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

Let n ∈ ℕ\{0, 1}. Let q be the n × n diagonal matrix with entries q₁₁,..., q_nn in] 0, +∞[. Then qℤⁿ is a q-periodic lattice in ℝⁿ with fundamental cell Q ≡ Πⁿ _j=0]0, q_jj[. 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 ν_p+εΩ denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition lim_ε→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.