This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!