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

Abstract

We study the following nonlinear and nonlocal elliptic equation in~${\mathbb{R}}^n$ $$ (-Δ)^s u = ε\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in (0,1)$, $n>2s$, $\epsilon >0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in (0,1)$, and~$h\in L^1({\mathbb{R}}^n)\cap L^{\infty }({\mathbb{R}}^n)$. 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~${\mathbb{R}}^n$ (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.

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