Concentration-compactness principle for nonlocal scalar field equations with critical growth

TL;DR

This paper develops a concentration-compactness principle for fractional Sobolev spaces and applies it to establish compactness and existence results for fractional scalar field equations with critical growth.

Contribution

It introduces a new concentration-compactness principle for fractional Sobolev spaces and proves existence of ground state solutions for critical growth nonlinearities.

Findings

Established Palais-Smale compactness for fractional scalar field equations.

Proved existence of ground state solutions in critical growth range.

Extended concentration-compactness methods to fractional Sobolev spaces.

Abstract

The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space $\mathcal{D}^{s,2} (\mathbb{R}^N)$ for $0<s<\min\{1,N/2\}.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional scalar field equation $(-\Delta)^{s} u = f(x,u)$ for $0<s<1.$ Moreover, using an analytic framework based on $\mathcal{D}^{s,2}(\mathbb{R}^N),$ we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.