  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.
