Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity

Abstract

In this article, we explore the fractional Kirchhoff-Choquard system given by $$\{\begin{array}{l}(a+b{\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{\frac{s}{2}}u{|}^{2}dx)(-\Delta {)}^{s}u={\lambda }_{1}u+(I_{\mu }\ast |v{|}^{2_{\mu ,s}^{\ast }})|u{|}^{2_{\mu ,s}^{\ast }-2}u+\alpha p(I_{\mu }\ast |v{|}^q)|u{|}^{p-2}u\text{in}{\mathbb{R}}^N,\\ (a+b{\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{\frac{s}{2}}v{|}^{2}dx)(-\Delta {)}^{s}v={\lambda }_{2}v+(I_{\mu }\ast |u{|}^{2_{\mu ,s}^{\ast }})|v{|}^{2_{\mu ,s}^{\ast }-2}u+\alpha q(I_{\mu }\ast |u{|}^p)|v{|}^{q-2}v\text{in}{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}|u{|}^2=d_1^2,{\int }_{{\mathbb{R}}^N}|v{|}^2=d_2^2.\end{array}$$ where $N>2s$, $s\in (0,1)$, $\mu \in (0,N)$, $\alpha \in \mathbb{R}$. Here, $I_{\mu }:{\mathbb{R}}^N\to \mathbb{R}$ denotes the Riesz potential. We denote by $2_{\mu ,\ast }:=\frac{2N-\mu }{N}$ and $\frac{2N-\mu }{N-2s}:=2_{\mu ,s}^{\ast }$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{\mu ,\ast }<p,q<2_{\mu ,s}^{\ast }$. Our primary focus is on the existence of normalized solutions for the case $\alpha >0$ in two scenarios: the $L^2$ subcritical case characterized by $22_{\mu ,\ast }<p+q<4+\frac{4s-2\mu }{N}$ and $L^2$ supercritical associated with $4+\frac{8s-2\mu }{N}<p+q<22_{\mu ,s}^{\ast }$.