A fractional elliptic problem in 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 R n (-Δ)su=εhuq+u2s∗-1in the convex case 1≤q<2s∗-1, where 2s∗=2n/(n-2s) is the critical fractional Sobolev exponent, (- Δ) s is the fractional Laplace operator, ε 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 with respect to the concave–convex case studied in Dipierro et al. (Fractional elliptic problems with critical growth in the whole of R n . Lecture Notes Scuola Normale Superiore di Pisa, vol 15. Springer, Berlin, 2017).