Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem

where $\partial^\pm\Omega$ represents the outer and inner boundaries of $\Omega$, respectively, $c$ is a constant. Let $\mu_k$ denote the $k$th Neumann eigenvalue of the Laplacian on $\Omega$ and $\Omega_h$ is the hole. We establish that if $\mu<\mu_8$, then $\Omega$ is an annulus.