Even homoclinic orbits for super quadratic Hamiltonian systems

Abstract

We study the existence of even homoclinic orbits for the second-order Hamiltonian systemü+V u (t, u) = 0. Let V(t, u) = −K(t, u)+W(t, u) ∈ C 1 (R×R n , R), where K is less quadratic and W is super quadratic in u at infinity. Since the system we considered is neither autonomous nor periodic, the (PS) condition is difficult to check when we use the Mountain Pass theorem. Therefore, we approximate the homoclinic orbits by virtue of the solutions of a sequence of nil-boundary-value problems.

Homoclinic Orbits for a Class of Subquadratic Second Order Hamiltonian Systems

“…It is not until the recent decade that variational principle has been widely used. The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). The main feature of the problem is the lack of global compactness due to unboundedness of domain.…”

“…Some papers treat the coercive case (see [5][6][7][8]). Recently, the symmetric case has been dealt with (see [9][10][11]). Compared with the superquadratic case, the case that ( , ) is subquadratic as | | → +∞ has been considered only by a few authors.…”