Marta García-Huidobro
Universidad Católica de Chile
Departamento de Matemática
email: mgarcia@mat.puc.cl

Abstract: Let $A,B:(0,\infty)\mapsto (0,\infty)$ be two given weight functions and consider the equation

$\displaystyle -\hbox{div}(A(\vert x\vert)\vert\nabla u\vert^{p-2}\nabla u)=B(\vert x\vert)\vert u\vert^{q-2}u,\quad x\in {\mathbb{R}}^n, \leqno( P)$

where

. By considering positive radial solutions to this equation that are bounded, we are led to study the initial value problem

$\begin{displaymath} \begin{cases}-(a(r)\vert u'\vert^{p-2}u')'=b(r)(u^+)^{q-1},\... ...ad\lim\limits_{r\to 0}a(r)\vert u'(r)\vert^{p-1}=0, \end{cases}\end{displaymath}$

where $a(r)=r^{(N-1)}A(r)$ and $b(r)=r^{(N-1)}B(r)$ . By means of two key functions

and

defined below, we obtain several new results that allow us to classify solutions to this initial value problem as being respectively crossing, slowly decaying, or rapidly decaying.