The Brezis-Nirenberg problem for nonlocal systems

TL;DR

This paper studies the existence, non-existence, and regularity of solutions for a nonlocal fractional Laplacian system with critical growth nonlinearities, using variational methods.

Contribution

It introduces new variational techniques to analyze a nonlocal system with critical nonlinearities involving the fractional Laplacian.

Findings

Established conditions for existence of solutions.

Proved non-existence results under certain parameters.

Analyzed regularity properties of solutions.

Abstract

By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.