Bounded in the mean solutions of \triangle𝑢=𝑃𝑢 on Riemannian manifolds

Source abstract

Let Φ \Phi be a convex positive increasing function and d = lim t → ∞ Φ ( t ) / t d = {\lim _{t \to \infty }}\Phi (t)/t . A harmonic function u u on a Riemann surface R R is called Φ \Phi -bounded if Φ ( | u | ) \Phi (|u|) is majorized by a harmonic function on R . M. Parreau has shown that if d > ∞ d > \infty ( d = ∞ d = \infty , resp.), then every positive (bounded, resp.) harmonic function on R R reduces to a constant if and only if every Φ \Phi -bounded harmonic function does. In this paper analogues of these results are given for the equation Δ u = P u ( P ≧ 0 ) \Delta u = Pu(P \geqq 0) on a Riemannian manifold.