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: $$Δ^2 u+(λV(x)-δ)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2 0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<δ<μ_0$ and $μ_0$ is the principle eigenvalue of $Δ^2$ 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 $λ>0$ large.