They make little clips that you can put on them underneath the mattress to make it stay. Look at Walmart or Bed Bath and Beyond
Velcro could work! Here's a simple set of instructions:
I had the same problem. I used clips, and they do work well. However, I later received a set of Casper Sheets, and the elastic is amazing! I do not need to use the clips with these sheets, though they are a bit pricey. They are available at Target online.