Asymmetric critical $p$-Laplacian problems

TL;DR

This paper establishes the existence of solutions for asymmetric critical $p$-Laplacian problems in bounded domains, employing topological linking methods suitable for nonlinear problems with critical growth.

Contribution

It introduces a linking theorem based on the ${f Z}_2$-cohomological index to find solutions without relying on a direct sum decomposition.

Findings

Existence of solutions for $p < N$ with critical Sobolev exponent.

Existence of solutions for $p = N$ with exponential nonlinearity.

Application of topological linking methods in nonlinear PDEs with critical growth.

Abstract

We obtain nontrivial solutions for two types of critical $p$-Laplacian problems with asymmetric nonlinearities in a smooth bounded domain in ${\mathbb R}^N,\, N \ge 2$. For $p < N$, we consider an asymmetric problem involving the critical Sobolev exponent $p^\ast = Np/(N - p)$. In the borderline case $p = N$, we consider an asymmetric critical exponential nonlinearity of the Trudinger-Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the ${\mathbb Z}_2$-cohomological index to obtain our solutions.