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(0.1)  i) φ ∈ C1((0,+∞)) ∩ C0([0,+∞)); ii) φ(0) = 0, φ(t) > 0 on (0,+∞); iii) φ(t) ≤ Atδ on [0,+∞) for some constants A, δ > 0. We state our main result in the form of the following Liouville type: Theorem A. Let (M, 〈〉), r(x) be as above and let φ satisfy (0.1). Let u ∈ C2(M) be a solution of the equation (0.2) div (|∇u|−1 φ(|∇u|)∇u) = a for some a ∈ R, such that (0.3) u(x) = o ( log r(x) ) as r(x) → +∞. Then u is constant (and a = 0) provided (0.4) vol (∂BR) ≤ cR for R 1 for some c > 0, where δ is as in (0.1) iii). For instance, the choices

A Liouville type theorem for a general class of operators on complete manifolds