Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem
This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ2u(n−1)+V′(u(n))=0, n∈ℤ(1,T), u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial so...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2010/907453 |
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| Summary: | This paper studies the existence of multiple solutions of the second-order difference
boundary value problem Δ2u(n−1)+V′(u(n))=0, n∈ℤ(1,T), u(0)=0=u(T+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 λk (k≥2) of linear difference problem
Δ2u(n−1)+λu(n)=0, n∈ℤ(1,T),
u(0)=0=u(T+1)
near infinity and the trivial solution of the first equation is a local minimizer under some assumptions
on V. |
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| ISSN: | 0161-1712 1687-0425 |