Liouville rigidity for higher-order elliptic operators under minimal assumptions

In this paper we prove that any distribution u∈D′(RN) satisfying u ≥ 0, −Δu≥0, and (−Δ)mu≥0, must be constant whenever N ≤ 2 m . The essential feature is that no requirement is made on the intermediate iterates (−Δ)ju for 2≤j≤m−1. We further prove that the reverse inequality (−Δ)mu≤0, together with u ≥ 0 and −Δu≥0, forces u to be constant, with no restriction on m or N . Both results extend to operators of the form P(−Δ), where P is a polynomial whose roots all lie in (−∞,0]. We also establish a more general version replacing the sign conditions on u and −Δu by a single averaged vanishing condition at infinity: any u∈Lloc1(RN) satisfying such a condition together with (−Δ)mu≥0 or ≤ 0 (and more generally P(−Δ)u≥0 or ≤ 0) must be constant, provided N ≤ 2 m .