On Coupled Dirac Systems under Boundary Condition

TL;DR

This paper investigates the existence of solutions for coupled Dirac systems on compact spin manifolds with boundary, employing fractional Sobolev spaces to handle superquadratic nonlinearities.

Contribution

It introduces an analytic framework using fractional Sobolev spaces to establish solution existence for coupled Dirac systems with superquadratic growth.

Findings

Existence results for solutions under superquadratic nonlinearities.

Application of fractional Sobolev space techniques.

Analysis on Dirac systems with boundary conditions.

Abstract

In this article we study the existence of solutions for the Dirac systems \begin{equation}\label{e:0.1} \left\{ \begin{array}{c} Pu=\frac{\partial H}{\partial v}(x,u,v) \quad\hbox{on} \ M, Pv=\frac{\partial H}{\partial u}(x,u,v) \quad\hbox{on} \ M, B_{\text{CHI}}u= B_{\text{CHI}}v=0\quad\hbox{on} \ \partial M \end{array} \right. \end{equation} where $M$ is an $m$-dimensional compact oriented Riemannian spin manifold with smooth boundary $\partial M$, $P$ is the Dirac operator under the boundary condition $B_{\text{CHI}}u= B_{\text{CHI}}v=0$ on $\partial M$, $ u,v\in C^{\infty}(M,\Sigma M)$ are spinors. Using an analytic framework of proper products of fractional Sobolev spaces, the solutions existence results of the coupled Dirac systems are obtained for nonlinearity with superquadratic growth rates.