Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls (Q464294)

The paper deals with existence and non-existence of positive singular solutions to second-order non-divergence type elliptic inequalities of the form \[ \sum_{i,j=1}^n a_{ij}(x)\dfrac{\partial^2u}{\partial x_i\partial x_j}+\sum_{i}^n b_{i}(x)\dfrac{\partial u}{\partial x_i}\geq K(x) u^p,\quad -\infty<p<\infty \] with measurable coefficients in a punctured ball \(B_R\setminus \{0\}\) of \(\mathbb{R}^N\) with \(N\geq 1\). Precisely, existence of a critical value \(p^*\) is proved, which separates the regions of existence and non-existence. In the critical case \(p = p^*,\) the authors show the existence of a singular solution depending on the rate at which the coefficients \(a_{ij}(x)\) and \(b_{i}(x)\) stabilize at zero, and some optimal conditions in this setting are provided as well.