Infinitely many solutions for a class of biharmonic equations with indefinite potentials

Abstract

In this paper, we consider the following sublinear biharmonic equations\begin{equation*} \Delta^2 u + V\left( x \right)u =K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*}where $N\geq5,~0<p<1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.

Keywords

• biharmonic equation
• indefinite potential
• symmetric mountain pass theorem