Marta García-Huidobro
Universidad Católica de Chile
Departamento de Matemática

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 $ q>p>1$. By considering positive radial solutions to this equation that are bounded, we are led to study the initial value problem

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

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