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Infinitely many homoclinic solutions for a second‐order Hamiltonian system
Author(s) -
Tang X. H.
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200253
Subject(s) - homoclinic orbit , mathematics , hamiltonian system , order (exchange) , hamiltonian (control theory) , pure mathematics , mathematical analysis , mathematical optimization , bifurcation , nonlinear system , physics , finance , quantum mechanics , economics
In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian systemu ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × Nand W : R × R N → R . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, L ( t ) is not necessary to be uniformly positive definite or coercive; through W ( t , x ) is still assumed to be superquadratic near | x | = ∞ , it is not assumed to be superquadratic near x = 0 .