On a class of critical $N$-Laplacian problems

TL;DR

This paper proves the existence of solutions for a class of critical N-Laplacian problems in bounded domains using an abstract linking theorem and cohomological index, addressing challenges without direct sum decomposition.

Contribution

It introduces a novel approach employing a ${ m Z}_2$-cohomological index and linking theorem to establish solutions where traditional methods fail.

Findings

Established existence of solutions for critical N-Laplacian problems

Developed an abstract linking framework based on cohomological index

Overcame the lack of direct sum decomposition in the analysis

Abstract

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in ${\mathbb R}^N$. In the absence of a suitable direct sum decomposition, we use an abstract linking theorem based on the ${\mathbb Z}_2$-cohomological index to obtain a nontrivial critical point.