Abstract
In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system △ [ p ( n ) △ u ( n − 1 ) ] −L(n)u(n)+∇W ( n , u ( n ) ) =0. Under the assumption that W(n,x) is of indefinite sign and subquadratic as |x|→+∞ and p(n) and L(n) are N×N real symmetric positive definite matrices for all n∈Z, and that lim inf | n | → + ∞ [ | n | ν − 2 inf | x | = 1 ( L ( n ) x , x ) ] >0 for some constant ν<2, we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark’s theorem in critical point theory.MSC:39A11, 58E05, 70H05.
Highlights
Consider the second-order self-adjoint discrete Hamiltonian system p(n) u(n – ) – L(n)u(n) + ∇W n, u(n) =, ( . )where n ∈ Z, u ∈ RN, u(n) = u(n + ) – u(n) is the forward difference operator, p, L : Z → RN ×N and W : Z × RN → R
We say that a solution u(n) of system ( . ) is homoclinic if u(n) → as n → ±∞
System ( . ) can be regarded as a special form of the Emden-Fowler equation appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting system, and many well-known results concerning properties of solutions of ( . ) are collected in [ ]
Summary
In papers [ – ], the authors studied the existence of homoclinic solutions of system Theorem A [ ] Assume that p(n) is an N × N real symmetric positive definite matrix for all n ∈ Z, and that L and W satisfy the following assumptions: (L) L(n) is an N × N real symmetric positive definite matrix for all n ∈ Z, and there exists a constant β > such that
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