Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. (September 2015) - Autuori, Giuseppina; Fiscella, Alessio; Pucci, Patrizia
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 L K and involving a critical nonlinearity. In particular, we consider the problem − M ( | | u | | 2 ) L K u = λ f ( x, u ) + | u | 2 s ∗ − 2 u in  Ω, u = 0 in  R n ∖ Ω, where Ω ⊂ R n is a bounded domain, 2 s ∗ is the critical exponent of the fractional Sobolev space H s ( R n ), 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.