Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

Bo Zheng and Huafeng Xiao

International Journal of Mathematics and Mathematical Sciences, 2010, vol. 2010, 1-21

Abstract:

This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ 2 𠑢 ( 𠑛 − 1 ) + 𠑉  ( 𠑢 ( 𠑛 ) ) = 0 , 𠑛 ∈ ℤ ( 1 , 𠑇 ) , 𠑢 ( 0 ) = 0 = 𠑢 ( 𠑇 + 1 ) . By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue 𠜆 𠑘 ( 𠑘 ≥ 2 ) of linear difference problem Δ 2 𠑢 ( 𠑛 − 1 ) + 𠜆 𠑢 ( 𠑛 ) = 0 , 𠑛 ∈ ℤ ( 1 , 𠑇 ) , 𠑢 ( 0 ) = 0 = 𠑢 ( 𠑇 + 1 ) near infinity and the trivial solution of the first equation is a local minimizer under some assumptions on 𠑉 .

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:907453

DOI: 10.1155/2010/907453

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