Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions
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arXiv
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| Natura: | Preprint |
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2025
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| author | Mauro, Simone Schiera, Delia Tavares, Hugo |
| author_facet | Mauro, Simone Schiera, Delia Tavares, Hugo |
| contents | We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -Δu + λ_1 u = u^3 + βuv^2, \ -Δv + λ_2 v = v^3 + βu^2 v \ \text{in } Ω,\qquad \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 \ \text{on } \partial Ω, \end{equation*} where $Ω\subset \mathbb{R}^N $ is a bounded $ C^2$ domain with $ N \leq 4 $, and $ ν$ denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative ($β> 0 $) and the competitive ($ β< 0 $) regimes, considering both the definite and the indefinite case, namely $λ_1,λ_2\in\mathbb R$. We emphasize that our analysis includes both the subcritical case $ N \leq 3 $ and the critical case $ N = 4 $.
Depending on the values of $β,λ_1,λ_2$, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions Mauro, Simone Schiera, Delia Tavares, Hugo Analysis of PDEs 35B33, 35B38, 35J47, 35J50, 35J57, 35J61 We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -Δu + λ_1 u = u^3 + βuv^2, \ -Δv + λ_2 v = v^3 + βu^2 v \ \text{in } Ω,\qquad \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 \ \text{on } \partial Ω, \end{equation*} where $Ω\subset \mathbb{R}^N $ is a bounded $ C^2$ domain with $ N \leq 4 $, and $ ν$ denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative ($β> 0 $) and the competitive ($ β< 0 $) regimes, considering both the definite and the indefinite case, namely $λ_1,λ_2\in\mathbb R$. We emphasize that our analysis includes both the subcritical case $ N \leq 3 $ and the critical case $ N = 4 $. Depending on the values of $β,λ_1,λ_2$, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions. |
| title | Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions |
| topic | Analysis of PDEs 35B33, 35B38, 35J47, 35J50, 35J57, 35J61 |
| url | https://arxiv.org/abs/2509.18835 |