  We explore the effect of mutual gravitational interaction between ultra-cold
gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude
oscillation of BEC is studied by applying variational technique to reduce the
Gross-Pitaevskii equation, with gravity included, to the equation of motion of
a particle moving in a potential. According to our analysis, if the s-wave
scattering length can be tuned to zero using Feshbach resonance for future BEC
with occupation numbers as high as $\approx 10^{20}$, there exists a critical
ground state occupation number above which the BEC is unstable, provided that
its constituents interact with a $1/r^3 $ gravity at short scales.
