On the number of critical points of solutions of semilinear equations in R^2

In this paper we construct families of bounded domains Ωε and solutions uε of ( −∆uε = 1 in Ωε uε = 0 on ∂Ωε such that, for any integer k ≥ 2, uε admits at least k maximum points for small enough . The domain Ωε is “not far” to be convex in the sense that it is starshaped, the curvature of ∂Ωε vanishes at exactly two points and the minimum of the curvature of ∂Ωε goes to 0 as ε → 0