# Rigidity results for the $p$-Laplace type equations on compact Riemannian manifolds

## TL;DR

This paper establishes rigidity results for p-Laplace and n-Laplace equations with exponential nonlinearity on compact Riemannian manifolds, showing solutions are constant under certain conditions, and derives an interpolation inequality.

## Contribution

It introduces new rigidity theorems for nonlinear PDEs on manifolds using nonlinear flow and carré du champ methods, with an application to interpolation inequalities.

## Key findings

- PDE solutions are constant under specific parameter ranges.
- Rigidity results are proved using nonlinear flow and carré du champ methods.
- An interpolation inequality is derived as an application.

## Abstract

In this paper, we obtain two rigidity results for $p$-Laplace type equation and $n$-Laplace equation with exponential nonlinearity on $n$-dimensional compact Riemannian manifolds by using of nonlinear flow and the carr\'e du champ methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.

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Source: https://tomesphere.com/paper/1904.02568