Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $\mu|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev upper critical exponent case $p=(N+\alpha)/(N-2)$. Our results are a nonlocal counterpart of the results in \cite{{Li 2021-4},{Soave JFA},{Wei-Wu 2021}}.