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 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. 
Giuseppina Autuori, Alessio Fiscella, Patrizia Pucci
math.AP