Perturbing singular solutions of the Gelfand problem

Juan Davila (1) , Louis Dupaigne (2)

1 DIM - Departamento de Ingenieria Matemàtica and CMM
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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.

Domaines: Equations aux dérivées partielles [math.AP]

Juan Davila, Louis Dupaigne. Perturbing singular solutions of the Gelfand problem. 2008. ⟨hal-00204945⟩