We study the effect of the density-dependent axial and vector form factors on the electro-neutrino ($ u_e$) and anti-neutrino $({bar u}_e)$ reactions for a nucleon in nuclear matter or in $^{12}$C. The nucleon form factors in free space are presumed to be modified for a bound nucleon in a nuclear medium. We adopt the density-dependent form factors calculated by the quark-meson coupling (QMC) model, and apply them to the $ u_e$ and ${bar u}_e$ induced reactions with the initial energy $E = $ 8 $sim$ 80 MeV. We find that the total ${ u}_e$ cross sections on $^{12}$C as well as a nucleon in nuclear matter are reduced by about 5% at the nuclear saturation density, $rho_0$. This reduction is caused by the modification of the nucleon structure in matter. Although the density effect for both cases is relatively small, it is comparable with the effect of Coulomb distortion on the outgoing lepton in the $ u$-reaction. In contrast, the density effect on the ${bar u}_e$ reaction reduces the cross section significantly in both nuclear matter and $^{12}$C cases, and the amount maximally becomes of about 35% around $rho_0$. Such large asymmetry in the $ u_e$ and ${bar u}_e$ cross sections, which seems to be nearly independent of the target, is originated from the difference in the helicities of ${bar u}_e$ and ${ u}_e$. It is expected that the asymmetry influences the r-process and also the neutrino-process nucleosynthesis in core-collapse supernovae.
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