Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity

Abstract

The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: $$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$ Δ 2 u − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = K ( x ) f ( u ) in R N , where $5\leq N\leq 7$ 5 ≤ N ≤ 7 , $\Delta ^{2}$ Δ 2 denotes the biharmonic operator, $K(x), V(x)$ K ( x ) , V ( x ) are positive continuous functions which vanish at infinity, and $f(u)$ f ( u ) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.