  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.
