Concentration phenomena for a class of fractional Kirchhoff equations in RN with general nonlinearities. (June 2020)

• Record Type: Journal Article
• Authors: Ambrosio, Vincenzo
• Is Part Of: Nonlinear analysis. Volume 195(2020)
• Publication Date: 2020-06
• Subjects: 47G20 -- 35R11 -- 35J20 -- 35J60 -- 35B33; Fractional Kirchhoff problems -- Extension method -- Pohozaev-identity -- Variational methods -- Critical exponent; 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.2020.111761
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the following class of fractional Kirchhoff problems: ε 2 s M ( ε 2 s − N [ u ] s 2 ) ( − Δ ) s u + V ( x ) u = f ( u ) in R N, u ∈ H s ( R N ), u > 0 in R N, where ε > 0 is a small parameter, s ∈ ( 0, 1 ), N ≥ 2, ( − Δ ) s is the fractional Laplacian, V : R N → R is a positive continuous function, M : [ 0, ∞ ) → R is a Kirchhoff function satisfying suitable conditions and f : R → R fulfills Berestycki–Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions ( u ε ) which concentrates at a local minimum of V as ε → 0 .