Critical fractional $p$-Laplacian problems with possibly vanishing potentials

TL;DR

This paper establishes the existence of solutions for a critical fractional p-Laplacian problem in unbounded domains with possibly vanishing potentials, overcoming compactness and decomposition challenges using a generalized linking method.

Contribution

It introduces a novel linking construction based on the Z2-cohomological index to handle the lack of compactness and decomposition in critical fractional p-Laplacian problems.

Findings

Existence of nontrivial solutions in the whole space

Handling of vanishing potentials in critical problems

Development of a generalized linking approach

Abstract

We obtain nontrivial solutions of a critical fractional $p$-Laplacian equation in the whole space and with possibly vanishing potentials. In addition to the usual difficulty of the lack of compactness associated with problems involving critical Sobolev exponents, the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments. We overcome this difficulty using a generalized linking construction based on the ${\mathbb Z}_2$-cohomological index.