Localized concentration of semi-classical states for nonlinear Dirac equations

The present paper studies concentration phenomena of semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: $$-i\hbar\alpha\cdot\nabla w+a\beta w+V(x)w=g(|w|)w \,.$$ 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 $\Lambda_j\subset\mathbb{R}^3$ such that $-a<\min_{\Lambda_j} V=V(x_j)<\min_{\partial\Lambda_j}V$, $x_j\in\Lambda_j$, then the $k$-families of solutions $w_\hbar^j$ concentrates around $x_j$ as $\hbar\to 0$, 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.