Asymptotic behavior of least energy nodal solutions for biharmonic Lane-Emden problems in dimension four

2023/06/07 by Zhijie Chen, Chen, Zhijie, Z. Cheng +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2306.04416

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

Abstract

In this paper, we study the asymptotic behavior of least energy nodal solutions u_p(x) to the following fourth-order elliptic problem \begincases Δ² u =|u|^p-1u amp;\hboxin Ω,
u=(∂ u)/(∂ ν)=0 amp;\hboxon ∂Ω, \endcases where Ω is a bounded C^4,α domain in ℝ⁴ and p>1. Among other things, we show that up to a subsequence of p→+∞, pu_p(x)→ 64π²√(e)(G(x,x⁺)-G(x,x^-)), where x⁺≠ x^-∈ Ω and G(x,y) is the corresponding Green function of Δ². This generalize those results for -Δu=|u|^p-1u in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincaré-AN, 30 (2013), 121-140) to the biharmonic case, and also gives an alternative proof of Grossi-Grumiau-Pacella's results without assuming their comparable condition p(‖u_p⁺‖_∞-‖u_p^-‖_∞)=O(1).

Related

• Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane-Emden equation in dimension two
• Nodal solutions for the weighted biharmonic equation with critical exponential growth
• On the least-energy solutions of the pure Neumann Lane-Emden equation
• Lane Emden problems with large exponents and singular Liouville equations
• Asymptotic analysis and energy quantization for the Lane-Emden problem in dimension two