Symmetry for a general class of overdetermined elliptic problems

Let $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{R}}^N$, and let $u\in C^1 (\overline{\Omega }) $beaweaksolutionofthefollowingoverdeterminedBVP:$-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$,$ u>0 $in$\Omega $and$u(x)=0, \ |\nabla u (x)| =\lambda (|x|)$on$\partial \Omega $,where$g\in C([0,+\infty ))\cap C^1 ((0,+\infty ) ) $with$g(0)=0$,$g’(t)>0$for$t>0$,$f\in C([0,+\infty )) \times [0, +\infty ) )$,$f$isnonincreasingin$|x|$,$\lambda \in C([0, +\infty )) $and$\lambda $ is positive and nondecreasing. We show that $\mathrm{\Omega }$ is a ball and $u$ satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.