Existence of Solutions for a Class of Kirchhoff-Type Equations with Indefinite Potential

This study explores the existence of solutions to the following Kirchhoff-type problem\[\left\{\begin{array}[c]{ll}%-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\u\in H^1(\mathbb{R}^3),%\end{array} %\right.\]where a and b are postive constants,...

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Bibliographic Details
Published inElectronic Journal of Applied Mathematics Vol. 2; no. 3; pp. 42 - 50
Main Authors Xiao, Linlian, Yuan, Jiaqian, Zhou, Jian, Wu, Yunshun
Format Journal Article
LanguageEnglish
Published 09.09.2024
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Summary:This study explores the existence of solutions to the following Kirchhoff-type problem\[\left\{\begin{array}[c]{ll}%-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\u\in H^1(\mathbb{R}^3),%\end{array} %\right.\]where a and b are postive constants, and the potential \(V(x)\) is continuous and indefinite in sign. With suitable assumptions on \(V(x)\) and \(f\), we establish the existence of solutions using the Symmetric Mountain Pass Theorem.
ISSN:2980-2474
2980-2474
DOI:10.61383/ejam.20242368