Multiplicity results for the assigned Gauss curvature problem in R 2
نویسندگان
چکیده
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.
منابع مشابه
F Ur Mathematik in Den Naturwissenschaften Leipzig Multiplicity Results for the Two-vortex Chern-simons Higgs Model on the Two-sphere Multiplicity Results for the Two-vortex Chern-simons Higgs Model on the Two-sphere
We consider a Ginzburg-Landau type functional on S 2 with a 6 th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices....
متن کاملConvex hypersurfaces of prescribed curvatures
For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski...
متن کاملCurvature Estimates for Submanifolds with Prescribed Gauss Image and Mean Curvature
We study that the n−graphs defining by smooth map f : Ω ⊂ R n → R m , m ≥ 2, in R m+n of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates sup DR(x)
متن کاملExact Multiplicity Result for the Perturbed Scalar Curvature Problem in R N ( N ≥ 3 )
Let D1,2(RN ) denote the closure of C∞ 0 (R N ) in the norm ‖u‖2 D1,2(RN ) = ∫ RN |∇u|2. Let N ≥ 3 and define the constants αN = N(N − 2) and CN = (N(N − 2)) N−2 4 . Let K ∈ C2(RN ). We consider the following problem for ε ≥ 0 : (Pε) ⎪⎨⎪⎩ Find u ∈ D1,2(RN ) solving : −∆u = αN (1 + εK(x))u N+2 N−2 , u > 0 } in RN . We show an exact multiplicity result for (Pε) for all small ε > 0.
متن کاملA New Modification of Legendre-Gauss Collocation Method for Solving a Class of Fractional Optimal Control Problems
In this paper, the optimal conditions for fractional optimal control problems (FOCPs) were derived in which the fractional differential operators defined in terms of Caputo sense and reduces this problem to a system of fractional differential equations (FDEs) that is called twopoint boundary value (TPBV) problem. An approximate solution of this problem is constructed by using the Legendre-Gauss...
متن کامل