It's clear that with the radial change of variables $r=\sqrt{x^{2}+y^{2}}$, the solutions of the form $\log\left(\frac{8b}{(1+\lambda b r^{2})^2}\right)$ solve the equation $-\Delta u=\lambda e^{u}$ with boundary conditions $u=0$ in $\partial B_{1}(0)\subset \mathbb{R}^{2}$, where $$b=\frac{4-\lambda\pm\sqrt{16-8\lambda}}{\lambda^2}.$$But I have to show that all the solutions of the PDE have that form. In the book I am following (Stable solutions of elliptic partial differential equations, L. Dupaigne, section 2.3, proposition 2.3.1) I am told that as the ODE $$-u''-\frac{N-1}{r}u'=\lambda e^{u}$$ has at most one solution, then it follows that all the solutions to $-\Delta u=\lambda e^{u}$ must have that logarithmic form. But I don't understand the reason of that ODE, just have no idea.
Any help with why that and why it gives the form of the solutions for the PDE? Thank you!