Normalized ground states solutions for nonautonomous Choquard equations

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-\mathrm{\Delta }u-\lambda u=(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}\ast A\mathrm{\mid }u{\mathrm{\mid }}^p)A\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,{\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c,u\in H^1({\mathbb{R}}^N,\mathbb{R}),$$ where $c>0$, $0<\mu <N$, $\lambda \in \mathbb{R}$, $A\in C^1({\mathbb{R}}^N,\mathbb{R})$. For $p\in (2_{\ast ,\mu },\bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2_{\mu }^{\ast })$, we find a normalized solution, which is not a global minimizer. $2_{\mu }^{\ast }$ and $2_{\ast ,\mu }$ are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. $\bar{p}$ is $L^2-$critical exponent. Our results generalize and extend some related results.