Fourth order weighted elliptic problem under exponential nonlinear growth

TL;DR

This paper investigates a weighted biharmonic problem in four dimensions with exponential nonlinear growth, establishing the existence of solutions using critical point theory despite compactness challenges.

Contribution

It introduces a new growth condition for exponential nonlinearities in weighted Sobolev spaces, ensuring the Palais-Smale condition for solution existence.

Findings

Existence of non-trivial solutions proven

New growth condition for exponential nonlinearity introduced

Overcomes compactness loss due to critical growth

Abstract

We deal with nonlinear weighted biharmonic problem in the unit ball of $\mathbb{R}^{4}$. The weight is of logarithm type. The nonlinearity is critical in view of Adam's inequalities in the weighted Sobolev space $W^{2,2}_{0}(B,w)$. We prove the existence of non trivial solutions via the critical point theory. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. We give a new growth condition and we point out its importance for checking the Palais-Smale compactness condition.