Homoclinic orbits for second order Hamiltonian system with quadratic growth

Existence of homoclinic solution for the second order Hamiltonian systems

“…, u + n − u + + C 8 u n − u L 2 for all n, which implies that u + n → u + as n → ∞ by (5), (15) and (16). From (16) and the equivalence of the norms on the finite-dimensional subspace E − ⊕ E 0 we obtain that u 0 n → u 0 and u − n → u − in W as n → ∞, which implies that u n → u in W as n → ∞.…”

“…With the aids of the variational methods, many authors extensively investigated the existence of homoclinic solutions for problem (1), see, e.g., [1][2][3][4][5][6][7][8][9][12][13][14][15][16]. Most of them treat the superquadratic case (see [1][2][3][4][5][6][7][8][9]12,13]), [15,16] consider the asymptotically quadratic case and [5,14] treat the subquadratic case. But except for [5] all known results are obtained under the following assumption that L(t) is uniformly positively definite for all t ∈ R, that is, there exists a constant d 0 > 0 such that…”

“…There are functions L and W which satisfy our Theorem 1 and do not satisfy the corresponding results in [1][2][3][4][5][6][7][8][9][12][13][14][15][16]. For example, let…”

“…for all n. From Lemma 1 we may assume, going to a subsequence if necessary, that u n → u weakly in E (15) and u n → u in L 2 (16) as n → ∞. By (W3) there is σ > 0 such that…”

Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition

“…With the variational methods, the existence and multiplicity of homoclinic orbits of problem ( 1) have been obtained by many papers (see [1][2][3][4][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). But in most superquadratic cases, there is a so-called global (AR) condition on W , that is, there exists a constant μ > 2 such that…”