Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains
Vladimir Kondratiev, Vitali Liskevich, Zeev Sobol · Transactions of the American Mathematical Society · 2008
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 ∑ i , j = 1 N a i j ( x ) ∂ 2 u ∂ x i ∂ x j + u p = 0 \sum _{i,j=1}^N a_{ij}(x)\frac {\partial ^2 u}{\partial x_i \partial x_j}+u^p=0 , − ∞ > p > ∞ -\infty >p>\infty , with measurable coefficients in exterior domains of R N \mathbb {R}^N . We prove that in a “generic” situation there is one critical value of p 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.