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

Abstract

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determined explicitly by $s$ and the geometry of the cone.