Dynamics of Partial Differential Equations

Volume 12 (2015)

Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

Pages: 379 – 403

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n4.a4

Authors

Y. Liu (School of Mathematics and Physics, University of Science and Technology, Beijing, China)

Y. Wang (Department of Mathematics and Physics, North China Electric Power University, Beijing, China; and Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland, Canada)

J. Xiao (Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland, Canada)

Abstract

In this paper we study nonnegative solutions of\begin{align}(\dagger) & &{\lvert g \rvert}^{\gamma}_{\mathbb{H}^n} u^p \leq (- \Delta_{\mathbb{H}^n})^{\frac{\alpha}{2}} u \text{ on } \mathbb{H}^n \text{,}\end{align}where $\mathbb{H}^n$ is the Heisenberg group; ${\lvert \cdot \rvert}_{\mathbb{H}^n}$ is the homogeneous norm; $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian; $(p, \alpha, \gamma) \in (1, \infty) \times (0, 2) \times [0, (p-1)Q)$; and $Q = 2n+ 2$ is the homogeneous dimension of $\mathbb{H}^n$. In particular, we prove that any nonnegative solution of $(\dagger)$ is zero if and only if $p \leq \frac{Q+\gamma}{Q-\alpha}$.

Keywords

Heisenberg group, nonnegative weak solution, fractional sub-Laplacian

2010 Mathematics Subject Classification

Primary 35R03. Secondary 35R11.

Published 10 December 2015