## The Asymptotic Behaviors and Phase Separation for a Class of Subcritical Bose-Einstein Condensation System

Zhang Jing,

 基金资助: 国家自然科学基金（11326098）、哈尔滨师范大学博士科研启动基金（XKB201311）和2018年黑龙江省普通本科高等学校青年创新人才培养计划（UNPYSCT-2018177）

 Fund supported: 国家自然科学基金（11326098）、哈尔滨师范大学博士科研启动基金（XKB201311）和2018年黑龙江省普通本科高等学校青年创新人才培养计划（UNPYSCT-2018177）

Abstract

In this paper, we study the phase separation phenomena of the limit profile as the coupling constant tending to minus infinity for some Bose-Einstein condensation system with subcritical exponent in a general smooth bounded domain via variational methods and elliptic equations theories.

Keywords： Bose-Einstein condensation system ; Subcritical exponent ; Variational methods ; Phase separation phenomena

Zhang Jing. The Asymptotic Behaviors and Phase Separation for a Class of Subcritical Bose-Einstein Condensation System. Acta Mathematica Scientia[J], 2019, 39(3): 441-450 doi:

## 1 引言

$\begin{eqnarray}\label{11} \left\{\begin{array}{ll} -\Delta u_1+\alpha_1 u_1 =\mu_1\, |u_1|^{p-1}u_1+\beta\, |u_1|^{\frac{p-3}{2}}u_1|u_2|^{\frac{p+1}{2}}, & x \in \Omega, \\ -\Delta u_2+\alpha_2 u_2 = \mu_2\, |u_2|^{p-1}u_2+\beta\, |u_1|^{\frac{p+1}{2}}\, |u_2|^{\frac{p-3}{2}}u_2 , &x \in \Omega, \\ u_1=u_2= 0, & x\in \partial \Omega, \end{array}\right. \end{eqnarray}$

$C$表示常数(不同地方取值可以不一样).记$H:= H_0^1(\Omega)\times H_0^1(\Omega)$,其上的范数为

## 2 非平凡解的渐近收敛行为

$\begin{eqnarray} \label{21} I_\beta(u_1, \, u_2) &:=& \frac{1}{2} (\| u_1\|_{\alpha_1}^2+\| u_2\|_{\alpha_2}^2)-\frac{1}{p+1}(\mu_1\, |u_1|_{p+1}^{p+1}+\mu_2\, |u_2|_{p+1}^{p+1})\\ &&+\frac{2|\beta|}{p+1} \int_{\Omega} |u_1|^{\frac{p+1}{2}}|u_2|^{\frac{p+1}{2}}\, {\rm d}x \end{eqnarray}$

(2)在开集$\{u_i\neq 0\}$, $i=1, \, 2$上,

$\begin{eqnarray}\label{22}-\Delta u_i+\alpha_i \, u_i=\mu_i\, u_i^p, \qquad i=1, \, 2;\end{eqnarray}$

(3) $u_1\cdot u_2\equiv 0$,当$\beta\rightarrow -\infty$时,有

(2)若$A \in\Gamma^{(k_1, \, k_2)}$,连续映射$\eta:\, A\rightarrow {\cal A}$满足$\eta \circ \sigma_i=\sigma_i\circ \eta, \, i=1, \, 2.$则有$\overline{\eta(A)}\in \Gamma^{(k_1, \, k_2)}$.

任取非空开集$\Omega_1, \, \Omega_2\subset \Omega$满足$\Omega_1\cap\Omega_2=\emptyset$.定义

(2)若$(k_1, \, k_2)=(2, \, 2)$,则有$\{u_i\neq 0\}$恰有两个连通分支,且$u_i$是方程

$\begin{eqnarray}\label{31} -\Delta u+\alpha_i \, u=\mu_i\, u^p, \qquad u\in H_0^1(\{u_i\neq 0\}) \end{eqnarray}$

(3)若$k_1\geq2$$k_2=1,则有u_1变号, \{u_1\neq 0\}至多有k_1个连通分支, u_2$$\{u_2\neq 0\}$上是正的;

(4)若$(k_1, \, k_2)=(2, \, 1)$,则有$\{u_1\neq 0\}$恰有两个连通分支, $\{u_2\neq 0\}$连通且$u_i$是方程(3.1)的最小能量变号解, $i=1, \, 2$.

(1)首先考虑$k_1, \, k_2\geq 2$的情形.由于$u_{i, \beta}\in C(\overline{\Omega})\cap C^2(\Omega)$$u_{i, \beta}是变号解,所以存在x^\pm_{i, \beta}\in \Omega满足 则有\Delta u_{i, \beta}(x^+_{i, \beta})\leq 0$$\Delta u_{i, \beta}(x^-_{i, \beta})\geq 0$.由于$(u_{1, \beta}, \, u_{2, \beta})$满足方程(1.1),对于$i\neq j, $$i, j=1, \, 2,可得 所以\alpha_i\, u_{i, \beta}(x^+_{i, \beta})\leq \mu_i\, u_{i, \beta}(x^+_{i, \beta})^{p},即 同理 所以\alpha_i\, u_{i, \beta}(x^-_{i, \beta})\geq - \mu_i\, |u_{i, \beta}(x^-_{i, \beta})|^{p-1}u_{i, \beta}(x^-_{i, \beta}),即 则由定理2.1(1)可知u_1$$u_2$变号,从而$\{u_i\neq 0\}$至少有两个连通分支, $i=1, \, 2$.

(2)若$(k_1, \, k_2)=(2, \, 2)$.假设$\{u_1\neq 0\}$至少有三个连通分支${\cal O}_1$, ${\cal O}_2$${\cal O}_3.假设在{\cal O}_1\cup{\cal O}_2上有u_1>0,在{\cal O}_3上有u_1 < 0.定义 其中 则有\omega^\pm, \, v\in H^1_0(\Omega)\backslash \{0\},且有 \begin{eqnarray}\label{32} \|u_1 \|_{\alpha_1}^2>\|\omega^+ \|_{\alpha_1}^2+\|\omega^-\|_{\alpha_1}^2. \end{eqnarray} 由于I_\beta'(u_1, \, u_2)(\omega^\pm, \, 0)=0$$I_\beta'(u_1, \, u_2)(0, \, u_{2}^\pm)=0$,可得

$$$\label{33} \|\omega^\pm\|_{\alpha_1}^2=\mu_1\, |\omega^\pm|_{p+1}^{p+1}-|\beta|\, \int_{\Omega} |\omega^\pm|^{\frac{p+1}{2}}|u_2|^{\frac{p+1}{2}}\, {\rm d}x,$$$

$$$\label{34} \|u_2^\pm\|_{\alpha_2}^2=\mu_2\, |u_2^\pm|_{p+1}^{p+1}-|\beta|\, \int_{\Omega} |u_1|^{\frac{p+1}{2}}|u_2^\pm|^{\frac{p+1}{2}}\, {\rm d}x.$$$

$A:=A_1\times A_2\in \Gamma ^{(2, 2)}$.对于任意的$(u, \, v)\in A$,存在$l_i\in{\mathbb R}, i=1, 2, 3, 4$,满足

$A\in \Gamma ^{(2, 2)}_\lambda$

$\begin{eqnarray}\label{35}\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\, (\|u_1 \|_{\alpha_1}^2+\|u_2\|_{\alpha_2}^2)&=&\lim\limits_{\beta\rightarrow -\infty}\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\, (\|u_{1, \beta} \|_{\alpha_1}^2+\|u_{2, \beta}\|_{\alpha_2}^2)\\&=&\lim\limits_{\beta\rightarrow -\infty}I_\beta(u_{1, \beta}, \, u_{2, \beta}) =\lim\limits_{\beta\rightarrow -\infty}\, d_{\beta, \delta_\beta}^{2, 2}\\&\leq& \lim\limits_{\beta\rightarrow -\infty}I_\beta(\omega^+-\omega^-, \, u_2)\\&=&\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\, (\|\omega^+ \|_{\alpha_1}^2+\|\omega^- \|_{\alpha_1}^2+\|u_2\|_{\alpha_2}^2)\\&<&\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\, (\|u_1\|_{\alpha_1}^2+\|u_2\|_{\alpha_2}^2).\end{eqnarray}$

$\|u_i\|_{\alpha_i}^2\leq \|v_i\|_{\alpha_i}^2$, $i=1, \, 2$,所以$(u_1, \, u_2)$是方程(3.1)的最小能量变号解.

(3)若$k_1\geq 2$$k_2=1.由参考文献[11]可知, u_{1, \beta}变号, u_{2, \beta}是正的,所以在\{u_2\neq 0\}$$u_2>0$.假设$\{u_1\neq 0\}$至少有$k_1+1$个连通分支.定义

$\begin{eqnarray}\label{36} I_\beta\bigg(\sum\limits_{k=1}^{k_1} t_k \, u_1\, \chi_{{\cal O}_k}, \, s\, u_2\bigg)&=&\frac{1}{2}\sum\limits_{k=1}^{k_1} t_k^2\|u_1\, \chi_{{\cal O}_k}\|_{\alpha_1}^2+ \frac{1}{2}s^{2}\|u_2\|_{\alpha_2}^2-\frac{\mu_1}{p+1}\sum\limits_{k=1}^{k_1} t_k^{p+1}|u_1\, \chi_{{\cal O}_k}|^{p+1}_{p+1}\\&& -\frac{\mu_2}{p+1} s ^{p+1}|u_2|^{p+1}_{p+1} + \frac{2|\beta|}{p+1} \int_{\Omega} \bigg|\sum\limits_{k=1}^{k_1} t_k \, u_1\, \chi_{{\cal O}_k}\bigg|^{\frac{p+1}{2}}|s\, u_2|^{\frac{p+1}{2}}\, {\rm d}x\\&\leq& \sum\limits_{k=1}^{k_1} \bigg(\frac{1}{2}t_k^2-\frac{1}{p+1}t_k^{p+1}\bigg) \|u_1\, \chi_{{\cal O}_k}\|_{\alpha_1}^2+ \bigg(\frac{1}{2}s^2-\frac{1}{p+1}s^{p+1}\bigg) \|u_2\|_{\alpha_2}^2\\&\leq&\bigg(\frac{1}{2}-\frac{1}{p+1}\bigg)\bigg(\sum\limits_{k=1}^{k_1}\|u_1\, \chi_{{\cal O}_k}\|_{\alpha_1}^2+ \|u_2\|_{\alpha_2}^2\bigg).\end{eqnarray}$

(4)若$(k_1, \, k_2)=(2, \, 1)$.$\{u_1\neq 0\}$恰有两个连通分支.假设$\{u_2\neq 0\}$至少有两个连通分支${\cal O}_1$${\cal O}_2$,则对任意的$\beta < 0$, $(u_1, \, u_2\chi_{{\cal O}_1})\in {\cal M}_\beta$.于是和(3.5)式类似讨论得到矛盾,所以$\{u_2\neq 0\}$连通.最后,和上面类似的讨论我们可证$u_1$是方程(3.1)当$i=1$的最小能量变号解,而$u_2$是方程(3.1)当$i=2$的最小能量变号解.

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