Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities

Abstract

In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3α_k\partial_k u+mβu=f(x,|u|)u+ωu, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*} where $u:{\mathbb{R}}^3\to {\mathbb{C}}^4$, $m>0$ is the mass of the Dirac particle, $\omega \in \mathbb{R}$ arises as a Lagrange multiplier, ${\partial }_k=\frac{\partial }{\partial x_k}$, ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are $4\times 4$ Pauli-Dirac matrices, $a>0$ is a prescribed constant, and $f(x,\cdot )$ has several physical interpretations that will be discussed in the Introduction. Under general assumptions on the nonlinearity $f$, we prove the existence of $L^2$-normalized solutions for the above nonlinear Dirac equations by using perturbation methods in combination with Lyapunov-Schmidt reduction. We also show the multiplicity of these normalized solutions thanks to the multiplicity theorem of Ljusternik-Schnirelmann. Moreover, we obtain bifurcation results of this problem.