Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems. (February 2015) - Sun, Juntao; Wu, Tsung-fang
Abstract: In this paper, we present some new results of homoclinic solutions for second-order Hamiltonian systems u ̈ − λ L ( t ) u + W u ( t, u ) = 0 ; here λ > 0 is a parameter, L ∈ C ( R, R N × N ) and W ∈ C 1 ( R × R N, R ) . Unlike most other papers on this problem, we require that L ( t ) is a positive semi-definite symmetric matrix for all t ∈ R, that is, L ( t ) ≡ 0 is allowed to occur in some finite interval T of R . Under some suitable assumptions on W, we prove the existence of two different homoclinic solutions u λ ( 1 ), u λ ( 2 ), which both vanish on R ∖ T as λ → ∞, and converge to u 0 ( 1 ), u 0 ( 2 ) in H 1 ( R ), respectively; here u 0 ( 1 ) ≠ u 0 ( 2 ) ∈ H 0 1 ( T ) are two nontrivial solutions of the Dirichlet BVP for Hamiltonian systems on the finite interval T .