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

Abstract

The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space Ds,2(RN) 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 (−Δ)su=f(x,u) for 0<s<1. Moreover, using an analytic framework based on Ds,2(RN), we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.