Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 2 / b / Solution

Suppose $u$ were positive somewhere. Since it tends to zero at infinity and to $-\infty$ at its vortex zero, it would attain a positive interior maximum away from the origin. At such a maximum the second-derivative test gives $\Delta u\le 0$, whereas the smooth Taubes equation givesThis contradiction with the maximum principle for subharmonic functions provesConsequently the Higgs magnitude of a vortex satisfies $|\varphi |=e^{u/2}\le 1$ everywhere.