  We study the problem of the existence and nonexistence of positive solutions
to a superlinear second-order divergence type elliptic equation with measurable
coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded
cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical
exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$
for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast
with exterior domains depends both on the geometry of the domain $G$ and the
coefficients $a$ of the equation.
