  In this paper, one of our aims is to investigate the instability of the
distribution of the critical point set $\mathcal{C}(u)$ of a solution $u$ to a
semilinear equation with Dirichlet boundary condition in the planar annular
domains. Precisely, we prove that $\mathcal{C}(u)$ in an eccentric circle
annular domain, or a petal-like domain, or an annular domain where the interior
and exterior boundaries are equally scaled ellipses contains only finitely many
points rather than a Jordan curve. This result indicates that the critical
point set $\mathcal{C}(u)$ is unstable when any boundary of planar concentric
circle annular domain $\Omega$ has some small deformation or minor
perturbation. Based on studying the distribution of the nodal sets
$u^{-1}_\theta(0)(u_\theta=\nabla u\cdot \theta)$ and $u^{-1}(0)$, we prove
that the solution $u$ on each symmetric axis has exactly two critical points
under some conditions. Meanwhile, we further obtain that $\mathcal{C}(u)$ only
has two critical points in an eccentric circle annular domain, has four
critical points in an exterior petal-like domain with the exterior boundary
$\gamma_E$ is an ellipse, and the maximum points are distributed on the long
symmetric semi-axis and the saddle points on the short symmetric semi-axis.
Moreover, we describe the geometric location of critical points of the solution
$u$ by the moving plane method.
