In this paper we gain some results on the regularity and also the blow-up rates and constants of solutions to the equation u − up = 0 under some different situations. The blow-up rate and blow-up constant of u(2n) are (p− 2n+ 2) and (±) (p− 2n+ 2)· Πn−1 i=0 (p−2i+ 2) (p− 2i+ 1)E (0)p/2 respectively; blow-up rate and blow-up constant of u(2n+1) are (p− 2n+ 1) and (p−2n+2) Πn−1 i=0 (p−2i+2)· (p−2i+1)E (0)p−n respectively, where E (0) = u (0)2 − 2p+1u(0)p+1
