Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3R3

We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem{−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=f(u)in R3,u∈H1(R3),u>0in R3, where ε>0ε>0 is a parameter and a,b>0a,b>0 are constants; V is a positive continuous potential satisfying some conditions and f is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where V attains its minimum. The results are proved by using the variational methods.