  We consider the $p$-Laplacian equation $-\Delta_p u=1$ for $1<p<2$, on a
regular bounded domain $\Omega\subset\mathbb R^N$, with $N\ge2$, 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 $\partial\Omega$ 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.
