Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds

It is proved that the Liouville function associated with the semilinear equation $\Delta u -g(x,u)=0$ is identical to zero if and only if there is only a trivial bounded solution of the semilinear equation on non-compact Riemannian manifolds. This result generalizes the corresponding result of S.A. Korolkov for the case of the stationary Schrödinger equation $ \Delta u-q (x) u = 0$. The concept of the capacity of a compact set associated with the stationary Schrödinger equation is also introduced and it is proved that if the capacity of any compact set is equal to zero, then the Liouville function is identically zero.