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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 PublicationBoundary Value Problems
ISSN1687-2770
Volume2019Issue:1
AbstractAbstractThis 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.
KeywordLiouville theorem Morse index Singular elliptic equation
DOI10.1186/s13661-019-1173-5
Language英语
WOS IDBMC:10.1186/s13661-019-1173-5
PublisherSpringer International Publishing
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/32528
Collection中国科学院数学与系统科学研究院
Affiliation1.
2.
Recommended Citation
GB/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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