Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $\mathbb {R}^{N}$

Abstract

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where $\Delta ^{2}:=\Delta (\Delta )$ is the biharmonic operator, $a,b>0$ are constants, $N\le 7,$ $p\in (4,2_{*})$ for $2_{*}$ defined below, and $V(x)\in C(\mathbb {R}^{N},\mathbb {R})$. Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.