Homoclinic orbits for the second-order Hamiltonian systems (Q518729)

The authors consider the following nonautonomous second-order Hamiltonian system \[ \ddot{u}-L(t)u+\nabla W(t,u)=0, \leqno(1) \] where \(L\in C(\mathbb{R},\mathbb{R}^{N^{2}})\) is a symmetric and positive definite matrix-valued function and \(W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})\) with gradient \(\nabla W(t,x)=\frac{\partial W}{\partial x}(t,x)\). They discuss two cases. First, by using the generalized Nehari manifold, they prove that System (1) possesses ground state homoclinic solutions when the potential \(W\) satisfies some suitable superquadratic conditions. Second, they treat the subquadratic case. For this end, they prove the existence of multiple classical homoclinic orbits for (1) by applying variational methods and the concentration-compactness principle.