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The aim of this paper is to study a concentration-compactness principle forinhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$As an application we establish Palais-Smale compactness for the Lagrangianassociated 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 nonnegativesolutions to this class of elliptic equations for a wide class of possiblesingular potentials $a(x)$; not necessarily bounded away from zero. We considerpossible oscillatory nonlinearities and that may not satisfy theAmbrosetti-Rabinowitz condition and for both cases; subcritical and criticalgrowth range which are superlinear at origin.

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