Bounded $\lambda$-harmonic functions in domains of ${\mathbb{H}}^n$ with asymptotic boundary with fractional dimension

Abstract

The existence and nonexistence of $\lambda$-harmonic functions in unbounded domains of ${\mathbb{H}}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then there is no bounded $\lambda$-harmonic function of $\Omega$ for $\lambda \in [0,{\lambda }_1({\mathbb{H}}^n)]$, where ${\lambda }_1({\mathbb{H}}^n)=(n-1{)}^2/4$. For these domains, we have comparison principle and some maximum principle. Conversely, for any $s>(n-1)/2,$ we prove the existence of domains with asymptotic boundary of dimension $s$ for which there are bounded ${\lambda }_1$-harmonic functions that decay exponentially at infinity.

arXiv comment15 pages

References in corpus (1)

• Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

Cited by in corpus (1)

• On nonlinear Schrödinger equations on the hyperbolic space