  We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$
(-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$
where~$s\in(0,1)$, $n>2s$, $\epsilon>0$ is a small parameter,
$p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and~$h\in L^1(\R^n)\cap L^\infty(\R^n)$.
The problem has a variational structure, and this allows us to find a positive
solution by looking at critical points of a suitable energy functional. In
particular, in this paper, we find a local minimum and a mountain pass solution
of this functional. One of the crucial ingredient is a
Concentration-Compactness principle.
  Some difficulties arise from the nonlocal structure of the problem and from
the fact that we deal with an equation in the whole of~$\R^n$ (and this causes
lack of compactness of some embeddings). We overcome these difficulties by
looking at an equivalent extended problem.
