Existence and nonrelativistic limit of ground states to nonlinear Dirac equation

Abstract

This paper explores the existence and properties of ground states, including both energy and action ground states, for nonlinear Dirac equations with power-type potentials. \begin{equation*} -i c\sum\limits_{k=1}^3α_k\partial_k u +mc^2 β{u}- |{u}|^{p-2}{u}=ω{u}. \end{equation*} We establish the existence of energy ground states and demonstrate that as the speed of light approaches infinity, both energy and action ground states converge to their counterparts in the nonlinear Schrödinger equation. Furthermore, we characterize the convergence rate of the ground state energy and investigate the equivalence between action and energy ground states.