Background: So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form, $$-\Delta u = u^p$$ where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{R}^n$. Using the method of moving planes one can show that every positive solution to the above equation is radially symmetric. Thus we reduce ourselves to the following case, $$-u''-\frac{n-1}{r}u'=u^p$$ where $u(x)=u(|x|)=u(r).$ I understand that by considering the transformation $u(r)=r^{1-n/2}v(r)$ one can obtain a first-order ODE which has explicit solutions of the form $u(r)=(1+r^2)^{1-n/2}$ up to scaling and translations.
Question: I am working with a similar equation $$-\Delta u +f(|x|)u=u^p$$ and I am looking at radial solutions of this equation. Thus I get the equation of the following form, $$-u''-\frac{n-1}{r}u'+f(r)u=u^p.$$ I know an explicit solution to the ODE above, say a function $g(r),$ however I think that one cannot solve the above ODE explicitly. So I was wondering if there are other methods to conclude that all the solutions must have the same form as the explicit solution $g$ up to scaling or translations?
