|MSC:||35Q40||Equations from quantum mechanics|
|35J50||Variational methods for elliptic systems|
|46E35||Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems|
We prove the critical Dirac-Sobolev inequality for p ∈ (1, 3). It follows that the Dirac Sobolev spaces are equivalent to classical Sobolev spaces if and only if p ∈ (1, 3). We prove the cocompactness of Lp* (R3 ) in H1,p (R3 ). As an application, we prove the existence of minimizers to a class of isoperimetric problems.