An infinite scale of incompressible twisting solutions to the nonlinear elliptic system L [u; A, B] = ?P and the discriminant ?(h, g)

In this paper we consider the second order nonlinear elliptic system div[A(|x|, |u| 2 , |?u| 2 )?u] + B(|x|, |u| 2 , |?u| 2 )u = [cof ?u]?P, where the unknown vector field u satisfies the incompressibility constraint det ?u = 1 a.e. along with suitable boundary conditions and P = P(x) is an a priori unknown hydrostatic pressure field. Here, A = A(r, s, ?) and B = B(r, s, ?) are sufficiently regular scalar functions satisfying natural structural properties. Most notably in the case of a finite symmetric annulus we prove the existence of a countably infinite scale of topologically distinct twisting solutions to the system in all even dimensions. In sharp contrast in odd dimensions the only twisting solution is the map u = x. We study a related class of systems by introducing the novel notion of a discriminant. Using this a complete and explicit characterisation of all twisting solutions for n = 2 is given and a curious dichotomy in the behaviour of the system and its solutions captured and analyse