Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 3 / 2 / Solution

Replacing a minimizer by its absolute value does not increase its gradient norm, so choose a nonnegative minimizer $P$. The Euler-Lagrange equation isfor a Lagrange multiplier $\mu$. Multiplication by $P$ and integration show thatso $\mu >0$. Set $P_{\eta }=\sqrt{\mu }P$. Then $P_{\eta }$ is nonzero, belongs to $H^1({\mathbb{R}}^2)$, and satisfies the trapped focusing cubic ground-state equationStandard elliptic regularity and the strong minimum principle for elliptic operators make the nonnegative solution positive.