# Existence and Multiplicity of Solutions for Nonhomogeneous Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations in RN

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2019-08-19

AUTHORS

Jiabin Zuo, Tianqing An, Yuanfang Ru, Dafang Zhao

ABSTRACT

In this article, we study the following nonhomogeneous Schrödinger–Kirchhoff-type equation Δ2u-(a+b∫RN|∇u|2dx)Δu+V(x)u=f(x,u)+h(x)inRN,u∈H2(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\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}\end{document}where a>0,b≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0,b\ge 0$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x)=0$$\end{document}. Moreover, we also show infinitely many radial solutions of this equation. More... »

PAGES

123

### References to SciGraph publications

• 1996. Minimax Theorems in NONE
• ### Journal

TITLE

Mediterranean Journal of Mathematics

ISSUE

5

VOLUME

16

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2

DOI

http://dx.doi.org/10.1007/s00009-019-1402-2

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1120409935

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