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Localized concentration of semi-classical states for nonlinear Dirac equations. (English) Zbl 1309.35102

Summary: The present paper studies concentration phenomena of the 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 \(w_\hbar^j\) concentrate 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.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81V10 Electromagnetic interaction; quantum electrodynamics
35P15 Estimates of eigenvalues in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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