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

Abstract

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.