  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.
