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Multiplicity of positive solutions of a nonlinear Schrodinger equation
Ding, YH; Tanaka, K
2003-09-01
Source PublicationMANUSCRIPTA MATHEMATICA
ISSN0025-2611
Volume112Issue:1Pages:109-135
AbstractWe consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation: -Deltau + (lambdaa(x) + b(x))u = u(p), u > 0 in R-N, (P-lambda) where p is an element of (1, N+2/N-2) if N greater than or equal to 3 and p is an element of (1, infinity) if N = 1, 2, and a (x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Omega := int a(-1) (0) consisting of k components Omega(1),..., Omega(k) and the first eigenvalues of -Delta + b(x) on Omega(j) under Dirichlet boundary condition are positive for all j = 1, 2,..., k. Under these conditions we show that (P-lambda) has at least 2(k) - 1 positive solutions for large lambda. More precisely we show that for any given non-empty subset J subset of {1, 2,... k}, (P-lambda) has a positive solutions u(lambda)(x) for large lambda. In addition for any sequence lambda(n) --> infinity we can extract a subsequence lambda(ni) along which ulambda(ni) converges strongly in H-1 (R-N). Moreover the limit function u(x) = lim(i-->infinity) ulambda(ni) satisfies (i) For j is an element of J the restriction u\Omega(j) of u(x) to Omega(j) is a least energy solution of -Deltav + b(x)v = v(p) in Omega(j) an v = 0 on partial derivativeOmega(j). (ii) u(x) = 0 for x is an element of R-N \ (boolean ORjis an element ofJ Omega(j)).
DOI10.1007/s00229-003-0397-x
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000185844200008
PublisherSPRINGER-VERLAG
Citation statistics
Cited Times:72[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/18086
Collection中国科学院数学与系统科学研究院
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Math, Beijing 100080, Peoples R China
2.Waseda Univ, Sch Sci & Engn, Dept Math, Shinjuku Ku, Tokyo 1698555, Japan
Recommended Citation
GB/T 7714
Ding, YH,Tanaka, K. Multiplicity of positive solutions of a nonlinear Schrodinger equation[J]. MANUSCRIPTA MATHEMATICA,2003,112(1):109-135.
APA Ding, YH,&Tanaka, K.(2003).Multiplicity of positive solutions of a nonlinear Schrodinger equation.MANUSCRIPTA MATHEMATICA,112(1),109-135.
MLA Ding, YH,et al."Multiplicity of positive solutions of a nonlinear Schrodinger equation".MANUSCRIPTA MATHEMATICA 112.1(2003):109-135.
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