We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P₁ and P₂ finite-element discretizations. Our analysis extends to the case of nonlinear eigenproblems the classical results about the comparative speeds of convergence of the eigenvalues with respect to the eigenvectors in the H¹- norm. We show that under some assumptions we recover a standard result for linear elliptic eigenvalue problems.