Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains

We study the problem of the existence and non-existence of positive super-solutions to a semi-linear second-order non-divergence type elliptic equation $\sum_{i,j=1}^N a_{ij}(x)\frac{\partial ^2 u}{\partial x_i \partial x_j}+u^p=0$, $-\infty<p<\infty$, with measurable coefficients in exterior domains of $\mathbb{R}^N$. We prove that in a ``generic'' situation there is one critical value of $p$ that separates the existence region from non-existence. We reveal the quantity responsible for the qualitative picture and for the numerical value of the critical exponent which becomes available under a mild stabilization condition at infinity.