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

Abstract

In this paper, the $L_p$-dual Minkowski problem of Monge-Ampère type were studied for different $p$ and $q$. Some new nonuniqueness results were obtained for the range $p\le q-n+1$, $p<q-{\lambda }_1(n,k)$ and $f\equiv 1$, where ${\lambda }_1(n,k)$ 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 $p\le -q,q\ge n$ on all dimensional spaces. The key ingredient of our proof was based on a generalization of Chou-Wang identity for $q=n$, $p$=$-q$ to a full range of $(p,q)$.

References in corpus (1)

• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems