On the Fragmen–Lindelof theorems for quasilinear elliptic equations of the second order

Abstract

Analogues are formulated of the well–known, in the theory of analytic functions, Phragmen–Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form $div(|\nabla u|^{\alpha–2}\nabla u)=f(x,u,\nabla u)$, where $f(x,u,\nabla u)$ is a function locally bounded in ${\mathbb R}^{2n+1}$. $f(x,u,\nabla u)  = 0$, $u f (x,u,\nabla u) \geq c| u |^{1+q}(1+|\nabla u|)^{\gamma}$, $\alpha > 1$, $c > 0$, $q > 0$, $\gamma$  is an arbitrary real number, and $n\geq 2$. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.