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

  • Jiabin ZuoEmail author
  • Tianqing An
  • Yuanfang Ru
  • Dafang Zhao


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(xu), 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.


Fourth-order elliptic equations Schrödinger Kirchhoff-type Fountain theorem symmetry mountain pass theorem 

Mathematics Subject Classification

35J20 35J65 35J60 



The authors thank the anonymous referees for invaluable comments and insightful suggestions which improved the presentation of this manuscript.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.School of Applied SciencesJilin Engineering Normal UniversityChangchunPeople’s Republic of China
  3. 3.College of ScienceChina Pharmaceutical UniversityNanjingPeople’s Republic of China
  4. 4.School of Mathematics and StatisticsHubei Normal UniversityHuangshiPeople’s Republic of China

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