  We study the matter density fluctuations in the running cosmological constant
(RCC) model using linear perturbations in the longitudinal gauge. Using this
observable we calculate the growth rate of structures and the matter power
spectrum, and compare them with the $SDSS$ data and other available data of the
linear growth rate. The distribution of collapsed structures may also
constraints models of dark energy. It is shown that RCC model enhances
departures from the $\Lambda CDM$ model for both cluster number and cumulative
cluster number predicted. In general increasing the characteristic parameter
$\nu$ leads to significant growth of the cluster number. In general, we found
that the theory of perturbations provides a good tool to distinguish the new
model $RCC$ of the standard cosmological model $\Lambda CDM$.
