Phragmen–Lindelöf theorems for second-order semilinear equations with nonnegative characteristic form

This article considers the qualitative properties of generalized (in the sense of an integral identity) solutions of equations of the form $Lu=f(x,u)$, where $L$ is a second-order linear homogeneous divergence operator with nonnegative characteristic form and bounded measurable coefficients, while $f(x,u)$ is a locally bounded (in $\mathbf{R}^{n+1}$) function such that $f(x,0)=0$, $uf(x,u)\geqslant a|u|^{1+q}$, $a>0$, $q\geqslant0$, $n\geqslant2$. The results of the article are a characterization of the behavior of solutions to the Dirichlet problem for the equation $Lu=f(x,u)$ in unbounded domains as a function of the geometric properties of the domains and the quantity $0\leqslant q<1$. The apparatus of capacity characteristics plays a fundamental role in the approach used here.