, Volume 96, Issue 1, pp 97-113

Minimum action solutions of some vector field equations

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Abstract

The system of equations studied in this paper is −Δu i =g i (u) on ℝ d ,d≧2, withu:ℝ d →ℝ n andg i (u)=∂G/∂u i . 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}.

Work partially supported by U.S. National Science Foundation Grant PHY-81-16101-A02
Communicated by A. Jaffe