On superquadratic periodic systems with indefinite linear part (Q1049056)

The authors consider the second order system of ordinary differential equations \[ -x''-A(t)x = \nabla F(t,x),\quad x\in\mathbb{R}^N, \] where \(A\) is an \(N\times N\) symmetric matrix with periodic entries, \(F\) is periodic in \(t\) and \(\nabla F(t,0)=0\). They show that, under a rather weak superquadraticity condition on \(F\) at infinity, this system has a nontrivial periodic solution. The proof uses Morse theory as follows. The superquadraticity condition implies that the Euler-Lagrange functional \(\varphi\) corresponding to the problem satisfies the Cerami condition (hence, the usual deformation lemma holds) and the critical groups at infinity \(C_k(\varphi,\infty)=0\) for all \(k\). The conditions on \(F\) at 0 imply that the critical group \(C_d(\varphi,0)\neq 0\) for a certain \(d\). Hence, there must exist a critical point \(x_0\neq 0\) for \(\varphi\) (it is also shown that \(\varphi(x_0)\neq 0\)).