  This paper deals with the existence and the asymptotic behavior of
non-negative solutions for a class of stationary Kirchhoff problems driven by a
fractional integro-differential operator $\mathcal L_K$ and involving a
critical nonlinearity. The main feature, as well as the main difficulty, of the
analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that
is the problem is degenerate. The adopted techniques are variational and the
main theorems extend in several directions previous results recently appeared
in the literature.
