Normalized solutions to critical Choquard systems with linear and nonlinear couplings
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arXiv
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2025
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| _version_ | 1866918171206746112 |
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| author | Pei, Wenliang Deng, Chonghao |
| author_facet | Pei, Wenliang Deng, Chonghao |
| contents | 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 $θ>θ_*,\;0<\varepsilon<\overline{\varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to critical Choquard systems with linear and nonlinear couplings Pei, Wenliang Deng, Chonghao Analysis of PDEs Functional Analysis 35J15, 35J60, 35Q55 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 $θ>θ_*,\;0<\varepsilon<\overline{\varepsilon}$. |
| title | Normalized solutions to critical Choquard systems with linear and nonlinear couplings |
| topic | Analysis of PDEs Functional Analysis 35J15, 35J60, 35Q55 |
| url | https://arxiv.org/abs/2510.22159 |