Normalized solutions to critical Choquard systems with linear and nonlinear couplings

TL;DR

Problem: existence of normalized (mass-constrained) solutions to the critical Choquard system with linear and nonlinear couplings in $\mathbb{R}^N$ ($N=3$ or $4$). Method: a constrained variational framework on $H=H^1_r(\mathbb{R}^N)\times H^1_r(\mathbb{R}^N)$ using the energy $\mathcal{J}_\theta$, the constraint set $\mathcal{S}(\alpha_1,\alpha_2)$, and the Pohozaev manifold $\mathcal{P}_\theta(\alpha_1,\alpha_2)$, with $L^2$-mass normalization; subcritical case uses Ekeland's variational principle, while the supercritical case uses minimax principles to obtain a ground state with $v_i>0$ and $\mu_i>0$. Findings: for $p+q<\dfrac{2N+2\omega+4}{N}$ there exist $\theta_0>0$ and $\varepsilon_*(\theta_0)>0$ such that a positive normalized ground state exists for $0<\theta<\theta_0$, $0<\varepsilon<\varepsilon_*$; for $p+q>\dfrac{2N+2\omega+4}{N}$ there exist $\theta_*>0$, $\overline{\varepsilon}>0$ such that a normalized ground state exists for $\theta>\theta_*$ and $0<\varepsilon<\overline{\varepsilon}$, with an improved result (Theorem 1.2) removing the upper bound on $\theta$. Significance: advances the theory of normalized solutions for nonlocal coupled PDE systems and clarifies how linear and nonlinear couplings interact with mass constraints under critical Choquard dynamics.

Abstract

We consider the critical Choquard system with both linear and nonlinear couplings $-Δv_1 + μ_1 v_1 = ( I_ω* |v_1|^{2_ω^*} ) |v_1|^{2_ω^* -2} v_1 + θp( I_ω* |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad in \,\, \mathbb{R}^N, -Δv_2 + μ_2 v_2 = ( I_ω* |v_2|^{2_ω^*} ) |v_2|^{2_ω^* -2} v_2 + θq( I_ω* |v_1|^p)|v_2|^{q-2} v_2 + \varepsilon v_1 , \quad in \,\, \mathbb{R}^N , \int_{\mathbb{R}^N} v_1^2 = α_1^2\, , \int_{\mathbb{R}^N} v_2^2 = α_2^2,$ where $N=3\,\, \text{or} \,\, 4$, $α_1,α_2 > 0 $, $θ> 0 $, $2_{ω,*} :=\frac{N+ω}{N} <p,q<2_ω^*:=\frac{N+ω}{N-2}$, $\varepsilon>0$, $0<ω<N$, $I_ω: \mathbb{R}^N \to \mathbb{R}$ represents the Riesz potential. For the $L^2$-subcritical case $p+q<\frac{2N+2ω+4}{N}$, we utilize the Ekeland's variational principle to obtain the existence of a positive normalized ground state for the system as $0<θ<θ_0,\;0<\varepsilon<\varepsilon_*$. For the $L^2$-supercritical case $p+q>\frac{2N+2ω+4}{N}$, we apply variational methods to establish the existence of a positive normalized ground state for the system as $θ>θ_*,\;0<\varepsilon<\overline{\varepsilon}$.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Proposition 2.1
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4