Homoclinic solutions for a class of the second order Hamiltonian systems

Abstract

We study the existence of homoclinic orbits for the second order Hamiltonian systemqis T-periodic in t. A map K satisfies the "pinching" condition b 1 |q| 2 K(t, q) b 2 |q| 2 , W is superlinear at the infinity and f is sufficiently small in L 2 (R, R n ). A homoclinic orbit is obtained as a limit of 2kT -periodic solutions of a certain sequence of the second order differential equations.

Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation

“…In [4], [14], the authors proved if (PS) condition is replaced by the weaker (C) c -condition, the deformation lemmas still hold. …”

Ground state homoclinic orbits of a class of superquadratic damped vibration problems

“…In the past two decades, many authors studied the homoclinic orbits for the Hamiltonian systems via the critical point theory and the variational methods. Assuming that L(t) and W (t, x) are independent of t or T −periodic in t, many authors have studied the existence of the homoclinic solutions for the Hamiltonian system (H S ), see, e.g., [3,5,[7][8][9]12] and the references therein. In this case, the existence of the homoclinic solutions can be obtained by going to the limit of the periodic solutions of the approximating problems.…”

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…”

“…which is very important to guarantee the boundedness of the (PS) c sequence (see [1], [2], [4], [6], [8], [14], [15]). Since the domain is unbounded, there is a lack of compactness of the Sobolev embedding.…”

“…Since the domain is unbounded, there is a lack of compactness of the Sobolev embedding. Many papers consider the periodic (autonomous, asymptotically periodic) problems (see [1], [2], [4], [7], [8], [14], [15]). Some papers treat the symmetric case (see [10], [11]).…”