  We study the existence of at least one conformal metric of prescribed
Gaussian curvature on a closed surface $\Sigma$ admitting conical singularities
of orders $\alpha_i$'s at points $p_i$'s. In particular, we are concerned with
the case where the prescribed Gaussian curvature is sign-changing. Such a
geometrical problem reduces to solving a singular Liouville equation. By
employing a min-max scheme jointly with a finite dimensional reduction method,
we deduce new perturbative results providing existence when the quantity
$\chi(\Sigma)+\sum_i \alpha_i$ approaches a positive even integer, where
$\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$.
