Existence and concentration of ground state solutions for a class of Kirchhoff-type problems. (June 2020)

• Record Type: Journal Article
• Authors: Lin, Xiaoyan; Wei, Jiuyang
• Is Part Of: Nonlinear analysis. Volume 195(2020)
• Publication Date: 2020-06
• Subjects: 35J20 -- 35J65 -- 35J60; Kirchhoff-type problem -- Semiclassical state -- Concentration behaviour -- 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.2019.111715
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with the following singularly perturbed Kirchhoff-type problem − ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x △ u + V ( x ) u = f ( u ), x ∈ R 3 ; u ∈ H 1 ( R 3 ), where ε > 0 is a small parameter, a, b > 0 are two constants, V ∈ C ( R 3, R ), and f ∈ C ( R, R ) is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition on f ( u ) ∕ u 3 . By using some new techniques and subtle analyses, we prove that there exists a constant ε 0 > 0 determined by V and f such that for ε ∈ ( 0, ε 0 ], the above problem has a ground state solution concentrating around global minimum of V in the semi-classical limit. Our results are available to the case that f ( u ) ∼ | u | s − 2 u for s ∈ ( 2, 6 ), and extend the existing results concerning the case that f ( u ) ∼ | u | s − 2 u for s ∈ [ 4, 6 ) .