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
where $\Omega$ is a smooth bounded domain in $\rn$, $N\geq8$, $2_*=2N/(N-4)$ 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 $\cat_{\Omega}(\Omega)$ solutions.