In this paper the centroaffine Minkowski problem, a critical case of the L p-Minkowski problem in the n+ 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.
Consider the existence of rotationally symmetric solutions to the L_p -Minkowski problem for L_p . Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in . In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
In this paper we study the prescribed centroaffine curvature problem in the Euclidean space R n+ 1. This problem is equivalent to solving a Monge鈥揂mp猫re equation on the unit sphere. It corresponds to the critical case of the Blaschke鈥揝antal贸 inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.