Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Abstract

In this paper, we prove the existence of nontrivial unbounded domains Ω⊂Rn+1,n≥1, bifurcating from the straight cylinder B×R (where B is the unit ball of Rn), such that the overdetermined elliptic problem{Δu+f(u)=0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, has a positive bounded solution. We will prove such result for a very general class of functions f:[0,+∞)→R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.