High energy solutions of modified quasilinear fourth-order elliptic equation

doi:10.1186/s13661-018-0970-6

	High energy solutions of modified quasilinear fourth-order elliptic equation
	Wang,Xiujuan; Mao,Anmin; Qian,Aixia
刊名	Boundary Value Problems
	2018-04-12
卷号	2018 期号:1
关键词	Super-quadratic High energy solutions Sign-changing potential Fountain theorem 35J25 35J20 35J60 35J61
ISSN号	1687-2770
DOI	10.1186/s13661-018-0970-6
英文摘要	AbstractThis paper focuses on the following modified quasilinear fourth-order elliptic equation: {△2u?(a+b∫R3\|?u\|2dx)△u+λV(x)u?12△(u2)u=f(x,u),in?R3,u(x)∈H2(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} \triangle^{2}u-(a+b\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\triangle u+\lambda V(x)u-\frac{1}{2}\triangle(u^{2})u=f(x,u),& \mbox{in }\mathbb{R}^{3}, \\ u(x)\in H^{2}(\mathbb{R}^{3}), \end{cases} $$\end{document} where △2=△(△)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\triangle^{2}=\triangle(\triangle)$\end{document} is the biharmonic operator, a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}, b≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\geq 0$\end{document}, λ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda\geq 1$\end{document} is a parameter, V∈C(R3,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V\in C(\mathbb{R}^{3},\mathbb{R})$\end{document}, f(x,u)∈C(R3×R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,u)\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})$\end{document}. V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} and f(x,u)u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,u)u$\end{document} are both allowed to be sign-changing. Under the weaker assumption lim\|t\|→∞∫0tf(x,s)ds\|t\|3=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim_{ \vert t \vert \rightarrow\infty}\frac{\int^{t}_{0}f(x,s)\,ds}{ \vert t \vert ^{3}}=\infty$\end{document} uniformly in x∈R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in\mathbb{R}^{3}$\end{document}, a sequence of high energy weak solutions for the above problem are obtained.
语种	英语
出版者	Springer International Publishing
WOS记录号	BMC:10.1186/S13661-018-0970-6
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/410]
专题	中国科学院数学与系统科学研究院
通讯作者	Qian,Aixia
作者单位
推荐引用方式 GB/T 7714	Wang,Xiujuan,Mao,Anmin,Qian,Aixia. High energy solutions of modified quasilinear fourth-order elliptic equation[J]. Boundary Value Problems,2018,2018(1).
APA	Wang,Xiujuan,Mao,Anmin,&Qian,Aixia.(2018).High energy solutions of modified quasilinear fourth-order elliptic equation.Boundary Value Problems,2018(1).
MLA	Wang,Xiujuan,et al."High energy solutions of modified quasilinear fourth-order elliptic equation".Boundary Value Problems 2018.1(2018).