Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights

M. Chuaqui; C. Cortázar; M. Elgueta; J. García-Melián

doi:10.3934/cpaa.2004.3.653

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2004, Volume 3, Issue 4: 653-662. Doi: 10.3934/cpaa.2004.3.653

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Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights

1.
Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, Correo 22 - Santiago
2.
Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna

Received: October 31, 2003

Revised: April 30, 2004

Published: November 2004

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Abstract

We consider the elliptic problems $\Delta u=a(x)u^m$, $m>1$, and $\Delta u=a(x)e^u$ in a smooth bounded domain $\Omega$, with the boundary condition $u=+\infty$ on $\partial\Omega$. The weight function $a(x)$ is assumed to be Hölder continuous, growing like a negative power of $d(x)=$ dist $(x,\partial\Omega)$ near $\partial\Omega$. We show existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.

Keywords:

Elliptic problems,
boundary blow up.

Mathematics Subject Classification: Primary 35J25, 35J60; Secondary 35B40.

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