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 of functional $G$ 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 $\rho=\Bigl(\frac{6}{5p||\,|x|^{p}\rho||_{1}}\Bigr)^{3/2}||\rho||_{5/3}^{5/2}[\lambda||\rho||_{5/3}^{-5p/6}-|x|^p]_{+}^{3/2}$
($[\cdot]_{+}$ denote 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.
EDIT: i forgot to say that $\rho\ge 0$