The general solution to this is you buy extra hardware that goes in front of the computer that looks at the incoming packets.
What that extra hardware does depends on what solution you want to use, you could have that hardware distribute the requests to many servers all running the same software (this would make the hardware you added a Load Balancer). You also could have the hardware detect that a unusually large number of packets coming from a single address, the hardware could then start dropping packets from that address instead of forwarding them on to the server (this would make the hardware you added a Stateful Firewall)
There are more options beyond those two but all solutions revolve around reducing the load on the server (usually shifting the load to another piece of hardware dedicated to taking the load). You could potentially upgrade your software to be more resilient to packet floods but unless your current software is written very poorly it won't buy you too much more capacity.