Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball
Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball
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球内指数增长椭圆方程分岔图的分类
DOI：
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发表时间：
2014
影响因子：
1
通讯作者：
Yasuhito Miyamoto
中科院分区：
文献类型：
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作者：
Yasuhito Miyamoto
让B⊂RNDocumentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathssy}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$Bsubset mathbb{R}^N$$end{Document}，N≥3DocumentClass[12pt]{Minimum}usepackage{amsmax}usepackage{amssymm}usepackage{amssymb}usepackage{maamssy}usepackage{upgresfs}usepackage{-dargemin{egpt}in$$nge$end}，单位长度为。我们研究了Δu+λf(U)=0inB，u=0 on∂B，u>；0inB，DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amssymb}usepackage{amsbsy}usepackage{upgreek}集长度{oddsidemarin}{-69pt}例如{∂}$egin{egin}{ll}Delta u+lambda f(U)=0&{}quad{in B；，\u=0&{}quad ext{on Partial}B，\u>；0&{}四进制数{in}B，结束{数组}
好的。}end{对准}$$end{Document}其中f(U)=EU+g(U)Documentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{amsbsy}usepackage{masbsy}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{document}$f(U)=e^u+g(U)$$end{DocumentClass[12pt]{Minimum}usepackage{usepackage{wa syysym}usepackage{amsssy}usepackage{amsb}usepackage{odsidemargin}{-69pt}例如{文档}$$f(U)=e^u+g(U)$$end{DocumentClass[12pt]{Minimum}usepackage{usepackage{wassym}usepackage{amsssy}usepackage{amsb}usepackage{dssidemar}setupackek{dsackack}{-69pt}例如{Document}$$g(U)$$end{Document}是一个较低阶项。解集是一个曲线CDocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amssymb}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{∞}$$mathcal{C}$-end{Document}在L的文档中参数化为[12pt]{Minimum}usepackage{amessym}usepackage{amsfonts}usepackage{amssymb}usepackage{maamsbsy}usepackage{masek}usepackage{upgreek}{setdsidemin{egpt}{egpt}L}参数化为解决方案的L文档类[12pt]{Minimum}usepackage{amssym}usepackage{amsfonts}usepackage{amssymb}usepackage{maamsbsy}usepackage{upsek}{setdsidemin{-end{Document}。我们证明了这个问题有奇异解(λ∗，u∗)DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$(lambda^*，U^*)$$end{Document}，并且曲线CDocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}中的{Document}$$mathcal{C}$$end{Document}围绕λ∗文档类[12pt]{minimum}usepackage{amsmax}usepackage{asysym}usepackage{amsbsy}usepackage{amsbsy}usepackage{madsidemargin}{-69pt}中有无限多个转折点}$$lambda^*$$end{Document}if 3≤N≤9DocumentClass[12pt]{Minimum}usepackage{amsmath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如在{Document}$$3le NLE 9$$End{Document}中。我们证明了在gDocumentclass[12pt]{minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$g$$end{document}上的特定条件下，如果N≥10DocumentClass[12pt]{最小}usepackage{waysym}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}egin{Document}$$mathcal{C}$end{Document}中的N DocumentClass[12pt]{Minimum}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{upsthorek}setsidememin$69pt}{Document}中的N DocumentCLASS[12pt]{Minimum}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{upstrek}setododemage{69pt}中的{Document}没有转折点。我们还研究了u∗文档类[12pt]{minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$u^*$$end{Document}的莫尔斯指数。
Let B⊂RNdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Bsubset mathbb {R}^N$$end{document}, N≥3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Nge 3$$end{document}, be the unit ball. We study the global bifurcation diagram of the solutions of Δu+λf(u)=0inB,u=0on∂B,u>0inB,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} {left{ egin{array}{ll} Delta u+lambda f(u)=0 &{}quad ext {in} B,\ u=0 &{} quad ext {on} partial B,\ u>0 &{} quad ext {in} B, end{array}
ight. } end{aligned}$$end{document}where f(u)=eu+g(u)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$f(u)=e^u+g(u)$$end{document} and g(u)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$g(u)$$end{document} is a lower order term. The solution set is a curve Cdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {C}$$end{document} parametrized by the L∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$L^{infty }$$end{document}-norm of the solution. We show that this problem has the singular solution (λ∗,u∗)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(lambda ^*,u^*)$$end{document} and that the curve Cdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {C}$$end{document} has infinitely many turning points around λ∗documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$lambda ^*$$end{document} if 3≤N≤9documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$3le Nle 9$$end{document}. We show that under a certain condition on gdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$g$$end{document}, the curve Cdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {C}$$end{document} has no turning point if N≥10documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Nge 10$$end{document}. We also study the Morse index of u∗documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$u^*$$end{document}.