# Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

## TL;DR

This paper studies the regularity of semi-stable, radially symmetric solutions to quasilinear elliptic equations on Riemannian manifolds, providing uniform bounds and estimates independent of boundary conditions and non-linearities.

## Contribution

It establishes regularity results for solutions of p-Laplace type equations on Riemannian models, including extremal solutions with zero Dirichlet conditions, regardless of boundary or non-linearity specifics.

## Key findings

- Proves uniform boundedness of solutions.
- Provides Lebesgue and Sobolev estimates.
- Shows regularity of extremal solutions.

## Abstract

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions.