American Institute of Mathematical Sciences

$J\dot u(t)+\nabla H(t,u(t))=0$ a.e. on $[0,T] $

$u(0)=u(T)$

where the function $H:[0,T]\times \mathbb R^{2N}\rightarrow \mathbb R$ is called Hamiltonian. Our attention will be focused upon the case in which the Hamiltonian H, besides being measurable on $t\in[0,T]$, is convex and continuously differentiable with respect to $u\in \mathbb R^{2N}$. Our basic assumption is that the Hamiltonian $H$ satisfies the following growth condition:
Let $1 < p < 2$ and $q=\frac{p}{p-1}$. There exist positive constants $\alpha,\delta$ and functions $\beta,\gamma \in L^q(0,T;\mathbb R^+)$ such that

$\delta|u|-\beta(t)\leq H(t,u)\leq\frac{\alpha}{q}|u|^q+\gamma(t),$

for all $u\in \mathbb R^{2N}$ and a.e. $t\in[0,T]$. Our main result assures that under suitable bounds on $\alpha,\delta$ and the functions $\beta,\gamma$, the problem above has at least a solution that belongs to $W_T^{1,p}$. Such a solution corresponds, in the duality, to a function that minimizes the dual action restricted to a subset of $\tilde{W}_T^{1,p}=${$v\in W_T^{1,p}: \int_0^{ T} v(t) dt=0$}.