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Title: Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems (English)
Author: Bai, Dingyong
Author: Chen, Yuming
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 60
Issue: 4
Year: 2015
Pages: 343-353
Summary lang: English
Category: math
Summary: We discuss the discrete $p$-Laplacian eigenvalue problem, \[ \begin {cases} \Delta (\phi _p(\Delta u(k-1)))+\lambda a(k)g(u(k))=0,\quad k\in \{1,2, \ldots , T\},\\ u(0)=u(T+1)=0, \end {cases} \] where $T>1$ is a given positive integer and $\phi _p(x):=|x|^{p-2}x$, $p > 1$. First, the existence of an unbounded continuum $\mathcal {C}$ of positive solutions emanating from $(\lambda , u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal {C}$ is a monotone continuous curve globally defined for all $\lambda >0$. (English)
Keyword: discrete $p$-Laplacian eigenvalue problem
Keyword: positive solution
Keyword: continuum
Keyword: Picone-type identity
Keyword: lower and upper solutions method
MSC: 34B09
MSC: 39A10
MSC: 39A12
idZBL: Zbl 06486915
idMR: MR3396469
DOI: 10.1007/s10492-015-0100-z
Date available: 2015-06-30T11:58:28Z
Last updated: 2020-07-02
Stable URL:
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