# Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$

## TL;DR

This paper proves the existence of infinitely many sign-changing solutions for a class of Kirchhoff type problems in -dimensional space, using variational methods, even when the nonlinearity is not 4-superlinear at infinity.

## Contribution

It introduces a novel approach combining invariant sets and minimax methods to find multiple sign-changing solutions for Kirchhoff problems with less restrictive nonlinear growth conditions.

## Key findings

- Established existence of infinitely many sign-changing solutions.
- Extended results to nonlinearities with subcritical growth and non-4-superlinear behavior.
- Applied variational methods to a class of problems with nonlocal terms.

## Abstract

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is continuous and coercive. For the case when $f$ is odd in $u$ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity $|u|^{p-2}u$ with $p\in(2,4]$.