Overdetermined elliptic problems in nontrivial contractible domains of the sphere

Abstract

In this paper, we prove the existence of nontrivial contractible domains $\Omega\subset\mathbb{S}^{d}$, $d\geq2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilon\Delta_{g} u +u-u^{p}=0 &\mbox{in $\Omega$, } u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=\mbox{constant} &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} admits a positive solution. Here $\Delta_{g}$ is the Laplace-Beltrami operator in the unit sphere $\mathbb{S}^{d}$ with respect to the canonical round metric $g$, $\varepsilon>0$ is a small real parameter and $1<p<\frac{d+2}{d-2}$ ($p>1$ if $d=2$). These domains are perturbations of $\mathbb{S}^{d}\setminus D,$ where $D$ is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.