ODE arising from best constant for Sobolev Embedding

Question

I was trying to solve the critical exponent PDE $$\Delta u = u^\frac{n+2}{n-2}$$ in $\mathbb{R}^n$ and ultimately reduced it to the following ODE $$\frac{dy}{dr}=ce^{\int_0^r{y}}-y^2-\frac{y}{r}$$ with the initial condition $y(0)=k \in \mathbb{R}$ for any $n\geq 3$. I want to prove existence and uniqueness in $[0,\epsilon)$. I tried to do this for the simpler case $$\frac{dy}{dr}=1-y^2-\frac{y}{r}$$ but even here I don't know how to deal with the $1/r$. Any hints?

Answer 1

Hint.

Making the variable change

$$ y = \frac{z'}{z}\Rightarrow r z''+z'-r z = 0 $$

which is linear. All this for $r z \ne 0$