I was reading a proof from Evans' book of partial differential equations, until I got stuck.
Until $(6)$ everything is clear, but then he assumes that $v$ is radial and then the ODE completely changes shape. Furthermore, I don't get the successive passages, until $(7)$. Is there anyone that can explain them to me? It would be of great help!
The reason for this "guess" is the fact that the laplacian $\Delta$ is rotation invariant, i.e. $\Delta u(x)=\Delta u(Tx)$ for any rotation $T:\Bbb R^n\to\Bbb R^n$, so one can seek a radial (or rotation invariant) solution. He then lets $v(y)=w(|y|)$ and plugs this into the PDE, which reduces to the ODE, now we are at the following part $$r^{n-1}w'+\frac{1}{2}r^nw=a.$$ We now assume $\lim_{r\to\infty}w'=\lim_{r\to\infty}w=0$, this condition just ensures nice behaviour of the solution at infinity, but if we take this limit in the above equation we see that we must have $a=0$ for this to be true.
Now by rearranging we have a nice easy to solve ODE, $$\frac{w'}{w}=-\frac{1}{2}r.$$