Normalized solutions to subcritical Choquard systems with double couplings

TL;DR

This work classifies normalized ground states for a coupled Choquard system with both linear and nonlinear couplings under mass constraints in dimensions $N\in\{3,4\}$ and nonlocal interactions governed by the Riesz potential with $2_{\alpha,*}=\frac{N+\alpha}{N}<p,q,r_1,r_2<2_{\alpha}^{*}=\frac{N+\alpha}{N-2}$. Employing a constrained variational framework on the $L^2$-torus $\mathcal{S}(\rho_1,\rho_2)$ and a Pohozaev manifold $\mathcal{P}_\beta(\rho_1,\rho_2)$ (with a decomposition into $\mathcal{P}_\beta^+, \mathcal{P}_\beta^-, \mathcal{P}_\beta^0$), the paper derives existence results for normalized ground states in both the symmetric case $r_1=r_2$ and the asymmetric case $r_1<r_2$. The analysis yields a rich classification across subcritical, critical, and supercritical regimes, including nonexistence in certain threshold scenarios and the realization of ground states as local minimizers or mountain-pass critical points, with positivity of the components and positive Lagrange multipliers. The results extend known single-Choquard and pure nonlinear-Coupled cases to a double-coupling nonlocal system, providing a comprehensive map of when normalized states exist under mass constraints and how the couplings influence the variational landscape.

Abstract

We consider the Choquard system with both linear and nonlinear couplings $-Δu + μ_1 u =λ_1 ( I_α* |u|^{r_1} ) |u|^{r_1-2} u + βp( I_α* |v|^q)|u|^{p-2} u + κv,$ $-Δv + μ_2 v =λ_2 ( I_α* |v|^{r_2} ) |v|^{r_2-2} v + βq( I_α* |u|^p)|v|^{q-2} v + κu , $ $\int_{\mathbb{R}^N} u^2 = ρ_1^2\, , \int_{\mathbb{R}^N} v^2 = ρ_2^2,$ where $N \in \{3,4\}$, $λ_1, λ_2, β, κ, ρ_1,ρ_2 > 0$, $2_{α,*} :=\frac{N+α}{N} <p,q , r_1, r_2 <2_α^*:=\frac{N+α}{N-2}$ 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 $(\frac{2N+2α}{N},\frac{2N+2α+4}{N})$, $\{\frac{2N+2α+4}{N}\}$, and $(\frac{2N+2α+4}{N},\frac{2N+2α}{N-2})$. Employing variational methods, we demonstrate the existence of a normalized ground state for the system in the mass subcritical, critical, and supercritical cases.

Theorems & Definitions (44)

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