Weak solutions for the Hamiltonian bifurcation problem (Q2812466)

The paper deals with the Hamiltonian system with the superquadratic nonlinearity and periodic condition NEWLINE\[NEWLINE\begin{cases} \dot p(t)=-\lambda q(t)-H_q(t,p(t),q(t)),\\ \dot q(t)=\lambda p(t)+H_p(t,p(t),q(t)),\end{cases}\leqno(1)NEWLINE\]NEWLINE where \(p,\,q\in\mathbb{R}^n\), \(H(t,z(t))\) is a \(C^1\) function defined on \(\mathbb{R}^1\times \mathbb{R}^{2n}\), \(n\geq 1\), which is \(2\pi\)-periodic with respect to the first variable \(t\), and \(\lambda\in\mathbb{R}\). By using the variational method and the critical point theory in terms of the \(S^1\)-invariant functions, the authors investigate the number of the \(2\pi\)-periodic weak solutions for the bifurcation problem of \((1)\).