Ground state normalized solutions to Schrödinger equation with inverse-power potential and HLS lower critical Hartree-type nonlinearity

Abstract

This paper is concerned with the following Schrödinger equation with inverse-power potential and HLS lower critical Hartree-type nonlinearity

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\Delta }u-\frac{u}{|x|^s}-\mu |u|^{q-2}u-(I_\alpha *h|u|^{2_\alpha })h|u|^{2_\alpha -2}u=\lambda u\ \ \text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N} u^2 dx = c, \end{array}\right. } \end{aligned}$$

where $\alpha \in (0,N)$, $N \ge 3$, $s\in (0,2)$, $\mu , c>0$, $2<q<2+\frac{4}{N}$, $\lambda \in {\mathbb {R}}$ is an unknown Lagrange multiplier and $h:{\mathbb {R}}^N\rightarrow (0,\infty )$ is a continuous function. Under reasonable assumptions on h(x), we investigate the existence and asymptotic behavior of ground state normalized solutions to the above problem. Compared with the existing references, we extend the results obtained by Li et al. (Comput Math Appl 79:303–316, 2020) to the HLS lower critical case.