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 $\mathbb{R}^n$ $$(-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1}$$ 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.