KMS Of Academy of mathematics and systems sciences, CAS
Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications | |
Liu, ZL; Li, SJ | |
2003 | |
Source Publication | NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS |
ISSN | 1021-9722 |
Volume | 10Issue:2Pages:133-170 |
Abstract | Assume that I is the functional defined on the Hilbert space H-0(1)(Omega) concerning the problem: -Deltau = f(u) in Omega and u = 0 on partial derivativeOmega, where f is sublinear at infinity and superlinear at 0, that is, lim sup(\t\ --> infinity) f (t)/t < lambda(1), lim inf(t-->0)f(t)/t > lambda(1), and lambda(1), is the first eigenvalue of -Delta in H-0(1)(Omega). Under very general conditions, I has at least two local minimizers u(1) and u(2) and one mountain pass point u(3), and max{I(u(1)), I(u(2))} < I(u(3)) < 0. Assuming that u(1), u(2) and u(3) are the only three nontrivial critical points of I, we prove that the level set I-b is contractible for all b greater than or equal to I (u(3)). Using this conclusion, we extend one of Hofer's result concerning existence of four nontrivial solutions of the above problem to the case where I is not C-2 and the trivial critical point 0 may be degenerate. Since I is not C-2, the local topological degree and the critical groups Of u(3) can not be clearly computed. The lack of topological information about 0 and u(3) makes it impossible to use topological degree theory or Morse theory in obtaining the fourth nontrivial solution. To overcome these difficulties, we explore a new technique in this paper. 2000 Mathematics Subject Classification: 35J20, 35J25. |
Keyword | critical point elliptic boundary value problem jumping nonlinearity four nontrivial solutions contractibility of level set |
DOI | 10.1007/S00030-003-1016-y |
Language | 英语 |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied |
WOS ID | WOS:000183520600001 |
Publisher | BIRKHAUSER VERLAG AG |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/18167 |
Collection | 中国科学院数学与系统科学研究院 |
Affiliation | 1.Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China 2.Acad Sinica, Acad Math & Syst Sci, Beijing 100080, Peoples R China |
Recommended Citation GB/T 7714 | Liu, ZL,Li, SJ. Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications[J]. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS,2003,10(2):133-170. |
APA | Liu, ZL,&Li, SJ.(2003).Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications.NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS,10(2),133-170. |
MLA | Liu, ZL,et al."Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications".NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 10.2(2003):133-170. |
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