The Nehari manifold for fractional systems involving critical nonlinearities

TL;DR

This paper investigates the existence of multiple positive solutions for a fractional system with critical nonlinearities using the Nehari manifold, revealing at least two solutions under certain parameter conditions.

Contribution

It introduces a novel application of the Nehari manifold to fractional systems with critical Sobolev exponents, establishing conditions for multiple solutions.

Findings

At least two positive solutions exist for certain parameter ranges.

The Nehari manifold approach effectively handles critical nonlinearities.

The results extend understanding of fractional systems with combined nonlinearities.

Abstract

We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $\R^2$.