Pengcheng Niu

Hardy type and Rellich type inequalities on the Heisenberg group

This paper contains some interesting Hardy type inequalities and Rellich type inequalities for the left invariant vector fields on the Heisenberg group.

Periodic solution and almost periodic solution of impulsive Lasota-Wazewska model with multiple time-varying delays

This paper deals with the impulsive Lasota-Wazewska model with multiple time-varying delays. Our results show that the system is uniformly persistent under some appropriate conditions. The sufficient condition for global exponential stability of the system is given. Applying Mawhins continuation theorem of coincidence degree, we prove that the periodic system has at least one strictly positive periodic solution. By employing hull theory of impulsive almost periodic system, the existence and uniqueness of strictly positive almost periodic solution of the almost periodic system is obtained. Two examples are provided to illustrate our results.

Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications☆

Abstract The purpose of this paper is to consider a class of p -degenerate subelliptic operators L p constructed by generalizing Greiners vector fields. Their fundamental solutions at the origin are established with the aid of the properties of radial functions. A Picone-type identity and a Hardy-type inequality with respect to vector fields are proved. Some Pohozaev-type identities and applications to nonlinear equations are given. Finally, a Carleman-type estimate and uniqueness of the operator L 2 are discussed.

Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space

In this paper, we consider the following weighted PDE system with Navier boundary conditions on a half space:1 When is any even number between and , we establish the equivalence between (1) and the following integral system2 under some very mild growth condition, where is the reflection of the point about the plane . Then, in the critical case of , we show that every pair of positive solutions of system (2) is rotationally symmetric about -axis. While in the subcritical case, we prove the non-existence of positive solutions. When dealing integral system (2), can be any real number between and .

Some theorems on existence and uniqueness of fixed points for decreasing operators

In this paper, we employ partial order method, cone theory and monotone iterative technique to prove several new theorems on the existence and uniqueness of fixed points for decreasing operators without compactness. Some applications are also given.

A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem

Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.

HARDY INEQUALITIES IN HALF SPACES OF THE HEISENBERG GROUP

In this paper we prove some Hardy inequalities in the half space of the Heisenberg group and indicate the sharp constant.

A Liouville theorem for the semilinear fractional CR covariant equation on the heisenberg group

ABSTRACT We obtain a Liouville theorem for the semiliear fractional CR covariant equation on the Heisenberg group . For this purpose, we extend the fractional equation to sub-Laplace Neumann problem on . Then the conclusion is derived by applying the CR inversion and moving plane method to the Neumann problem. Our result is a generalization for the corresponding one to

A Liouville type theorem to an extension problem relating to the Heisenberg group

We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.

Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

Suppose that are left invariant real vector fields on the homogeneous group with being homogeneous of degree two and homogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kind where and is a constant matrix satisfying the elliptic condition on . By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Holder estimates for .