Multiple solutions of second-order ordinary differential equation via Morse theory (Q1947897)

The authors consider the second-order ordinary differential equation \[ -\ddot{x}=f(t,x(t)) \] subject to the periodic boundary conditions \[ x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0, \] where \(f\in C([0,2\pi]\times\mathbb{R},\mathbb{R})\), \(f(t,x)\) is continuously differentiable in \(x\) for a.e. \(t\in [0,2\pi]\) and satisfies \(f(t,0)\equiv 0\) for all \(t\in[0,2\pi]\). The operator \(K=(-\frac{d^{2}}{dt^{2}}+I)^{-1}\) plays an important role. By using Morse index, Leray-Schauder degree and the Morse index theorem of Lazer-Solimini type, they obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.