Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type

2018/03/07 by Zhenluo Lou, Lou, Zhenluo, Tobias Weth +3
Computer Science · Mathematics · #35B06 #35J50 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1803.02712

openalex publication_date 2018/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Dirichlet problem for the Schrödinger-Hénon system -Δu + μ₁ u = |x|^α∂_u F(u,v), -Δv + μ₂ v = |x|^α∂_v F(u,v) in the unit ball Ω⊂ ℝ^N, N≥ 2, where α>-1 is a parameter and F: ℝ² → ℝ is a p-homogeneous C²-function for some p>2 with F(u,v)>0 for (u,v) \not = (0,0). We show that, as α→ ∞, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar Hénon equation and extends a previous result by Moreira dos Santos and Pacella for the case N=2. In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an α-independent variational minimax principle.

Citations

• On Phase-Separation Models: Asymptotics and Qualitative Properties
• Dynamics of strongly competing systems with many species

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