Strongly localized semiclassical states for nonlinear Dirac equations

We study semiclassical states of the nonlinear Dirac equation $$-i\mathrm{\hslash }{\mathrm{\partial }}_t\psi =ic\mathrm{\hslash }\sum _{k=1}^3{\alpha }_k{\mathrm{\partial }}_k\psi -m{c}^2\beta \psi -M(x)\psi +f(\mathrm{\mid }\psi \mathrm{\mid })\psi ,t\in \mathbb{R},x\in {\mathbb{R}}^3,$$ where $V$ is a bounded continuous potential function and the nonlinear term $f(\mathrm{\mid }\psi \mathrm{\mid })\psi$ 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.