## Relative Morse Index and Multiple Solutions for Asymptotically Linear Dirac Equation

Shan Yuan,

School of Mathematics, Nanjing Audit University, Nanjing 211815

 基金资助: 国家自然科学基金.  11701285江苏省自然科学基金.  BK20161053

 Fund supported: The NSFC.  11701285National Science Foundation of Jiangsu Province.  BK20161053

Abstract

This paper is concerned with the existence and multiplicity of periodic solutions for Dirac equation. We will establish a relative Morse index theory to classify the associated linear Dirac equation. Under a general twist condition for the nonlinear part via the relative Morse index, existence of multiple solutions are obtained.

Keywords： Relative Morse index ; Periodic solution of Dirac equation ; Twist condition

Shan Yuan. Relative Morse Index and Multiple Solutions for Asymptotically Linear Dirac Equation. Acta Mathematica Scientia[J], 2023, 43(1): 69-81 doi:

## 1 引言

$$$\label{orignal Dirac equation} -{\rm i}\Sigma_{k=1}^3\alpha_k\partial_k x+a\beta x +V(t) x =G'(t,x ),$$$

$\beta= \left(\begin{array}{ccccc} I&&0{} \\ 0&~&-I{} \\ \end{array}\right),\ \ \alpha_k= \left(\begin{array}{ccccc} 0~&\sigma_k{} \\ \sigma_k~&0{} \\ \end{array}\right), \ \ k=1,2,3,$

$\sigma_1= \left(\begin{array}{ccccc} 0~&1{} \\ 1~&0{} \\ \end{array}\right),\ \ \sigma_2= \left(\begin{array}{ccccc} 0~&-{\rm i}{} \\ i~&0{} \\ \end{array}\right),\ \ \sigma_3= \left(\begin{array}{ccccc} 1~&0{} \\ 0~&-1{} \\ \end{array}\right).$

$\begin{matrix}\label{wave equation} -{\rm i}\hbar \partial_s\psi=-{\rm i}c \hbar\Sigma_{k=1}^3 \alpha_k \partial_k \psi -mc^2\beta \psi - M(t)\psi +F_{\psi}(t, \psi). \end{matrix}$

$\begin{matrix} \alpha=(\alpha_1, \alpha_2, \alpha_3), \ \ \alpha\cdot \nabla =\Sigma_{k=1}^3\alpha_k\partial_k, \end{matrix}$

$\begin{matrix}\label{Dirac equation}-{\rm i}\alpha\cdot \nabla x + a \beta x +V(t)x=G'(t,x). \end{matrix}$

Ding和Liu[10,12]建立了Dirac方程周期解的变分框架, 并利用文献[3] 中所建立的临界点定理得到非线性系统解的存在性和多重性. 其中, Ding 和Liu[10]研究了超二次和次临界系统(1.3)的周期解. 随后, 在文献[12]中研究了渐近线性情形, 他们主要做出了如下假设

($G_1$) 存在 $b_0\in C(Q, [0,\infty))$$b_0(t)$$1$ -周期的, 使得

$G'(t,x)-b_0(t)x=o(|x|), (|x|\rightarrow 0), \hbox{关于$t\in Q$一致},$

($G_2$) 存在$b_\infty\in C(Q, [0,\infty))$$b_\infty(t)是1 -周期的, 使得 G'(t,x)-b_\infty(t)x=o(|x|) (|x|\rightarrow \infty), \hbox{关于t\in Q一致}, 其中Q=[0,1]\times [0,1] \times [0,1]. 受上述工作的启发, 我们主要考虑更为一般的渐近线性条件. 首先给出记号: 记L_s(\Bbb R^4)$$4\times4$实对称矩阵的集合. 对于$L_s(\Bbb R^4)$中的任意两个矩阵$A_1$$A_2, 若A_2-A_1是半正定的, 则记A_1\leq A_2; 若A_2-A_1是正定的, 则记A_1<A_2. 对于L^\infty(Q, L_s(\Bbb R^4))中的任意两个矩阵函数A_1(t), A_2(t), 若A_1(t)\leq A_2(t) 对于几乎所有的t\in Q成立, 则记A_1\leq A_2; 若A_1\leq A_2且在Q的非零测度子集上成立A_1(t)<A_2(t)则记A_1< A_2. 考虑如下假设 (G_0) 存在A_1(t), A_2(t)\in L^\infty(Q, L_s(\Bbb R^4))$$B_{0}(t,x)\in L^\infty(Q\times \Bbb C^4, L_s(\Bbb R^4))$, 满足

$\begin{matrix}\label{g0} A_1(t)\leq B_{0}(t,x) \leq A_2(t), \ \ \forall x\in \Bbb C^4\ \hbox{和几乎所有的}\ t\in Q, \end{matrix}$

$G'(t,x)=B_{0}(t,x)x+o(|x|), ( |x|\rightarrow 0)\ \hbox{关于$Q$一致}.$

$(G_\infty)$ 存在$B_1(t)$, $B_2(t)\in L^\infty(Q, L_s(\Bbb R^4))$$B_{\infty}(t,x)\in L^\infty(Q\times \Bbb C^4, L_s(\Bbb R^4)), 满足 \begin{matrix}\label{g infty} B_1(t)\leq B_{\infty}(t,x) \leq B_2(t), \ \ \forall x\in \Bbb C^4\ \hbox{和几乎所有的}\ t\in Q, \end{matrix} 使得 G'(t,x)=B_{\infty}(t,x)x+o(|x|), (|x|\rightarrow \infty)\ \hbox{关于Q一致}. 条件(G_1)-(G_2)或(G_0)-(G_\infty)常被运用于哈密顿系统周期解的研究中, 并被称为系统非线性项的渐近线性条件. 指标理论是测量(G_1)-(G_2)或(G_0)-(G_\infty)中扭转的定量的工具. 在文献[17]中, Ekeland利用对偶变分以及凸分析针对凸哈密顿系统建立了指标理论. Conley, Zehnder和Long在文献[5,23-25]中针对辛道路建立了指标理论. Long和Zhu[26,34]利用线性算子的谱流定义了两个线性算子之间的相对Morse指标, 并重新定义了辛道路的Maslov指标. Abbondandolo[1]针对带有紧扰动的Fredholm算子建立了相对Morse指标. 对于带有紧预解集的自伴算子方程, Dong[6]利用对偶变分建立了指标理论. Liu[21]利用代数方法建立了辛道路的指标理论, 并在文献[22]中研究了几类指标理论之间的关系. 上述指标理论被广泛的应用于各类微分方程的研究中. 例如: 哈密顿系统, 椭圆微分方程, 时滞微分方程等. 在利用变分法研究的过程中, 上述系统满足 \begin{matrix}\label{333} Ax=F'(x), \ \ x\in D(A). \end{matrix} 其中A为自伴算子. 特别地, A满足如下的谱性质 \begin{matrix}\label{555} \sigma(A)=\sigma_d(A). \end{matrix} 这里\sigma(A)$$\sigma_d(A)$分别代表$A$的谱和离散谱(有限重特征值). 将该谱性质与修改后的鞍点约化相结合, Dong和Shan[15]对相应的线性方程建立了指标理论. 随后, 我们在文献[30]中利用文献[15]的指标理论研究了非线性算子方程的扭转性条件. 作为应用, 我们得到了带有Sturm-Liouvillean边值条件的渐近线性一阶哈密顿系统和渐近线性时滞微分方程解的存在性条件.

$A=-{\rm i}\alpha\cdot \nabla + a \beta +V(t)$

$I(b_0(t), b_\infty(t))\geq d(q_0^+, q_\infty).$

### 2.1 变分结构

$L^q$ -范数记为$\|\cdot\|_q$, $L^2$ -内积记为$(\cdot, \cdot)_2$.$Q=[0,1]\times [0,1] \times [0,1]$

$\begin{matrix} L^q(Q):=\{u\in L^q (\Bbb R^3,\Bbb C^4):u(x+\hat{e}_i)=u(x)\ \hbox{对几乎所有的$x$都成立},i=1, 2, 3 \}, \end{matrix}$

$A=-{\rm i}\alpha\cdot \nabla +a \beta + V$

$L^2(Q)$上的自伴算子. 记$A$的谱、离散谱和本质谱分别为$\sigma(A), \ \ \sigma_d(A)$, $\sigma_e(A)$. 由文献[12], 可得

$\sigma(A)= \sigma_d(A).$

$\begin{matrix}\label{projection 1} P_{\beta,B_0}^+=\int_0^{+\infty}{\rm d}F_{\lambda},\ \ P_{\beta,B_0}^0=\int_{-\beta}^{0}{\rm d}F_{\lambda}, \ \ P_{\beta,B_0}^-=\int_{-\infty}^{-\beta}{\rm d}F_{\lambda}. \end{matrix}$

$\begin{matrix}\label{L2decomposiition} &&L^2(Q)=L_{\beta,B_0}^+\oplus L_{\beta,B_0}^0 \oplus L_{\beta,B_0}^-, \\ &&L_{\beta,B_0}^0=P_{\beta,B_0}^0L^2(Q), L_{\beta,B_0}^{\pm}=P_{\beta,B_0}^{\pm}L^2(Q). \end{matrix}$

$|A_{B_0}|$$A_{B_0}的绝对值. 取E_{B_0}=D(|A_{B_0}|^{\frac{1}{2}})并在E_{B_0}上引入内积 (z,w)_{B_0}=(|A_{B_0}|^{\frac{1}{2}}z,|A_{B_0}|^{\frac{1}{2}}w)_2 及其导出的范数\|z\|_{B_0}=(z,w)_{B_0}^{\frac{1}{2}}. E_{B_0}有关于(\cdot, \cdot)_2$$(\cdot, \cdot)_{B_0}$的正交分解

$\begin{matrix} &&E_{B_0}=E_{\beta,B_0}^+\oplus E_{\beta,B_0}^0 \oplus E_{\beta,B_0}^-,\\ &&E_{\beta, B_0}^{\pm}=E_{B_0}\cap L_{\beta,B_0}^{\pm}\ \hbox{和}\ E_{\beta, B_0}^{0}=E_{B_0}\cap L_{\beta,B_0}^{0}.\nonumber \end{matrix}$

$E_{B_0}$上定义泛函

$\begin{matrix}\label{original functional} I_{\beta,B_0}(x)=\frac{1}{2}\|x^+\|_{B_0}^2-\frac{1}{2}\|x^0\|_{B_0}^2-\frac{1}{2}\|x^-\|_{B_0}^2 -\Phi_{B_0}(x), \ \ x\in E_{B_0}, \end{matrix}$

