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 $$u_2'=r^{m-1}|u_1'|^p \ \textrm{satisfies} \ u_2(0)=0, u_2(\infty)=1 \ \textrm{and that} \ u_1(\infty)=0$$ \ 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 $\mathbb{R}$.\

\ 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. $f(\xi)=r^{p-m}u_1^{-p}u_2$

\

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.

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."
$$p\lambda\frac{\partial}{\partial_X}(r^{m-1}|X_1|^{p-1})=qr^{m-1}u_1^{q-1}$$

Sorry for the very long question and the formating.

I'm reading the paper by Giorgio Talenti on the best constant for the Sobolev inequality. 
The main theorem states that for $u:\mathbb{R}^m\rightarrow \mathbb{R}$ sufficiently smooth (eg. Lipschitz) and fast enough decay at $\infty$ we have $$ \|u\|_q\leq C\|Du\|_p\quad 1<p<m,\; q=\frac{mp}{m-p} $$ 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: $$\big\{\Phi(r)=a(1+br^{p'})^{1-m/p}: a,b>0\big\}$$ 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

• $u=o(r^{1-m/p})$ as $r\rightarrow 0$ or $\infty$ and
• $u'=o(r^{-m/p})$ as $r\rightarrow 0$ or $\infty$

is equivalent to the problem of maximizing $$ \int_0^{\infty} r^{m-1}|u_1|^q dr $$ under the condition that

• $u_2'=r^{m-1}|u_1'|^p$ satisfies $u_2(0)=0, u_2(\infty)=1$ and that
• $u_1(\infty)=0$.

Using the extremizers of the original problem, we obtain exremizing pairs for the reformulated problem: $$ \begin{align} \Phi_1(r) &=a(1+br^{p'})^{1-m/p},\ a,b>0\\ \Phi_2(r) &=\int_0^r t^{m-1}|\Phi_1'(t)|^p dt=r^{m-1}\Phi_1(r)^p f\left(\frac{br^{p'}}{1+br^{p'}}\right) \end{align} $$ 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 $\mathbb{R}$. 
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 $(0,\infty)\rightarrow \mathbb{R}^3_1$, $r\mapsto \big(r,\Phi_1(r), \Phi_2(r)\big)$ and these paths are trajectories of a smooth vector field $X$ on $\mathbb{R}^3_1$: $$ \begin{cases} X_0(r,u_1,u_2)=1, \\ \\ X_1(r,u_1,u_2)=-\dfrac{m-p}{p-1}\dfrac{u_1}{r}\xi, \end{cases}\qquad X_0(r,u_1,u_2)=r^{m-1}|X_1(r,u_1,u_2)|^p $$ where $\xi$ is the unique root in $(0,1)$ s.t. $f(\xi)=r^{p-m}u_1^{-p}u_2$.
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 the Lagrange multiplier principle, gives $$ \begin{align} \partial_rW(r,u_1,u_2) &=r^{m-1}u_1^q+\lambda(p-1)r^{m-1}X_1|^p \\ \partial_{u_1}W(r,u_1,u_2) &=\lambda p r^{m-1}X_1|^{p-1} \\ \partial_{u_2}W(r,u_1,u_2) &=\lambda \end{align} $$ where $\lambda$ is a $C^1$ function to be determined.

Questions.
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: $$ \begin{pmatrix} 1 & 0 & -(p-1)(r^{m-1}|X_1|^p) \cr 0 & 1 & -p(r^{m-1}|X_1|^{p-1}) \cr p(r^{m-1}|X_1|^{p-1}) & (p-1)(r^{m-1}|X_1|^p) & 0 \cr \end{pmatrix} \begin{pmatrix} \partial_r\lambda \cr \partial_{u_1}\lambda \cr \partial_{u_2}\lambda \cr \end{pmatrix} \\ = \begin{pmatrix} 0\cr 0\cr q r^{m-1}u_1^{q-1}\cr \end{pmatrix} -p\lambda \begin{pmatrix} X_1\partial_{u_2}\cr -\partial_{u_2}\cr \partial_r+X_1\partial_{u_1}\cr \end{pmatrix}\big(r^{m-1}|X_1|^{p-1}\big) $$

claimed compatability conditions.

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."
$$p\lambda\frac{\partial}{\partial_X}(r^{m-1}|X_1|^{p-1})=qr^{m-1}u_1^{q-1}$$

Sorry for the very long question and the formating.