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 $\u(epsilon,cdot)_epsilonin]0,epsilon'[$ uniquely determined (for $epsilon$ small) by its limiting behavior as $epsilon o 0$. 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 $\u(epsilon,cdot)_epsilonin]0,epsilon'[$ in order to prove an analogous real analytic continuation result for the dependence of the corresponding energy integral $mathcalE(u(epsilon,cdot))$ upon the parameter $epsilon$. Then we focus our attention on the limiting behavior of $\u(epsilon,cdot)_epsilonin]0,epsilon'[$ as $epsilon o 0$. 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.