  We aim to constrain mass-varying neutrino models using large scale structure
observations and produce forecast for the Euclid survey. We investigate two
models with different scalar field potential and both positive and negative
coupling parameters \beta. These parameters correspond to growing or decreasing
neutrino mass, respectively. We explore couplings up to |\beta|<5. In the case
of the exponential potential, we find an upper limit on $\Omega_\nu h^2$<0.004
at 2-$\sigma$ level. In the case of the inverse power law potential the null
coupling can be excluded with more than 2-\sigma significance; the limits on
the coupling are \beta>3 for the growing neutrino mass and \beta<-1.5 for the
decreasing mass case. This is a clear sign for a preference of higher
couplings. When including a prior on the present neutrino mass the upper limit
on the coupling becomes |\beta|<3 at 2-$\sigma$ level for the exponential
potential. Finally, we present a Fisher forecast using the tomographic weak
lensing from an Euclid-like experiment and we also consider the combination
with the cosmic microwave background (CMB) temperature and polarisation spectra
from a Planck-like mission. If considered alone, lensing data is more efficient
in constraining $\Omega_\nu$ with respect to CMB data alone. There is, however,
a strong degeneracy in the \beta-$\Omega_\nu h^2$ plane. When the two data sets
are combined, the latter degeneracy remains, but the errors are reduced by a
factor ~2 for both parameters.
