Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms
Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms
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节点气泡塔解决带有 Hardy 项的轻微亚临界椭圆问题
DOI：
10.1007/s42985-020-00029-9
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Qianqiao Guo
中科院分区：
文献类型：
--
作者：
T. Bartsch;Qianqiao Guo
We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term -Δu-μu|x|2=|u|2∗-2-εuinΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\mu \frac{u}{|x|^2} = |u|^{2^{*}-2-\varepsilon }u &{} \text {in } \Omega , \\ u = 0&{} \text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$\end{document}in a bounded domain Ω⊂RN(N≥7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N (N\ge 7)$$\end{document} with 0∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \Omega$$\end{document}, as μ,ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ,\epsilon \rightarrow 0^+$$\end{document}. In [6], we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ=O(ϵα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =O(\epsilon ^\alpha )$$\end{document} with α>N-4N-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >\frac{N-4}{N-2}$$\end{document}, ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0^+$$\end{document}. Here we prove the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as μ=O(ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =O(\epsilon )$$\end{document}, ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0^+$$\end{document}.
We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term -Δu-μu|x|2=|u|2∗-2-εuinΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\mu \frac{u}{|x|^2} = |u|^{2^{*}-2-\varepsilon }u &{} \text {in } \Omega , \\ u = 0&{} \text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$\end{document}in a bounded domain Ω⊂RN(N≥7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N (N\ge 7)$$\end{document} with 0∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \Omega$$\end{document}, as μ,ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ,\epsilon \rightarrow 0^+$$\end{document}. In [6], we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ=O(ϵα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =O(\epsilon ^\alpha )$$\end{document} with α>N-4N-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >\frac{N-4}{N-2}$$\end{document}, ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0^+$$\end{document}. Here we prove the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as μ=O(ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =O(\epsilon )$$\end{document}, ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0^+$$\end{document}.
DOI：
10.1007/s00526-017-1140-x
发表时间：
2017-04-01
影响因子：
2.1
作者：
Frank, Rupert L.;Lemm, Marius;Simon, Barry
通讯作者：
Simon, Barry