Normalized Solutions for the Fractional Choquard Equations with Hardy–Littlewood–Sobolev Upper Critical Exponent

Abstract

In the present paper, we study the existence of normalized ground states for the following nonlinear fractional Choquard equations with Hardy–Littlewood–Sobolev upper critical exponent:

$$\begin{aligned} (-\Delta )^su=\lambda u+\mu (I_\alpha *|u|^p)|u|^{p-2}u+(I_\alpha *|u|^{2^*_{\alpha ,s}}) |u|^{2^*_{\alpha ,s}-2}u,\quad x\in \mathbb R^{N}, \end{aligned}$$

having prescribed mass

$$\begin{aligned} \int _{\mathbb R^N}|u|^2dx=a>0, \end{aligned}$$

where $N>2s$, $s\in (0,1)$, $\alpha \in (0,N)$, $\underline{p}:= \frac{N+\alpha }{N}<p<\overline{p}:=\frac{N+2s+\alpha }{N}<2^*_{\alpha ,s}$, $2^*_{\alpha ,s}:=\frac{N+\alpha }{N-2s}$ is the upper Hardy–Littlewood–Sobolev critical exponent, $\mu >0$ and $\lambda \in \mathbb R$. Furthermore, the qualitative behavior of the ground states as $\mu \rightarrow 0^+$ is also studied.