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35r45 (secondary)

Bulletin of the London Mathematical Society (1)

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A CRITICAL PHENOMENON FOR SUBLINEAR ELLIPTIC EQUATIONS IN CONE-LIKE DOMAINS
VLADIMIR KONDRATIEV, VITALI LISKEVICH, VITALY MOROZ, ZEEV SOBOL
Journal:

Bulletin of the London Mathematical Society / Volume 37 / Issue 4 / August 2005

Published online by Cambridge University Press:

02 August 2005, pp. 585-591

Print publication:

August 2005
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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.