Energy functionals of Kirchhoff-type problems having multiple global minima

Abstract

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: Let $Ω\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b, ν\in {\bf R}$, with $a\geq 0$ and $b>0$, and let $p\in \left ] 0,{{n+2}\over {n-2}}\right [$. Then, for each $λ>0$ large enough and for each convex set $C\subseteq L^2(Ω)$ whose closure in $L^2(Ω)$ contains $H^1_0(Ω)$, there exists $v^*\in C$ such that the problem $$\cases {-\left ( a+b\int_Ω|\nabla u(x)|^2dx\right )Δu =ν|u|^{p-1}u+λ(u-v^*(x)) & in $Ω$\cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)$ of the corresponding energy functional.