# Joint boundedness of solutions of a family of Sturm-Liouville ODE

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