Nonlinear Dirac Equation on Compact Spin Manifold with Chirality Boundary Condition

In this paper, we study the following nonlinear boundary value problem D \(\begin{aligned} \left\{ \begin{array}{ll} D\psi -a(x)\psi =f(x,\psi )+\epsilon h(x,\psi )&{} \quad \hbox {on }M \\ B_{CHI}\psi =0 &{} \quad \hbox {on }\partial M \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}D\psi -a(x)\psi =f(x,\psi )+ϵh(x,\psi )\hfill & \text{on}M\hfill \\ B_{\mathit{CHI}}\psi =0\hfill & \text{on}\partial M\hfill \end{array}\right)\end{array}$$ where M is a compact Riemannian spin manifold of dimension \(m\ge 2\) $m\ge 2$ and the boundary \(\partial M\) $\partial M$ has non-negative mean curvature, and D is the Dirac-Atiyah-Singer operator. Let S(M) denote the spinor bundle on M and \(\psi :M\rightarrow S(M)\) $\psi :M\to S(M)$ be a section. a(x) is a scalar field on M, \(\epsilon \in \mathbb {R}\) $ϵ\in \mathbb{R}$ . \(f(x,\psi )\) $f(x,\psi )$ , \(h(x,\psi ):M\times S_{m}\rightarrow S_{m}\) $h(x,\psi ):M\times S_m\to S_m$ are two nonlinear map, and \(B_{CHI}\) $B_{\mathit{CHI}}$ is the chirality boundary operator. Under some mild assumptions on a, f, and h, we obtain the ground state solution of (D) with \(\epsilon =0\) $ϵ=0$ and infinitely many large norm solutions of (D) with \(\epsilon \in \mathbb {R}\) $ϵ\in \mathbb{R}$ and infinitely many small energy solutions of (D) with \(\epsilon >0\) $ϵ>0$ by using variational methods.