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 | 2 d x ) Δ u = λ f ( u ) in Ω , u | ∂ Ω = 0 . Here, Ω is a bounded open set in R n with smooth boundary, a, b, λ are positive real numbers and f : R → R is a continuous function. In particular, we give an answer to an open problem recently proposed by B. Ricceri.