The <NLS ground state> is the positive radial solution of
$$
-\Delta Q+Q-Q^3=0.
$$
It is, up to phase, translation, and scaling, the optimizer of the <Sharp Gagliardo-Nirenberg inequality>
$$
\|w\|_4^4
\leq\frac2{\|Q\|_2^2}\|w\|_2^2\|\nabla w\|_2^2.
$$
Consequently
$$
E(w)
\geq\frac12\left(1-\frac{\|w\|_2^2}{\|Q\|_2^2}\right)
\|\nabla w\|_2^2.
$$
Equality in the sharp inequality occurs precisely for the symmetry orbit of $Q$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 3 / 1 / Solution

{c}
{wiki=Gagliardo–Nirenberg_interpolation_inequality}

In two dimensions,
$$
\|u\|_4^4\leq
\frac2{\|Q\|_2^2}\|u\|_2^2\|\nabla u\|_2^2,
$$
and equality holds exactly on the phase, translation, and scaling orbit of the <NLS ground state> $Q$.

Sharp Gagliardo-Nirenberg inequality

{c}
{title2=$Q$}

For the two-dimensional focusing cubic nonlinear Schrödinger equation, the NLS ground state is the positive radial solution of
$$
-\Delta Q+Q-Q^3=0.
$$
Its symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.


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