On the concentration of semi-classical states for a nonlinear Dirac-Klein-Gordon system

TL;DR

This paper investigates the semi-classical behavior of a nonlinear Dirac-Klein-Gordon system, demonstrating the existence and concentration of ground states around potential maxima as the semi-classical parameter approaches zero.

Contribution

It establishes the existence of a family of ground states for small semi-classical parameter and shows their concentration behavior, using a variational approach with a novel cutoff technique.

Findings

Ground states exist for small semi-classical parameter.

Ground states concentrate around maxima of the nonlinear potential.

Method overcomes lack of convexity in nonlinearities.

Abstract

In the present paper, we study the semi-classical approximation of a Yukawa-coupled massive Dirac-Klein-Gordon system with some general nonlinear self-coupling. We prove that for a constrained coupling constant there exists a family of ground states of the semi-classical problem, for all $\hbar$ small, and show that the family concentrates around the maxima of the nonlinear potential as $\hbar\to0$. Our method is variational and relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.