$\Phi_{B_0}(x)=\Phi(x)-\frac{1}{2}(B_0x,x)_2\hbox{ 和 }\ \Phi(x)=\int_Q G(t,x).$

$$$\label{abstract original functional} Ax=\Phi_{B_0}'(t,x).$$$

$x\in E_{B_0}$为方程(2.7)的解, 并令$u=|A_{B_0}|^{\frac{1}{2}}x$, 则有$u\in L^2(Q)$$u满足 \begin{matrix}\label{abstract linear DDE 1} u^+-u^0-u^-|A_{B_0}|^{-\frac{1}{2}}\Phi'_{B_0}(|A_{B_0}|^{-\frac{1}{2}}u)=0, \end{matrix} 其中u=u^+ + u^0 + u^-\in L_{\beta, B_0}^+\oplus L_{\beta,B_0}^0 \oplus L_{\beta,B_0}^-. 由(2.8)式, 可推导出 u^+-u^0-(P^+_{\beta, B_0}+P^0_{\beta, B_0})|A_{B_0}|^{-\frac{1}{2}}\Phi'_{B_0}(|A_{B_0}|^{-\frac{1}{2}}u)=0, $$\label{negative system L2} -u^- -P^-_{\beta, B_0}|A_{B_0}|^{-\frac{1}{2}}\Phi'_{B_0}(|A_{B_0}|^{-\frac{1}{2}}u)=0.$$ 此外, 根据投影算子P^-_{\beta, B_0}的定义, 可得 \|P^-_{\beta, B_0}|A_{B_0}|^{-\frac{1}{2}}\|\leq\frac{1}{\sqrt{\beta}}. 根据压缩映像原理, \forall$$u^{\ast}\in L_{\beta, B_0}^{\ast}=L_{\beta, B_0}^{+} \oplus L_{\beta, B_0}^{0}$, 方程(2.9)具有唯一解. 因此, 存在自伴算子

$$$\label{operator L} T_{\beta, \Phi}: L_{\beta, B_0}^{\ast}\rightarrow L_{\beta, B_0}^{-},$$$

$-T_{\beta, \Phi}u^{\ast} -P^-_{\beta, B_0}|A_{B_0}|^{-\frac{1}{2}}\Phi'_{B_0}(|A_{B_0}|^{-\frac{1}{2}}(u^{\ast}+T_{\beta, \Phi}u^{\ast}))=0.$

$i_{\beta,B_0}(B)=\hbox{dim } E^{-}_{\beta,B_0}(B),\nu_{\beta,B_0}(B)=\hbox{dim } E^{0}_{\beta,B_0}(B),$

$i_{\beta,B_0}(B)$$B的指标, \nu_{\beta,B_0}(B)$$B$的零维数.

(2) $i_{\beta,B_0}(B)$$\tilde{q}_{\beta, B_0, B}的Morse指标; \nu_{\beta,B_0}(B)是方程(1.3)解空间的维数, 即\nu_{\beta,B_0}(B)=\dim \ker (A-B). (3) 存在\epsilon_0>0使得\forall \epsilon\in (0, \epsilon_0], \begin{matrix} &&\nu_{\beta, B_0}(B+\epsilon)=0=\nu_{\beta, B_0}(B-\epsilon), \\ &&i_{\beta, B_0}(B-\epsilon)=i_{\beta, B_0}(B), \\ &&i_{\beta, B_0}(B+\epsilon)=i_{\beta, B_0}(B)+\nu_{\beta, B_0}(B). \end{matrix} (4) 对于任意的B\in L^\infty(Q, L_s(\Bbb R^4))$$\nu_{\beta, B_0}(B)=0$$\beta>0充分大, 则有 ( \tilde{q}_{\beta, B_0,B}(x^{\ast},x^{\ast}))^{1\over 2}\ \mbox{和}\ (- \tilde{q}_{\beta, B_0,B}(x^{\ast},x^{\ast}))^{1\over 2} 分别为E^+_{\beta, B_0}(B)$$E^-_{\beta, B_0}(B)$上的等价范数.

(5) 对于任意的$B_1, B_2\in L^\infty(Q, L_s(\Bbb R^4))$满足 $B_1\leq B_2$,

$i_{\beta,B_0}(B_2)-i_{\beta,B_0}(B_1)=\Sigma_{\lambda\in [0,1)}\nu_{\beta,B_0}(B_1+\lambda(B_2-B_1)).$

$I(B_1, B_2)=\sum_{\lambda\in [0,1)}\nu((1-\lambda)B_1+\lambda B_2),$

$I(B_1, B_2)=I(B_1, Kid)-I(B_2, Kid),$

$i_{\beta,B_0}(B_2)-i_{\beta,B_0}(B_1)=I(B_1, B_2).$

$i_{\beta,B_0}(Kid)= i_{\beta,B_0}(B_1)+I(Kid, B_1) = i_{\beta,B_0}(B_2)+I(Kid, B_2).$

