$N$-Laplacian problems with critical Trudinger-Moser nonlinearities

TL;DR

This paper establishes existence and multiplicity of solutions for N-Laplacian problems with critical exponential nonlinearities, extending known results from the semilinear case to all dimensions N ≥ 2 using novel abstract critical point theorems.

Contribution

It introduces new abstract critical point theorems based on the ${\mathbb Z}_2$-cohomological index applicable to N-Laplacian problems without linear eigenspaces.

Findings

Proved existence of solutions for N-Laplacian with critical nonlinearities.

Extended results from N=2 to all N ≥ 2.

Developed new critical point theorems for nonlinear operators.

Abstract

We prove existence and multiplicity results for a $N$-Laplacian problem with a critical exponential nonlinearity that is a natural analog of the Brezis-Nirenberg problem for the borderline case of the Sobolev inequality. This extends results in the literature for the semilinear case $N = 2$ to all $N \ge 2$. When $N > 2$ the nonlinear operator $- \Delta_N$ has no linear eigenspaces and hence this extension requires new abstract critical point theorems that are not based on linear subspaces. We prove new abstract results based on the ${\mathbb Z}_2$-cohomological index and a related pseudo-index that are applicable here.