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

João Marcos do Ó; Diego Ferraz

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arXiv · 1610.04724

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

João Marcos do Ó·Diego Ferraz

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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.

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João Marcos do Ó, Diego Ferraz. 2016-10-15. Concentration-compactness at the mountain pass level for nonlocal Schr\"{o}dinger equations. https://arxiv.org/abs/1610.04724

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