## Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity

Zhang Qingye,1, Xu Bin,2

 基金资助: 国家自然科学基金.  11761036国家自然科学基金.  11201196江西省自然科学基金.  20171BAB211002

 Fund supported: the NSFC.  11761036the NSFC.  11201196the NSF of Jiangxi Province.  20171BAB211002

Abstract

In this paper, we mainly study multiple periodic solutions for a class of stationary Dirac equations with local nonlinearity via variational methods. Under some fairly mild conditions on the nonlinearity, we show that the existence of a sequence of nontrivial periodic solutions is rather a local phenomenon, which is only forced by the sublinearity of the nonlinearity near the origin. Meanwhile, some recent results from the literature are generalized and essentially improved.

Keywords： Dirac equation ; Periodic solution ; Variational method ; Local nonlinearity

Zhang Qingye, Xu Bin. Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity. Acta Mathematica Scientia[J], 2021, 41(4): 1013-1023 doi:

## 1 引言与主要结果

$$$-{\rm i}\sum\limits_{k = 1}^3\alpha_k\partial_k u+a\beta u+V(x)u = H_u(x, u),$$$

$$$-{\rm i}\alpha\cdot \nabla u+a\beta u+V(x)u = H_u(x, u).$$$

(V$_0$) $V\in C({\Bbb R}^3, {\Bbb R})$并且$V(x)$关于每个$x_k$ ($k = 1, \, 2, \, 3$)$1$ -周期的.

(H$_1$) $H\in C^1({\Bbb R}^3\times B_\delta(0), {\Bbb R})$, 其中$B_\delta(0): = \{u\in {\Bbb C}^4\mid |u|< \delta\}$, 并且$H(x, u)$关于$u$是偶的且关于每个$x_k$ ($k = 1, \, 2, \, 3$)$1$ -周期的.

(H$_2^{\pm}$) $\lim\limits_{|u|\to 0}H(x, u)/|u|^2 = \pm\infty$关于$x\in Q = [0, 1]\times[0, 1]\times[0, 1]$一致地成立.

(H$_3^{\pm}$) 对任意$(x, u)\in Q\times (B_\delta(0)\setminus \{0\}$), 都有$\pm (2H(x, u)-H_u(x, u)\cdot u)>0$.

$$${\it\Phi}(u) = \int_Q\Big[\frac{1}{2}(-{\rm i}\alpha\cdot\nabla u+a\beta u+V(x)u)\cdot u-H(x, u)\Big]{\rm d}x.$$$

$$$\|u\|_p\le \tau_p\|u\|.$$$

$$$\big|\widetilde{H}_u(x, u)\big|\le c_1$$$

$$$\big|\widetilde{H}(x, u)\big|\le c_2.$$$

$$${\it\Psi}(u): = \int_Q\widetilde{H}(x, u){\rm d}x$$$

$$$\langle{\it\Psi}'(u), v\rangle = \int_Q\widetilde{H}_u(x, u)\cdot v{\rm d}x,$$$

首先, 由式(2.4)和引理2.1可知, 通过式(2.5)所给的泛函${\it\Psi} $$E 上是良好定义的. 现设在 E$$ u_n\rightharpoonup u$, 于是由引理2.1可知在$L_{T}^1(Q) $$u_n\to u . 如有需要过渡到一个子列, 可设在 Q 中几乎处处有 u_n\to u . 由式(2.4) 和Lebesgue控制收敛定理可知, 下式成立. 因此, {\it\Psi} 是弱连续的. 接下来证明 {\it\Psi}$$ E$上是Gâteaux可微的. 对任意给定的$u\in E$, 定义一个相关的线性算子${\cal J}(u):E\to {\Bbb R}$如下

$$$\langle{\cal J}(u), v\rangle = \int_Q\widetilde{H}_u(x, u)\cdot v{\rm d}x$$$

$$$|\langle{\cal J}(u), v\rangle|\leq \int_Q\left|\widetilde{H}_u(x, u)\right||v|{\rm d}x\leq c_1\|v\|_1\leq c_1\tau_1\|v\|$$$

