Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation \(\begin{aligned} {\left\{ \begin{array}{ll} -{\Delta }u-\mu (I_\alpha *[h|u|^p])h|u|^{p-2}u-(I_\alpha *|u|^{2^*_\alpha })|u|^{2^*_\alpha -2}u=\lambda u\ \ \text{ in }\ \mathbb {R}^N, \\ \int _{\mathbb {R}^N} u^2 dx = c, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u-\mu (I_{\alpha }{\ast [h|u|}^p{])h|u|}^{p-2}u-(I_{\alpha }{\ast |u|}^{2_{\alpha }^{\ast }}{)|u|}^{2_{\alpha }^{\ast }-2}u=\lambda u\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{u}^{2}dx=c,\hfill \end{array}\right)\end{array}$$ where \(\mu ,c>0\) $\mu ,c>0$ , \(N \ge 3\) $N\ge 3$ , \(0<\alpha <N\) $0<\alpha <N$ , \(2_\alpha :=\frac{N+\alpha }{N}<p<2^*_\alpha :=\frac{N+\alpha }{N-2}\) $2_{\alpha }:=\frac{N+\alpha }{N}<p<2_{\alpha }^{\ast }:=\frac{N+\alpha }{N-2}$ , \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ is a Lagrange multiplier, \(I_\alpha \) $I_{\alpha }$ is the Riesz potential and \(h:\mathbb {R}^N\rightarrow (0,\infty )\) $h:{\mathbb{R}}^N\to (0,\infty )$ is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case \(\frac{N+\alpha +2}{N}\le p<\frac{N+\alpha }{N-2}\) $\frac{N+\alpha +2}{N}\le p<\frac{N+\alpha }{N-2}$ and discuss its asymptotical behaviors as \(\mu \rightarrow 0^+\) $\mu \to 0^{+}$ and \(c\rightarrow 0^+\) $c\to 0^{+}$ respectively. When \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\) $\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}$ , we obtain the existence of one local minimizer after considering a suitable minimization problem.