Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where is a positive parameter, a and b are positive constants, and 3<p<5. Let A(s) denotes the ground energy function associated with -(a+bR3|delta u|2dx)u+V(s)u=K(s)up, xR3, where sR3 is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|(-) with 0<2, we prove the existence of positive solutions u belonging to H1(R3) for vanishing or unbounded K(x) when > 0 small. Furthermore, we show that the solution u concentrates at the minimum points of A(s) as 0(+).