The extremal solution of a boundary reaction problem

Juan Dávila; Louis Dupaigne; Marcelo Montenegro

doi:10.3934/cpaa.2008.7.795

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2008, Volume 7, Issue 4: 795-817. Doi: 10.3934/cpaa.2008.7.795

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The extremal solution of a boundary reaction problem

1.
Departamento de Ingeniería Matemática, CMM (UMI CNRS 2807), Universidad de Chile, Casilla 170/3, Correo 3, Santiago
2.
LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039Amiens Cedex 1
3.
Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 13083-970, Campinas, SP

Received: February 28, 2007

Revised: February 29, 2008

Published: June 2008

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Abstract

We consider

$\Delta u = 0$ in $ \Omega$, $\qquad \frac{\partial u}{\partial \nu} =\lambda f(u)$ on $\Gamma_1, \qquad u = 0$ on $\Gamma_2$

where $\lambda>0$, $f(u) = e^u$ or $f(u) = (1+u)^p$, $\Gamma_1$, $\Gamma_2$ is a partition of $\partial \Omega$ and $\Omega\subset \mathbb R^N$. We determine sharp conditions on the dimension $N$ and $p>1$ such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest $\lambda$ for which a solution exists.

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Mathematics Subject Classification: 26D07, 35J25, 35J65.

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