Fractional elliptic problems with critical growth in the whole of Rn

Abstract

We study a nonlinear and nonlocal elliptic equation. 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 the space (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.