  The observed rotation curves of low surface brightness (LSB) galaxies play an
essential role in studying dark matter, and indicate that there exists a
central constant density dark matter core. However, the cosmological N-body
simulations of cold dark matter predict an inner cusped halo with a power-law
mass density distribution, and can't reproduce a central constant-density core.
This phenomenon is called cusp-core problem. When dark matter is quiescent and
satisfies the condition for hydrostatic equilibrium, using the equation of
state can get the density profile in the static and spherically symmetric
space-time. To solve the cusp-core problem, we assume that the equation of
state is independent of the scaling transformation. Its lower order
approximation for this type of equation of state can naturally lead to a
special case, i.e. $p=\zeta\rho+2\epsilon V_{rot}^{2}\rho$, where $p$ and
$\rho$ are the pressure and density, $V_{rot}$ is the rotation velocity of
galaxy, $\zeta$ and $ \epsilon$ are positive constants. It can obtain a density
profile that is similar to the pseudo-isothermal halo model when $\epsilon$ is
around $0.15$. To get a more widely used model, let the equation of state
include the polytropic model, i.e. $p= \frac{\zeta}{\rho_{0}^{s}}\rho^{1+s}+
2\epsilon V_{rot}^{2}\rho$, we can get other kinds of density profiles, such as
the profile that is nearly same with the Burkert profile, where $s$ and
$\rho_{0}$ are positive constants.
