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

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 $\mathrm{\hslash }$ small, and show that the family concentrates around the maxima of the nonlinear potential as $\mathrm{\hslash }\to 0$. Our method is variational and relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.