Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity

Abstract

In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} u|^2\;dx)(-\Delta)^su=\lambda_1u+(I_{\mu}*|v|^{{2^*_{\mu,s}}})|u|^{{2^*_{\mu,s}}-2}u +\alpha p (I_{\mu}*|v|^{q})|u|^{p-2}u \;\text{in}\;\mathbb{R}^N,\\ (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v|^2\;dx)(-\Delta)^sv=\lambda_2v+ (I_{\mu}*|u|^{{2^*_{\mu,s}}})|v|^{{2^*_{\mu,s}}-2}u +\alpha q(I_{\mu}*|u|^{p})|v|^{q-2}v \;\;\text{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=d_1^2,\;\;\int_{\mathbb{R}^N}|v|^2=d_2^2. \end{array} \right. $$ 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 $2_{\mu,*}:=\frac{2N-\mu}{N}$ and $\frac{2N-\mu}{N-2s}:={2^*_{\mu,s}}$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{\mu,*} < p,q< {2^*_{\mu,s}}$. 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 $22_{\mu,*}<p + q < 4 + \frac{4s-2\mu}{N}$ and $L^2$ supercritical associated with $4+\frac{8s-2\mu}{N}< p + q < 2{2^*_{\mu,s}}$.