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 (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 ).