A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem

We study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of Kirchhoff type{−(a+b∫Ω|∇u|2dx)Δu=λf(u)in Ω,u|∂Ω=0. Here, Ω   is a bounded open set in RnRn with smooth boundary, a, b, λ   are positive real numbers and f:R→Rf:R→R is a continuous function. In particular, we give an answer to an open problem recently proposed by B. Ricceri.