A note on multiple solutions for Kirchhoff-type equations with a Neumann condition

TL;DR

This paper establishes a multiplicity theorem for a class of Kirchhoff-type equations with Neumann boundary conditions, utilizing a recent minimax inequality result to demonstrate the existence of multiple solutions.

Contribution

It introduces a new multiplicity theorem for Kirchhoff-type equations with Neumann conditions, based on a recent minimax inequality, expanding the understanding of solution multiplicity in such problems.

Findings

Proves the existence of multiple solutions for the Kirchhoff-type problem.

Utilizes a recent minimax inequality to establish the multiplicity theorem.

Provides a theoretical framework for analyzing nonlinear PDEs with nonlocal terms.

Abstract

Using as a main tool our recent result on the strict minimax inequality proved in [5], in this note we establish a multiplicity theorem for a problem of the type $$\cases{-K\left(\int_{\Omega}|\nabla u(x)|^2dx\right)\Delta u = h(x,u) & in $\Omega$\cr & \cr {{\partial u}\over {\partial\nu}}=0 & on $\partial\Omega$.\cr}$$