Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth

In this paper we concern with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation{−(ε2a+bε∫R3|∇u|2)Δu+M(x)u=λf(u)+|u|4u,x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0ε>0 is a small parameter, a, b   are positive constants and λ>0λ>0 is a parameter, and f   is a continuous superlinear and subcritical nonlinearity. Suppose that M(x)M(x) has at least one minimum. We first prove that the system has a positive ground state solution uεuε for λ>0λ>0 sufficiently large and ε>0ε>0 sufficiently small. Then we show that uεuε converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of M(x)M(x) in certain sense as ε→0ε→0. Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik–Schnirelmann theory.