I'm studying the Willem's book Minimax Theorems. And there is the following result (My question is on below):
Theorem 1.42(Aubin, Talenti, 1976) The instanton $$U(x):=\dfrac{[N(N-2)]^{(N-2)/4}}{[1+|x|^2]^{(N-2)/2}}$$
is a minimizer for $S:=\inf_{u\in D^{1,2}(R^N),|u|^{2^*}=1}|\nabla u|_2^2>0$.
Proof. 1) By the preceding theorem, there exists a minimizer $u\in D^{1,2}(R^N)$ for $S$. By theorem C.4, $u$ is radially symmetric. Replacing $u$ by $|u|$, we may assume that $u$ is non-negative.
- It follows from Lagrange multiplier rule that, for some $\lambda>0$, $u$ is a solution of $$-\Delta u=\lambda u^{\frac{N+2}{N-2}}$$.
By the argument of Lemma 1.30, $u\in C^2(R^N).$ The strong maximum principle implies that $u$ is positive.
- After scaling, we may assume $$-\Delta u= u^{\frac{N+2}{N-2}}$$
Moreover we can choose $\varepsilon>0$ such that $$U_\varepsilon(x):=\varepsilon^{(2-N)/2}U(x/\varepsilon)$$
satisfies $$U_{\varepsilon}(0)=u(0)$$.
But then $u$ and $U_{\varepsilon}$ are solutions of the problem $$\partial_r(r^{N-1}\partial_r v)=r^{N-1}v^{\frac{N+2}{N-2}}, r>0$$ $$v(0)=u(0), \partial_rv(0)=0$$.
It follows easily that $u=U_{\varepsilon}$. By invariance, $U$ is a minimizer for $S$.
So I'm stuck in proving that the function $u$ in fact satisfies the ODE $\partial_r(r^{N-1}\partial_r v)=r^{N-1}v^{\frac{N+2}{N-2}}$. Maybe I even do not know what that means. I think that since $u$ is radially symmetric then we can see $u$ as a function of $r$ where $r$ is the lenght of a vector ($\|x\|=r$). But how to differentiate such a function, using the hypotesis on $u$ and in fact show that such ODE is satisfied? How to interpret such ODE? Thanks in advance.