A singularly perturbed nonlinear Robin problem in a periodically perforated domain: A functional analytic approach

Let n ∈ ℕ, 1. Let q be the n × n diagonal matrix with entries q11,..., qnn in] 0, +∞[. Then qℤn is a q-periodic lattice in ℝn with fundamental cell Q ≡ Πnj=0]0, qjj[. Let p ∈ Q. Let Ω be a bounded open subset of ℝn containing 0. Let G be a (nonlinear) map from ∂Ω × ℝ to ℝ. Let γ be a positive-valued function defined on a right neighbourhood of 0 in the real line. Then we consider the problem for ε > 0 small, where νp+εΩ denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition limε→0+γ(ε)-1ε ∈ ℝ, we prove that the above problem has a family of solutions u(ε, ·)ε∈]0, ε′[ for ε′ sufficiently small, and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis. © 2013 Copyright Taylor and Francis Group, LLC.