Periodic solutions of second-order systems with subquadratic convex potential

Abstract

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: \begin{equation} \begin{cases} \ddot u(t)+m^2\omega^2u(t)+\nabla F(t,u(t))=0\qquad \mbox{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0, \end{cases} \end{equation} where $m>0$, the potential $F(t,x)$ is convex in $x$ and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem [Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989].

Keywords

• second order hamiltonian systems; critical points; variational methods; sobolev's inequality