Normalized Ground States for a Fractional Choquard System in \(\mathbb {R}\)

In this paper, we study the following fractional Choquard system \(\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{1/2}u=\lambda _1 u+(I_\mu *F(u,v))F_u (u,v), \quad \text{ in }\ \ \mathbb {R}, \\ (-\Delta )^{1/2}v=\lambda _2 v+(I_\mu *F(u,v)) F_v(u,v), \quad \text{ in }\ \ \mathbb {R}, \\ \displaystyle \int _{\mathbb {R}}|u|^2\textrm{d}x=a^2,\quad \displaystyle \int _{\mathbb {R}}|v|^2\textrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb {R}), \end{array} \right. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{(-\mathrm{\Delta })}^{1/2}u={\lambda }_{1}u+(I_{\mu }\ast F(u,v))F_u(u,v),\text{in}\mathbb{R},\hfill \\ {(-\mathrm{\Delta })}^{1/2}v={\lambda }_{2}v+(I_{\mu }\ast F(u,v))F_v(u,v),\text{in}\mathbb{R},\hfill \\ {\displaystyle {\int }_{\mathbb{R}}{|u|}^2\text{d}x=a^2,{\int }_{\mathbb{R}}{|v|}^2\text{d}x=b^2,u,v\in H^{1/2}(\mathbb{R}),}\hfill \end{array}\right)\end{array}\end{array}$$ where \((-\Delta )^{1/2}\) ${(-\mathrm{\Delta })}^{1/2}$ denotes the 1/2-Laplacian operator, \(a,b>0\) $a,b>0$ are prescribed, \(\lambda _1,\lambda _2\in \mathbb {R}\) ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ , \(I_\mu (x)=\frac{{1}}{{|x|^\mu }}\) $I_{\mu }(x)=\frac{1}{{|x|}^{\mu }}$ with \(\mu \in (0,1)\) $\mu \in (0,1)$ , \(F_u,F_v\) $F_u,F_v$ are partial derivatives of F and \(F_u,F_v\) $F_u,F_v$ have exponential critical growth in \(\mathbb {R}\) $\mathbb{R}$ . By using a minimax principle and analyzing the monotonicity of the ground state energy with respect to the prescribed masses, we obtain at least one normalized ground state solution for the above system.