  The current generation of Cherenkov telescopes, together with Fermi-LAT, have
greatly increased our knowledge of $\gamma$-ray blazars. Among them, the
high-frequency-peaked BL Lacertae object (HBL) PKS1424+240 (z $\simeq$ 0.6) is
the farthest persistent emitter of very-high-energy (VHE; E $\geq$ 100 GeV)
$\gamma$-ray photons. Current emission models can satisfactorily reproduce
typical blazar emission assuming that the dominant emission process is
synchrotron-self-Compton (SSC) in HBLs; and external-inverse-Compton (EIC) in
LBLs and FSRQs. Hadronic models are also able to correctly reproduce the
$\gamma$-ray emission from blazars, although they are in general disfavoured
for bright quasars and rapid flares.The blazar PKS1424+240 is a rare example of
a luminous HBL, and we aim to determine which is the emission process most
likely responsible for its $\gamma$-ray emission. This will impact more
generally our comprehension of blazar emission models, and how they are related
to the luminosity of the source and the peak frequency. We investigate
different blazar emission models applied to the SED of PKS1424+240. Among
leptonic models, we study a 1-zone SSC model, a 2-zone SSC model, and an EIC
model. We then investigate a blazar hadronic model, and finally a scenario in
which the $\gamma$-ray emission is associated with cascades in the
line-of-sight produced by cosmic rays from the source. After a systematic study
of the parameter space of the one-zone SSC model, we conclude that this
scenario is not compatible with $\gamma$-ray observations of PKS1424+240. A
two-zone SSC scenario can alleviate this issue, as well as an EIC solution. For
the latter, the external photon field is assumed to be the infra-red radiation
from the dusty torus. Alternatively, hadronic models can satisfactorily
reproduce the $\gamma$-ray emission from PKS1424+240, both as in-source
emission and as cascade emission.
