Strongly localized semiclassical states for nonlinear Dirac equations

Abstract

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_tψ= ic\hbar\sum_{k=1}^3α_k\partial_kψ- mc^2βψ- M(x)ψ+ f(|ψ|)ψ,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] 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 Schrödinger 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.