Positive solutions for the fractional p-Laplacian via mixed topological and variational methods

TL;DR

This paper establishes the existence of two positive solutions for a nonlinear nonlocal fractional p-Laplacian problem using a novel combination of topological and variational methods, under certain local conditions.

Contribution

It introduces a mixed approach combining topological degree theory and variational methods to prove positive solutions for fractional p-Laplacian problems.

Findings

Existence of two positive solutions in coercive cases

Existence of two positive solutions in noncoercive cases

Application of combined topological and variational techniques

Abstract

We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We assume local conditions ensuring the existence of sub- and supersolutions. So we prove existence of two positive solutions, in both the coercive and noncoercive cases.