In　multi-beam　low-earth　orbit　(LEO)　satellite　networks,　frequent　handovers　between　intra-satellite　and　inter-satellite　beams　are　inevitable.　In　this　letter,　we　design　a　beam　handover　strategy　based　on　the　multi-objective　reinforcement　learning　(MORL)　method　to　achieve　seamless　and　effective　handover　between　multiple　beams　of　LEO　satellites.　We　first　model　the　handover　optimization　problem　of　the　multi-beam　LEO　satellite　networks　as　a　multi-objective　optimization　(MOO)　problem　to　jointly　maximize　throughput,　minimize　the　handover　frequency,　and　keep　the　network　load　balanced.　On　this　basis,　we　convert　the　MOO　problem　into　a　multi-objective　Markov　decision　process　(MOMDP),　and　utilize　an　MORL　method,　called　multi-objective　deep　Q-learning　network　(MODQN),　to　learn　and　achieve　the　optimal　solution.　Simulation　results　show　the　effectiveness　and　superiority　of　the　proposed　handover　scheme.