Existence and non existence results for the singular Nirenberg problem Articles uri icon

authors

  • De Marchis, Francesca
  • LOPEZ SORIANO, RAFAEL

publication date

  • March 2016

start page

  • 1

end page

  • 35

issue

  • 55

International Standard Serial Number (ISSN)

  • 0944-2669

Electronic International Standard Serial Number (EISSN)

  • 1432-0835

abstract

  • In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793821, 1991), of prescribing the Gaussian curvature under a conformal change of the metric on surfaceswithconicalsingularities.Suchgeometricalproblemcanbereducedtothesolvability 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. Whenthesurface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K,concerningmainlytheregularity ofits nodalline andthetopology 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 S2 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.

subjects

  • Mathematics