Abstract We study global transverse Poincare sections and give topological obstructions to their existence. prove that any energy hypersurface equipped with a section has an induced cosymplectic structure. family of Hamiltonian systems all possible topologies. Finally, we address the question when compact symplectic manifold possesses

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions the higher order Hardy-H\'enon equations \[ (-\Delta)^m u = |x|^\sigma u^p \] in $\mathbf R^n$ with $p > 1$. We show that condition n - 2m \frac{2m+\sigma}{p-1} >0 necessary for existence distributional solutions. For $n \geq 2m$ and $\sigma -2m$, we prove any solution satisfies an ...