On the number of critical points of stable solutions in bounded strip-like domains

In this paper we show that there exists a family of domains Ω_e ⊂ R^N with N≥ 2, such that the stable solution of the problem -∆u= g(u) in Ω_e u>0 in Ω_e u=0 on ∂Ω_e admits k critical points with k≥ 2. Moreover the sets Ω_e are star-shaped and ``close'' to a strip as e→0. Next, if g(u)=1 and N≥ 3 we exhibit a family of domains Ω_e with positive mean curvature and solutions u_e which have k critical points with k≥ 2. In this case, the domains Ω_e turn out to be ``close'' to a cylinder as e→0.