Perturbing singular solutions of the Gelfand problem
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2008
- DOI:
- 10.48550/arXiv.0801.2441
- arXiv:
- arXiv:0801.2441
- Bibcode:
- 2008arXiv0801.2441D
- Keywords:
-
- Mathematics - Analysis of PDEs