Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents

Abstract

In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Amp\`ere type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-\lambda_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $\lambda_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincar\'e inequality on $\mathbb{S}^{n-1}$ with k-symmetricity. The second part of this paper is devoted to prove some new nonexistence results for the supercritical range $\mathit{p}\leq-\mathit{q}, \mathit{q}\geq\mathit{n}$ on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for $\mathit{q}=\mathit{n}$, $\mathit{p}$=$-\mathit{q}$ to a full range of $(\mathit{p},\mathit{q})$.