$\begin{eqnarray} \lim\limits_{t\to 0} \frac{{\it\Psi}(u+tv)-{\it\Psi}(u)}{t}& = &\lim\limits_{t\to 0}\int_Q\widetilde{H}_u(x, u+\theta(x)tv)\cdot v{\rm d}x{} \\ & = &\int_Q\widetilde{H}_u(x, u)\cdot v{\rm d}x = \langle{\cal J}(u), v\rangle, \end{eqnarray}$

$$$\int_Q\left|\widetilde{H}(x, u_n)-\widetilde{H}(x, u)\right|^2{\rm d}x\to 0.$$$

$$$-{\rm i}\alpha\cdot \nabla u+a\beta u+V(x)u = \widetilde{H}_u(t, u),$$$

$\begin{eqnarray} {\it\Phi}(u)& = &\frac{1}{2}\big(\|u^+\|^2-\|u^-\|^2\big)-{\it\Psi}(u){} \\ & = &\frac{1}{2}\big(\|u^+\|^2-\|u^-\|^2\big)-\int_Q\widetilde{H}(x, u){\rm d}x, \end{eqnarray}$

$E$是一个Banach空间且$A $$E 的一个子集. 称 A 为对称的, 若 u\in A 蕴含 -u\in A . \Gamma$$ E$中所有不含$0$的闭对称子集所构成的集族. 对任意的$A\in \Gamma$, 定义$A$的亏格$\gamma(A)$为使得存在从$A $${\Bbb R}^m\setminus\{0\} 的奇连续映射的最小的整数 m . 若不存在这样的整数 m , 则定义 \gamma(A) = \infty . 此外, 规定 \gamma(\phi) = 0 . 定理 2.1[23, Theorem 1.2] 设 E 是一个 \rm Banach 空间, \{E_n\}_{n = 0}^{\infty}$$ E$的一列无穷维闭子空间使得$E_0\subset E_1\subset E_2\subset\cdots$, $E_0 $$E_n 中的余维数 d_n 有限且 E = \overline{\bigcup\limits_{n = 0}^{\infty}E_n} , 并且 {\it\Phi}\in C^1(E, {\Bbb R}) . 假设 {\it\Phi} 是偶的且关于 \{E_n\}_{n = 0}^{\infty} 满足 {\rm (PS)}^* 条件, {\it\Phi}|_{E_0} 下有界且满足 \rm(PS) 条件, 并且 {\it\Phi}(0) = 0 . 若存在 n_0>0 使得对任意 k\in {\Bbb N} , 存在 \epsilon_k>0 , \rho_k>0 满足 \rho_k\to 0 , 以及对称子集 A_k\subset \{u\in E\mid\|u\| = \rho_k\} 使得 \gamma(E_n\cap A_k) = d_n+k 且对所有的 n\geq n_0 都有 { }\sup_{E_n\cap A_k}{\it\Phi}<-\epsilon_k , 则以下结论至少有一个成立. (i) 存在一列临界点 \{u_k\} 使得对所有 k\in {\Bbb N} 都有 {\it\Phi}(u_k)<0 , 并且当 k\to \infty 时, \|u_k\|\to 0 . (ii) 存在 r>0 使得对任意 0<b<r 都存在临界点 u 满足 \|u\| = b$$ {\it\Phi}(u) = 0$.

## 3 主要结果的证明

$E = E^+\oplus E^0\oplus E^-$为式(2.1)中所给的分解式, 其中$E^0$是有限维的. 令

$$$E_n = E_1\oplus {\rm{span}} \{e_{-1}, e_{-2}, \cdots, e_{-n+1}\}, \quad n = 2, 3, \cdots,$$$

$\{E_n\}_{n = 0}^{\infty} $$E 的一列无穷维闭子空间使得 E_0\subset E_1\subset E_2\subset\cdots , 对每个 n\in {\Bbb N} , E_0$$ E_n$中的余维数$d_n = \dim E^0+n-1$, 并且$E = \overline{\bigcup\limits_{n = 0}^{\infty}E_n}$.

因为$E_0 = E^+$, 于是由式(2.4)和(2.12)可知

$$${\it\Phi}(u) = \frac{1}{2}\|u\|^2-\int_Q\widetilde{H}(x, u){\rm d}x\ge \frac{1}{2}\|u\|^2-c_2$$$

$$$u_{n_k}\rightharpoonup u_0$$$

$$$\langle{\it\Psi}'(u_{n_k})-{\it\Psi}'(u_0), u_{n_k}-u_0\rangle\to 0.$$$

${\it\Phi}|_{E_0}$满足(PS)条件.证毕.

