We consider viscosity solutions to a one-phase free boundary problem for a nonlinear elliptic PDE with non-zero right hand side. We obtain regularity results for solutions and their free boundaries. The operator under consideration is a model case in the class of partial differential equations with non-standard growth. This type of operators have been used in the modelling of non-Newtonian fluids, such as electrorheological or thermorheological fluids, also in non-linear elasticity and image reconstruction. We also obtain some new results for the governing operator that are of independent interest. This is joint work with Fausto Ferrari (University of Bologna, Italy)