We study the maximal branch of solutions for a bifurcation phenomena in variational inequalities. We give optimal results in L∞norm for the approximation by finite element method and sharp estimates for the localization of the continuous and discreate free boundary as the bifurcation parameter goes to infinity. We also obtain lipchitz regularity results for the maximal branch. We establish in some cases a stability condition which has appeared of great importance in the description of the bifurcation phenomena and in the numerical analysis study.