Critical nonlocal systems with concave-convex powers

TL;DR

This paper proves the existence of multiple solutions for a fractional p-Laplacian system with critical nonlinearities using advanced variational methods, despite the lack of a classification for optimizers in the critical fractional Sobolev embedding.

Contribution

It introduces a novel combination of fibering and Nehari manifold techniques to handle critical fractional systems with concave-convex powers.

Findings

Multiple solutions established under smallness conditions on parameters.

Application of fibering and Nehari manifold methods to fractional systems.

Overcomes challenges due to non-classification of Sobolev optimizers.

Abstract

By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional $p$-Laplacian system involving critical concave-convex nonlinearities provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained despite there is no general classification for the optimizers of the critical fractional Sobolev embedding.