I am trying to recalculate 'Stability of Matter' Paper from Lieb.

I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem so far.

I have the following functional to be minimized on $\rho\in L^1(\mathbb R^d)$. Here $\lambda$ is a Lagrange multiplier and $\rho\geq 0$.

$h(\rho) = \frac{1}{C_d}\left(\int_{\mathbb R^d} dx \rho(x)^\frac{d}{d-2}\right)^\frac{d-2}{d} - Z \int_{\mathbb R^d} dx \frac{\rho (x)}{|x|} +\lambda\left( \int_{\mathbb R^d} dx \rho(x) -1\right)$

My idea is the following:

$\frac{\text{d}}{\text{d}t}h(\rho_m +t \eta)|_{t=0} =0$, where $\rho_m$ is the minimizer and $\eta\in C_0^\infty(\mathbb R^d)$ arbitrary.

Then I obtain the following:

$0 = \int dx \,\eta(x) \left(\frac{1}{C_d}\|\rho_m\|_\frac{d}{d-2}^\frac{-2}{d-2}\rho_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda\right)$

Then by the fundamental lemma of variations it follows that:

$\frac{1}{C_d}\|\rho_m\|_\frac{d}{d-2}^\frac{-2}{d-2} \rho_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda = 0$

Now I use that $\rho_m\geq 0$ and therefore $\frac{Z}{|x|} - \lambda\geq 0$. Since one also sees that $\lambda$ has to be bigger or equal 0 as otherwise $\rho_m$ wouldn't be in $L^1$. I conclude that $\rho_m$ has the following form:

$\rho_m (x) =\begin{cases} C_d^\frac{d-2}{2} \|\rho_m\|_\frac{d}{d-2} \left(\frac{Z}{|x|} - \lambda \right)^\frac{d-2}{2} & \text{ if} |x|\leq \frac{Z}{\lambda} \\ 0 & \text{ if} |x|> \frac{Z}{\lambda} \end{cases}$

Now comes the point of my concerns: This $\rho_m$ has compact support and therefore cannot really satisfy $0 = \int dx \,\eta(x) \left(\frac{1}{C_d}\|\rho_m\|_\frac{d}{d-2}^\frac{-2}{d-2}\rho_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda\right)$. Is there a reason why this $\rho_m$ is however the correct minimizer?

Best wishes :)