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 Liouville type theorem for a singular elliptic equation with finite Morse index Xiu,Zonghu1; Zhao,Jing1; Chen,Jianyi1; Yang,Hongwei2 2019-03-19 Source Publication Boundary Value Problems ISSN 1687-2770 Volume 2019Issue:1 Abstract AbstractThis paper considers the nonexistence of solutions for the following singular quasilinear elliptic problem: 0.1{?div(|x|?ap|?u|p?2?u)=f(|x|)|u|r?1u,x∈R+N,|x|?ap|?u|p?2?u?ν=g(|x|)|u|q?1u,on??R+N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{aligned} \textstyle\begin{cases} -\operatorname{div} ( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= f( \vert x \vert ) \vert u \vert ^{r-1}u, \quad x\in {\mathbb {R}} ^{N}_{+}, \\ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\frac{\partial u}{\partial \nu }=g( \vert x \vert ) \vert u \vert ^{q-1}u, \quad \text{on } \partial {\mathbb {R}} ^{N}_{+}, \end{cases}\displaystyle \end{aligned} \end{document} where R+N={x=(x′,xN)|x′∈RN?1,xN>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb {R}} ^{N}_{+}=\{x=(x',x_{N})| x'\in {\mathbb {R}} ^{N-1}, x_{N}>0 \}$\end{document} and ?R+N={x=(x′,xN)|x′∈RN?1,xN=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial {\mathbb {R}} ^{N}_{+}=\{x=(x',x_{N})| x'\in {\mathbb {R}} ^{N-1}, x_{N}=0\}$\end{document}. When the weight functions satisfy some suitable assumptions, we prove that problem (0.1) has no nontrivial bounded solutions with finite Morse index. Keyword Liouville theorem Morse index Singular elliptic equation DOI 10.1186/s13661-019-1173-5 Language 英语 WOS ID BMC:10.1186/s13661-019-1173-5 Publisher Springer International Publishing Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/32528 Collection 中国科学院数学与系统科学研究院 Corresponding Author Xiu,Zonghu Affiliation 1.2. Recommended CitationGB/T 7714 Xiu,Zonghu,Zhao,Jing,Chen,Jianyi,et al. Liouville type theorem for a singular elliptic equation with finite Morse index[J]. Boundary Value Problems,2019,2019(1). APA Xiu,Zonghu,Zhao,Jing,Chen,Jianyi,&Yang,Hongwei.(2019).Liouville type theorem for a singular elliptic equation with finite Morse index.Boundary Value Problems,2019(1). MLA Xiu,Zonghu,et al."Liouville type theorem for a singular elliptic equation with finite Morse index".Boundary Value Problems 2019.1(2019).
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