  This paper extends the definition of Rabinowitz Floer homology to non-compact
hypersurfaces. We present a general framework for the construction of
Rabinowitz Floer homology in the non-compact setting under suitable compactness
assumptions on the periodic orbits and the moduli spaces of Floer trajectories.
We introduce a class of hypersurfaces being the level sets of specific
Hamiltonians: strongly tentacular Hamiltonians, for which the compactness
conditions are satisfied, thus enabling us to define the Rabinowitz Floer
homology for this class. Rabinowitz Floer homology in turn serves as a tool to
address the Weinstein conjecture and establish existence of closed
characteristics.
