A critical phenomenon for sublinear elliptic equations in cone-like domains

Abstract

We study positive supersolutions to an elliptic equation $(\ast )$: $-\Delta u=c|x{|}^{-s}{u}^p$, $p,s\in \mathbf{R}$ in cone-like domains in ${\mathbf{R}}^{\mathbf{N}}$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_{\ast }<1$ such that equation $(\ast )$ 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.

6 pages, 2 figures

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