Equivalence between radial solutions of different non-homogeneous $p$-Laplacian type equations

Abstract

We study radial viscosity solutions to the equation \[ -\ |Du\ |^{q-2}Δ_{p}^{N}u=f(\ |x\ |)\quad\text{in }B_{R}\subset\mathbb{R}^{N}, \] where $f\in C[0,R)$, $p,q\in (1,\infty )$ and $N\ge 2$. Our main result is that $u(x)=v(|x|)$ is a bounded viscosity supersolution if and only if $v$ is a bounded weak supersolution to $-\kappa {\Delta }_q^{d}v=f$ in $(0,R)$, where $\kappa >0$ and ${\Delta }_q^d$ is heuristically speaking the radial $q$-Laplacian in a fictitious dimension $d$. As a corollary we obtain the uniqueness of radial viscosity solutions. However, the full uniqueness of solutions remains an open problem.