,\\u,\quad \Delta u = 0 & \text{on } \partial\Omega,\end{cases}$$where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq8$, $2_*=2N/(N-4)$ and $\mu_1(\Omega)$ is the first eigenvalue of $\Delta^2$ in $H^2(\Omega)\cap H_{0}^{1}(\Omega)$. We prove that there exists $0 < \overline{\mu}< \mu_1(\Omega)$ such that, for each $0 < \mu < \overline{\mu}$, the problem has at least ${\rm cat}_{\Omega}(\Omega)$ solutions.