Banach spaces (cf., *e.g.*, Deimling [@De85 Theorem 15.3]), there exist an open neighborhood $\mathcal{U} \subseteq \tilde{\mathcal{U}}$ of $(0,r_{\ast})$ in $\mathbb{R}^2$ and a real analytic map $(\Theta^{i}_1,\Theta^{i}_2, \Theta^{o}, \Xi)$ from $\mathcal{U}$ to $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$ such that $$\label{eq:ansol}
\Lambda\bigl[\varrho_1, \varrho_2, \Theta^{i}_1[\varrho_1,\varrho_2],\Theta^{i}_2[\varrho_1,\varrho_2], \Theta^{o}[\varrho_1,\varrho_2], \Xi[\varrho_1,\varrho_2]\bigr]=0 \qquad \forall (\varrho_1,\varrho_2) \in  \mathcal{U}\, .$$ In particular, by Proposition \[prop:finteq\] and Lemma \[lem:lim\], we have $$\begin{split}
(\theta^{i}_1[\varrho_1,\varrho_2],\theta^{i}_2[\varrho_1,\varrho_2], \theta^{o}[\varrho_1,\varrho_2], \xi[\varrho_1,\varrho_2])=(\Theta^{i}_1[\varrho_1,\varrho_2],\Theta^{i}_2[ &\varrho_1,\varrho_2], \Theta^{o}[ \varrho_1,\varrho_2], \Xi[ \varrho_1,\varrho_2])\\ &  \forall (\varrho_1,\varrho_2) \in \mathcal{U} \cap ]0,+\infty[^2\, ,
\end{split}$$ and $$(\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2, \tilde{\theta}^{o}, \tilde{\xi})=(\Theta^{i}_1[0,r_\ast], \Theta^{i}_2[0,r_\ast], \Theta^{o}[0,r_\ast], \Xi[0,r_\ast])\, ,$$ and thus the proof is complete.

A functional analytic representation theorem for the solution of problem  {#fure}
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In the following theorem, we exploit the analyticity result for the solutions of equation in order to prove representation formulas for $u[\varrho_1,\varrho_2]$ in terms of real analytic maps. Then, by the analysis of the behavior of $u[\varrho_1,\varrho_2]$ for $(\varrho_1,\varrho_2)$ close to the degenerate value $(0,r_\ast)$, we will be able to answer questions (i), (ii), (iii) asked in the introduction and concerning the behavior of the solution $u_\epsilon$ of problem .

\[thm:rep\] Let $\alpha\in]0,1[$. Let $\Omega^i_1$, $\Omega^i_2$, $\Omega^o$ be as in . Let $p^1$, $p^2$ be as in . Let $r_\ast \in [0,+\infty[$. Let hold. 