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

• Date: 1992
• Main Authors: Kurta , V. V., Курта , В. В.
• Format: Article
• Language: Russian
• Published: Institute of Mathematics, NAS of Ukraine 1992
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/8234
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.