An optimization problem with volume constraint in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω), with a constraint on the volume of {u>0}. The conditions on the function G allow for a different behavior at 0 and at ∞. 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, ∂{u>0}∩Ω is smooth.