Overdetermined elliptic problems in nontrivial contractible domains of the sphere

Abstract

In this paper, we prove the existence of nontrivial contractible domains Ω⊂Sd, d≥2, such that the overdetermined elliptic problem{−εΔgu+u−up=0in Ω, u>0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, admits a positive solution. Here Δg is the Laplace-Beltrami operator in the unit sphere Sd with respect to the canonical round metric g, ε>0 is a small real parameter and 1<p<d+2d−2 (p>1 if d=2). These domains are perturbations of Sd∖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.