  Quantum chromodynamics (QCD) is the theory of strong interaction and accounts
for the internal structure of hadrons. Physicists introduced phe- nomenological
models such as the M.I.T. bag model, the bag approximation and the soliton bag
model to study the hadronic properties. We prove, in this paper, the existence
of excited state solutions in the symmetric case and of a ground state solution
in the non-symmetric case for the soliton bag and the bag approximation models
thanks to the concentration compactness method. We show that the energy
functionals of the bag approximation model are Gamma -limits of sequences of
soliton bag model energy functionals for the ground and excited state problems.
The pre- compactness, up to translation, of the sequence of ground state
solutions associated with the soliton bag energy functionals in the
non-symmetric case is obtained combining the Gamma -convergence theory and the
concentration-compactness method. Finally, we give a rigorous proof of the
original derivation of the M.I.T. bag equations done by Chodos, Jaffe, Johnson,
Thorn and Weisskopf via a limit of bag approximation ground state solutions in
the spherical case. The supersymmetry property of the Dirac operator is the key
point in many of our arguments.
