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.