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

TL;DR

This paper proves the existence and multiplicity of normalized solutions for nonlinear Dirac equations with non-autonomous nonlinearities, using perturbation methods, Lyapunov-Schmidt reduction, and Ljusternik-Schnirelmann theory.

Contribution

It introduces new techniques to establish existence, multiplicity, and bifurcation of solutions for Dirac equations with non-autonomous nonlinearities, expanding prior results.

Findings

Existence of normalized solutions under general nonlinear assumptions

Multiple solutions established via topological methods

Bifurcation phenomena identified for the equations

Abstract

In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*} where $u: \mathbb{R}^{3}\rightarrow \mathbb{C}^{4}$, $m>0$ 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.