Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

Abstract

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space ${\mathbb{H}}^N$, $N\in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -Δ_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha }\left(\Omega \right)\cap H^1\left(\Omega \right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.

26 pages