  We analyze the isoscalar response related to breathing modes with particular
attention being paid to low-lying excitations in neutron--rich nuclei. We use
the subtracted second random--phase approximation (SSRPA) to describe
microscopically the response. By increasing the neutron excess, we study the
evolution of the response in Ca isotopes going from $^{40}$Ca to $^{48}$Ca and
to $^{60}$Ca as well as in $N=20$ isotones going from $^{40}$Ca to $^{36}$S and
to $^{34}$Si. Finally, the case of $^{68}$Ni is investigated. We predict soft
monopole modes in neutron--rich nuclei which are driven by neutron excitations.
At variance with dipole pygmy modes, these neutron excitations are not only
strongly dominant at the surface of the nucleus but over its entire volume. The
effect of the mixing with two particle-two hole configurations induced by the
SSRPA model is analyzed. The properties of such soft neutron modes are
investigated in terms of their excitation energies, transition densities and
wave--function components. Their collectivity is also discussed as a function
of the isospin asymmetry and of the mass of the nucleus. The link between such
low--energy compression modes and a compressibility modulus introduced for
neutron--rich infinite matter is finally studied.