设$\{u_{n_j}\}\subset E$为一有界点列使得$n_j\to \infty$, $u_{n_j}\in E_{n_j} $$({\it\Phi}|_{E_{n_j}})'(u_{n_j})\to 0 . 注意到 E^0 是有限维的. 因而, 如有需要过渡到一个子列, 可设 $$u_{n_j}\rightharpoonup u, \;u_{n_j}^\pm\rightharpoonup u^\pm \ ={ 且 }\ u_{n_j}^0\to u^0$$ 对某个 u = u^++u^0+u^-\in E 成立. 对任意 n\in {\Bbb N} , 记 P_n:E\to E_n 为投影算子, E_n^*$$ E_n$的对偶空间, 并记$\langle\cdot, \cdot\rangle_n : E_n^*\times E_n\to {\Bbb R}$为相应的对偶作用. 显然, 由定义有

$$$\big\|P_{n_j}u^–u^-\big\|\to 0.$$$

$\begin{eqnarray} \|u_{n_j}^–P_{n_j}u^-\|^2& = &\big\langle{\it\Phi}'(u), u_{n_j}^–P_{n_j}u^-\big\rangle- \big\langle({\it\Phi}|_{E_{n_j}})'(u_{n_j}), u_{n_j}^–P_{n_j}u^-\big\rangle_n{} \\ &&-\big\langle{\it\Psi}'(u_{n_j})-{\it\Psi}'(u), u_{n_j}^–P_{n_j}u^-\big\rangle. \end{eqnarray}$

$$$\big\langle{\it\Phi}'(u), u_{n_j}^–P_{n_j}u^-\big\rangle\to 0.$$$

$$$\big\langle({\it\Phi}|_{E_{n_j}})'(u_{n_j}), u_{n_j}^–P_{n_j}u^-\big\rangle_n\to 0.$$$

$$$\big\langle{\it\Psi}'(u_{n_j})-{\it\Psi}'(u), u_{n_j}^–P_{n_j}u^-\big\rangle\to 0.$$$

$$$\big\|u_{n_j}^–P_{n_j}u^-\big\|\to 0.$$$

$$$\|u_{n_j}^–u^-\|\to 0.$$$

$$$\|u_{n_j}^+-u^+\|\to 0.$$$

$$$\widetilde{H}(x, u)\geq M_k|u|^2-C_k|u|^3$$$

$\begin{eqnarray} {\it\Phi}(u)&\leq & \frac{1}{2}\|u^+\|^2-\frac{1}{2}\|u^-\|^2-M_k\int_Q|u|^2{\rm d}x+C_k\int_Q|u|^3{\rm d}x{} \\ &\leq & \frac{1}{2}\|u^++u^0\|^2-\frac{1}{2}\|u^-\|^2-M_k\|u^++u^0\|_2^2+C_k\|u\|_3^3{} \\ &\leq & \frac{1}{2}\|u^++u^0\|^2-\frac{1}{2}\|u^-\|^2-\|u^++u^0\|^2+C_k\tau_3^3\|u\|^3{} \\ &\leq &-\frac{1}{2}\|u\|^2+C_k\tau_3^3\|u\|^3, \end{eqnarray}$

由于对任意$(x, u)\in {\Bbb R}^3\times ({\Bbb C}^4\setminus B_{2\delta}(0))$, 都有$\widetilde{H}_u(x, u) = 0$, 因而由文献[19]中命题2.5的证明可知, 存在某个$p_0>3$使得$\{u_k\} $$W^{1, p_0}(Q, {\Bbb C}^4) 中有界. 注意到 W^{1, p_0}(Q, {\Bbb C}^4) 能紧嵌入 E = H^{1/2}(Q, {\Bbb C}^4) 中并且在 E$$ u_k\to 0$, 于是$\{u_k\} $$W^{1, p_0}(Q, {\Bbb C}^4) 中的任一弱收敛子列的极限必为 0 . 由于 W^{1, p_0}(Q, {\Bbb C}^4) 也能紧嵌入 L^{\infty}(Q, {\Bbb C}^4) 中, 因此, 在 L^{\infty}(Q, {\Bbb C}^4)$$ u_k\to 0$. 另一方面, 因为$u_k$是方程(2.11)的解, 所以由Lebesgue控制收敛定理可知, 当$k\to \infty$时, 有

$$$\|{\cal A}_0 u_k\|_2^2 = \int_Q \left|\widetilde{H}_u(x, u_k)-a\beta u_k-V(x)u_k\right|^2{\rm d}x\to 0.$$$

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