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.

We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions ${\mathbb{R}}^N(N⩾4)$. In particular, in dimension $N=4$, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $2+\frac{8}{N}$ is equal to the energy critical exponent $\frac{2N}{N-2}$, which leads to the fact that the equation no longer has a variational structure in dimensions $N⩾4$ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.

This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $(-\mathrm{\Delta }{)}^s$

$$\{\begin{array}{ll}\left(1+\alpha {\int }_{{\mathbb{R}}^3}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u{|}^{2}dx\right)(-\mathrm{\Delta }{)}^{s}u+\mu K(x)u=g(x)|u{|}^{p-2}u,& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ u\in H^s({\mathbb{R}}^3),\end{array}$$

where $\alpha ,\mu >0$, $s\in [\frac{3}{4},1)$, $2<p<4$. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on $K$ and $g$.