Critical Dimension for Stable Self-gravitating Stars in AdS

We study the self-gravitating stars with a linear equation of state, $P=a rho$, in AdS space, where $a$ is a constant parameter. There exists a critical dimension, beyond which the stars are always stable with any central energy density; below which there exists a maximal mass configuration for a certain central energy density and when the central energy density continues to increase, the configuration becomes unstable. We find that the critical dimension depends on the parameter $a$, it runs from $d=11.1429$ to 10.1291 as $a$ varies from $a=0$ to 1. The lowest integer dimension for a dynamically stable self-gravitating configuration should be $d=12$ for any $a in [0,1]$ rather than $d=11$, the latter is the case of self-gravitating radiation configurations in AdS space.