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 $$\Sigma $$Σ admitting conical singularities of orders $$\alpha _i$$αi’s at points $$p_i$$pi’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 $$\chi (\Sigma )+\sum _i \alpha _i$$χ(Σ)+∑iαi approaches a positive even integer, where $$\chi (\Sigma )$$χ(Σ) is the Euler characteristic of the surface $$\Sigma $$Σ.

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