Monotonicity of the inverse of weakly elliptic differential operators

Abstract

We consider a weakly elliptic differential operator \( Lu:= \sum\limits_{j,k=1}^{n} a_{j,k}D_{j}D_{k}u + \sum\limits_{j=1}^{n} b_{j}D_{j}u + cu \) in the Frechet space F of all \( C^\infty \) functions \( u: \mathbb{R}^n \to \mathbb{R} \) with u and all derivatives of u bounded. We prove in case c < 0 that \( L : F \rightarrow F \) is invertible and L-1 is monotone decreasing with respect to each ordering on F which is defined by a shift invariant wedge. This result can be applied to obtain informations on the behaviour as \( \|x\| \to \infty \) of bounded entire solutions of Lu = v and related nonlinear problems.