Normalized ground states for a fractional Choquard system in $\mathbb{R}$

Abstract

In this paper, we study the following fractional Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} (-Δ)^{1/2}u=λ_1 u+(I_μ*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}, (-Δ)^{1/2}v=λ_2 v+(I_μ*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2\mathrm{d}x=a^2,\quad \displaystyle\int_{\mathbb{R}}|v|^2\mathrm{d}x=b^2,\quad u,v\in H^{1/2}(\mathbb{R}), \end{array} \right. \end{split} \end{align*} 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.

arXiv admin note: substantial text overlap with arXiv:2306.02963, arXiv:2307.06602