  We investigate the instabilities that may lead to the breakdown of the
Luttinger liquid in the small-diameter (5,0) nanotubes, paying attention to the
competition between the effective interaction mediated by phonon-exchange and
the Coulomb interaction. We use bosonization methods to achieve an exact
treatment of the Coulomb interaction at small momentum-transfer, and apply next
renormalization group methods to analyze the low-energy behavior of the
electron system. This allows us to discern the growth of several response
functions for charge-density-wave modulations and for superconducting
instabilities with different order parameters. We show that, in the case of
single nanotubes exposed to screening by external gates, the Luttinger liquid
is unstable against the onset of a strong-coupling phase with very large
charge-density-wave correlations. The temperature of crossover to the new phase
depends crucially on the dielectric constant $\kappa $ of the environment,
ranging from $T_{C}\sim 10^{-4} {\rm K}$ (at $\kappa \approx 1$) up to a value
$T_{C} \sim 10^{2} {\rm K}$ (reached from $\kappa \approx 10$). The physical
picture is however different when we consider the case of a large array of
nanotubes, in which there is a 3D screening of the Coulomb interaction over
distances much larger than the intertube separation. The electronic instability
is then triggered by the divergence of one of the charge stiffnesses in the
Luttinger liquid, implying a divergent compressibility and the appearance of a
regime of phase separation into spatial regions with excess and defect of
electron density.
