= 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 $\Delta u\leq0$, whereas the smooth Taubes equation gives
$$
\Delta u=e^u-1>0.
$$
This contradiction with the <maximum principle for subharmonic functions> proves
$$
\boxed{u\leq0\quad\text{on }\mathbb R^2}.
$$
Consequently the Higgs magnitude of a vortex satisfies $|\phi|=e^{u/2}\leq1$ everywhere.
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