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

Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let $q$ be the $n\times n$ diagonal matrix with entries $q_{11}$, \dots, $q_{nn}$ in $]0,+\infty[$. Then $q{\mathbb{Z}}^{n}$ is a $q$-periodic lattice in ${\mathbb{R}}^{n}$ with fundamental cell $Q\equiv\Pi_{j=1}^{n}]0,q_{jj}[$. Let $p\in Q$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ containing $0$. Let $G$ be a (nonlinear) map from $\partial\Omega\times{\mathbb{R}}$ to ${\mathbb{R}}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Then we consider the problem \[ \left\{ \begin{array}{ll} \Delta u=0 & {\mathrm{in}}\ {\mathbb{R}}^{n}\setminus \bigcup_{z\in {\mathbb{Z}}^{n}}\left( qz+p+\epsilon{\mathrm{cl}} \Omega\right)\,, \\ u\ {\mathrm{is}}\ $q$-{\mathrm{periodic }}\,, \\ -\frac{\partial u}{\partial\nu_{ p+\epsilon\Omega }}(x) = \frac{1}{\gamma(\epsilon)}G((x-p)/\epsilon, u(x))& \forall x\in p+\epsilon\partial\Omega\,, \end{array} \right. \] for $\epsilon>0$ small, where $\nu_{p+\epsilon\Omega }$ denotes the outward unit normal to $p+\epsilon\partial\Omega$. Under suitable assumptions and under condition $\lim_{\epsilon\to 0+}\gamma(\epsilon)^{-1}\epsilon \in {\mathbb{R}}$, we prove that the above problem has a family of solutions $\{u(\epsilon, \cdot)\}_{ \epsilon\in ]0,\epsilon'[ }$ for $\epsilon'$ sufficiently small, and we analyze the behaviour of such a family as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis.