Symmetry for a general class of overdetermined elliptic problems

Let ({Omega}) a bounded domain in ({mathbb{R} ^N }), and let ({uin C^1 (overline{Omega})}) a weak solution of the following overdetermined BVP: ({-nabla (g(|nabla u|)|nabla u|^{-1}nabla u)=f(|x|,u)}), ({ u > 0 }) in ({Omega }) and ({u=0, |nabla u(x)|=lambda (|x|)}) on ({partial Omega }), where ({gin C([0,+infty)cap C^1 ((0,+infty ) ) }) with ({g(0)=0}), ({g'(t) > 0}) for ({t > 0}), ({fin C([0,+infty ) times [0, +infty ) )}), f is nonincreasing in ({|x|}), ({lambda in C([0, +infty )) }) and ({lambda }) is positive and nondecreasing. We show that ({Omega }) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.