Sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions

2023/06/01 by Qing Guo, Shuangjie Peng, Guo, Qing +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2306.00663

openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begincases -Δu₁=|u₂|^p_ε-1u₂, amp;in Ω,
-Δu₂=|u₁|^q_ε-1u₁, amp;in Ω,
∂_νu₁=∂_νu₂=0, amp;on ∂Ω\endcases where Ω=B₁(0) is the unit ball in ℝⁿ (n≥4) centered at the origin, p_ε=p+αε, q_ε=q+βε with α,β>0 and \frac1p+1+\frac1q+1=\fracn-2n. We show the existence and multiplicity of concentrated solutions based on the Lyapunov-Schmidt reduction argument incorporating the zero-average condition by certain symmetries. It is worth noting that we simultaneously consider two cases: p>\frac nn-2 and p<\frac nn-2. The coupling mechanisms of the system are completely different in these different cases, leading to significant changes in the behavior of the solutions. The research challenges also vary. Currently, there are very few papers that take both ranges into account when considering solution construction. Therefore, this is also the main feature and new ingredient of our work.

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