  Aims: The nearby galaxy clusters Abell 496 and Abell 85 are studied in the
very high energy (VHE, E > 100 GeV) band to investigate VHE cosmic rays (CRs)
in this class of objects which are the largest gravitationally bound systems in
the Universe. Methods: H.E.S.S., an array of four Imaging Atmospheric Cherenkov
Telescopes (IACT), is used to observe the targets in the range of VHE gamma
rays. Results: No significant gamma-ray signal is found at the respective
position of the two clusters with several different source size assumptions for
each target. In particular, emission regions corresponding to the high density
core, to the extension of the entire X-ray emission in these clusters, and to
the very extended region where the accretion shock is expected, are
investigated. Upper limits are derived for the gamma-ray flux at energies E>570
GeV for Abell 496 and E>460 GeV for Abell 85. Conclusions: From the
non-detection in VHE gamma rays, upper limits on the total energy of hadronic
CRs in the clusters are calculated. If the cosmic-ray energy density follows
the large scale gas density profile, the limit on the fraction of energy in
these non-thermal particles with respect to the total thermal energy of the
intra-cluster medium (ICM) is 51% for Abell 496 and only 8% for Abell 85 due to
its larger mass and higher gas density. These upper limits are compared with
theoretical estimates. They predict about ~10% of the thermal energy of the ICM
in non-thermal particles. The observations presented here can constrain these
predictions especially for the case of the Abell 85 cluster.
