Existence and Asymptotical Behavior of $$L^2$$ -Normalized Standing Wave Solutions to HLS Lower Critical Choquard Equation with a Nonlocal Perturbation

Abstract

This paper is concerned with the following HLS lower critical Choquard equation with a nonlocal perturbation

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

where \(\alpha \in (0,N)\), \(N \ge 3\), \(\mu , c>0\), \(\frac{N+\alpha }{N}<q<\frac{N+\alpha +2}{N}\), \(\lambda \in {\mathbb {R}}\) is an unknown Lagrange multiplier and \(h:{\mathbb {R}}^N\rightarrow (0,\infty )\) is a continuous function. The novelty of this paper is that we not only investigate autonomous case but also handle nonautonomous situation for the above problem. For both cases, we prove the existence and discuss asymptotic behavior of ground state normalized solutions. Compared with the existing references, we extend the recent results obtained by Ye et al. (J Geom Anal 32:242, 2022) to the HLS lower critical case.