Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities

In this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function \begin{document}$ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}u+|u|^{p-2}u, \end{equation*} $\end{document} where $ u\in H^2(\mathbb{R}^N)(4 < N < 8) $, $ \lambda > 0, 1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4)) $, $ f(x) $ satisfy suitable conditions, and $ f(x) $ may change sign in $ \mathbb{R}^N $. Using Nehari manifold and fibering maps, the existense of multiple solutions is established. Moreover, the existence of sign-changing solution is obtained for $ f(x)\equiv0 $. Our results generalize some recent results in the literature.