Concentration-compactness at the mountain pass level for nonlocal Schr\"{o}dinger equations

TL;DR

This paper develops a concentration-compactness principle for fractional Sobolev spaces and applies it to establish existence results for nonlocal Schrödinger equations with singular potentials and oscillatory nonlinearities.

Contribution

It introduces a new concentration-compactness framework for inhomogeneous fractional Sobolev spaces and proves existence of solutions under broad conditions, including unbounded and oscillatory nonlinearities.

Findings

Established Palais-Smale compactness for fractional Schrödinger equations.

Proved existence of nontrivial nonnegative solutions with singular potentials.

Handled nonlinearities without the Ambrosetti-Rabinowitz condition.

Abstract

The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schr\"{o}dinger equation $(-\Delta)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the Ambrosetti-Rabinowitz condition and for both cases; subcritical and critical growth range which are superlinear at origin.