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

We study positive supersolutions to an elliptic equation $(\ast )$: $-\Delta u=c|x|^{-s}u^p$,$p,s\in\bf R$incone-likedomainsin$\bf R^N$($N\ge 2$). 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 $-\mathrm{\infty }<p<p_{\ast }$. The value of $p_{\ast }$ is determined explicitly by $s$ and the geometry of the cone.