Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

Abstract

Abstract. In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to R l , where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M . We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands. §1. IntroductionThe classical Liouville theorem, which states that any bounded harmonic function on R 2 must be constant, has long been an interesting topic of study to analysts and geometers. In 1975, Yau [20] proved a remarkable result that any complete Riemannian manifold with nonnegative Ricci curvature has no nonconstant positive harmonic functions. On the other hand, the validity of the Liouville property means that the space of positive harmonic functions on the manifold is one dimensional. In this view point, it is natural to regard the finite dimensionality of the space of positive harmonic functions as a generalized version of the Liouville property. Later, such a theory is, in this line, well developed by the works of Donnelly [6], Grigor'yan [7], Li and Tam [15], [16], Kim and the present author [14], and others.In this paper, we consider, in line with the above viewpoint, the generalized version of the Liouville property for solutions for a nonlinear elliptic