Normalized solutions to subcritical Choquard systems with double couplings

TL;DR

This paper studies a coupled nonlinear system involving the Choquard equation with both linear and nonlinear interactions, classifies solution existence based on parameter ranges, and proves the existence of normalized ground states across different criticality regimes.

Contribution

It provides a comprehensive classification of solution existence for the Choquard system with double couplings depending on parameter ranges and establishes the existence of normalized ground states variationally.

Findings

Existence of normalized ground states in subcritical, critical, and supercritical regimes.

Classification of parameter ranges for solution existence.

Variational methods successfully applied to prove ground state existence.

Abstract

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha * |v|^{r_2} ) |v|^{r_2-2} v + \beta q( I_\alpha * |u|^p)|v|^{q-2} v + \kappa u , $ $\int_{\mathbb{R}^N} u^2 = \rho_1^2\, , \int_{\mathbb{R}^N} v^2 = \rho_2^2,$ where $N \in \{3,4\}$, $\lambda_1, \lambda_2, \beta, \kappa, \rho_1,\rho_2 > 0$, $2_{\alpha,*} :=\frac{N+\alpha}{N} <p,q , r_1, r_2 <2_\alpha^*:=\frac{N+\alpha}{N-2}$ and $p+q\leq 2r_1 \leq 2r_2$ . We investigate a classification result as the parameters $p+q$, $2r_1$ and $2r_2$ vary across the ranges $(\frac{2N+2\alpha}{N},\frac{2N+2\alpha+4}{N})$, $\{\frac{2N+2\alpha+4}{N}\}$, and $(\frac{2N+2\alpha+4}{N},\frac{2N+2\alpha}{N-2})$. Employing variational methods, we demonstrate the existence of a normalized ground state for the system in the mass subcritical, critical, and supercritical cases.