Normalized solutions to critical Choquard systems with linear and nonlinear couplings

Abstract

We consider the critical Choquard system with both linear and nonlinear couplings $-\hat{I}”v_1+\hat{I}{¼}_1{v}_1=(I_{\ddot{I}}‰\ast |v_1{|}^{2_{\ddot{I}}{‰}^{\ast }})|v_1{|}^{2_{\ddot{I}}{‰}^{\ast }-2}{v}_1+\hat{I}¸p(I_{\ddot{I}}‰\ast |v_2{|}^q)|v_1{|}^{p-2}{v}_1+\epsilon v_2,in{\mathbb{R}}^N,-\hat{I}”v_2+\hat{I}{¼}_2{v}_2=(I_{\ddot{I}}‰\ast |v_2{|}^{2_{\ddot{I}}{‰}^{\ast }})|v_2{|}^{2_{\ddot{I}}{‰}^{\ast }-2}{v}_2+\hat{I}¸q(I_{\ddot{I}}‰\ast |v_1{|}^p)|v_2{|}^{q-2}{v}_2+\epsilon v_1,in{\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}{v}_1^2=\hat{I}{\pm }_1^2,{\int }_{{\mathbb{R}}^N}{v}_2^2=\hat{I}{\pm }_2^2,$ where $N=3\text{or}4$, $\hat{I}{\pm }_1,\hat{I}{\pm }_2>0$, $\hat{I}¸>0$, $2_{\ddot{I}‰,\ast }:=\frac{N+\ddot{I}‰}{N}<p,q<2_{\ddot{I}}{‰}^{\ast }:=\frac{N+\ddot{I}‰}{N-2}$, $\epsilon >0$, $0<\ddot{I}‰<N$, $I_{\ddot{I}}‰:{\mathbb{R}}^N\to \mathbb{R}$ represents the Riesz potential. For the $L^2$-subcritical case $p+q<\frac{2N+2\ddot{I}‰+4}{N}$, we utilize the Ekeland's variational principle to obtain the existence of a positive normalized ground state for the system as $0<\hat{I}¸<\hat{I}{¸}_0,0<\epsilon <{\epsilon }_{\ast }$. For the $L^2$-supercritical case $p+q>\frac{2N+2\ddot{I}‰+4}{N}$, we apply variational methods to establish the existence of a positive normalized ground state for the system as $\hat{I}¸>\hat{I}{¸}_{\ast },0<\epsilon <\overline{\epsilon }$.

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