$i_{\beta,B_0}(B_2)- i_{\beta,B_0}(B_1)= I(Kid, B_1)-I(Kid, B_2)=I(B_1, B_2).$

$I(A_1,A_2)=0, \ \ \nu(A_2)=0; \ \ I(B_1,B_2)=0, \ \ \nu(B_2)=0.$

$\begin{matrix} &&i_{\beta,B_0}(A_1)=i_{\beta,B_0}(A_2), \ \ \nu(A_1)=\nu(A_2)=0, \\ &&i_{\beta,B_0}(B_1)=i_{\beta,B_0}(B_2), \ \ \nu(B_1)=\nu(B_2)=0. \end{matrix}$

$I(A_i, B_j)=I(A_k, B_l), \ \ \forall \ \ i,j,k,l=1,2.$

$|R_1'(t,x)|\leq \epsilon |x|+C_\epsilon |x|^{p-1},$

$|R_1(t,x)|\leq \frac{1}{2}\epsilon |x|^2+\frac{C_\epsilon}{p}|x|^{p},$

$\begin{matrix}\label{inequality at 0} &&\int_{Q}R_1(t,x){\rm d}t\geq\frac{1}{2}(A_2(t)x,x)_2-\frac{1}{2}\epsilon \|x\|_2^2-\frac{C_\epsilon}{p}\|x\|_p^{p}, \\ &&\int_{Q}R_1(t,x){\rm d}t\leq \frac{1}{2}(A_2(t)x,x)_2+\frac{1}{2}\epsilon \|x\|_2^2+\frac{C_\epsilon}{p}\|x\|_p^{p}. \end{matrix}$

$\begin{matrix} \tilde{I}_{\beta, A_2+\epsilon}(x^{\ast})&=&I_{\beta, A_2+\epsilon}(x^{\ast}+\tilde{T}_{\beta, \Phi}x^{\ast}) \\ &\geq& \frac{1}{2}\|x^+\|^2-\frac{1}{2}\|x^0\|^2 -\frac{1}{2}\|\tilde{L}_{\beta, \Phi}x^{\ast}\|^2 -C\|x^{\ast}+\tilde{L}_{\beta, \Phi}x^{\ast}\|^p. \end{matrix}$

$E_2= E^+_{A_2+\epsilon}$, 可得$x^{\ast}=x^+$对于任意的$x^{\ast}\in E_2$都成立. 又易得$\|\tilde{T}_{\beta, \Phi}\|\leq \frac{M}{\beta+M}$, 其中$M$在条件($G_0$)中给出. 则有

$\tilde{I}_{\beta, A_2+\epsilon}(x^{\ast})\geq \frac{1}{2}\|x^{\ast}\|^2 - \frac{1}{2}(\frac{M}{\beta+M})^2\|x^{\ast}\|^2-C(1+(\frac{M}{\beta+M})^2) \|x^{\ast}\|^p.$

$\begin{matrix}\label{inequation at infinity} R(t,x)\geq \frac{1}{2}((B_1(t)-2\epsilon)x,x)-C_1. \end{matrix}$

$\hbox{当}\ \|x^{\ast}\|\rightarrow \infty\ \hbox{时}, \tilde{I}_{\beta, A_2+\epsilon}(x^{\ast})\rightarrow -\infty.$

$\begin{matrix} \tilde{I}_{\beta, A_2+\epsilon}(x^{\ast})&\leq& \frac{1}{2}\|x^+\|^2-\frac{1}{2}\|x^0\|^2-\frac{1}{2}\|\tilde{T}_{\beta, \Phi}x^{\ast}\|^2 \\ &&-\frac{1}{2}((B_1-2\epsilon)x^{\ast}+\tilde{L}_{\beta, \Phi}x^{\ast},x^{\ast}+\tilde{L}_{\beta, \Phi}x^{\ast})_2 \\ &&+\frac{1}{2}((A_2+\epsilon)x^{\ast}+\tilde{L}_{\beta, \Phi}x^{\ast},x^{\ast}+\tilde{L}_{\beta, \Phi}x^{\ast})_2+C_1 \\ &=&q_{\beta,A_2+\epsilon,B_1-2\epsilon }(x^{\ast}+\tilde{T}_{\beta,\Phi}x^{\ast},x^{\ast}+\tilde{T}_{\beta,\Phi}x^{\ast})+C_1 \\ &\leq &\tilde{q}_{\beta,A_2+\epsilon,B_1-2\epsilon }(x^{\ast}, x^{\ast})+C_1. \end{matrix}$

