  We present the results obtained with VLT/MUSE on the faint-end of the
Lyman-alpha luminosity function (LF) based on deep observations of four lensing
clusters. The precise aim of the present study is to further constrain the
abundance of Lyman-alpha emitters (LAEs) by taking advantage of the
magnification provided by lensing clusters. We blindly selected a sample of 156
LAEs, with redshifts between $2.9 \le z \le 6.7$ and magnification-corrected
luminosities in the range $ 39 \lesssim \log L_{Ly_{\alpha}}$ [erg s$^{-1}$]
$\lesssim 43$. The price to pay to benefit from magnification is a reduction of
the effective volume of the survey, together with a more complex analysis
procedure. To properly take into account the individual differences in
detection conditions (including lensing configurations, spatial and spectral
morphologies) when computing the LF, a new method based on the 1/Vmax approach
was implemented. The LAE LF has been obtained in four different redshift bins
with constraints down to $\log L_{Ly_{\alpha}} = 40.5$. From our data only, no
significant evolution of LF mean slope can be found. When performing a
Schechter analysis including data from the literature to complete the present
sample a steep faint-end slope was measured varying from $\alpha =
-1.69^{+0.08}_{-0.08}$ to $\alpha = -1.87^{+0.12}_{-0.12}$ between the lowest
and the highest redshift bins. The contribution of the LAE population to the
star formation rate density at $z \sim 6$ is $\lesssim 50$% depending on the
luminosity limit considered, which is of the same order as the Lyman-break
galaxy (LBG) contribution. The evolution of the LAE contribution with redshift
depends on the assumed escape fraction of Lyman-alpha photons, and appears to
slightly increase with increasing redshift when this fraction is conservatively
set to one. (abridged)
