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