Solutions of nonlinear periodic Dirac equations with periodic potentials

Abstract

This paper is concerned with the nonlinear Dirac equation $ -i\sum_{k = 1}^{3}\alpha_{k}\partial_{k}u + [V(x)+a]\beta u + \omega u = f(x, u) $ in $ \mathbb{R}^3 $, where $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ f(x, u) $ is asymptotically linear and superlinear as $ |u|\rightarrow \infty $. Under weaker assumptions on $ f $, we obtain the existence of one nontrivial solution for the above equation.