  Let $\Omega\subset\mathbb{C}$ be a bounded domain. In this note, we use
complex variable methods to study the number of critical points of the function
$v=v_\Omega$ that solves the elliptic problem $\Delta v = -2$ in $\Omega,$ with
boundary values $v=0$ on $\partial\Omega.$ This problem has a classical flavor
but is especially motivated by recent studies on localization of
eigenfunctions. We provide an upper bound on the number of critical points of
$v$ when $\Omega$ belongs to a special class of domains in the plane, namely,
domains for which the boundary $\partial\Omega$ is contained in $\{z:|z|^2 =
f(z) + \overline{f(z)}\},$ where $f'(z)$ is a rational function. We furnish
examples of domains where this bound is attained. We also prove a bound on the
number of critical points in the case when $\Omega$ is a quadrature domain, and
conclude the note by stating some open problems and conjectures.
