Bounded in the mean solutions of $\triangle u=Pu$ on Riemannian manifolds

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