Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity - Wen Guan; Hua-Bo Zhang (Journal of Inequalities and Applications, 2021)
Abstract The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: 0.1 Δ 2 u − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = K ( x ) f ( u ) in R N , $$ Δ ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) Δ u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$ where 5 ≤ N ≤ 7 $5≤ N≤ 7$ , Δ 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.