Phragmén-Lindelöf theorems for second-order semilinear equations with nonnegative characteristic form
Let \(L = \sum^n_{i,j = 1} {\partial \over \partial x_i} (a_{ij} (x) {\partial \over \partial x_j})\) be an operator with bounded measurable coefficients \(a_{ij}\) satisfying \(\sum^n_{i,j = 1} a_{ij} (x) \xi_i \xi_j \geq 0\), \(\forall \xi_1, \ldots, \xi_n \in \mathbb{R}\). The paper deals with qualitative properties of generalized solutions \(u\) of the equation \(Lu=f(x,u)\), where \(f\) is a locally bounded function of its \((n + 1)\) variables \(x_1, \ldots, x_n,u\) satisfying \(f(x,0) = 0\), \(uf(x,u) \geq a |u |^{1 + q}\), where \(0 \leq q < 1\), \(a > 0\). The behaviour of the solutions \(u(x)\) as \(|x |\to \infty\) is studied in unbounded domains in dependence on the geometric properties of the domains which are specified in terms of appropriate capacities.
scientific article; zbMATH DE number 148615
scientific article