ABSTRACT. In this paper we study the problem, posed by Troyanov in [47], of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities.Besides, we find a class of functions on S 2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp.

ABSTRACT. In this paper we consider a mean field problem on a compact surface without boundary in presence of conical singularities. The corresponding equation, named after Liouville, appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics. Our contribution focuses on the case of sign-changing potentials, and gives results on compactness, existence and multiplicity of solutions.

We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange energy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have diverging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up analysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.

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a b s t r a c tWe consider the following problem: given a bounded domain Ω ⊂ R n and a vector field ζ : Ω → R n , find a solution to −∆ ∞ u − ⟨Du, ζ ⟩ = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1-homogeneous infinity Laplace operator that is formally given by ∆ ∞ u = ⟨D 2 u Du |Du| , Du |Du| ⟩ and f a Lipschitz boundary datum. If we assume that ζ is a continuous gradient vector field then we obtain the existence and uniqueness of a viscosity solution by an L p -approximation procedure. Also we prove the stability of the unique solution with respect to ζ . In addition when ζ is more regular (Lipschitz continuous) but not necessarily a gradient, using tug-ofwar games we prove that this problem has a solution.