The equation to be solved is
$r^2 u''(r)-A(r^2\ u(r)u'(r)+2\ r\ u(r)^2)+2\ r\ u'(r)-2\ u(r)=0$
with boundary conditions $u(0)=u_0$ and $u(\infty)=0$. Additionally it is assumed that $0<A<<1$. A closed-form or "analytical" solution is sought--though it seems unlikely one can be found. It is noted that under the transformation $y(r)=r^2\ u(r)$ the above equation transforms to,
$r^2\ y''(r)-(A\ y(r)+2\ r)y'(r)=0$
with boundary conditions appropriately modified.
This apparently simpler equation may be helpful in finding an closed-form approximate solution by treating the non-linear term perturbatively or via some other technique unknown to me.
Also, this problem may not be properly posed, in which case I am open to suggestions that would make it so. It is apparently related to an engineering problem but, not being an engineer, I cannot say which one.
Your thoughts?
Thanks!