Normalized solutions to lower critical Choquard equation with a local perturbation

Abstract

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 \text{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, R\v{a}dulescu 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.