Where can you catch uxie mesprit and azelf in Pokemon diamond?

You can catch them in their caverns.Each lake has it's own cavern,and after you free Uxie,Mesprit and Azelf you will find them in there caverns at each lake. but only after you've got Dialga/Palkia.

You can find Uxie at Lake Acuity in Snowpoint city and Azelf in Lake Verity by Pastoria and Sunnyshore City. Mespirit is somewhat different. The Last lake, is where you will find Mespirit but once you contact with it it runs. then it is released in the area of Sinnoh. Then you can use the App Marking Map and the thing that looks like a pikachu head is the Pokemon or another rare type Pokemon about in Sinnoh. Good Luck catching because your best bet to cath either one is to have a Pokemon with a move that won;t let your opponent run away. Otherwise, Mespirit will run like heck over and over and over and over. Good Luck. That's why i traded mine. I hope this helped some!!!