How close to LA?
If you go live in around Pomona (about 20 miles away from LA) then rent is cheaper. There's a few Asian markets there, notable a large Vietnamese supermarket (but it caters to all Asians).
Apartments with one or two bedrooms range from around 700 to 900 (average). It depends on how nice the place is.
By the way, Pomona is like in the center of the LA area, you have easy access to Los Angeles, Orange Country and the Inland Empire (Ontario, Riverside, San Bern.). It's right next to the San Gabriel Valley which has a huge Asian population and lots of Asian stores.