Equivalent Norms of Solutions to Hyperbolic Poisson’s Equations

Abstract

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.