Existence and concentration of ground state solutions for a class of Kirchhoff-type problems

Abstract

This paper is concerned with the following singularly perturbed Kirchhoff-type problem −ε2a+εb∫R3|∇u|2dx△u+V(x)u=f(u),x∈R3;u∈H1(R3),where ε>0 is a small parameter, a,b>0 are two constants, V∈C(R3,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)∕u3. By using some new techniques and subtle analyses, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], 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−2u for s∈(2,6), and extend the existing results concerning the case that f(u)∼|u|s−2u for s∈[4,6).