Write
$$
A=\|\nabla u\|_2^2,\qquad B=\|u\|_2^2,\qquad C=\|u\|_p^p.
$$
Differentiating $\log J(u+th)$ at $t=0$ for a real <space of test functions>[test function] $h$ gives
$$
\frac2A\int\nabla u\mathbin{\cdot}\nabla h
+\frac4{dB}\int uh-\frac pC\int u^{p-1}h=0.
$$
After <integration by parts>,
$$
\Delta u-\lambda u+\mu u^{1+4/d}=0,
\qquad
\lambda=\frac{2A}{dB}>0,\qquad
\mu=\frac{pA}{2C}>0.
$$
Rescaling the dependent and independent variables reduces this <Euler-Lagrange equation> to the ground-state equation for $Q$. The uniqueness of its positive radial solution and the equality cases in rearrangement show that all minimizers are
$$
u(x)=aQ(b(x-x_0)),
\qquad
a\in\mathbb C\setminus\{0\},\quad b>0,\quad x_0\in\mathbb R^d.
$$
This is the <classification of Weinstein-functional minimizers>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2 / 2 / 3 / Solution

Every minimizer of the <Weinstein functional> has the form
$$
u(x)=aQ(b(x-x_0)),
\qquad
a\in\mathbb C\setminus\{0\},\quad b>0,\quad x_0\in\mathbb R^d,
$$
where $Q$ is the positive radial <NLS ground state>. The <Euler-Lagrange equation> first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

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 Incoming links: Classification of Weinstein-functional minimizers