Least energy sign-changing solutions for fourth-order Kirchhoff-type equation with potential vanishing at infinity

Abstract

In this paper, we study the following fourth-order Kirchhoff-type equation $$\begin{aligned} \Delta ^{2}u-\left( a+ b\int _{\mathbb {R}^{N}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=K(x)f(u), ~x~\text {in}\ \mathbb {R}^{N}, \end{aligned}$$ with the potential V(x) vanishing at infinity. Under suitable conditions, by using constraint variational method and the quantitative deformation lemma, we obtain a least energy sign-changing (or nodal) solution to this problem. Moreover, we prove that this least energy sign-changing solution has precisely two nodal domains.