Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds

We prove that if the s-harmonic boundary of a complete Riemannian manifold consists of finitely many points, then the set of bounded energy finite solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to R^l, where l is the cardinality of the s-harmonic boundary. We also prove that the finiteness of cardinality of s-harmonic boundary is a rough isometric invariant, moreover, in this case, the cardinality is preserved under rough isometries between complete Riemannian manifolds. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Kim and the present author, of Holopainen, and of the present author, but with different techniques which are demanded by the peculiarity of nonlinearity.