Normalized solutions for a class of nonlinear Choquard equations

Abstract

We prove the existence of a least energy solution to the problem $$\begin{aligned} -\Delta u-(I_{\alpha }*F(u))f(u)=\lambda u\ \text { in }\ {\mathbb {R}}^{N},\quad \int _{{\mathbb {R}}^N}u^2(x)dx = a^2, \end{aligned}$$ where $$N\ge 1$$ , $$\alpha \in (0,N)$$ , $$F(s):=\int _{0}^{s}f(t)dt$$ , and $$I_{\alpha }:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ is the Riesz potential. If f is odd in u then we prove the existence of infinitely many normalized solutions.