On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains

TL;DR

This paper investigates the distribution and stability of critical points of solutions to semilinear elliptic equations in various planar annular domains, revealing finite critical points and their geometric arrangements under perturbations.

Contribution

It demonstrates that the critical point set is finite and unstable under boundary perturbations, and characterizes the exact number and location of critical points in different annular geometries.

Findings

Critical point set is finite in certain annular domains.

Solutions have exactly two critical points on symmetric axes under specific conditions.

Maximum and saddle points are distributed on symmetric semi-axes depending on domain shape.

Abstract

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.