Nonexistence of stable solutions to quasilinear elliptic equations on Riemannian manifolds

TL;DR

This paper proves that under certain geometric conditions, there are no nontrivial stable solutions to a class of quasilinear elliptic equations on Riemannian manifolds, highlighting the influence of manifold geometry on solution existence.

Contribution

It establishes nonexistence results for stable solutions to quasilinear elliptic equations on Riemannian manifolds with specific volume growth conditions, extending previous Euclidean results.

Findings

No nontrivial stable solutions exist under given conditions

Stable solutions must be trivial on manifolds with certain volume growth

Results apply to equations with potential on Riemannian manifolds

Abstract

We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.