### Abstract

The system of equations studied in this paper is -Δu_{i}=g^{i}(u) on ℝ^{d}, d≧2, with u:ℝ^{d}→ℝ^{n} and g^{i}(u)=∂G/∂u_{i}. Associated with this system is the action, S(u)=ε{1/2|∇u|^{2}-G(u)}. Under appropriate conditions on G (which differ for d=2 and d≧3) it is proved that the system has a solution, u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, v ≢0}.

Original language | English (US) |
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Pages (from-to) | 97-113 |

Number of pages | 17 |

Journal | Communications In Mathematical Physics |

Volume | 96 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1984 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Brezis, H., & Lieb, E. H. (1984). Minimum action solutions of some vector field equations.

*Communications In Mathematical Physics*,*96*(1), 97-113. https://doi.org/10.1007/BF01217349