Existence and concentration of ground state solutions for critical Kirchhoff-type equation with steep potential well

In this paper, we study the nonlinear Kirchhoff type equation {-(a + b integral(R3) vertical bar del u vertical bar(2) dx) Delta u + lambda V (x)u = vertical bar u vertical bar(4) u + f(u), x is an element of R-3, u is an element of H-1 (R-3), where a, b are positive constants and lambda > 0. Suppose that the non- negative continuous potential V represents a potential well with the bottom V-1 (0) and f is an element of C(R, R) satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the concentration behavior of the ground state solutions as lambda -> infinity is also worth considering.