Three solutions for a nonlocal problem with critical growth

TL;DR

This paper proves the existence of three distinct solutions to a nonlocal fractional p-Laplacian problem with critical growth, using advanced variational methods and concentration compactness techniques.

Contribution

It introduces new solutions for a critical nonlocal problem by extending concentration compactness and variational principles to the fractional p-Laplacian setting.

Findings

Established existence of three solutions with different signs.

Extended concentration compactness principle to fractional p-Laplacian.

Applied Ekeland's variational principle successfully.

Abstract

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where $(-\Delta_p)^s$ is the well known $p$-fractional Laplacian and $p^*_s=\frac{np}{n-sp}$ is the critical Sobolev exponent for the non local case. The proof is based in the extension of the Concentration Compactness Principle for the $p$-fractional Laplacian and Ekeland's variational Principle.