A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities

TL;DR

This paper proves the existence of positive solutions for a fractional elliptic equation with critical growth and convex nonlinearities in inity, using variational methods, harmonic extension, and concentration-compactness techniques.

Contribution

It introduces a novel approach to handle critical fractional problems with convex nonlinearities via harmonic extension and variational methods.

Findings

Existence of positive solutions established for the fractional elliptic problem.

Application of the Mountain-Pass Theorem in a nonlocal setting.

Use of concentration-compactness to overcome loss of compactness.

Abstract

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $\mathbb{R}^n$ \[ (-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1} \] in the convex case $1\leq q<2_s^*-1$, where $ 2_s^*={2n}/({n-2s}) $ is the critical fractional Sobolev exponent, $(-\Delta)^s$ is the fractional Laplace operator, $\varepsilon$ is a small parameter and $h$ is a given bounded, integrable function. The problem has a variational structure and we prove the existence of a solution by using the classical Mountain-Pass Theorem. We work here with the harmonic extension of the fractional Laplacian, which allows us to deal with a weighted (but possibly degenerate) local operator, rather than with a nonlocal energy. In order to overcome the loss of compactness induced by the critical power we use a Concentration-Compactness principle. Moreover, a finer analysis of the geometry of the energy functional is needed in this convex case.