The corners on my rugs are starting to curl up because they are getting old.. and loved! They go perfectly with the rest of my decor so I don't what to have to go find new ones.
I was in the same position a few months ago! Some people told me to use double-sided tape but that only worked on hard floors.. and that wasn't even every good. I ended up getting Ruggies or Ruggie Rug Grippers - they have worked great for me to secure mats on both hard floors and even on carpets. They're also reusable- if you just rinse it off, they will easily restick to a new rug.
You can go for various rug pads available online.Another option would be use Two sided tape and paste in on floor and the next step is stick a Velcro on the tape and the other part of the Velcro on bottom edge of the rug. This will surely help your old rug from curling up from edges.
