Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Abstract

In this paper, we prove the existence of nontrivial unbounded domains Ω ⊂ R n+1 , n ≥ 1, bifurcating from the straight cylinder B × R (where B is the unit ball of R n ), such that the overdetermined elliptic problemhas 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.