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 -ε ^2Δ u+V(x)u=ε ^-αQ(x)(I_α *|u|^2^*_α) |u|^2^*_α-2u+ f(u) in ℝ^N, where N≥ 3 , (N-4)_+<α < N , I_α is the Riesz potential, ε is a small parameter, V(x)∈ C(ℝ^N)∩ L^∞ (ℝ^N) is a positive potential, f∈ C^1(ℝ^+,ℝ) is a subcritical nonlinear term and 2^*_α=N+α/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 ε >0 small.