对数薛定谔系统解的存在性

对数薛定谔系统解的存在性

张晶^,^*, 修明威

哈尔滨师范大学数学科学学院哈尔滨 150025

Existence of Solutions for Schrödinger Systems with Logarithmic Terms

Zhang Jing^,^*, Xiu Mingwei

School of Mathematics, Harbin Normal University, Harbin 150025

通讯作者: *张晶, E-mail:zhjmath11@163.com

收稿日期: 2025-03-13 修回日期: 2025-06-27

基金资助:

国家自然科学基金(12561040)
国家自然科学基金(12061040)

黑龙江省省属本科高校基本科研业务费

Basic Scientific Research Project of Provincial Undergraduate Universities in Heilongjiang Province(2025-KYYWF-ZR0124)

Received: 2025-03-13 Revised: 2025-06-27

Fund supported:

NSFC(12561040)
NSFC(12061040)(2025-KYYWF-ZR0124)

摘要

对数薛定谔系统在物理学中有广泛的应用, 因此研究其解的存在性具有重要的意义. 该文研究了一类带有临界扰动的对数薛定谔系统解的存在性, 即在 Sobolev 临界条件下, 通过构造系统的能量泛函并应用变分方法, 临界点理论和集中紧性原理得到系统基态解的存在性及相应解的性质.

关键词： 对数薛定谔系统; 基态解; 变分方法; 临界扰动

Abstract

In this paper, we study the existence of solutions for a class of logarithmic Schrödinger systems with critical perturbations. We obtain the existence of ground state solutions by constructing auxiliary functional and applying variational method, critical point theory and concentrated compactness principle.

Keywords： logarithmic Schrödinger system; ground state solution; variational method; critical perturbations.

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本文引用格式

张晶, 修明威. 对数薛定谔系统解的存在性[J]. 数学物理学报, 2026, 46(3): 1092-1104

Zhang Jing, Xiu Mingwei. Existence of Solutions for Schrödinger Systems with Logarithmic Terms[J]. Acta Mathematica Scientia, 2026, 46(3): 1092-1104

1 引言与主要结果

本文研究如下具有临界扰动项的对数薛定谔系统解的存在性

(1.1)$\begin{cases}-\Delta u_1+\omega_1 u_1=\mu _1 u_1 \log u_1^2+\frac{2p}{p+q} \left|u_2\right|^q \left|u_1\right|^{p-2}u_1+\frac{m\varepsilon}{m+n} \left|u_1\right|^{m-2}u_1\left|u_2\right|^n, \\-\Delta u_2+\omega_2 u_2=\mu _2 u_2 \log u_2^2+\frac{2q}{p+q} \left|u_1\right|^p \left|u_2\right|^{q-2}u_2+\frac{n\varepsilon }{m+n} \left|u_2\right|^{n-2}u_2\left|u_1\right|^m, \\\int_\Omega \left|{u_i}^2\right| \mathrm{d}x=\rho_i, i=1,2,\\(u_1,u_2) \in H_0 ^1(\Omega,\mathbb{R}^2),\end{cases}$

其中 $ \Omega\subset\mathbb{R}^N(N\geq3) $ 是一个有界光滑区域, $ \omega _i \in \mathbb{R} $, $ \mu_i $, $ \rho_i>0 $, $ i=1 $, $ 2 $. $ p $, $ q $, $ m $, $ n\geq1 $, $ 2+\frac{2}{N}<p+q<2^* $, $ m+n=2^* $, 其中 $ 2^*=\frac{2N}{N-2} $ 为临界 Sobolev 指数.

对数薛定谔方程是一种非线性薛定谔方程, 它是薛定谔方程的一种推广, 在凝聚态物理和量子信息领域中被广泛应用. Brézis 和 Nirenberg 在文献 [1] 中提出了方程

$-\Delta u=|u|^{2^{*}-2} u+\lambda u,$

解的存在性取决于 $ \lambda $ 和 $ N $ 的值. Noris, Tavares 和 Verzini 在文献 [2] 中考虑了非线性方程

(1.2)$\left\{\begin{array}{l}-\Delta u+\lambda u=|u|^{p-1} u, u \in \Omega, \\\int_{\Omega} u^{2} \mathrm{~d} x=\rho>0,\end{array}\right.$ (1.2)

