Energy functionals of Kirchhoff-type problems having multiple global minima

TL;DR

This paper investigates a class of non-local Kirchhoff-type problems with multiple solutions, demonstrating the existence of at least three weak solutions including two global minima, for large parameters and specific convex sets.

Contribution

It introduces new results on the existence of multiple solutions for non-local Kirchhoff problems using advanced variational methods and energy functional analysis.

Findings

Existence of at least three weak solutions for large λ

Two solutions are global minima of the energy functional

Solutions exist under convex set conditions in L^2()

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.