  We naturally obtain a power-law total mass-density profile for the inner
regions (within several effective radius) of early-type galaxies(ETGs) from
space-time structures, which are described by the static spherically-symmetric
solutions of Einstein equations with the perfect fluid matters. We point out
that, in the inner region of an early type galaxy, the average value of the
equation of state $\omega$ determines the power-law index $\gamma$ of the total
mass-density profile given by \cite{2006ApJ...649..599K}. We then use 119
galaxy-galaxy strong gravitational lensing systems from SLACS, BELLS, BELLS
GALLERY and SL2S, to constrain the equation of state $\omega$ for the inner
regions of ETGs. We find that the average value of $\omega$ for totally 119
ETGs is $\omega=-0.315\pm0.073$ (68\% CL), which is very close to $-1/3$. If
$\omega=-1/3$ is the best value to describe the equation of state of the inner
regions of the dark matter in galaxies, the deviation from $-1/3$ could
represent the mixture of baryonic matter and dark matter in galaxies.
