Fractional elliptic problems with critical growth in the whole of ⁿ

This is the authors' abstract. We don't add key points for this paper.

We study the following nonlinear and nonlocal elliptic equation in~ⁿ (-Δ)ˢ u = ε\,h\,uq + uᵖ \ in ⁿ, where~s∈(0,1), $n>2s$, $ε>0$ is a small parameter, p=n+2s/n-2s, q∈(0,1), and~h∈ L¹(ⁿ)∩ L^∞(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~ⁿ (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.