The interplay between two Euler-Lagrange operators relating to the nonlinear elliptic system S[(u, P), ?]

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain SO(n)-valued matrix fields, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector fields and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.