Solutions of a certain nonlinear elliptic equation on Riemannian manifolds

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 $\cal A$-harmonic functions on $M$ is one to one corresponding to $\mathbf{R}^l$, where $\cal A$ 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 $\cal A$-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.