Existence and nonrelativistic limit of ground states to nonlinear Dirac equation

TL;DR

This paper studies the existence of ground states for nonlinear Dirac equations with power potentials and shows their convergence to nonlinear Schrödinger equation states as the speed of light increases.

Contribution

It establishes the existence of energy ground states for nonlinear Dirac equations and analyzes their convergence to Schrödinger ground states in the nonrelativistic limit.

Findings

Existence of energy ground states for nonlinear Dirac equations.

Convergence of ground states to nonlinear Schrödinger equation states as c→∞.

Characterization of convergence rate and ground state equivalence.

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}^3\alpha_k\partial_k u +mc^2 \beta {u}- |{u}|^{p-2}{u}=\omega {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 Schr\"odinger equation. Furthermore, we characterize the convergence rate of the ground state energy and investigate the equivalence between action and energy ground states.