Localized concentration of semi-classical states for nonlinear Dirac equations

Abstract

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 ${\Lambda }_j\subset {\mathbb{R}}^3$ 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 $w_{\hslash }^j$ concentrates around $x_j$ as $\hslash \to 0$, 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.

References in corpus (2)

• On the concentration of semi-classical states for a nonlinear Dirac-Klein-Gordon system
• On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system