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Global continuum of positive solutions for discrete p-Laplacian eigenvalue problems

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Abstract

We discuss the discrete p-Laplacian eigenvalue problem,

$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$

where T > 1 is a given positive integer and φ p (x):= |x|p−2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, 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 λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.

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Correspondence to Dingyong Bai.

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The research of Bai is supported partially by PCSIRT of China (No. IRT1226) and NSF of China (No. 11171078). The research of Chen is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Bai, D., Chen, Y. Global continuum of positive solutions for discrete p-Laplacian eigenvalue problems. Appl Math 60, 343–353 (2015). https://doi.org/10.1007/s10492-015-0100-z

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  • DOI: https://doi.org/10.1007/s10492-015-0100-z

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