Perturbing singular solutions of the Gelfand problem

@inproceedings{Dvila2007PerturbingSS,
  title={Perturbing singular solutions of the Gelfand problem},
  author={Juan D{\'a}vila and Louis Dupaigne and Ignacio Guerra and Marcelo Montenegro},
  year={2007}
}
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… CONTINUE READING

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