Solutions for biharmonic equations with steep potential wells

Abstract

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $${\Delta }^{2}u+(\lambda V(x)-\delta )u=|u{|}^{p-2}uin{\mathbb{R}}^N$$ where $N\ge 5,2<p\le \frac{2N}{N-4},\lambda >0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<\delta <{\mu }_0$ and ${\mu }_0$ is the principle eigenvalue of ${\Delta }^2$ in the zero sets $\mbox{int} V^{-1}(0)$ of $V(x)$. Here $\mbox{int} V^{-1}(0)$ denotes the interior part of the set $V^{-1}(0):=\{x\in {\mathbb{R}}^N:V(x)=0\}$. We prove that the above equation admits a least energy solution which is trapped near the zero sets $\mbox{int} V^{-1}(0)$ for $\lambda >0$ large.

23 pages