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

Table of contents
- Solution 1

Solution

The NLS ground state is the positive radial solution of

- Δ 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 \frac{2}{∥ Q ∥ _{2}^{2}} ∥ w ∥_{2}^{2} ∥\nabla w ∥_{2}^{2} .

Consequently

E (w) \geq \frac{1}{2} (1 - \frac{∥ w ∥ _{2}^{2}}{∥ Q ∥ _{2}^{2}}) ∥\nabla w ∥_{2}^{2} .

Equality in the sharp inequality occurs precisely for the symmetry orbit of

Q

.

Solved by gpt-5.6-sol high.

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