Asymptotic behavior for the solution of a nonlinear ODE In a nutshell, if $u$ is a solution to
$$
\partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\
\lim_{r \to \infty} u(r) = 0, 
\quad \text{and} \quad 
u(r_0) = u_0 > 0 .
$$
I'd like to know the behavior of $u(r)$ as $r \to \infty$.
Motivation: $u$ is the steady-state concentration of a substance whose dynamics are determined by a reaction-diffusion equation obtained by adding a time derivative to the first line (the Fisher-KPP), and there is an influx of the substance at $r_0$.  The setting is two-dimensional, but assumed to be radially symmetric; if you want to be more general replace $1/r$ by $(d-1)/r$.  
Without the first derivative term the equation can be written explicitly with elliptic functions,
having $u(r) \sim \exp(-r)$ as $r \to \infty$; but I have good reason to think $u(r) \sim \exp(-r)/\sqrt{r}$ in this case.
 A: I do not have a complete solution, just a comment. As $u(r)\to 0$, one may try to neglect
the quadratic term. Then the equation becomes linear, and its solution tending to $0$
is called the modified Bessel function $K_0$. $K_0(x)=Y_0(ix)$, where $Y_0$ is the Weber
function ("second" solution of the Bessel equation). It has asymptotic behavior
$$K_0(x)=cr^{-1/2}e^{-r}(1+O(1/r)).$$
This is consistent with what you wrote. 
Now I think that rejection of the quadratic term can be justified in this case with some standard perturbative method.
Some evidence that such an approximation can be justified comes from 
$u^{\prime\prime}-u(1-u)=0$, which can be exactly solved using elliptic functions. The linearlization
that I propose would give $u^{\prime\prime}-u=0$, whose solution tending to $0$ is $e^{-r}$ and
this is confirmed by exact solution in elliptic functions. 
I believe that a rigorous justification of this asymptotics can be obtained from the papers:
J. Malmquist, Acta Math. 73 (1940), 87–129; 74 (1941), 1–64, 109–128; MR0003898 and
M. Iwano, 
Intégration analytique d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier. I.
Ann. Mat. Pura Appl. (4) 44 1957 261–292, MR0096838.
In English: W. Wasow, Asymptotic expansions for ordinary differential equations, John Wiley and sons, NY 1965, he has a brief chapter IX on non-linear equations, explaining the results of Iwano and Malmquist.
