Existence of planar H-loops via Hardy's inequality

Given a continuous function on $\mathbb{R}^2$, we study the existence of non-constant, $2\pi$-periodic solutions to the $2$-dimensional Hamiltonian system
\begin{equation} \label{eq:problem} u''=|u'|~\!{\rm H}(u) iu'. \end{equation}
Any non-constant solution to \eqref{eq:problem} parametrizes a closed planar curve having prescribed curvature ${\rm H}$ at each point. Our interest in \eqref{eq:problem} is motivated also by its relations with Arnold's problem on ${\rm H}$-magnetic geodesics.
Problem \eqref{eq:problem} admits a variational formulation; under reasonable assumptions on the prescribed (non-constant) curvature ${\rm H}$, the associated energy functional has a nice Mountain-Pass geometry. However, due to the groups of dilations and translations in $\mathbb{R}^2$, the Palais-Smale condition fails to hold, and in fact there could exists unbounded Palais-Smale sequences.
We will present an existence result which is strongly based on a Hardy type inequality for functions of two variables.
This is joint work with Gabriele Cora (Università di Torino, Italy).