## Abstract

We prove that for any continuous function f on the s-harmonic (1 < s < ∞) boundary of a complete Riemannian manifold M, there exists a solution, which is a limit of a sequence of bounded energy finite solutions in the sense of supremum norm, for a certain elliptic operator A on M whose boundary value at each s-harmonic boundary point coincides with that of f. If E_{1}, E_{2},..., E_{l} are M-nonparabolic ends of M, then we also prove that there is a one to one correspondence between the set of bounded energy finite solutions for A on M and the Cartesian product of the sets of bounded energy finite solutions for A on E_{i} which vanish at the boundary ∂E_{i} for i = 1, 2,..., l.

Original language | English |
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Pages (from-to) | 807-819 |

Number of pages | 13 |

Journal | Journal of the Korean Mathematical Society |

Volume | 45 |

Issue number | 3 |

DOIs | |

State | Published - May 2008 |

## Keywords

- A-harmonic function
- End
- s-harmonic boundary