An optimization problem with volume constrain for a degenerate quasilinear operator

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$withaconstrainonthevolumeof${u>0}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial{u>0}\cap \Omega$, is smooth.