On some methods of the obtaining of a priori pointwise estimates of Dirichlet problem solution for Emden-Fouler equation

Abstract

In this paper we propose some approaches for finding of pointwise estimates of a solution of the Dirichlet boundary value problem $-\Delta u \pm |u|^{q-1} u = 0 $, $|u|=k$ when $|x|=d<1$ and $|u|=0$ when $|x|=1$ where $x\in \Omega = \{x| d<|x|<1\}$. Along with these we consider the same boundary conditions for the Laplace equation and get appropriate estimates for this easier case. We indicate some way what permit to find upper and lower estimates of a solution with explicit constants in turns.