On a class of critical $(p,q)$-Laplacian problems

TL;DR

This paper establishes the existence of nontrivial solutions for a class of critical $(p,q)$-Laplacian problems in bounded domains, overcoming compactness issues and lack of classical decomposition methods.

Contribution

It introduces an abstract critical point theorem based on cohomological index and constructs minimax levels to find solutions.

Findings

Existence of nontrivial solutions for critical $(p,q)$-Laplacian problems.

Development of a new critical point theorem using cohomological index.

Solutions obtained despite loss of compactness and absence of classical decomposition.

Abstract

We obtain nontrivial solutions of a critical $(p,q)$-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, 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.