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Normalized solutions for the Choquard equation with mass supercritical nonlinearity

Na XuShiwang Ma
Oct 2022
摘要
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 Lagarangemultiplier, and $I_\alpha$ is the Riesz potential. Under general assumptions on the nonlinearity $F,$ we prove the existence andmultiplicity of normalized solutions.
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