Homoclinic orbits for Lagrangian systems (Q1917818)

The Lagrangian system \[ {d \over dt} \left( {\partial L(t,x,y)\over \partial y} \right) - {\partial L(t,x,y) \over \partial x} = 0 \tag{LS} \] with \(L(t, x, y) = {1 \over 2} \sum^N_{i,j = 1} a_{ij} (x) y_i y_j - V(t,x)\), where \(x, y \in \mathbb{R}^N\), is considered. The following assumptions are made: (1) \(a(x) = \{a_{ij} (x)\}\) is a positive definite symmetric \(C^2\) matrix. (2) The potential \(V(t,x)\) is globally superquadratic in \(x\) and \(T\)-periodic in \(t\). The existence of at least two homoclinic orbits for the Lagrangian system (LS) is proved. The concentration-compactness lemma and minimax argument are used to prove the existences.