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.

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.