Fractional elliptic problems with critical growth in the whole of $\R^n$

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.