Positive and sign-changing solutions for Kirchhoff equations with indefinite potential

Yan-Fei Yang; Chun-Lei Tang; Yan-Fei Yang; Chun-Lei Tang

doi:10.3934/cam.2025008

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 159-187. doi: 10.3934/cam.2025008

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Research article

Positive and sign-changing solutions for Kirchhoff equations with indefinite potential

Yan-Fei Yang ,
Chun-Lei Tang ^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: 09 May 2024 Revised: 15 January 2025 Accepted: 07 February 2025 Published: 03 March 2025
35J20, 35J60

We deal with the nonlinear Kirchhoff problem
$ \begin{equation} \notag -\left( a+b\displaystyle {\int}_{\mathbb{R}^{3}}|\nabla u|^2\mathrm{d}x\right) \Delta u+ V(x)u = f(u), \ \ \ \ x \in \mathbb{R}^{3}, \end{equation}$
where $ a $ is a positive constant, $ b > 0 $ is a parameter, the potential function $ V $ is allowed to change its sign, and the nonlinearity $ f\in C(\mathbb{R}, \mathbb{R}) $ exhibits subcritical growth. Under some suitable conditions on $ V $, we first prove that the problem has a positive ground state solution for all $ b > 0 $. Then, by using a more general global compactness lemma and a sign-changing Nehari manifold, combined with the method of constructing a sign-changing $ ({PS})_c $ sequence, we show the existence of a least energy sign-changing solution for $ b > 0 $ that is sufficiently small. Moreover, the asymptotic behavior $ b\searrow 0 $ is established.
- Kirchhoff problem,
- positive solutions,
- sign-changing solutions,
- variational methods,
- indefinite potential
Citation: Yan-Fei Yang, Chun-Lei Tang. Positive and sign-changing solutions for Kirchhoff equations with indefinite potential[J]. Communications in Analysis and Mechanics, 2025, 17(1): 159-187. doi: 10.3934/cam.2025008

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Abstract

We deal with the nonlinear Kirchhoff problem

$ \begin{equation} \notag -\left( a+b\displaystyle {\int}_{\mathbb{R}^{3}}|\nabla u|^2\mathrm{d}x\right) \Delta u+ V(x)u = f(u), \ \ \ \ x \in \mathbb{R}^{3}, \end{equation}$

where $ a $ is a positive constant, $ b > 0 $ is a parameter, the potential function $ V $ is allowed to change its sign, and the nonlinearity $ f\in C(\mathbb{R}, \mathbb{R}) $ exhibits subcritical growth. Under some suitable conditions on $ V $, we first prove that the problem has a positive ground state solution for all $ b > 0 $. Then, by using a more general global compactness lemma and a sign-changing Nehari manifold, combined with the method of constructing a sign-changing $ ({PS})_c $ sequence, we show the existence of a least energy sign-changing solution for $ b > 0 $ that is sufficiently small. Moreover, the asymptotic behavior $ b\searrow 0 $ is established.

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