A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Let f : [ 0 , + ∞ ) → ℝ be a (locally) Lipschitz function and Ω ⊂ ℝ 2aC 1 , αdomain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem[ {Δ u + f ( u ) = 0in  Ω ,u = 0on       Ω ,∂ u ∂ ν →= 1on   Ω ,we prove that Ω is a half‐plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli, and L. Nirenberg in 1997.© 2017 Wiley Periodicals, Inc.