Ground state solutions for weighted biharmonic problem involving non linear exponential growth

TL;DR

This paper establishes the existence of ground state solutions for a weighted biharmonic equation with exponential nonlinearity in four dimensions, employing variational methods and degree theory.

Contribution

It introduces a novel approach to handle singular weights and critical exponential growth in biharmonic problems using constrained minimization and topological tools.

Findings

Existence of ground state solutions proven.

Handling of singular logarithmic weights in biharmonic equations.

Application of Nehari set and degree theory techniques.

Abstract

In this article, we study the following problem $$\Delta(w(x)\Delta u) = \ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R}^{4}$ and $ w(x)$ a singular weight of logarithm type. The reaction source $f(x,u)$ is a radial function with respect to $x$ and it is critical in view of exponential inequality of Adams' type. The existence result is proved by using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory results.