Existence and non existence results for the singular Nirenberg problem

Abstract

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.

Uniform bounds for solutions to elliptic problems on simply connected planar domains

“…• Finally, assuming K(x) to be positive is essential. In fact, in [32,17,18] the authors show existence of solutions to the Liouville equation (1.1) with sign-changing potential even in the case of simply connected domain; here, a crucial role seems to be played not by the topology of Ω but rather of the set {x ∈ Ω : K(x) > 0}.…”

Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case

“…Thus (14) holds and the proof of Step 2 terminates. One can easily see that the conclusion of the lemma follows from (12).…”

A double mean field equation related to a curvature prescription problem

“…On the other hand, blow-up analysis seems much more difficult in the case of sign-changing potentials, because in case concentration occurs at a point where K or h changes sign some compensation phenomena may occur (see for instance [18,19]). Some results concerning blow-up analysis with sign-changing potentials have been provided in [19].…”