Determine infimum of the energy of the hydrogen atom By Heisenberg's uncertainty principle we can bound the energy of the hydrogen atom (with a wave function u: $\mathbb{R}^3 \rightarrow \mathbb{C}$) from below by
\begin{equation}  \mathcal{E}(u) = \frac{9}{4} \left(\int_{\Bbb{R}^3}\Vert\mathbf{x}\Vert^2\ \vert u(\mathbf{x}) \vert^{2}\ d^3\mathbf{x}\right)^{-1} - \int_{\mathbb{R}^{3}} \frac{ \vert u(\mathbf{x}) \vert^{2}}{\Vert\mathbf{x}\Vert}\ d^3\mathbf{x}  
\end{equation}
Determine
$$
\inf \left \{\mathcal{E} (u): u \in C_{c}^{1}(\mathbb{R}^3), \int_{\mathbb{R}^{3}} \vert u(\mathbf{x}) \vert^{2} d^3\mathbf{x} = 1\right\}
$$
I think the answer should be $-\frac{1}{4}$ but nothing I have tried so far seems to work out.
 A: I will elaborate on the answer given by achille hui in the comments of the OP.
The your functional is indeed unbounded, i.e. $\inf \{ \mathcal{E}(u)\} = - \infty$. This is what is usually meant by "the Heisenberg uncertainty is not enough to account for the stability of the hydrogen atom". See for example in this paper by Lieb: http://www.pas.rochester.edu/~rajeev/phy246/lieb.pdf
The basic argument goes as follows: Suppose $u(x)$ is spherically symmetric and suppose it has half of its weight concentrated in a thin shell in the vicinity of the origin, say at a radius $\epsilon$. The other half is concentrated far away from the origin in a thin shell at distance $R$. Then the first term 
$$\left(\int \left\lvert x \right\rvert^2 \left\lVert u(x) \right\rVert^2\right)^{-1} \sim (R^2)^{-1} \to 0,$$
for sufficiently large $R$, as the contribution from the shell at $R$ dominates the one from the other one at the origin. For the second term, it is the other way around: 
$$\int \frac{\left\lVert u(x) \right\rVert^2}{\left\lvert x \right\rvert} \sim \frac{1}{\epsilon} \to \infty $$ 
This implies, that $$\inf \{ \mathcal{E}(u)\} = - \infty.$$
If you want, you can explicitly construct such a function by using ever better approximations of dirac "functions", i.e. $$u(x) = \frac{1}{2}d(\left\lvert x \right\rvert - \epsilon) + \frac{1}{2}d(\left\lvert x \right\rvert - R),$$
where $d(\cdot)$ are such approximations. (Although, I guess you would have to smoothly massage them to $u(x) = 0$ for some arbitrary large $\left\lvert x \right\rvert = r$ and at the origin, such that $u$ is in the set of continously differentiable functions with compact support, $C^1_c(\mathbb{R}^3)$.)
Edit: Minor misspellings.
