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We present the results obtained with VLT/MUSE on the faint-end of theLyman-alpha luminosity function (LF) based on deep observations of four lensingclusters. The precise aim of the present study is to further constrain theabundance of Lyman-alpha emitters (LAEs) by taking advantage of themagnification provided by lensing clusters. We blindly selected a sample of 156LAEs, with redshifts between $2.9 \le z \le 6.7$ and magnification-correctedluminosities 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 ofthe effective volume of the survey, together with a more complex analysisprocedure. To properly take into account the individual differences indetection conditions (including lensing configurations, spatial and spectralmorphologies) when computing the LF, a new method based on the 1/Vmax approachwas implemented. The LAE LF has been obtained in four different redshift binswith constraints down to $\log L_{Ly_{\alpha}} = 40.5$. From our data only, nosignificant evolution of LF mean slope can be found. When performing aSchechter analysis including data from the literature to complete the presentsample 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 lowestand the highest redshift bins. The contribution of the LAE population to thestar formation rate density at $z \sim 6$ is $\lesssim 50$% depending on theluminosity limit considered, which is of the same order as the Lyman-breakgalaxy (LBG) contribution. The evolution of the LAE contribution with redshiftdepends on the assumed escape fraction of Lyman-alpha photons, and appears toslightly increase with increasing redshift when this fraction is conservativelyset to one. (abridged)

Faint end of the $z \sim 3-7$ luminosity function of Lyman-alpha emitters behind lensing clusters observed with MUSE