  Some dwarf galaxies are within the Mondian regime at all radii, i.e., the
gravitational acceleration provided by the observed baryons is always below the
threshold of $g_\dag\simeq 1.2\times 10^{-10}\,{\rm m\,s^{-2}}$. These dwarf
galaxies often show cores, in the sense that assuming Newton's gravity to
explain their rotation curves, the total density profile $\rho(r)$ presents a
central plateau or {\em core} ($d\log \rho/d\log r\rightarrow 0$ when
$r\rightarrow 0$). Here we show that under MOND gravity, the existence of this
core implies a baryon content whose density $g_{\rm bar}$ must decrease toward
the center of the gravitational potential ($g_{\rm bar}\rightarrow 0$ when
$r\rightarrow 0$). Such drop of baryons toward the central region is neither
observed nor appears in numerical simulations of galaxy formation following
MOND gravity. We analyze the problem posed for MOND as well as possible
workarounds.

Astrophysics of Galaxies