Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

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