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Minimum action solutions of some vector field equations. (English) Zbl 0579.35025
The authors study the system of equations $$-\Delta u_ i=g^ i(u)$$ on $${\mathbb{R}}^ d$$ (d$$\geq 2)$$, where $$u: {\mathbb{R}}^ d\to {\mathbb{R}}^ n$$ and $$g^ i(u)=\partial G/\partial u_ i$$. Under appropriate conditions on G they show that the system has a non-trivial solution with finite action $$S(u):=\int \{(1/2)| \nabla u|^ 2-G(u)\}$$ and that this solution minimises the action within the class of non-trivial solutions with finite action. The proof is ingenious, and the paper contains numerous technical results of interest in their own right.
Reviewer: D.Edmunds

##### MSC:
 35J60 Nonlinear elliptic equations 49S05 Variational principles of physics
##### Keywords:
vector field equations; minimum action; finite action
Full Text:
##### References:
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