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

Abstract

In this paper, we study the following fractional Choquard system

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{1/2}u=\lambda _1 u+(I_\mu *F(u,v))F_u (u,v), \quad \text{ in }\ \ \mathbb {R}, \\ (-\Delta )^{1/2}v=\lambda _2 v+(I_\mu *F(u,v)) F_v(u,v), \quad \text{ in }\ \ \mathbb {R}, \\ \displaystyle \int _{\mathbb {R}}|u|^2\textrm{d}x=a^2,\quad \displaystyle \int _{\mathbb {R}}|v|^2\textrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb {R}), \end{array} \right. \end{aligned} \end{aligned}$$

where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(a,b>0\) are prescribed, \(\lambda _1,\lambda _2\in \mathbb {R}\), \(I_\mu (x)=\frac{{1}}{{|x|^\mu }}\) with \(\mu \in (0,1)\), \(F_u,F_v\) are partial derivatives of F and \(F_u,F_v\) have exponential critical growth in \(\mathbb {R}\). 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.