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 Σ admitting conical singularities of orders α 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 χ ( Σ ) + ∑ i α i approaches a positive even integer, where χ ( Σ ) is the Euler characteristic of the surface Σ .