We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form −∂i(aij∂ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +∞. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as −aij∂iju = f are also considered, along with various extensions. The case of general advection-diffusion equations −aij∂iju + bj∂ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].