We show that the total energy of the pure displacement problem in nonlinear elasticity possesses a unique global minimizer for a large class of hyperelastic materials, including that of Saint Venant - Kirchhoff, pro- vided the density of the applied forces are small in Lp-norm. We also establish a nonlinear Korn inequality with boundary showing that the H1-distance between two deformation fields is bounded, up to a multi- plicative constant, by the L2-distance between their Cauchy-Green strain tensors.