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Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
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Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
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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} -\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\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.
Forward citations
Cited by 3 Pith papers
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