= Solution
Let $\chi$ be the angle between $\widehat{\mathbf p}$ and $\widehat{\mathbf n}$. Then
$$
p_iQ_{ij}p_j
=\lambda p^2(2\cos^2\chi-1)
=\lambda p^2\cos2\chi.
$$
For $\zeta,\lambda>0$, the coupling $-\zeta\lambda p^2\cos2\chi/2$ is minimized by $\cos2\chi=1$, so
$$
\boxed{\widehat{\mathbf p}=\pm\widehat{\mathbf n}}.
$$
The uniform terms depending on $p$ reduce to
$$
\frac12(a-\zeta\lambda)p^2+\frac b4p^4.
$$
Their nonzero minimum is
$$
\boxed{p^2=-\frac{a-\zeta\lambda}{b}}.
$$
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