Strongly localized semiclassical states for nonlinear Dirac equations

TL;DR

This paper investigates semiclassical standing wave solutions of the nonlinear Dirac equation that concentrate near critical points of the potential, introducing a novel variational approach for strongly indefinite functionals.

Contribution

It develops a new variational method to find localized solutions of the nonlinear Dirac equation, overcoming limitations of traditional techniques.

Findings

Existence of semiclassical states concentrating near potential critical points

Development of a variational approach for strongly indefinite functionals

Handling of superlinear and critical growth nonlinearities

Abstract

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_t\psi = ic\hbar\sum_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|\psi|)\psi$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schr\"odinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=s^p$. We develop a variational method for the strongly indefinite functional associated to the problem.