Normalized solutions to subcritical Choquard systems with double couplings

Abstract

We consider the Choquard system with both linear and nonlinear couplings $-\hat{I}”u+\hat{I}{¼}_{1}u=\hat{I}{»}_1(I_{\hat{I}}\pm \ast |u{|}^{r_1})|u{|}^{r_1-2}u+{\hat{I}}^{2}p(I_{\hat{I}}\pm \ast |v{|}^q)|u{|}^{p-2}u+\hat{I}ºv,$ $-\hat{I}”v+\hat{I}{¼}_{2}v=\hat{I}{»}_2(I_{\hat{I}}\pm \ast |v{|}^{r_2})|v{|}^{r_2-2}v+{\hat{I}}^{2}q(I_{\hat{I}}\pm \ast |u{|}^p)|v{|}^{q-2}v+\hat{I}ºu,$ ${\int }_{{\mathbb{R}}^N}{u}^2=\ddot{I}{}_1^2,{\int }_{{\mathbb{R}}^N}{v}^2=\ddot{I}{}_2^2,$ where $N\in \{3,4\}$, $\hat{I}{»}_1,\hat{I}{»}_2,{\hat{I}}^2,\hat{I}º,\ddot{I}{}_1,\ddot{I}{}_2>0$, $2_{\hat{I}\pm ,\ast }:=\frac{N+\hat{I}\pm }{N}<p,q,r_1,r_2<2_{\hat{I}}{\pm }^{\ast }:=\frac{N+\hat{I}\pm }{N-2}$ and $p+q\le 2r_1\le 2r_2$ . We investigate a classification result as the parameters $p+q$, $2r_1$ and $2r_2$ vary across the ranges $(\frac{2N+2\hat{I}\pm }{N},\frac{2N+2\hat{I}\pm +4}{N})$, $\{\frac{2N+2\hat{I}\pm +4}{N}\}$, and $(\frac{2N+2\hat{I}\pm +4}{N},\frac{2N+2\hat{I}\pm }{N-2})$. Employing variational methods, we demonstrate the existence of a normalized ground state for the system in the mass subcritical, critical, and supercritical cases.

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