Minimizzare funzionale integrale

[franzu]

I want to minimize the functional \[F(\rho)=(\int_{\mathbb{R}^{3}}\rho^{5/3}d x)^{p/2}\int_{\mathbb{R}^{3}}|x|^{p}\rho dx\] ($p>0$) for \(L^{1}\)-normalized functions. Then I considered the minimization problem \[G(\rho)=(\int_{\mathbb{R}^{3}}\rho^{5/3}dx)^{p/2}\int_{\mathbb{R}^{3}}|x|^{p}\rho dx-\lambda\Bigl(\int_{\mathbb{R}^{3}}\rho dx-1\Bigr)\]

with no condition on \(\rho\). I have found that the minimum is \[\frac{6(5p)^{p/2}}{(5p+6)^{p/2+1}}\Bigl(\frac{p}{4\pi}\frac{\Gamma(3/p+5/2)}{\Gamma(3/p)\Gamma(5/2)}\Bigr)^{p/3}\] (here \(\Gamma\) denotes the gamma function).
Can anyone confirm that? I need to know if I am correct. If I am wrong I will write my attempt of proof, to see where is the problem (in all honesty i am not 100% about it).
Thank you all for the answers

I will briefly descrive the method i used. I took \(\phi\) smooth and compactly supported and \(\varepsilon\ge 0\). And considered \(G(\rho+\varepsilon\phi)-G(\rho)\). I linearized the non-linear part keeping only terms of order one in \(\varepsilon\). Imposing the vanishing of derivative with respect to \(\varepsilon\) leads to\[\int_{\mathbb{R}^{3}}dx\phi(x)\biggl(\frac{5}{6}p\Bigl(\int_{\mathbb{R}^{3}}dy\rho(y)^{5/3}\Bigr)^{p/2-1}\int_{\mathbb{R}^{3}}dy|y|^{p}\rho(y)\,\rho^{2/3}(x)+\Bigl(\int_{\mathbb{R}}dy\rho^{5/3}(y)\Bigr)^{p/2}|x|^{p}-\lambda\biggl)=0\].

The arbitrary choice of \(\phi\) implies that \[\rho=\Bigl(\frac{6}{5p||\,|x|^{p}\rho||_{1}}\Bigr)^{3/2}||\rho||_{5/3}^{5/2}[\lambda||\rho||_{5/3}^{-5p/6}-|x|^p]_{+}^{3/2}\]

(\([\cdot]_{+}\) denotes the positive part). So i took a function of the form \[\rho=[a-b|x|^p]_{+}^{3/2}\] with \(a\) and \(b\) constant, i imposed the normalizatin condition and hence reducing to just one parameter (that i chose to be \(a\)). I then plugged the function in the functional, in order to express the functional in terms of \(a\), and then imposing \(\frac{dG}{da}=0\) to find the optimal \(a\). But when I introduce \(\rho\) in the functional, \(a\) cancels out. So, i find an optimal value for the functional, but \(\rho\) seems to be optimal whenever has the form above and is normalized. This is my main concern with my attempt.

(Potete rispondere in italiano, avevo giá scritto tutto in inglese e mi sembrava poco utile ritraddure)
Eventualmente mi va bene anche che qualcuno risolva il problema numericamente e mi confermi/smentisca
EDIT: i forgot to say that \(\rho\ge 0\)