Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: $$\begin{array}{cccc}& -i\epsilon c{\sigma }_1{\mathrm{\partial }}_{x}u+m{c}^2{\sigma }_{3}u+V(x)u=f(\mathrm{\mid }u\mathrm{\mid })u,x\in \mathcal{G},& & \text{(P)}\end{array}$$ where $V:\mathcal{G}\to \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ are continuous, $\epsilon >0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here ${\sigma }_1,{\sigma }_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\epsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\epsilon \to 0$, their concentration points approach the set of global minima of $V$.