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

Shi-Zhong Du, YanNan Liu, Jian Lu

Abstract

In this paper, the $\mathit{L}_{\mathit{p}}$-dual Minkowski problem of Monge-Ampère type were studied for different $\mathit{p}$ and $\mathit{q}$. Some new nonuniqueness results were obtained for the range $\mathit{p}\le\mathit{q}-\mathit{n}+1$, $\mathit{p}\lt\mathit{q}-λ_{1}(\mathit{n},\mathit{k})$ and $\mathit{f}\equiv1$, where $λ_{1}(\mathit{n},\mathit{k})$ is the best constant of the Poincaré 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 $(\mathit{p},\mathit{q})$.

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

Abstract

In this paper, the -dual Minkowski problem of Monge-Ampère type were studied for different and . Some new nonuniqueness results were obtained for the range , and , where is the best constant of the Poincaré inequality on with k-symmetricity. The second part of this paper is devoted to prove some new nonexistence results for the supercritical range on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for , = to a full range of .

Paper Structure

This paper contains 8 sections, 28 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. $k-$symmetricity on a discrete sub-group of $SO(n)$
  3. New variational scheme on the $k-$symmetricity
  4. Instability of constant solution for $q>p+\lambda_1(n,k)$
  5. Best constant of Poincaré inequality on ${{\mathcal{G}}_{n,k}}$
  6. Multiple solutions of \ref{['e1.3']}
  7. Chou-Wang type identity for \ref{['e1.2']}
  8. Nonexistence result for $p\leq-q, q\geq n$

Key Result

Theorem 1.1

Considering e1.3 on dimension $n\geq2$ and assuming $p\leq-n+1$, we have the following result of multiple solutions which are different from each other up to rotations: (1) On the planar case $n=2$, if $q-p>\tau^2$ hold for some integer $\tau\geq3$, then e1.3 admits at least $\tau-1$ solutions. (2)

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Lemma 3.1
  • ...and 24 more