Nonlinear Dirac Equation on Compact Spin Manifold with Chirality Boundary Condition

Abstract

In this paper, we study the following nonlinear boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} D\psi -a(x)\psi =f(x,\psi )+\epsilon h(x,\psi )&{} \quad \hbox {on }M \\ B_{CHI}\psi =0 &{} \quad \hbox {on }\partial M \end{array} \right. \end{aligned}$$

(D)

where M is a compact Riemannian spin manifold of dimension$m\ge 2$ and the boundary $\partial M$ has non-negative mean curvature, and D is the Dirac-Atiyah-Singer operator. Let S(M) denote the spinor bundle on M and $\psi :M\rightarrow S(M)$ be a section. a(x) is a scalar field on M, $\epsilon \in \mathbb {R}$. $f(x,\psi )$, $h(x,\psi ):M\times S_{m}\rightarrow S_{m}$ are two nonlinear map, and $B_{CHI}$ is the chirality boundary operator. Under some mild assumptions on a, f, and h, we obtain the ground state solution of (D) with $\epsilon =0$ and infinitely many large norm solutions of (D) with $\epsilon \in \mathbb {R}$ and infinitely many small energy solutions of (D) with $\epsilon >0$ by using variational methods.