Let us fix $0 neq lambda in mathbb{R}$. Let us consider the following ODE, on $(0,infty)$: $$ y^{prime prime} (x) + frac{r e^{-x}}{(1+e^{-x})^2} y(x) = -lambda^2 y(x).$$ Here $r ge 1$ is a parameter. Let us consider the solution $e_r (x)$ which satisfies $e_r (x) sim e^{i lambda x}$ as $x to +infty$. How would one approach showing (if this is indeed true) that $$sup_{substack{r ge 1\ x in (0,infty)}} |e_r (x)| < infty.$$ Ideally, I am interested in some wider class of examples, so I am less interested in a "trick" that happens to work in this very particular case, and more in some conceptual approach, but still would like to hear all approaches.

Thank you