= Solution
Replacing a minimizer by its absolute value does not increase its gradient norm, so choose a nonnegative minimizer $P$. The <Euler-Lagrange equation> is
$$
-\Delta P+P+\frac\eta4|x|^2P=\mu P^3
$$
for a <Lagrange multiplier> $\mu$. Multiplication by $P$ and integration show that
$$
\mu M=\int|\nabla P|^2+\int|P|^2+\frac\eta4\int|x|^2|P|^2>0,
$$
so $\mu>0$. 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 equation>
$$
\boxed{\Delta P_\eta-P_\eta-\frac\eta4|x|^2P_\eta+P_\eta^3=0}.
$$
Standard elliptic regularity and the <strong minimum principle for elliptic operators> make the nonnegative solution positive.
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