## Abstract

In this paper, we investigate solutions of the hyperbolic Poisson equation Δ _{h}u(x) = ψ(x) , where ψ∈ L^{∞}(B^{n}, R^{n}) and (Formula Presented.)Δhu(x)=(1-|x|2)2Δu(x)+2(n-2)(1-|x|2)∑i=1nxi∂u∂xi(x)is the hyperbolic Laplace operator in the n-dimensional space R^{n} for n≥ 2. We show that if n≥ 3 and u∈ C^{2}(B^{n}, R^{n}) ∩ C(B^{n}¯ , R^{n}) is a solution to the hyperbolic Poisson equation, then it has the representation u= P_{h}[ ϕ] - G_{h}[ ψ] provided that u∣Sn-1=ϕ and ∫Bn(1-|x|2)n-1|ψ(x)|dτ(x)<∞. Here P_{h} and G_{h} denote Poisson and Green integrals with respect to Δ _{h}, respectively. Furthermore, we prove that functions of the form u= P_{h}[ ϕ] - G_{h}[ ψ] are Lipschitz continuous.

Original language | English |
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Article number | 13 |

Pages (from-to) | 1-32 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Feb 2018 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Primary: 31B05
- Secondary: 31C05