Subharmonic Solutions and Minimal Periodic Solutions of First-order Hamiltonian Systems with Anisotropic Growth

TL;DR

This paper proves the existence of subharmonic and minimal periodic solutions in non-autonomous Hamiltonian systems with anisotropic growth, using variational methods and Maslov index inequalities.

Contribution

It introduces a novel approach combining homologically link theorem and iteration inequalities to handle anisotropic growth in Hamiltonian systems.

Findings

Established existence of subharmonic solutions under anisotropic growth conditions.

Analyzed minimal period problems for autonomous systems with anisotropic growth.

Applied variational methods and Maslov index techniques to non-autonomous systems.

Abstract

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.