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

$${\mathrm{\Delta }}^{2}u-\left(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u+V(x)u=\lambda f(x)|u{|}^{q-2}u+|u{|}^{p-2}u,$$

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 $f(x)\equiv 0$. Our results generalize some recent results in the literature.