Existence of a Normalized Mountain Pass Solution for a Coupled Cubic Schrödinger System with Partial Confinement
Abstract
This paper studies a mass supercritical Schrödinger system with partial confinement: −Δu1+|y|2u1 =λ1u1+μ1u13+2βu1u22, x∈R3, −Δu2+|y|2u2 =λ2u2+μ2u23+2βu12u2, x∈R3, subject to the mass constraints $ \int_{\R^3}u_1^2\dd x=a_1$, ∫R3u22dd x=a2, where x=(y,z)∈R2×R, the parameters μ1,μ2,β are positive, and λ1,λ2∈R are unknown Lagrange multipliers. The masses satisfy (a1,a2)=eps(ρ1,ρ2) for a fixed vector ρ=(ρ1,ρ2)∈ (0,∞)2. For all sufficiently small eps>0, we establish the existence of a positive normalized solution ueps- at a mountain pass level on the full product mass sphere. This level coincides with the infimum of the energy over the negative part of the Pohozaev set. The main difficulty arises from the lack of compactness due to partial confinement. To overcome this difficulty, we develop a compactness argument that preserves the two independent mass constraints and show that every minimizing sequence for this infimum admits a strongly convergent subsequence after suitable common translations along the unconfined axial direction. We further establish the asymptotic relation eps E(ueps-)→ m0(ρ) as ε→0, where m0(ρ)>0 denotes the infimum of the free energy under the corresponding Pohozaev constraint. In addition, we derive the asymptotic behavior of the kinetic and potential energies. The solution obtained here is distinct from the normalized ground states constructed by Jia et al. [Discrete Contin. Dyn. Syst., 40(2020), 2739-2766] through local minimization for sufficiently small masses.
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BibTeX
@misc{tang2026existence,
title = {{Existence of a Normalized Mountain Pass Solution for a Coupled Cubic Schrödinger System with Partial Confinement}},
author = {Wan-Yue Tang and Hong-Rui Sun and Jianlun Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05805},
eprint = {2611.05805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05805},
}
RIS
TY - UNPB
TI - Existence of a Normalized Mountain Pass Solution for a Coupled Cubic Schrödinger System with Partial Confinement
AU - Wan-Yue Tang
AU - Hong-Rui Sun
AU - Jianlun Liu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05805
UR - https://arxiv.org/abs/2611.05805
AB - This paper studies a mass supercritical Schrödinger system with partial confinement: −Δu_1+|y|^2u_1 =λ_1u_1+μ_1u_1^3+2βu_1u_2^2, x∈R^3, −Δu_2+|y|^2u_2 =λ_2u_2+μ_2u_2^3+2βu_1^2u_2, x∈R^3, subject to the mass constraints $ \int_{\R^3}u_1^2\dd x=a_1$, ∫_R^3u_2^2dd x=a_2, where x=(y,z)∈R^2×R, the parameters μ_1,μ_2,β are positive, and λ_1,λ_2∈R are unknown Lagrange multipliers. The masses satisfy (a_1,a_2)=eps(ρ_1,ρ_2) for a fixed vector ρ=(ρ_1,ρ_2)∈ (0,∞)^2. For all sufficiently small eps>0, we establish the existence of a positive normalized solution u_eps^- at a mountain pass level on the full product mass sphere. This level coincides with the infimum of the energy over the negative part of the Pohozaev set. The main difficulty arises from the lack of compactness due to partial confinement. To overcome this difficulty, we develop a compactness argument that preserves the two independent mass constraints and show that every minimizing sequence for this infimum admits a strongly convergent subsequence after suitable common translations along the unconfined axial direction. We further establish the asymptotic relation eps E(u_eps^-)→ m_0(ρ) as ε→0, where m_0(ρ)>0 denotes the infimum of the free energy under the corresponding Pohozaev constraint. In addition, we derive the asymptotic behavior of the kinetic and potential energies. The solution obtained here is distinct from the normalized ground states constructed by Jia et al. [Discrete Contin. Dyn. Syst., 40(2020), 2739-2766] through local minimization for sufficiently small masses.
ER -