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

DOI: http://dx.doi.org/10.12775/TMNA.2015.026

#### 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.

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.

#### Keywords

Biharmonic equation; Critical exponent; Lusternik-Schnirelman category; Positive solutions.

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