在球面域 $ \Omega =B_1 $ 上的解的存在性和轨道稳定性. Tavares 和 Verzini 在文献 [3] 中考虑了方程 (1.2) 在有界域上正规化解的存在性. Squassina 和 Szulkin 在文献 [5] 中研究了带有周期势的对数薛定谔方程 $ -\Delta u+V(x)u=u\log u^2, x \in \mathbb{R}^N $正基态解的存在性. 薛艳昉, 韩建新在文献 [5] 中研究了一类薛定谔方程正解的存在性问题. 景丽, 滕凯民在文献 [6] 中利用扰动方法证明了一类拟线性薛定谔方程基态规范解的存在性并分析了解的收敛性. Deng, He, Pan 和 Zhong 在文献 [7] 中证明了方程 $ -\Delta u+\omega u=\mu u\log u^2+\left|u\right|^{p-2} u $ 在 $ \mathbb{{R}}^N $ 上正解的存在性.

为了得到 (1.1) 式的解, 寻找下列泛函

(1.3)$\begin{aligned}I(u_1,u_2)=&\frac{1}{2} \int_\Omega(\left|\nabla u_1\right|^2+\left|\nabla u_2\right|^2)\mathrm{d}x-\frac{1}{2}\int_\Omega\Bigg(\mu_1 u_1^2(\log u_1^2-1)+\frac {4}{p+q}\left| u_1\right|^p \left| u_2\right|^q\\&+\mu_2 u_2^2(\log u_2^2-1)+\frac {2\varepsilon}{m+n}\left| u_1\right|^m \left| u_2\right|^n\Bigg)\mathrm{d}x\end{aligned}$

在 $ M:=\{(u_1,u_2)\in H_0^1(\Omega;\mathbb{R}^2):\int_\Omega \left|{u_i}^2\right| \mathrm{d}x=\rho_i,\,\,\,i=1,2\} $ 上的局部极小值. 主要结论如下

定理 1.1 令 $ \Omega\subset \mathbb{R}^N $ 是一个光滑有界域, $ N\geq3, 2+\frac{2}{N}<p+q<2^*, m+n=2^* $, 记 $ a:=\frac{N(p+q-2)}{4} $, $ r:=\frac{p+q}{4}-\frac{N(p+q-2)}{8} $. 假设 $ \alpha>\max\{\frac{1}{\rho_1+\rho_2},\lambda_1 (\Omega)\} $, 使得

(1.4)$(\rho_1+\rho_2)(\alpha-\lambda_1(\Omega))<\min\Bigg\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\Bigg\},$

其中

$\begin{align*} \Lambda=\Lambda(\rho_1,\rho_2):=\underset{\{x^2+y^2=1\}}{\max}\Bigg\{\frac{N-2}{2e}S(N)((\rho_1+\rho_2)\alpha)^{\frac{2^*}{2}-a}(\mu_1 |x|^{2^*}+ \mu_2 |y|^{2^*}) +\frac{4}{p+q} \\ C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}|x|^{\frac{2ap}{p+q}}|y|^{\frac{2aq}{p+q}}+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\varepsilon S(N)|x|^{m}|y|^{n} \Bigg\}, \end{align*}$

则在 $ B_\alpha $ 中任何与 $ c_\alpha $ 相关的序列相对紧. 特别地, $ c_\alpha $ 可达.

在定理 1.1 的基础上, 引入如下集合

(1.5)$A:=\left\{\left(\rho_{1}, \rho_{2}\right) \in(0, \infty)^{2}: \begin{array}{l}\alpha \geq \max \left\{\frac{1}{\rho_{1}+\rho_{2}}, \lambda_{1}(\Omega)\right\}, c_{\alpha}<\hat{c_{\alpha}}, \\\left(\rho_{1}+\rho_{2}\right)\left(\alpha-\lambda_{1}(\Omega)\right)<\min \left\{\frac{1}{\Lambda^{\frac{1}{a-1}}}, \frac{1}{\Lambda^{\frac{N-2}{2}}}\right\}\end{array}\right\} \cup\{(0,0)\},$

$ A $ 取决于 $ \Omega $, $ N $, $ p $, $ q $, $ \mu_1 $, $ \mu_2 $. 此外, 若 $ (\rho_1,\rho_2)\in A $, 对于某些 $ (\omega_1,\omega_2)\in \mathbb{R}^2 $, 可以找到一组局部极小值 $ (u_1,u_2)\in M $ 作为系统 (1.1) 的解.

