  Massive planetary cores ($\sim 10$ Earth masses) trigger rapid gas accretion
to form gas giant planets \rev{such as} Jupiter and Saturn. We investigate the
core growth and the possibilities for cores to reach such a critical core mass.
At the late stage, planetary cores grow through collisions with small
planetesimals. Collisional fragmentation of planetesimals, which is induced by
gravitational interaction with planetary cores, reduces the amount of
planetesimals surrounding them, and thus the final core masses. Starting from
small planetesimals that the fragmentation rapidly removes, less massive cores
are formed. However, planetary cores acquire atmospheres that enlarge their
collisional cross section before rapid gas accretion. Once planetary cores
exceed about Mars mass, atmospheres significantly accelerate the growth of
cores. We show that, taking into account the effects of fragmentation and
atmosphere, initially large planetesimals enable formation of sufficiently
massive cores. On the other hand, because the growth of cores is slow for large
planetesimals, a massive disk is necessary for cores to grow enough within a
disk lifetime. If the disk with 100\,km-sized initial planetesimals is 10 times
as massive as the minimum mass solar nebula, planetary cores can exceed 10
Earth masses in the Jovian planet region ($>5\,$AU).
