Regularization of radial solutions of $p$-Laplace equations, and computations using infinite series

TL;DR

This paper studies radial solutions of p-Laplace equations, introduces a variable change to handle singularities, expresses solutions as infinite series, and implements an exact computational algorithm using Mathematica.

Contribution

It provides a novel change of variables to analyze solutions and develops an explicit infinite series representation with an implementation in Mathematica.

Findings

Solutions are $C^2$ functions of $r^{p/(2(p-1))}$

Explicit formulas for series coefficients are derived

Exact computations are enabled by Mathematica implementation

Abstract

We consider radial solutions of equations with the $p$-Laplace operator in $R^n$. We introduce a change of variables, which in effect removes the singularity at $r=0$. While solutions are not of class $C^2$, in general, we show that solutions are $C^2$ functions of $\displaystyle r^{\frac{p}{2(p-1)}}$. Then we express the solution as an infinite series in powers of $\displaystyle r^{\frac{p}{p-1}}$, and give explicit formulas for its coefficients. We implement this algorithm, using Mathematica software. Mathematica's ability to perform the exact computations turns out to be crucial.