定理 1.2 令 $ \Omega\subset \mathbb{R}^N $ 是一个光滑有界域, 且 $ 2+\frac{2}{N}<p+q<2^*, m+n=2^* $. 若 $ A $ 如上定义, 则 $ A $ 关于 $ (0,0) $ 是星形的. 此外, 存在一个正常数 $ R=R(\Omega,N,p+q) $, 使得

$ \frac{N-2}{2e}S(N)((\rho_1+\rho_2)\alpha)^{\frac{2^*}{2}-a} \max\{\mu_1,\mu_2\}+\frac{|\Omega|}{e}(\mu_1+\mu_2)((\rho_1+\rho_2)\alpha)^{a-1} \leq R(\Omega,N,p+q), $

则 $ (\rho_1,\rho_2)\in A $. 其中

定理 1.3 令 $ \Omega\subset \mathbb{R}^N $ 是一个光滑有界域, $ 2+\frac{2}{N}<p+q<2^*, m+n=2^* $, 且 $ (\rho_1,\rho_2)\in A $. 令$ \overline{\alpha} \geq \max\{\frac{1}{\rho_1+\rho_2},\lambda_1 \Omega\} $, 使得 $ c_{\overline{\alpha}}<\hat{c_{\overline{\alpha}}} $, 且 $ \overline{\alpha} $ 满足 (1.4) 式, 则局部基态集合

$G_{\bar{\alpha}}:=\left\{\left(u_{1}, u_{2}\right) \in H_{0}^{1}\left(\Omega ; \mathbb{C}^{2}\right):\left(\left|u_{1}\right|,\left|u_{2}\right|\right) \in B_{\bar{\alpha}}, I\left(u_{1}, u_{2}\right)=c_{\bar{\alpha}}\right\}$

是轨道稳定的.

2 预备知识

对于 $ \alpha\geq \max\{\lambda_1(\Omega),\frac{1}{\rho_1+\rho_2}\} $, 令 $ \Omega \subset \mathbb{R}^N(N\geq3) $ 是一个 Lipschitz 有界域. 定义

(2.1)$\begin{array}{l}B_{\alpha}:=\left\{\left(u_{1}, u_{2}\right) \in M: \int_{\Omega}\left(\left|\nabla u_{1}\right|^{2}+\left|\nabla u_{2}\right|^{2}\right) \mathrm{d} x \leq\left(\rho_{1}+\rho_{2}\right) \alpha\right\}, \\U_{\alpha}:=\left\{\left(u_{1}, u_{2}\right) \in M: \int_{\Omega}\left(\left|\nabla u_{1}\right|^{2}+\left|\nabla u_{2}\right|^{2}\right) \mathrm{d} x=\left(\rho_{1}+\rho_{2}\right) \alpha\right\},\end{array}$

由文献 [3] 可知, 当 $ (\sqrt {\rho_1}\varphi_1,\sqrt {\rho_2}\varphi_2)\in B_\alpha $, 则 $ B_\alpha \neq \emptyset $. 此外, 令

$\hat{c_{\alpha}}:=\inf _{U_{\alpha}} I, \quad c_{\alpha}:=\inf _{B_{\alpha}} I.$

当 $ m+n=2^* $ 时, 即在 Sobolev 临界条件下, $ H_0^1(\Omega)\hookrightarrow L^{2^*} (\Omega) $ 不再是紧嵌入, 此时 $ I|_M $ 不是弱下半连续, 无法直接得出 $ {c_\alpha} $ 可达. 为了克服这一困难, 可以通过讨论 $ \rho $ 和 $ \alpha $ 取值, 从而得到与 $ {c_\alpha} $ 相关的极小序列的紧性. 为了简便, 整篇文章引入下列记号

$ \lambda_{1}(\Omega) $ 为Dirichlet Laplacian算子在 $ \Omega $ 中的第一特征值, $ \varphi_1 $ 是相应的第一特征函数, $ \varphi_1 $ 在 $ L^2(\Omega) $ 中是正规的并且在 $ \Omega $ 中是正的, $ \left|\Omega\right| $ 为 $ \Omega $ 的测度.

定义 $ L^q {(\Omega)} (1\leq q< \infty) $, $ H_0^1(\Omega) $, $ H^1(\mathbb{R}^N) $ 中的范数

$ \left|u\right|_{L^q{(\Omega)}} ^q:=\int_\Omega \left|u\right|^q \mathrm{d} x,\ \|u\|_{H_0^1(\Omega)}^2:=\int_\Omega\left|\nabla u\right|^2 \mathrm{d} x,\ \|u\|_{H^1(\mathbb{R}^N)}^2:=\int_{\mathbb{R}^N}(\left|\nabla u\right|^2+\left|u\right|^2) \mathrm{d} x, $

其中 $ \left| u \right|_{L^q{(\Omega)}} $ 可简记为 $ \left|u\right|_{q} $.

