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Variational approach to solutions for a class of fractional Hamiltonian systems
Author(s) -
Zhang Ziheng,
Yuan Rong
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2941
Subject(s) - mathematics , hamiltonian system , hamiltonian (control theory) , infinity , novelty , class (philosophy) , pure mathematics , mathematical physics , combinatorics , mathematical analysis , mathematical optimization , philosophy , theology , artificial intelligence , computer science
In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: FHS{t D ∞ α ( − ∞ D t α u ( t ) )+ L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) ,u ∈ H α ( ℝ , ℝ n ) ,( FHS )where α ∈ (1 ∕ 2,1), t ∈ R , u ∈ R n , and L ∈ C ( R , R n 2 ) are symmetric and positive definite matrices for all t ∈ R , W ∈ C 1 ( R × R n , R ) , and ∇ W is the gradient of W at u . The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.