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

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.