Communications in Mathematical Physics

, Volume 96, Issue 1, pp 97–113

Minimum action solutions of some vector field equations

  • Haim Brezis
  • Elliott H. Lieb

DOI: 10.1007/BF01217349

Cite this article as:
Brezis, H. & Lieb, E.H. Commun.Math. Phys. (1984) 96: 97. doi:10.1007/BF01217349


The system of equations studied in this paper is −Δui=gi(u) on ℝd,d≧2, withu:ℝd→ℝn andgi(u)=∂G/∂ui. Associated with this system is the action,S(u)=ε{1/2|∇u|2G(u)}. Under appropriate conditions onG (which differ ford=2 andd≧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}.

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Haim Brezis
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Département de MathématiquesUniversité Paris VIParis, Cédex 05France
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA