Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well

In this paper, we consider the following Kirchhoff type equations(0.1){−(a+b∫R3|∇u|2)Δu+λV(x)u=q(x)f(u) in R3,u∈H1(R3), where a, b, λ>0, V∈C(R3,R) is a potential well, q(x) is a positive bounded function, f(s) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V, q and f, the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26], which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well.