On critical points of solutions of elliptic equations

In this thesis we deal with qualitative properties of solutions of the semilinear elliptic problem −∆u = f(u) in Ω u = 0 on ∂Ω, where Ω ⊆ R^N , N ≥ 2 is a smooth domain and f : R → R is a smooth function. A classical problem concerns the study of the shape of u related to the one of the domain. In particular we investigate the number of critical points of u with respect to the convexity of Ω. Both the cases of positive and sign-changing solutions are treated.