Multiplicity and concentration behavior of solutions to the critical Kirchhoff-type problem

In this paper, we study the multiplicity and concentration phenomenon of positive solutions to the critical Kirchhoff-type problem: -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2)dx)Delta u + V(x)u = f(u) + h(x)u(5) in R-3 where epsilon is a small positive parameter, a, b are positive constants, V is an element of C(R-3, R) is a positive potential, f is an element of C-1(R+, R) is a subcritical nonlinear term, h(x)u(5) is a critical nonlinearity. When epsilon > 0 small, we establish the relationship between the number of positive solutions and the profile of V and h. The concentration behavior and some further properties of positive solutions are also obtained.