Existence and Multiplicity of Solutions for Nonhomogeneous Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations in $\mathbb {R}^{N}$

Abstract

In this article, we study the following nonhomogeneous Schrödinger–Kirchhoff-type equation

$$\begin{aligned} \left\{ \begin{array}{cl} &{}\displaystyle \Delta ^{2}u-(a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm{d}x)\Delta u+V(x)u=f(x,u)+h(x)\;\; \text {in }\;\; \mathbb {R}^{N}, \\ &{}\displaystyle u \in H^{2}(\mathbb {R}^{N}), \end{array}\right. \end{aligned}$$

where $a>0,b\ge 0$. Under the suitable assumptions of V(x), f(x, u), and h(x), we prove the existence of nontrivial solution using the Mountain Pass Theorem. In addition, infinitely many high-energy solutions are obtained by two kinds of methods (i.e., Symmetry Mountain Pass Theorem and Fountain Theorem) when $h(x)=0$. Moreover, we also show infinitely many radial solutions of this equation.