Existence and concentration of ground state solutions for a class of Kirchhoff-type problems. (June 2020) - Lin, Xiaoyan; Wei, Jiuyang
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 ) .