This research project aims at finding evidence of physics beyond the Standard Model of particle physics (SM) by testing lepton flavour universality in kaon decays. This goal can be reached through the measurement of the branching fraction ratio and of the form factors parameters of the decays K+ ! p+m+m? and K+ ! p+e+e?. The SM has proven to be an extremely successful theory so far, but several theoretical arguments and experimental evidences point towards the existence of physics beyond the SM. Hints of possible existence of new physics come from LHCb, that has recently obtained the indication of a lepton flavour universality violation in B decays. Theoretical models developed to explain these flavour anomalies predict also effects in rare kaon decays like K+ !p+m+m? and K+ !p+e+e?. A precise measurement of these decays could therefore provide a strong experimental evidence complementary to the B anomaly, crucial to solve the current puzzle in the flavour sector.