Existence of multi-bump solutions to biharmonic operator with critical exponential growth in $\mathbb{R}^4$

TL;DR

This paper proves the existence of multiple localized solutions for a biharmonic PDE with critical exponential growth in four-dimensional space using variational methods.

Contribution

It establishes the existence of multi-bump solutions for a class of biharmonic equations with critical exponential growth, extending previous results to this complex setting.

Findings

Multi-bump solutions exist for the biharmonic problem.

Variational methods are effective in handling critical exponential growth.

Results contribute to understanding higher-order PDEs with critical nonlinearities.

Abstract

Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{4}, u \in H^{2}(\mathbb{R}^{4}), \end{array} \right. $$ where $\Delta^2$ is the biharmonic operator, $f$ is a continuous function with critical exponential growth and $V : \mathbb{R}^4 \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.