Solution
= Solution
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 $Q$.