  In this paper we are concerned with the number of critical points of
solutions of nonlinear elliptic equations. We will deal with the case of
non-convex, contractile and non-contractile planar domains. We will prove
results on the estimate of their number as well as their index. In some cases
we will provide the exact calculation. The toy problem concerns the multi-peak
solutions of the Gel'fand problem, namely $$\begin{cases}
  -\Delta u=\lambda e^{u}&\mbox{ in }\Omega
  u=0 & \mbox{ on }\partial\Omega, \end{cases} $$ where
$\Omega\subset\mathbb{R}^2$ is a bounded smooth domain and $\lambda>0$ is a
small parameter.
