Perturbing singular solutions of the Gelfand problem

TL;DR

This paper investigates the stability and boundedness of singular solutions to the Gelfand problem under domain deformations, revealing conditions under which solutions remain singular or become bounded.

Contribution

It demonstrates the persistence of singular solutions under small domain deformations for dimensions N≥4 and identifies conditions where large deformations lead to bounded extremal solutions.

Findings

Small deformations preserve singular solutions in N≥4.

In N≥11, singular solutions relate to extremal solutions.

Large deformations can cause extremal solutions to become bounded in many cases.

Abstract

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