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, 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.