$(N,q)$-Laplacian problems with critical Trudinger-Moser nonlinearities

TL;DR

This paper establishes the existence of nontrivial solutions for a $(N,q)$-Laplacian problem with critical Trudinger-Moser nonlinearities, overcoming compactness issues and lacking classical sum decomposition methods.

Contribution

It introduces an abstract critical point theorem based on cohomological index to find solutions in the presence of critical nonlinearities and non-standard problem structure.

Findings

Existence of nontrivial solutions for the problem.

Development of a new variational framework using cohomological index.

Solutions obtained at energy levels below a specific threshold.

Abstract

We obtain nontrivial solutions of a $(N,q)$-Laplacian problem with a critical Trudinger-Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.