Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations

The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: -iħα·∇w+aβw+V(x)w=g(|w|)w.Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains Λj⊂R3 such that -a<minΛjV=V(xj)<min∂ΛjV , xj∈Λj , then the k-families of solutions wħj concentrate around x_j as ħ→0 , respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions.