Where are exactly NP-complete problems if P=NP? They will be definitely in P, but will they be P-complete?
Most NP-complete problems are NP-complete under LOGSPACE reductions; that is, you can take an arbitrary problem in NP, and using a LOGSPACE algorithm, reduce it to your problem. Any NP-complete problem under LOGSPACE reductions will also be P-complete under LOGSPACE reduction, even if P=NP, as you can use the same procedure to reduce a P-complete problem to your problem with a LOGSPACE reduction (since P $\in$ NP).
Some problems are only NP-complete under P reductions. In this case, it is not clear whether or not they are P-complete. I don't know of any NP-complete problems which are not also known to be P-complete, but I haven't tried searching to see whether anybody has investigated this.