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

Abstract 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 LK and involving a critical nonlinearity. In particular, we consider the problem -M(||u||2)LKu=λf(x,u)+|u|2s∗|-2u in Ω,u=0in ℝn Ω, where Ω⊂ℝn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. 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. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.