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

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in ${\mathbb{R}}^n$ $$(-\mathrm{\Delta }{)}^{s}u=\epsilon h{u}^q+u^{2_s^{\ast }-1}$$ in the convex case $1\le q<2_s^{\ast }-1$, where $2_s^{\ast }=2n\mathrm{/}(n-2s)$ is the critical fractional Sobolev exponent, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator, $\epsilon$ 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.