= Solution
For nonzero $\lambda$, zero potential requires both <F-flatness> and <D-flatness>:
$$
\varphi_+\varphi_-=0,\quad\varphi_0\varphi_-=0,\quad
\varphi_0\varphi_+=0,\qquad |\varphi_+|=|\varphi_-|.
$$
The product condition and equality of charged magnitudes together force $\varphi_+=\varphi_-=0$. There is no condition on the neutral scalar. Consequently the global minima form
$$
\boxed{\varphi_+=\varphi_-=0,\qquad\varphi_0=v\in\mathbb C,\qquad V_{\min}=0.}
$$
Only the neutral field can acquire a <vacuum expectation value>. It does not give the gauge vector a mass: its scalar <kinetic term> has no charged covariant derivative. Hence \b[the gauged $U(1)$ remains unbroken in every global minimum for $\lambda\ne0$]. This is a <neutral flat direction with oppositely charged chiral fields>; a continuous vacuum family is not automatically gauge-symmetry breaking.
The exceptional uncoupled case $\lambda=0$ should be separated. Then only <D-flatness> remains, allowing $|\varphi_+|=|\varphi_-|=r$ and arbitrary $\varphi_0$. For $r>0$, the charged expectations Higgs the $U(1)$; for $r=0$ it remains unbroken. The usual interacting answer assumes $\lambda\ne0$.
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