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Existence and multiplicity of solutions for a weakly coupled radial system in a ball
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, Chile |
| 2. | LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039Amiens Cedex 1, France |
| 3. | Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 13083-970, Campinas, SP, Brazil |
$\Delta u = 0$ 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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