  We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in
various two dimensional settings. We assume that $V>0$ is a given function,
$\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$
as $\lambda\to 0$. In a recent paper we prove the existence of the maximal
solutions for a particular choice $V\equiv 1$, $\rho=0$ when the problem is
posed in doubly connected domains under Dirichlet boundary conditions. We
related the maximal solutions with a novel free boundary problem. The purpose
of this note is to derive the corresponding free boundary problems in other
settings. Solvability of such problems is, viewed formally, the necessary
condition for the existence of the maximal solution.
