Multiplicity results for the assigned Gauss curvature problem in R2

Tarantello, Gabriella; Esteban, Maria J.; Dolbeault, Jean

Tarantello, Gabriella; Esteban, Maria J.; Dolbeault, Jean (2009), Multiplicity results for the assigned Gauss curvature problem in R2, Nonlinear Analysis: Theory, Methods & Applications, 70, 8, p. 2870-2881. http://dx.doi.org/10.1016/j.na.2008.12.040

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00323700/en/

Date

2009

Journal name

Nonlinear Analysis: Theory, Methods & Applications

Volume

Number

Publisher

Elsevier

Pages

2870-2881

Abstract (EN)

To study the problem of the assigned Gauss curvature with conical singularities on Riemanian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce the problem to a Liouville equation on the euclidean plane. We prove new multiplicity results for bounded radial solutions, which improve on earlier results of C.-S. Lin and his collaborators. Based on numerical computations, we also present various conjectures on the number of unbounded solutions. Using symmetries, some multiplicity results for non radial solutions are also stated.

Subjects / Keywords

Self-dual gauge field vortices; Onsager equation; Liouville equation; Riemannian manifolds; Conical singularities; Gauss curvature; Multiplicity; Uniqueness; Radial symmetry; Blow-up; Emden–Fowler transformation; Stereographic projection