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 ${\mathcal{D}}^{s,2}({\mathbb{R}}^N)$ 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.

References in corpus (1)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian

Cited by in corpus (1)

• The Nehari manifold for fractional p-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions