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The minimal periodic solutions for superquadratic autonomous Hamiltonian systems without the Palais-Smale condition

Yuming Xiao, Gaosheng Zhu

Abstract

In this paper, we prove the existence of periodic solutions with any prescribed minimal period $T>0$ for even second order Hamiltonian systems and convex first order Hamiltonian systems under the weak Nehari condition instead of Ambrosetti-Rabinowitz's. To this end, we shall develop the method of Nehari manifold to directly deal with a frequently occurring problem where the Nehari manifold is not a manifold.

The minimal periodic solutions for superquadratic autonomous Hamiltonian systems without the Palais-Smale condition

Abstract

In this paper, we prove the existence of periodic solutions with any prescribed minimal period for even second order Hamiltonian systems and convex first order Hamiltonian systems under the weak Nehari condition instead of Ambrosetti-Rabinowitz's. To this end, we shall develop the method of Nehari manifold to directly deal with a frequently occurring problem where the Nehari manifold is not a manifold.
Paper Structure (4 sections, 8 theorems, 112 equations)

This paper contains 4 sections, 8 theorems, 112 equations.

Table of Contents

  1. Introduction and the main results
  2. A generalization of the method of Nehari manifold
  3. Application of Theorem \ref{['abstract1']} to the second order case
  4. Application of Theorem \ref{['abstract2']} to the first order case

Key Result

Theorem 1.1

Suppose that $V\in C^{2}(\mathbb{R}^{N},\mathbb{R})$ satisfies the following conditions ${\rm (V1)}$$\lim_{x\to0}{V(x)\over|x|^{2}}=0$, ${\rm (V2)}$$\lim_{x\to\infty}{V(x)\over|x|^{2}}=+\infty,$${\rm (V3)}$$0< V'(x)\cdot x \leq V"(x)x\cdot x,\ \forall x\in\mathbb{R}^{N}\backslash\{0\},$${\rm (V4)}$$

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • ...and 8 more