Normalized solutions to critical Choquard systems with linear and nonlinear couplings

TL;DR

This paper studies the existence of normalized solutions for critical Choquard systems with linear and nonlinear couplings, employing variational methods to handle subcritical and supercritical cases in dimensions 3 and 4.

Contribution

It introduces new existence results for normalized ground states of critical Choquard systems with both linear and nonlinear couplings, covering subcritical and supercritical regimes.

Findings

Existence of positive normalized ground states in subcritical case using Ekeland's variational principle.

Existence of positive normalized ground states in supercritical case via variational methods.

Results depend on parameters , heta, and \u03b5, with specific thresholds identified.

Abstract

We consider the critical Choquard system with both linear and nonlinear couplings $-\Delta v_1 + \mu_1 v_1 = ( I_\omega * |v_1|^{2_\omega^*} ) |v_1|^{2_\omega^* -2} v_1 + \theta p( I_\omega * |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad in \,\, \mathbb{R}^N, -\Delta v_2 + \mu_2 v_2 = ( I_\omega * |v_2|^{2_\omega^*} ) |v_2|^{2_\omega^* -2} v_2 + \theta q( I_\omega * |v_1|^p)|v_2|^{q-2} v_2 + \varepsilon v_1 , \quad in \,\, \mathbb{R}^N , \int_{\mathbb{R}^N} v_1^2 = \alpha_1^2\, , \int_{\mathbb{R}^N} v_2^2 = \alpha_2^2,$ where $N=3\,\, \text{or} \,\, 4$, $\alpha_1,\alpha_2 > 0 $, $\theta > 0 $, $2_{\omega,*} :=\frac{N+\omega}{N} <p,q<2_\omega^*:=\frac{N+\omega}{N-2}$, $\varepsilon>0$, $0<\omega<N$, $I_\omega: \mathbb{R}^N \to \mathbb{R}$ represents the Riesz potential. For the $L^2$-subcritical case $p+q<\frac{2N+2\omega+4}{N}$, we utilize the Ekeland's variational principle to obtain the existence of a positive normalized ground state for the system as $0<\theta<\theta_0,\;0<\varepsilon<\varepsilon_*$. For the $L^2$-supercritical case $p+q>\frac{2N+2\omega+4}{N}$, we apply variational methods to establish the existence of a positive normalized ground state for the system as $\theta>\theta_*,\;0<\varepsilon<\overline{\varepsilon}$.