Minimum action solutions of some vector field equations (Q1067090)

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.