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: $-i\hbarα\cdot\nabla w+aβw+V(x)w=g(|w|)w \,.$ 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 $Λ_j\subset\mathbb{R}^3$ such that $-a<\min_{Λ_j} V=V(x_j)<\min_{\partialΛ_j}V$, $x_j\inΛ_j$, 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.

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