$\tilde{q}_{\beta,A_2+\epsilon,B_1-2\epsilon }(x^{\ast}, x^{\ast})\leq -C_2\|x^{\ast}\|^2, \ \ \forall \ \ x^{\ast}\in E^-_{A_2+\epsilon}(B_1-2\epsilon),$

$\tilde{I}_{\beta, A_2+\epsilon}(x^\ast)\leq -C_2\|x^{\ast}\|^2+C_1.$

$\begin{matrix}\label{12} -x_j^-= P_{\beta, A_2+\epsilon}^- \Phi'_{A_2+\epsilon}(x_j). \end{matrix}$

$\begin{matrix}\label{13} o(1)=(x_j^+-x_j^0-x_j^-, y ) -(\Phi'_{A_2+\epsilon}(x_j), y), \ \ \forall y\in E_{A_0+\epsilon}. \end{matrix}$

$\hbox{在$E_{A_2+\epsilon}$中}, v_j\rightharpoonup v,\ \hbox{且在$L^2(Q) $中}\ v_j\rightarrow v.$

$\begin{matrix} o(1)&&=\frac{\tilde{I}'_{\beta, A_0+\epsilon}(x_j^{\ast})}{\|x_j \|}y \\ &&=(v_j^+-v_j^0-v_j^-, y) -\int_Q \frac{G'(t,x_j)y}{\|x_j \|}+((A_0+\epsilon)v_j, y )_2. \end{matrix}$

$\int_Q \frac{G'(t,x_j)\varphi}{\|x_j \|} \rightarrow \int_Q A_{\infty}(t)v\varphi, \ \ \forall \varphi \in L^2(Q).$

$A_{A_2+\epsilon}v-A_\infty v +(A_2+\epsilon)v=0,$

$(-{\rm i}\alpha\cdot \nabla +a \beta + V)v -A_\infty v =0.$

$\nu(A_\infty)=0.$

$\begin{matrix} o(1)&&=(\frac{I_{\beta,A_2+\epsilon}'(x_j)}{\|x_j\|}, v_j^+-v_j^0) \\ &&=o(1)+1-\int_{Q}\frac{G'(t,x_j)}{|x_j|}|v_j|(v_j^+-v_j^0-v_j^-)+ ((A_2+\epsilon)v_j,v_j^+-v_j^0-v_j^- )_2 \\ && \geq o(1)+ 1-C\|v_j\|_2^2=1+o(1). \end{matrix}$

$-x_j^-=P_{\beta, A_2+\epsilon}^{-}\tilde{I}'_{\beta, A_2+\epsilon}(x_j),$

$\begin{matrix} o(1)&=&(\tilde{I}_{\beta,A_2+\epsilon}'(x_j^{\ast}), w_j^+-w_j^0) \\ &=&o(1)+\|w_j\|^2-\int_{Q}G'(t,x_j)(w_j^+-w_j^0-w_j^-) \\ && + ((A_2+\epsilon)x_j,w_j^+-w_j^0-w_j^- )_2 \\ &=& o(1)+\|w_j\|^2-\int_{Q}\frac{G'(t,x_j)}{|x_j|}|x_j|(w_j^+-w_j^0-w_j^-) \\ && + ((A_2+\epsilon)w_j,w_j^+-w_j^0-w_j^- )_2 \\ &=&o(1)+\|w_j\|^2, \end{matrix}$

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