Prescribed Gauss curvature problem on singular surfaces

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