Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions

2025/09/23 by Mauro, Simone, Schiera, Delia, Tavares, Hugo · 2 citations
#35B33 #35B38 #35J47 #35J50 #35J57 #35J61 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.18835

Abstract

We study the following gradient elliptic system with Neumann boundary conditions -Δu + λ₁ u = u³ + βuv², -Δv + λ₂ v = v³ + βu² v in Ω, (∂ u)/(∂ ν) = (∂ v)/(∂ ν) = 0 on ∂ Ω, where Ω⊂ ℝ^N is a bounded C² domain with N ≤ 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 λ₁,λ₂∈\mathbb R. We emphasize that our analysis includes both the subcritical case N ≤ 3 and the critical case N = 4 . Depending on the values of β,λ₁,λ₂, 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.

Citations

• On Brezis' First Open Problem: A Complete Solution
• On the least-energy solutions of the pure Neumann Lane-Emden equation
• Solutions to indefinite weakly coupled cooperative elliptic systems
• Spiked solutions for Schr "odinger systems with Sobolev critical\n exponent: the cases of competitive and weakly cooperative interactions
• Schauder estimate for solutions of Poisson's equation with Neumann\n boundary condition
• Existence and symmetry results for competing variational systems
• Existence and bounds of positive solutions for a nonlinear Schroedinger system

Cited by

• On the emergence of dead cores in elliptic systems with sublinear competitive interactions
• Quasilinear Elliptic Cooperative and Competitive Systems

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