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 the 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} $, and $ \lambda\in \mathbb{R} $ is an unknown parameter that appears as a Lagrange multiplier. The results of this paper about this equation answer some questions proposed by Yao, Chen, Rǎdulescu and Sun [Siam J. Math. Anal., 54(3) (2022), 3696-3723]. Moreover, based on the results obtained, we study the multiplicity of normalized solutions to the non-autonomous Choquard equation

$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = (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 \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx = c^2, \end{cases} \end{equation*} $

where $ \epsilon>0 $, $ 2<q<2+\frac{4}{N} $, and $ h $ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $ h $ when $ \epsilon $ is small enough.