Solutions for biharmonic equations with steep potential wells

TL;DR

This paper proves the existence of least energy solutions for certain biharmonic equations with steep potential wells, showing solutions concentrate near zero potential regions as the parameter increases.

Contribution

It establishes the existence and localization of least energy solutions for biharmonic equations with steep potential wells, a novel result in higher-order PDEs with variable potentials.

Findings

Solutions exist for large mbda

Solutions concentrate near zero potential regions

The approach applies to 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.