Some results related to Schiffer’s problem

We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain Ω with analytic boundary ∂Ω having at least one bounded connected component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = g(u)}\;\;\;\;\;\;\;&{\text{in}\;\Omega ,} \\ {\tfrac{{\partial u}}{{\partial v}} = 0\;\text{and}\;u = c}&{\text{on}\;\partial \Omega ,} \end{array}} \right.$$\end{document} where c is a constant. When g(c) = 0 the constant solution u ≡ c is the unique solution. For g(c) ≠ 0, we show that the boundary is a circle if and only if the problem admits a solution that has a constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.