Normalized solutions for the Choquard equation with mass supercritical nonlinearity

Abstract

We consider the nonlinear Choquard equation $$\{\begin{array}{llc}& -\Delta u=(I_{\alpha }\ast F(u))F^{\prime }(u)-\mu u\text{in}{\mathbb{R}}^N,& u\in H^1({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=m,\end{array}$$ 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.

arXiv admin note: text overlap with arXiv:2002.03973 by other authors