Multiplicity of Concentrating Solutions for Choquard Equation with Critical Growth

In this paper, we consider the multiplicity and concentration phenomenon of positive solutions to the following Choquard equation -epsilon 2 Delta u + V(x)u = epsilon-alpha Q(x)(I alpha * |u|2*alpha)|u|2*alpha-2u + integral(u) in RN, where N >= 3, (N - 4)+ < alpha < N, I alpha is the Riesz potential, epsilon is a small parameter, V(x) is an element of C(RN) boolean AND L infinity(RN) is a positive potential, f is an element of C1(R+, R) is a subcritical nonlinear term and 2*alpha = N +alpha/N -2 is the upper-critical exponent in the sense of Hardy- Littlewood-Sobolev inequality. By means of variational methods and delicate energy estimates, we establish the relationship between the number of solutions and the profiles of potentials V and Q, and the concentration behavior of positive solutions is also obtained for epsilon > 0 small.