The existence of positive solutions with prescribed L2-norm for nonlinear Choquard equations

In this paper, we study the existence of positive solutions with prescribed L2-norm to a class of nonlinear Choquard equation −Δu − λu = (Iα∗F(u)) F′(u) in ℝN, where λ ∈ ℝ, N ≥ 3, α ∈ (0, N), Iα:ℝN → ℝ is the Riesz potential. Under some conditions imposed on F, by using a minimax procedure and the concentration compactness of P. L. Lions, we show that for any c > 0, the equation possesses at least a couple (uc, λc) ∈ H1(ℝN) × ℝ− of weak solution such that ‖u‖L2(RN)2=c.