Weighted biharmonic equations involving continous potentiel under exponential nonlinear growth

TL;DR

This paper investigates a weighted biharmonic equation with exponential nonlinear growth in the unit ball, establishing the existence of positive solutions using variational methods and concentration compactness techniques.

Contribution

It introduces a new approach to handle critical exponential growth in weighted biharmonic problems with continuous potentials, proving existence of solutions.

Findings

Existence of a nontrivial positive weak solution is established.

A concentration compactness result is proved to overcome loss of compactness.

The problem is addressed under critical exponential growth conditions.

Abstract

We deal with a weighted biharmonic problem in the unit ball of $\mathbb{R}^{4}$. The non-linearity is assumed to have critical exponential growth in view of Adam's type inequalities. The weight $w(x)$ is of logarithm type and the potential $V$ is a positive continuous function on $\overline{B}$. It is proved that there is a nontrivial positive weak solution to this problem by the mountain Pass Theorem. We avoid the loss of compactness by proving a concentration compactness result and by a suitable asymptotic condition.