Title: 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: $$\Delta^2 u+(\lambda
V(x)-\delta)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5,
2<p\leq\frac{2N}{N-4}, \lambda>0$ 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.