Existence and uniqueness of correctors for nonlinear homogenization in uniformly local Sobolev spaces

We prove the existence and uniqueness of correctors for nonlinear homogenization problems associated with monotone operators of p -growth. The corrector equation is posed in the full space Rᵈ , as required in the non-periodic deterministic setting, and is solved within the framework of uniformly local Sobolev spaces. Our method relies on a localization procedure based on a sequence of increasing balls combined with the large mass trick, together with Caccioppoli-type estimates to establish compactness and stability. The results provide a foundation for extending deterministic homogenization theory beyond the periodic framework, ensuring that correctors exist and are uniquely defined in general heterogeneous media.