Some monotonicity results for general systems of nonlinear elliptic PDEs

TL;DR

This paper proves that minima and stable solutions of general nonlinear elliptic PDE systems exhibit certain monotonicity properties under specific growth conditions, extending known rigidity results in the field.

Contribution

It introduces general monotonicity results for solutions of nonlinear elliptic PDE systems with broad applicability and includes new rigidity theorems.

Findings

Minima and stable solutions have monotonicity properties.

Results apply to a wide class of energy functionals.

Includes new rigidity results in the literature.

Abstract

In this paper we show that minima and stable solutions of a general energy functional of the form $$ \int_{\Omega} F(\nabla u,\nabla v,u,v,x)dx $$ enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature.