This subject belongs to a weak topology. Generally Sobolev space is unbounded and not compact. But it is compactly embedded in L^p space. This is a big topic and I think a good references :

1. Lawrence Evans , PDES

2. Haim Brezis, Functional Analysis, Sobolev spaces and PDEs

Hi Habeeb, If it is compactly embedded in L^p space, can we apply Shauder's Fixed Point Theorem for a continuous operator defined on it?