Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. (September 2015)

• Record Type: Journal Article
• Authors: Autuori, Giuseppina; Fiscella, Alessio; Pucci, Patrizia
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 699
• Page End: 714
• Publication Date: 2015-09
• Subjects: primary 35J60 35R11 35B09 -- secondary 35J20 49J35 35S15; Stationary Kirchhoff Dirichlet problems -- Existence of solutions -- Fractional Sobolev spaces -- Critical nonlinearities -- Variational methods; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.014
• Languages: English
• ISSNs: 0362-546X

Abstract

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.