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

TL;DR

This paper establishes an equivalence between radial viscosity solutions of certain non-homogeneous p-Laplacian equations and weak solutions of related equations, providing insights into solution uniqueness.

Contribution

It demonstrates a novel equivalence between viscosity and weak solutions for radial non-homogeneous p-Laplacian equations, advancing understanding of solution properties.

Findings

Radial viscosity supersolutions correspond to weak supersolutions in a fictitious dimension.

Established conditions for the equivalence of solutions.

Proved uniqueness of radial viscosity solutions under certain conditions.

Abstract

We study radial viscosity solutions to the equation \[ -\ |Du\ |^{q-2}\Delta_{p}^{N}u=f(\ |x\ |)\quad\text{in }B_{R}\subset\mathbb{R}^{N}, \] where $f\in C[0,R)$, $p,q\in(1,\infty)$ and $N\geq2$. 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.