Let $p>3/2 $ and define the energy functional $\mathcal E$ by $$\mathcal E(\psi) := \int_{\mathbb R^3} G \lvert \psi(x) \rvert^p - Z \frac{\lvert \psi(x) \rvert^2}{\lvert x \rvert} \, dx \quad \forall \psi \in H^1(\mathbb R^d),$$ where $G, Z $ are positive constants. Determine the minimizer of the variational problem $$E_0 := \inf\{\mathcal E(\psi): \lVert \psi \rVert _2 = 1\}.$$
I don't know how to properly start doing this. I know that I can carry out an ansatz like $$\frac{d}{dt}(\psi_0 + t f) = 0$$ if $\psi_0$ is a minimizer, but I wasn't able to use this here in a nice way. I would really appreciate a nice explanation on how to work with this kind of problem.