Nodal solutions for Logarithmic weighted $N$-Laplacian problem with exponential nonlinearities

TL;DR

This paper establishes the existence of nodal solutions for a logarithmic weighted N-Laplacian problem with exponential nonlinearities in a unit ball, using variational methods and degree theory.

Contribution

It introduces a novel approach combining constrained minimization, deformation lemma, and degree theory to find nodal solutions for this class of nonlinear PDEs.

Findings

Existence of nodal solutions proven

Solutions are characterized via variational methods

Applicable to problems with exponential nonlinearities and singular weights

Abstract

In this article, we study the following problem $$-{\rm div} (\omega(x)|\nabla u|^{N-2} \nabla u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R^{N}}$, $N\geq2$ and $ w(x)$ a singular weight of logarithm type. The reaction source $f(x,u)$ is a radial function with respect to $x$ and is subcritical or critical with respect to a maximal growth of exponential type. By using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions.