Can a LAN be implemented just with a switch?
For hosts on the same LAN, a router serves no purpose. Routers route packets between networks, not from a network back to the same network.
Hosts on a LAN can get IP addressing from a DHCP server, be statically configured, or configure themselves with link-local IP addressing.
just by MAC addresses?
No. Traffic delivered on a LAN is delivered by the layer-2 LAN addressing (MAC addresses for IEEE LAN protocols, e.g. ethernet), but the network stack will still have addressing for the other layers in the stack, such as the layer-3 (IPv4, IPv6, etc.) and layer-4 (TCP, UDP, etc.) protocols.
When sending a frame from the source to the destination,and inbetween
there is a switch,what will the MAC address for destination be? (Again,no router in the LAN)
Again, a router plays no part in traffic sourced and destined for the same LAN. Frames are addressed with the source and destination host layer-2 addresses. Switches are transparent devices. On ethernet, the source MAC address on a frame is the source host's MAC address, and the destination MAC address on a frame is the destination host's MAC address. Only frames destined to the management interface of a switch (as a host on the LAN) would be addressed with a MAC address of the switch, otherwise, the hosts do not even know the switch exists.