= Solution
The <Derrick scaling of the Abelian Higgs energy> must rescale the <gauge potential> as well as the <Higgs field>. Set
$$
\Phi_\lambda(x)=\Phi(\lambda x),\qquad A_{j,\lambda}(x)=\lambda A_j(\lambda x).
$$
Then $B_\lambda(x)=\lambda^2B(\lambda x)$ and
$$
(\partial_j-iA_{j,\lambda})\Phi_\lambda(x)=\lambda(D_j\Phi)(\lambda x).
$$
Let $E_B=\int B^2$, $E_D=\int|D\Phi|^2$ and $E_P=\frac14\int(1-|\Phi|^2)^2$, with all integrals over the plane. Changing variables gives
$$
V(\lambda)=\lambda^2E_B+E_D+\lambda^{-2}E_P.
$$
At a static solution, $V'(1)=2E_B-2E_P=0$. Consequently
$$
\boxed{\int_{\mathbb R^2}B^2\,d^2x=\frac14\int_{\mathbb R^2}(1-|\Phi|^2)^2\,d^2x.}
$$
The covariant-gradient contribution is scale invariant, while the magnetic and potential terms balance. This is why the scalar-only nonexistence conclusion does not extend to the <Abelian Higgs model>.
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