DCDS
Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth
Chungen Liu Xiaofei Zhang

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

keywords: Maslov-type index Morse index homologically link subharmonic solution minimal period
DCDS
Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems
Chungen Liu
In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\R^{2n}, \R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.
keywords: Hamiltonian systems minimal period problem. index iteration theory
DCDS
Symmetrical symplectic capacity with applications
Chungen Liu Qi Wang
In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed characteristic (brake orbit and $S$-invariant brake orbit are two examples) on prescribed symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.
keywords: Symmetrical symplectic manifolds brake orbits. symmetrical symplectic capacity
CPAA
Existence results for the fractional Q-curvature problem on three dimensional CR sphere
Chungen Liu Yafang Wang

In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.

keywords: CR manifolds CR fractional sub-Laplacian Yamabe problem critical points at infinity fractional Q-curvature

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