A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach†
Massimo Lanza de Cristoforis, Паоло Мусоліно · Complex Variables and Elliptic Equations · 2012
Let n ∈ ℕ∖{0, 1}. Let q be the n × n diagonal matrix with entries q 11, … , q nn in] 0, +∞[. Then qℤ n is a q-periodic lattice in ℝ n with fundamental cell . Let p ∈ Q. Let Ω be a bounded open subset of ℝ n 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 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.