Perturbing singular solutions of the Gelfand problem

2008/01/16 by Juan Davila, Juan Dávila, Louis Dupaigne +6
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0801.2441

arxiv created 2008/01/16 · openalex publication_date 2008/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

he equation -Δu = λe^u posed in the unit ball B ⊆ \R^N, with homogeneous Dirichlet condition u|_{∂ B} = 0, has the singular solution U=log\frac1|x|² when λ= 2(N-2). If N≥ 4 we show that under small deformations of the ball there is a singular solution (u,λ) close to (U,2(N-2)). In dimension N≥ 11 it corresponds to the extremal solution -- the one associated to the largest λ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when N≥ 10, the extremal solution remains bounded in many cases.

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