Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff Dirichlet problems driven by a fractional integro-differential operator and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. We also treat in a new way the so called non-degenerate case and arise an open problem. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.