Normalized solutions for the Choquard equation with mass supercritical nonlinearity

Abstract

We consider the nonlinear Choquard equation $$\begin{cases} & - \Delta u = (I_\alpha \ast F(u))F'(u) -\mu u \ \text{in}\ \mathbb{R}^N, & u \in \ H^1(\mathbb{R}^N), \ \int_{\mathbb{R}^N} |u|^2 dx=m, \end{cases} $$ where $\alpha\in(0,N)$, $m>0$ is prescribed, $\mu \in \mathbb{R}$ is a Lagarange multiplier, and $I_\alpha$ is the Riesz potential. Under general assumptions on the nonlinearity $F,$ we prove the existence and multiplicity of normalized solutions.