Existence and Concentration Solutions for a class of elliptic PDEs involving $p$-biharmonic Operator

TL;DR

This paper establishes the existence of solutions for a class of elliptic PDEs involving the p-biharmonic operator and studies how these solutions concentrate around the zero set of the potential as a parameter grows large.

Contribution

It provides new existence results for solutions to elliptic PDEs with p-biharmonic operators and analyzes their concentration behavior as a parameter tends to infinity.

Findings

Existence of nontrivial solutions under certain conditions.

Solutions concentrate on the set where the potential vanishes as λ increases.

Theoretical framework for p-biharmonic elliptic equations.

Abstract

In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \begin{align*} \Bigg\{\begin{split} & \Delta^2_p u -\Delta_p u + \lambda V(x)|u|^{p-2}u = f(x,u)\,,\,x\in \mathbb{R}^N, & u \in W^{2,p}(\mathbb{R}^N), \end{split} \end{align*} where $\lambda>0$, $p>1, N>2p$ and $V\in C(\mathbb{R}^N, \mathbb{R}^+)$, $f\in C(\mathbb{R}^N \times \mathbb{R},\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $\lambda \rightarrow \infty$.