On a biharmonic equations with steep potential well and indefinite potential

TL;DR

This paper investigates biharmonic equations with steep and indefinite potentials, establishing new existence results for solutions when a parameter is large, especially in cases previously unexplored, and analyzes their concentration behavior.

Contribution

The paper introduces a novel variational framework for biharmonic equations with negative coefficient parameters, leading to the first known solutions in this setting and detailed concentration analysis.

Findings

Existence of nontrivial solutions for large mbda.

New variational method applicable for a_0<0.

Concentration behavior as mbda 

Abstract

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&\Delta^2u-a_0\Delta u+(\lambda b(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\lambda})}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $\lambda>0$ is a parameter, $b(x)\geq0$ is a potential well and $f(t)\in C(\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $\Delta^2-a_0\Delta+\lambda b(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}_{\lambda})$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\mathcal{P}_{\lambda})$ for all $a_0, b_0\in\bbr$ with $\lambda$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $\lambda\to+\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\mathcal{P}_{\lambda})$ is obtained in the case of $a_0<0$.