  We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla
u|) dx$ in the class of functions $W^{1,G}(\Omega)$, with a constrain on the
volume of $\{u>0\}$. The conditions on the function $G$ allow for a different
behavior at
  0 and at $\infty$. 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.
