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

Abstract 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 { − Δ 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 .