### Abstract

Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: q¨+V_{q}(q,t)=0q(t+T)=q(t).}We assume V (q, t) < 0 for all q, t and V (q, t), V_{q}(q, t) → 0 as |q| → ∞. Moreover we assume V (q, t) is of a form: V(q,t)=−1|q|α+U(q,t)where 0 < α <2 and U(q, t) ∈ C^{2} ((R^{N}\{0}) × R, R) is a T-periodic funetion in t such that |q|^{α} U (q, t), |q|^{α + 1} U_{q}(q, t), |q|^{α+2} U_{qq}, (q, t), |q|^{α} U_{t}, (q, t) → 0 as |q| → 0. For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

Original language | English |
---|---|

Pages (from-to) | 215-238 |

Number of pages | 24 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Mar 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Hamiltonian systems
- minimax methods
- morse index
- Periodic solutions
- singular potentials

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics

### Cite this

**Non-collision solutions for a second order singular Hamiltonian system with weak force.** / Tanaka, Kazunaga.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Non-collision solutions for a second order singular Hamiltonian system with weak force

AU - Tanaka, Kazunaga

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: q¨+Vq(q,t)=0q(t+T)=q(t).}We assume V (q, t) < 0 for all q, t and V (q, t), Vq(q, t) → 0 as |q| → ∞. Moreover we assume V (q, t) is of a form: V(q,t)=−1|q|α+U(q,t)where 0 < α <2 and U(q, t) ∈ C2 ((RN\{0}) × R, R) is a T-periodic funetion in t such that |q|α U (q, t), |q|α + 1 Uq(q, t), |q|α+2 Uqq, (q, t), |q|α Ut, (q, t) → 0 as |q| → 0. For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

AB - Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: q¨+Vq(q,t)=0q(t+T)=q(t).}We assume V (q, t) < 0 for all q, t and V (q, t), Vq(q, t) → 0 as |q| → ∞. Moreover we assume V (q, t) is of a form: V(q,t)=−1|q|α+U(q,t)where 0 < α <2 and U(q, t) ∈ C2 ((RN\{0}) × R, R) is a T-periodic funetion in t such that |q|α U (q, t), |q|α + 1 Uq(q, t), |q|α+2 Uqq, (q, t), |q|α Ut, (q, t) → 0 as |q| → 0. For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.

KW - Hamiltonian systems

KW - minimax methods

KW - morse index

KW - Periodic solutions

KW - singular potentials

UR - http://www.scopus.com/inward/record.url?scp=85011599319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011599319&partnerID=8YFLogxK

U2 - 10.1016/S0294-1449(16)30219-0

DO - 10.1016/S0294-1449(16)30219-0

M3 - Article

VL - 10

SP - 215

EP - 238

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 2

ER -