Stability and intersection properties of solutions to the nonlinear biharmonic equation

2007/07/23 by Paschalis Karageorgis, Karageorgis, Paschalis
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0707.3450

arxiv created 2007/07/23 · openalex publication_date 2007/07/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation Δ² ϕ= ϕ^p. First, we show that there exists a critical value p_c, depending on the space dimension, such that the solutions are linearly unstable if p<p_c and linearly stable if p≥ p_c. Then, we focus on the supercritical case p≥ p_c and we show that the graphs of no two solutions intersect one another.

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