Abstract:

In this paper, we study the following biharmonic equation with Navier boundary condition:

$\begin{equation} \left\{ \begin{array}{ll} \Delta^2u=|u|^{p-1}u+f &\mbox{ in } \Omega,\\ \Delta u=u=0 &\mbox{ on } \partial\Omega, \end{array}\tag{$0.1_f$} \right. \end{equation}$

where $1<p<\frac{N+4}{N-4}$ ($1<p<\infty$ for $N=1,\,2,\,3,\,4$), and $\Omega$ is a smooth and bounded domain in $\mathbb{R}^N$ with boundary $\partial\Omega$. We prove that there exists an open dense subset of $L^2(\Omega)$ such that for any $f$ belongs to this set, (0.1f) has infinitely many solutions. This is an application of the topological result for a certain class of functionals developed by [Bahri A. J Funct Anal,1981, 41(3): 397--427].

Key words: biharmonic equation, Navier boundary condition, infinitely many solutions, topological result.