Overdetermined elliptic problems in nontrivial contractible domains of the sphere

Abstract

In this paper, we prove the existence of nontrivial contractible domains $\Omega \subset S^d$, $d\ge 2$, such that the overdetermined elliptic problem$\{\begin{array}{cc}-\epsilon {\Delta }_{g}u+u-u^p=0& \text{in Ω, }\\ u>0& \text{in Ω, }\\ u=0& \text{on ∂Ω, }\\ {\partial }_{\nu }u=\text{constant}& \text{on ∂Ω, }\end{array}$ admits a positive solution. Here ${\Delta }_g$ is the Laplace-Beltrami operator in the unit sphere $S^d$ 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 $S^d\setminus D$, 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.