Liouville theorems for elliptic systems of arbitrary order

Abstract

We prove Liouville-type theorems for solutions of linear homogeneous elliptic systems of partial differential equations, with variable coefficients and of arbitrary order. We require certain growth and smoothness conditions on the coefficients, and coerciveness and ellipticity condtions on the differential operators. Under various hypotheses it is established that solutions defined in all of Euclidean space, of a prescribed growth or decay at infinity, are necessarily constant, identically zero, or polynomials

Keywords:

• Liouville theorem
• elliptic systems
• polynomial growth

AMS:

• 35J45
• 35B05