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Multiplicity results for the assigned Gauss curvature problem in \(\mathbb R^2\). (English) Zbl 1165.35366

Summary: To study the problem of the assigned Gauss curvature with conical singularities on Riemannian 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.

MSC:

35J60 Nonlinear elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05 Elliptic equations on manifolds, general theory
58J70 Invariance and symmetry properties for PDEs on manifolds
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