The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

We consider the p-Laplacian equation -Delta(p)u = 1 for 1 < p < 2, on a regular bounded domain Omega subset of R-N, with N >= 2, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature H of partial derivative Omega is constant, then Omega is a ball and the unique solution of the Dirichlet p-Laplacian problem is radial. The main tools used are integral identities, the P-function, and the maximum principle.