Summary: This paper deals with a free boundary problem for both Laplacian and \(p\)-Laplacian operators. We begin by proving the existence of solution (which is of class \(C^2\)) for the associated shape optimization problem. Then, after performing the shape derivative we will present two approaches in order to get sufficient conditions of existence of the free boundaries. The first one needs the use of some maximum principle. The second one uses the monotonicity of the mean curvature and can be applied for general divergence operators.