We assume that \(n\ge 3\), \(u \in C^{2}( \mathbb {B}^{n},\mathbb {R}^{n}) \cap C(\overline{\mathbb {B}}^{n},\mathbb {R}^{n})\) is a solution to the hyperbolic Poisson equation \(\Delta _{h}u=\psi \) in \(\mathbb {B}^{n}\) with the boundary condition \(u|_{\mathbb {S}^{n-1}}=\phi \), where \(\Delta _{h}\) is the hyperbolic Laplace operator and \(\psi \in C( \mathbb {B}^{n},\mathbb {R}^{n})\). In Chen et al. (Calc Var Partial Differ Equ 57:32, 2018), the first, the second, and the last author of this paper, together with Rasila, studied expressions of u, and proved that \(u=P_{h}[\phi ]-G_{h}[\psi ]\), where \(P_{h}[\phi ]\) and \(G_{h}[\psi ]\) denote the Poisson integral of \(\phi \) and the Green integral of \(\psi \) with respect to \(\Delta _h\), respectively. With the assumption \(|\psi (x)|\le M(1-|x|^{2})\) \((M\ge 0\) is a constant), the Lipschitz-type continuity of u was also investigated. As a continuation, in this paper, we first consider the existence of the solutions, and demonstrate that if \(\phi \in L^{\infty }(\mathbb {S}^{n-1}, \mathbb {R}^{n})\), \(\psi \in L^{\infty }(\mathbb {B}^{n}, \mathbb {R}^{n})\), \(\int _{\mathbb {B}^{n}} (1-|x|^2)^{n-1} |\psi (x)|\mathrm{d}\tau (x)<\infty \), and if the mapping \(u=P_{h}[\phi ]-G_{h}[\psi ]\in C^{2}(\mathbb {B}^{n}, \mathbb {R}^{n})\cap C (\overline{\mathbb {B}}^{n}, \mathbb {R}^{n})\), then u is a solution to the above Dirichlet problem. Then, by using fast majorants, we get several equivalent norms related to the solutions. The proofs are mainly based on the relationships of the Lipschitz-type continuity between the solutions u and the boundary mappings \(\phi \), which are of independent interest. As an application, we have a counterpart of the main results in Cho et al. (Taiwan J Math 12:741–751, 2008) and Dyakonov (Acta Math 178:143–167, 1997) in the setting of the solutions u.

我们假设\（n \ ge 3 \），\（u \ in C ^ {2}（\ mathbb {B} ^ {n}，\ mathbb {R} ^ {n}）\ cap C（\ overline { \ mathbb {B}} ^ {N}，\ mathbb {R} ^ {N}）\）是双曲泊松方程的溶液\（\德尔塔_ {H} U = \ PSI \）在\（\ mathbb {B} ^ {n} \）的边界条件为\（u | _ {\ mathbb {S} ^ {n-1}} = \ phi \），其中\（\ Delta _ {h} \）是双曲Laplace运算符和\（\ psi \ in C（\ mathbb {B} ^ {n}，\ mathbb {R} ^ {n}）\）。在Chen等。（Calc Var Partial Differ Equ Equ 57:32，2018），本文的第一，第二和最后作者与Rasila一起研究了u的表达式，并证明\（u = P_ {h} [\ phi ] -G_ {h} [\ psi] \）其中\（P_ {H} [\ PHI] \）和\（G_ {H} [\ PSI] \）表示的泊松积分\（\披\）和绿积分\（\ PSI \）与关于\（\ Delta _h \）。假设\（| \ psi（x）| \ le M（1- | x | ^ {2}）\） \（（M \ ge 0 \）是一个常数），则u的Lipschitz型连续性为还进行了调查。作为继续，在本文中，我们首先考虑解的存在，并证明如果\（\ phi \ in L ^ {\ infty}（\ mathbb {S} ^ {n-1}，\ mathbb {R } ^ {n}）\），\（\ psi \ in L ^ {\ infty}（\ mathbb {B} ^ {n}，\ mathbb {R} ^ {n}）\），\（\ int _ {\ mathbb {B} ^ {n}}（1- | x | ^ 2）^ {n-1} | \ psi（x）| \ mathrm {d} \ tau（x）<\ infty \），如果映射\（u = P_ {h} [\ phi] -G_ {h} [\ psi] \ in C ^ {2}（\ mathbb {B} ^ {n}，\ mathbb { R} ^ {n}）\ cap C（\ overline {\ mathbb {B}} ^ {n}，\ mathbb {R} ^ {n}）\），则u是上述Dirichlet问题的解决方案。然后，通过使用快速主成分，我们获得了与解决方案有关的几个等效准则。证明主要基于解u与边界映射\（\ phi \）之间的Lipschitz型连续性的关系，这是具有独立利益的。作为应用程序，我们与Cho等人的主要结果相对应。（Taiwan J Math 12：741–751，2008）和Dyakonov（Acta Math 178：143–167，1997）设置解u。