Asymptotic Analysis of Periodically Perforated Nonlinear Media Close to the Critical Exponent

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with p-growth for p converging to the space dimension n. We prove that for p close to the critical exponent n there are three regimes, two with a non-trivial size of the perforations (exponential and mixed polynomial-exponential) and one where the Γ-limit is always trivial.