TY - JOUR

T1 - Solutions of a certain nonlinear elliptic equation on Riemannian manifolds

AU - Lee, Yong Hah

PY - 2001/6

Y1 - 2001/6

N2 - 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 Rl, 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.

AB - 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 Rl, 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.

UR - http://www.scopus.com/inward/record.url?scp=0039251834&partnerID=8YFLogxK

U2 - 10.1017/S0027763000007844

DO - 10.1017/S0027763000007844

M3 - Article

AN - SCOPUS:0039251834

VL - 162

SP - 149

EP - 167

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

ER -