Some monotonicity results for general systems of nonlinear elliptic PDEs

In this paper we show that minima and stable solutions of a general energy functional of the form $${\int }_{\mathrm{\Omega }}F(\mathrm{\nabla }u,\mathrm{\nabla }v,u,v,x)dx$$ 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.