### Abstract

We consider the existence of periodic solutions of a Hamiltonian system q ̈ + ∇V(q) = 0 (HS) such that 1 2| q ̇(t)|^{2}+ V(q(t)) = H for all t, where q ∈ R^{N} (N ≥ 3), H <0 is a given number, V(q) ∈ C^{2}(R^{N}\{0}, R) is a potential with a singularity, and ∇V/(q) denotes its gradient. We consider a potential V(q) which behaves like -l/|q|^{α} (α ∈ (0, 2)). In particular, in case V(q) satisfies ∇V(q) q ≤ -α_{1}V(q) for all q ∈ R^{N}\{0}, and ∇V(q) q ≤ -α_{2}V(q) for all 0 < |q| ≤ R_{0} for α_{1}, α_{2} ∈ (0, 2), R_{0} > 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = - 1 |q|^{α} + W(q), where 0 < α < 2 and |q|^{α}W(q), |q|^{α + 1} ∇W(q), |q|^{α + 2} ∇^{2}W(q) → 0 as |q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical (non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.

Original language | English |
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Pages (from-to) | 351-390 |

Number of pages | 40 |

Journal | Journal of Functional Analysis |

Volume | 113 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1993 May 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis

### Cite this

**A Prescribed Energy Problem for a Singular Hamiltonian System with a Weak Force.** / Tanaka, Kazunaga.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 113, no. 2, pp. 351-390. https://doi.org/10.1006/jfan.1993.1054

}

TY - JOUR

T1 - A Prescribed Energy Problem for a Singular Hamiltonian System with a Weak Force

AU - Tanaka, Kazunaga

PY - 1993/5/1

Y1 - 1993/5/1

N2 - We consider the existence of periodic solutions of a Hamiltonian system q ̈ + ∇V(q) = 0 (HS) such that 1 2| q ̇(t)|2+ V(q(t)) = H for all t, where q ∈ RN (N ≥ 3), H <0 is a given number, V(q) ∈ C2(RN\{0}, R) is a potential with a singularity, and ∇V/(q) denotes its gradient. We consider a potential V(q) which behaves like -l/|q|α (α ∈ (0, 2)). In particular, in case V(q) satisfies ∇V(q) q ≤ -α1V(q) for all q ∈ RN\{0}, and ∇V(q) q ≤ -α2V(q) for all 0 < |q| ≤ R0 for α1, α2 ∈ (0, 2), R0 > 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = - 1 |q|α + W(q), where 0 < α < 2 and |q|αW(q), |q|α + 1 ∇W(q), |q|α + 2 ∇2W(q) → 0 as |q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical (non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.

AB - We consider the existence of periodic solutions of a Hamiltonian system q ̈ + ∇V(q) = 0 (HS) such that 1 2| q ̇(t)|2+ V(q(t)) = H for all t, where q ∈ RN (N ≥ 3), H <0 is a given number, V(q) ∈ C2(RN\{0}, R) is a potential with a singularity, and ∇V/(q) denotes its gradient. We consider a potential V(q) which behaves like -l/|q|α (α ∈ (0, 2)). In particular, in case V(q) satisfies ∇V(q) q ≤ -α1V(q) for all q ∈ RN\{0}, and ∇V(q) q ≤ -α2V(q) for all 0 < |q| ≤ R0 for α1, α2 ∈ (0, 2), R0 > 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = - 1 |q|α + W(q), where 0 < α < 2 and |q|αW(q), |q|α + 1 ∇W(q), |q|α + 2 ∇2W(q) → 0 as |q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical (non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.

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U2 - 10.1006/jfan.1993.1054

DO - 10.1006/jfan.1993.1054

M3 - Article

VL - 113

SP - 351

EP - 390

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -