### Abstract

In this paper, we prove that the Faddeev energy E _{1} at the unit Hopf charge is attainable. The proof is based on utilizing an important inequality called the substantial inequality in our previous paper which describes how the Faddeev energy splits into its sublevels in terms of energy and topology when compactness fails. With the help of an optimal Sobolev estimate of the Faddeev energy lower bound and an upper bound of E _{1}, we show that E _{1} is attainable. For the two-dimensional Skyrme model, we prove that the substantial inequality is also valid, which allows us to greatly improve the range of the coupling parameters for the existence of unit-charge solitons previously guaranteed in a smaller range of the coupling parameters by the validity of the concentration-compactness method.

Original language | English (US) |
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Pages (from-to) | 137-152 |

Number of pages | 16 |

Journal | Communications In Mathematical Physics |

Volume | 269 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2007 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics