Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains $\mathrm{\Omega }\subset {\mathbb{S}}^d$, $d\ge 2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilonΔ_{g} u +u-u^{p}=0 &\mbox{in $\mathrm{\Omega }$, } u>0 &\mbox{in $\mathrm{\Omega }$, } u=0 &\mbox{on $\mathrm{\partial }\mathrm{\Omega }$, } \partial_ν u=\mbox{constant} &\mbox{on $\mathrm{\partial }\mathrm{\Omega }$, } \end{cases} \end{equation*} admits a positive solution. Here ${\mathrm{\Delta }}_g$ is the Laplace-Beltrami operator in the unit sphere ${\mathbb{S}}^d$ with respect to the canonical round metric $g$, $\epsilon >0$ 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.