Multiple solutions of the $$L_{p}$$ L p -Minkowski problem

• DOI: 10.1007/s00526-016-1063-y
• Publication Date: 2016-09-22T11:51:26Z
• Authors: Yan He; Qi-Rui Li; Xu-Jia Wang
• Keywords: 0101 mathematics; 01 natural sciences

In this paper, we study the existence of multiple solutions to the \(L_{p}\)-Minkowski problem. We prove if \(p 0\), there exists a smooth positive function f on \(\mathbb S^n\) such that the \(L_{p}\)-Minkowski problem admits at least N different smooth solutions. We also construct nonsmooth, positive function f for which the \(L_{p}\)-Minkowski problem has infinitely many \(C^{1,1}\) solutions.