On a fractional system of NLS-KDV equations with Hardy potentials

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger-Korteweg-De Vries (NLS-KdV, in short) equations with Hardy potentials: \begin{equation*} \left{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{}-1} &= 2νh(x) u^{}v^{} & \quad \mbox{in} ~ $\mathbb{R}$^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{}-1} &= νh(x) u^{2} & \quad \mbox{in} ~ $\mathbb{R}$^{N}, u,v >0 \quad \mbox{in} ~ $\mathbb{R}$^{N} \setminus {0}, \end{aligned} \right. \end{equation*} where $s_1,s_2\in (0,1)\text{ and }{\lambda }_i\in (0,{\mathrm{\Lambda }}_{N,s_i})$ with ${\mathrm{\Lambda }}_{N,s_i}=2{\pi }^{N\mathrm{/}2}\frac{{\mathrm{\Gamma }}^2(\frac{N+2s_i}{4})\mathrm{\Gamma }(\frac{N+2s_i}{2})}{{\mathrm{\Gamma }}^2(\frac{N-2s_i}{4})\mathrm{\mid }\mathrm{\Gamma }(-s_i)\mathrm{\mid }},(i=1,2)$. By imposing certain assumptions on the parameter $\nu$ and on the function $h$, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.