Dambrine, and Lacave [@BoDaLa], where the authors exploit the method of multiscale asymptotic expansions. More precisely, in [@BoDaToVi09] they carefully analyze the case when $\eta(\epsilon)=\epsilon^\beta$ for $\beta \in ]0,1[$ and they provide asymptotic expansions.

Here, instead, we answer the questions in (i), (ii), (iii) by representing the maps of (i), (ii), (iii) in terms of real analytic maps and in terms of known functions of $\epsilon$ (such as $\eta(\epsilon)$, $\epsilon/\eta(\epsilon)$, $\log \eta(\epsilon)$, [*etc*]{}.). We observe that our approach does have its advantages. Indeed, if for example we know that the function in (i) equals for $\epsilon>0$ a real analytic function defined in a whole neighborhood of $\epsilon=0$, then we know that such a map can be expanded in power series for $\epsilon$ small. Moreover, we emphasize that we do not make any assumption on the form of the function $\eta(\epsilon)$ and that, by setting $\varrho_1=\eta(\epsilon)$ and $\varrho_2=\epsilon/\eta(\epsilon)$, we can treat $\varrho_1$ and $\varrho_2$ as two independents variables and prove real analyticity results for the solution upon the pair $(\varrho_1,\varrho_2)$. In particular, one can deduce asymptotic expansions in the new variable $(\varrho_1,\varrho_2)$ around $(0,r_\ast)$.

Such an approach has been carried out for problems for the Laplace operator in a domain with 