Normalized solutions to lower critical Choquard equation with a local perturbation

In this paper, we study the existence and non-existence of normalized solutions to lower critical Choquard equation with a local perturbation $ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \gamma (I_{\alpha}\ast|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u+\mu |u|^{q-2}u, \quad {\rm{in}}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx c^2, \end{cases} \end{equation*} where \gamma, \mu, c>0 $, 2<q\leq 2+\frac{4}{N} \lambda\in \mathbb{R} is an unknown parameter that appears as Lagrange multiplier. The results paper about answer some questions proposed by Yao, Chen, Rǎdulescu Sun [Siam J. Math. Anal., 54(3) (2022), 3696-3723]. Moreover, based on obtained, multiplicity non-autonomous equation$ (I_\alpha\ast [h(\epsilon x)|u|^{\frac{N+\alpha}{N}}])h(\epsilon x)|u|^{\frac{N+\alpha}{N}-2}u+\mu|u|^{q-2}u, \ x\in $where \epsilon>0 2<q<2+\frac{4}{N} h positive continuous function. It proved numbers are at least global maximum points when \epsilon small enough.