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A CRITICAL PHENOMENON FOR SUBLINEAR ELLIPTIC EQUATIONS IN CONE-LIKE DOMAINS

Published online by Cambridge University Press:  02 August 2005

VLADIMIR KONDRATIEV ,
VITALI LISKEVICH ,
VITALY MOROZ  and
ZEEV SOBOL
VLADIMIR KONDRATIEV
Affiliation:
Department of Mathematics and Mechanics, Moscow State University, Moscow 119 899, Russiakondrat@vnmok.math.msu.su
VITALI LISKEVICH
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdomv.liskevich@bristol.ac.uk, v.moroz@bristol.ac.uk
VITALY MOROZ
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdomv.liskevich@bristol.ac.uk, v.moroz@bristol.ac.uk
ZEEV SOBOL
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdomz.sobol@swansea.ac.uk
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Abstract

The authors of this paper study positive supersolutions to the elliptic equation $-\Delta U=c|x|^{-s}u^p$ in Cone-like domains of $\mathbb{R}^N$ ($N\ge 2$), where $p,s\in\mathbb{R}$ and $c>0$. They prove that in the sublinear case $p<1$ there exists a critical exponent $p_\ast<1$ such that the equation has a positive supersolution if and only if $-\infty<p<p_\ast$. The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.

Keywords

35J60 (primary)35B0535R45 (secondary)
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 37 , Issue 4 , August 2005 , pp. 585 - 591
Copyright
© The London Mathematical Society 2005

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