Energy functionals of Kirchhoff-type problems having multiple global minima

In this paper, using the theory developed in Ricceri (2012), we obtain some results of a totally new type about a class of non-local problems. Here is a sample:Let Ω⊂Rn be a smooth bounded domain, with n≥4n≥4, let a,b,ν∈R, with a≥0a≥0 and b>0b>0, and let p∈]0,n+2n−2[.Then, for each λ>0λ>0 large enough and for each convex set C⊆L2(Ω)C⊆L2(Ω) whose closure in L2(Ω)L2(Ω) contains H01(Ω), there exists v∗∈Cv∗∈C such that the problem {−(a+b∫Ω|∇u(x)|2dx)Δu=ν|u|p−1u+λ(u−v∗(x))in  Ωu=0on  ∂Ω has at least three weak solutions, two of which are global minima in H01(Ω) of the corresponding energy functional.