Localized concentration of semi-classical states for nonlinear Dirac equations

TL;DR

This paper investigates how solutions to a semi-classical nonlinear Dirac equation concentrate around local minima of the potential, especially when nonlinearities are super-linear or asymptotically linear, using variational methods and penalization techniques.

Contribution

It demonstrates concentration of solutions around potential minima for nonlinear Dirac equations with general nonlinearities, employing novel penalization methods for strongly indefinite functionals.

Findings

Solutions concentrate around local minima of the potential.

Multiple solution families localize at different potential minima.

The penalization technique effectively handles the indefinite energy functional.

Abstract

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.