The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials

Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions

“…In last decades, the existence and multiplicity of homoclinic orbits have been intensively studied by many mathematicians with variational methods [26][27][28][29][30][31][35][36][37] and the reference therein.…”

“…This condition is well known as the global Ambrosetti-Rabinowitz condition which can help prove the compact condition. In recent years, there are many papers [7,8,28,30,31,35] obtained the existence and multiplicity of homoclinic solutions of problem (1) with some other superquadratic conditions on W instead of ðA 1 Þ. Subsequently, we set f W ðt; xÞ ¼ ðrWðt; xÞ; xÞ À 2Wðt; xÞ:…”

“…It is easy to see that (4) satisfies the conditions of Theorem 1.2, but not the results in [8,13,21,[27][28][29][30][31] since (4) satisfies (3).…”

Multiplicity of Homoclinic Solutions for Fractional Hamiltonian Systems with Subquadratic Potential

“…The existence of homoclinic orbits of (2) has been studied by several researchers by using critical point theory. Examples and details can be found in a series of papers [2][3][4][5][6][7][8][9][10][11][12][13][14] and the references cited therein.…”

Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

“…For homoclinic solutions (HS), in the last thirty years the existence and multiplicity of homoclinic solutions have been extensively investigated via critical point theory and variational methods. Under the assumption that L(t) and W (t, u) are either independent of t or periodic in t, many authors have investigated the existence of (HS), see, for instance, [6,9,12,36,51] and the references therein, and some more general Hamiltonian systems are considered in the recent papers [19,27,45]. For the case that L(t) and W (t, u) are neither autonomous nor periodic in t, the existence of homoclinic solutions of (HS) is in sharp contrast to the periodic systems, because of the lack of compactness of Sobolev embedding [2,24,33,37].…”