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December  2004, 3(4): 653-662. doi: 10.3934/cpaa.2004.3.653

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

M. Chuaqui 1, , C. Cortázar 1, , M. Elgueta 1, and  J. García-Melián 2,

1. 

Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, Correo 22 - Santiago, Chile, Chile, Chile
2. 

Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna, Spain

Received  November 2003 Revised  May 2004 Published  September 2004

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 35B4.
Citation: M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653
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