Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in $\mathbb{R}^2$

TL;DR

This paper proves the existence of multiple multi-bump nodal solutions for a class of elliptic equations with exponential critical growth in two-dimensional space, expanding understanding of solution multiplicity in nonlinear PDEs.

Contribution

It establishes the existence and multiplicity of multi-bump nodal solutions for elliptic problems with exponential critical growth in , under certain conditions on the potential and nonlinearity.

Findings

Multiple multi-bump nodal solutions are proven to exist.

Solutions exhibit complex nodal structures with multiple bumps.

Results extend the theory of elliptic equations with exponential critical growth.

Abstract

In this paper, we establish the existence and multiplicity of multi-bump nodal solutions for the following class of problems $$ -\Delta u+(\lambda V(x)+1)u=f(u),~~\mbox{in}~~\mathbb{R}^2, $$ where $\lambda\in(0,\infty)$, $f$ is a continuous function with exponential critical growth and $V:\mathbb{R}^2\to \mathbb{R}$ is a continuous function verifying some hypotheses.