We first show how the displacement-traction problem of nonlinear three-dimensional elasticity can be recast either as a boundary value problem or as a minimization problem over a Banach manifold, where the unknown is the Cauchy–Green strain tensor instead of the deformation as is customary. We then consider the pure displacement problem, and we show that, under appropriate smoothness assumptions on the data, either problem recast in this fashion possesses at least a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. In particular, the minimization problem provides an example where the functional is not coercive.