On a singularly perturbed periodic nonlinear Robin problem

This paper shows some applications of a functional analytic approach to the analysis of a nonlinear Robin problem in a periodically perforated domain with small holes of size proportional to a positive parameter $\epsilon$. The second and third authors have proved in a previous paper the existence of a particular family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon'[}$ uniquely determined (for $\epsilon$ small) by its limiting behavior as $\epsilon\to 0$. Also, the dependence of $u(\epsilon,\cdot)$ upon the parameter $\epsilon$ can be described in terms of real analytic operators of $\epsilon$ defined in a open neighborhood of $0$ and of completely known functions of $\epsilon$. Here, we exploit such a result for the family $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon'[}$ in order to prove an analogous real analytic continuation result for the dependence of the corresponding energy integral $\mathcal{E}(u(\epsilon,\cdot))$ upon the parameter $\epsilon$. Then we focus our attention on the limiting behavior of $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon'[}$ as $\epsilon\to 0$. To do so, we introduce some specific families of solutions which display a suitable property of convergence in the vicinity of the boundary of the holes. First we show that their limit is the solution of a certain ``limiting boundary value problem'' and then we prove a local uniqueness result for such converging families.