Energy functionals of Kirchhoff-type problems having multiple global minima

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample:par Let $Omegasubset {f R}^n$ be a smooth bounded domain, with $ngeq 4$, let $a, b, uin {f R}$, with $ageq 0$ and $b>0$, and let $pin left ] 0,{{n+2}over {n-2}} ight [$.par Then, for each $lambda>0$ large enough and for each convex set $Csubseteq L^2(Omega)$ whose closure in $L^2(Omega)$ contains $H^1_0(Omega)$, there exists $v^*in C$ such that the problem $$cases {-left ( a+bint_{Omega}| abla u(x)|^2dx ight )Delta u = u|u|^{p-1}u+lambda(u-v^*(x)) & in $Omega$cr & cr u=0 & on $partialOmega$cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Omega)$ of the corresponding energy functional.