Local behavior for the solutions of a second order nonlinear differential equation

Abstract

In this paper we study the local behavior near zero for the solutions of a class of second order differential equations which include, as a particular case, the radial version of the partial nonlinear differential equation, involving the m-Laplacian operator,

div(|∇u|^m−2∇u)+f(|x|,u)=0, x∈Rⁿ\{0},

where n>m>1. Some important examples of f are given by

f(t,u)=Ct^νu|u|^q−2  and  f(u)=C₁u|u|^p−2+C₂u|u|^q−2.

As an application of our general results to the first case we extend to m>1 and ν>−m some estimates given in the literature for the Laplacian case m=2 or for ν=0.