I'm reading the paper by Giorgio Talenti on the best constant for the Sobolev inequality.
The Theorem states that for $u:\mathbb{R}^m\rightarrow \mathbb{R}$ sufficiently smooth (eg. Lipschitz) and decaying fast enough at $\infty$, with $1<p<m$, $q=\frac{mp}{m-p}$ we have
$$||u||_q\leq C||Du||_p$$
where $C=\pi^{-1/2}m^{-1/2}(\frac{p-1}{m-p})^{1-1/p} \left(\frac{\Gamma(1+m/2)\Gamma(m)}{\Gamma(m/p)\Gamma(1+m-m/p)}\right)^{1/m}$ and equality holds if $u(x)=\left(a+b|x|^{\frac{p}{p-1}}\right)^{1-m/p}$

In Lemma 1 it is shown that $u^* $, the radial rearrangement of u, satisfies $\frac{||u^* ||_q}{||Du^* ||_p}\geq\frac{||u||_q}{||Du||_p}$ and hence we can restrict to radial functions when we look for maximizers of this ratio.

By the Euler-Lagrange Equations we find that extremizers are a two-parameter family of functions:
$$\{\Phi(r)=a(1+br^{p'})^{1-m/p}: a,b>0\}$$
Where $p'=\frac{p}{p-1}$.

The problem of maximizing $\frac{(\int_0^{\infty} r^{m-1}|u|^q dr)^{1/q}}{(\int_0^{\infty} r^{m-1}|u'|^p dr)^{1/p}}$ under the conditions
$$u=o(r^{1-m/p})\ \textrm{as} \ r\rightarrow 0 \ \textrm{or} \ \infty \ \textrm{and} \ u'=o(r^{-m/p}) \ \textrm{as} \ r\rightarrow 0 \ \textrm{or} \ \infty$$
is equivalent to the problem of maximizing $\int_0^{\infty} r^{m-1}|u_1|^q dr$ under the condition that $$u_2'=r^{m-1}|u_1'|^p \ \textrm{satisfies} \ u_2(0)=0, u_2(\infty)=1 \ \textrm{and that} \ u_1(\infty)=0$$
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Using the extremizers of the original problem, we obtain exremizing pairs for the reformulated problem:
$$\Phi_1(r)=a(1+br^{p'})^{1-m/p},\ a,b>0$$
$$\Phi_2(r)=\int_0^r t^{m-1}|\Phi_1'(t)|^p dt=r^{m-1}\Phi_1(r)^p f(\frac{br^{p'}}{1+br^{p'}})$$

Where $f(\xi)=\frac{1}{p'}\frac{m-p}{p-1}^p\xi^p\int_0^1 (1-t)^{m/p'}(1-\xi t)^{-m}dt$ , i.p. $f$ maps $(0,1)$ injectively to $\mathbb{R}$.\\

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Now these pairs of extremizers form a vector field on $\mathbb{R}^3_1=\{r,u_1,u_2\in\mathbb{R}^3: r,u_1,u_2>0\}$:

Every pair $(\Phi_1,\Phi_2)$ gives a path $(0,\infty)\rightarrow \mathbb{R}^3_1$, $r\mapsto (r,\Phi_1(r), \Phi_2(r))$ and these paths are trajectories of a smooth vector field $X$ on $\mathbb{R}^3_1$:
$$X_0(r,u_1,u_2)=1, \ X_1(r,u_1,u_2)=-\frac{m-p}{p-1}\frac{u_1}{r}\xi, \ X_0(r,u_1,u_2)=r^{m-1}|X_1(r,u_1,u_2)|^p$$
where $\xi$ is the unique root in $(0,1)$ s.t. $f(\xi)=r^{p-m}u_1^{-p}u_2$

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To show that the extremizing pairs are indeed maximizers, the goal is to construct an exact differential $dW$, s.t. along any path $r\mapsto (r,u_1(r), u_2(r))$ which statisfies $u_2'(r)=r^{m-1}|u_1'(r)|^p$:
$$\int_0^{\infty}dW\geq\int_0^\infty r^{m-1}|u_1(r)|^q dr$$
\\
\\For any $(r,u_1,u_2)\in\mathbb{R}^3_1$ define a linear function:
$$\Psi_{r,u_1,u_2}(\xi_0,\xi_1\xi_2)=u_2'(r)\xi_0-\nabla W(r,u_1,u_2).(\xi_0,\xi_1\xi_2)^T$$
restricted to the cone of all directions issuing from $(r,u_1,u_2)$ s.t. $\xi_0>0$ and $\xi_0^{p-1}\xi_2=r^{m-1}|\xi_1|^p$.
\\We want $X(r,u_1,u_2)$  to be a critical point of $\Psi_{r,u_1,u_2}$ (i.e. the component of the gradient which is parallel to the cone vanishes).\\
This by Lagrange multiplier principle gives
$$\partial_rW(r,u_1,u_2)=r^{m-1}u_1^q+\lambda(p-1)r^{m-1}X_1|^p$$
$$\partial_{u_1}W(r,u_1,u_2)=\lambda p r^{m-1}X_1|^{p-1}$$
$$\partial_{u_2}W(r,u_1,u_2)=\lambda$$
where $\lambda$ is a $C^1$ function to be determined.

This is done by examining compatibility conditions of the above equations, I guess this means that we want the equations to hold for any $(r,u_1,u_2)$, but I really don't see how to derive the following:
$\left( \matrix{ 1 & 0 & -(p-1)(r^{m-1}|X_1|^p) \cr
0 & 1 & -p(r^{m-1}|X_1|^{p-1}) \cr
p(r^{m-1}|X_1|^{p-1}) & (p-1)(r^{m-1}|X_1|^p) & 0 \cr} \right)
\left( \matrix{ \partial_r\lambda \cr
\partial_{u_1}\lambda \cr
\partial_{u_2}\lambda \cr} \right) \=
\left( \matrix{ 0\cr
0\cr
q r^{m-1}u_1^{q-1}\cr} \right)-p\lambda
\left( \matrix{ X_1\partial_{u_2}\cr
-\partial_{u_2}\cr
\partial_r+X_1\partial_{u_1}\cr} \right)(r^{m-1}|X_1|^{p-1})$

[claimed compatability conditions][1].

The other thing which is unclear to me, is how this system for $\lambda$ is solved:
"Since the matrix on the lefthand side of has rank 2, we must impose orthogonality between the right-hand side and the eigenvectors of the transposed matrix, i.e." \
$$p\lambda\frac{\partial}{\partial_X}(r^{m-1}|X_1|^{p-1})=qr^{m-1}u_1^{q-1}$$


  [1]: https://i.sstatic.net/kUGXy.png

Sorry for the very long question and the formating.