Periodic solutions for a class of non-coercive Hamiltonian systems - Morched Boughariou (Electronic Journal of Differential Equations, 2001)
We prove the existence of non-constant $T$-periodic orbits of the Hamiltonian system $$displaylines{ dot q =H_p (t, p(t), q(t))cr dot p =-H_q (t, p(t), q(t)), }$$ where $H$ is a $T$-periodic function in $t$, non-convex and non-coercive in $(p,q)$, and has the form $H(t,p,q)sim |q|^{alpha}(|p|^{Beta}-1)$ with $alpha>Beta>1$.