If you are on the lookout for unique things for your home, try QL inredning. They have everything from kitchenware to bathroom rugs to furniture...not the cheapest place to buy these things of course, but it's definitely contemporary design from Sweden and other countries.
What to pay: From very little to very much, depending on what you buy.
Kupan is run by the Red Cross, and its Luleå store recently (spring '05) moved to a better location. Now it is a quite large second-hand/thrift shop, with everything from clothes to books. If you need to pick up cheap kitchen equipment this could be a good place to start, if you don't mind things that look like they're straight out of the 70s...
What to pay: Between 1 and 50 crowns, depending on what you buy. Some clothes could be slightly more expensive.
Europe's first indoor shopping mall was built in Luleå in 1955, and the building is still there...it's called Shopping, and it's located on Storgatan right in the center of Luleå. The architect behind it was the well-known Ralph Erskine, and he was inspired by boat/ship design, so inside Shopping you will see several materials and designs that remind you of boats.
Inside the mall are many different shops. For example the Swedish chains Indiska, H&M and Kapp-Ahl, but also smaller shops that sell clothes, shoes or accessories.
Yeah, Luleå's "Chinatown" is actually one shop with that name...though it is located next to a Chinese restaurant, so maybe it will grow. :-)
The shop imports things from China and has a variety of items such as jewellery, clothes, home decor and accessories. Prices are quite low by Swedish standards.
This little store in the old city has very interesting items locally made and original. Some clothing, decoration or just souvenirs.