An overdetermined problem for a weighted Poisson's equation

In this paper, we discuss an overdetermined problem for a weighted Poisson's equation. We prove that if there exists a solution of the weighted Poisson's equation △u+∇logw⋅∇u=−1 on a smooth bounded domain Ω with both Dirichlet and Neumann constant boundary condition, and the weight function w satisfies some conditions in Ω, then Ω is a ball. We also study some applications of the overdetermined problems and some overdetermined problems with nonconstant Neumann boundary condition.