引理 2.1^[8] ($Hölder不等式$) 设 $ p>1 $, $ q>1 $, 且 $ \frac{1}{p}+\frac{1}{q}=1 $. 若 $ f\in L^p(\mathbb{R}^2) $, $ g\in L^q(\mathbb{R}^2) $, 则有 $ f\cdot g\in L^1(\mathbb{R}^2) $, 且

$\int _{\mathbb{R}^2}\left | f\left ( x \right )g\left ( x \right ) \right | \mathrm{d}x\le\left \| f \right \| _{L^p(\mathbb{R}^2)}\cdot \left \| g \right \| _{L^q(\mathbb{R}^2)}.$

引理 2.2^[9] (Gagliardo-Nirenberg不等式) 对于任意的 $ u\in H^1(\mathbb{R}^N) $,% $ r=\frac{p+q}{4}-\frac{N(p+q-2)}{8} $

$\begin{align*} \int_{\mathbb{R}^N}|u|^{p+q}\mathrm{d}x\leq C_{N,p+q}\bigg(\int_{\mathbb{R}^N}u^2\mathrm{d}x\bigg)^{\frac {p+q}{2}-a}\bigg(\int_{\mathbb{R}^N}|\nabla u|^2\mathrm{d}x\bigg)^a, \end{align*}$

其中

$\begin{align*} \hspace{-2.4cm}C_{N,p+q}=\underset {u\in H^1_{0} (\mathbb{R}^N) \backslash \{0\}}{\inf}\frac {\|u\|_{L^{p+q}(\mathbb{R}^N)}^{p+q}}{\|\nabla u\|_{L^2 (\mathbb{R}^N)}^{2a} \|u\|_{L^2 (\mathbb{R}^N)}^{4r}}, \end{align*}$

$ S_N:=C_{N,2^*}, a:=\frac{N(p+q-2)}{4}, r:=\frac{p+q}{4}-\frac{N(p+q-2)}{8}. $

下面给出对数项 $ s^2 \log s^2 $ 的一些性质.

引理 2.3 函数 $ s^2 \log s^2 $ 满足如下性质

(1) $ |s^2 \log s^2|\leq \frac{1}{e} $, $ 0\leq s\leq1 $;

(2) $ 0<|s^2 \log s^2|\leq \frac{(N-2)}{2e}s^{2^*} $, $ s>1 $.

接下来, 分别引入对数项 $ {u}^2 \log {u}^2 $ (见文献 [10]) 和对偶项 $ |u_{1}|^p |u_{2}|^q $ (见文献 [11])

的 Brezis-Lieb 型引理.

引理 2.4 令 $ \{u_n\} $ 在 $ H_0^1(\Omega) $ 中弱收敛于 $ u $, 若 $ \{u_n\} $ 在 $ \Omega $ 中几乎处处收敛于 $ u $, 且有 $ \{{u_n}^2 \log {u_n}^2\} $ 在 $ L^1(\Omega) $ 中有界, 则 $ {u}^2 \log {u}^2\in L^1(\Omega) $, 且

$ \int_\Omega {u_n}^2 \log {u_n}^2\mathrm{d}x=\int_\Omega {u}^2 \log {u}^2\mathrm{d}x+\int_\Omega |u_n-u|^2 \log |u_n-u|^2\mathrm{d}x+o(1)\,\,\,, \,\,\, n\rightarrow\infty. $

引理 2.5 令 $ \{u_{1,n}\},\{u_{2,n}\} $ 在 $ H_0^1(\Omega) $ 中分别弱收敛于 $ u_1,u_2 $, 且 $ \{u_{1,n}\},\{u_{2,n}\} $ 在 $ \Omega $ 中几乎处处收敛于 $ u_1,u_2 $, 则

$ \underset{n \to \infty}{\lim}\int_\Omega |u_{1,n}|^p |u_{2,n}|^q\mathrm{d}x=\underset{n \to \infty}{\lim}\int_\Omega |u_{1,n}-u_1|^p |u_{2,n}-u_2|^q\mathrm{d}x. $

引理 2.6 令 $ \{u_n\} $ 是 $ H_0^1(\Omega) $ 中的有界序列, $ u_n\rightharpoonup u $, 且当 $ n\rightarrow\infty $ 时 $ u_n $ 在 $ \Omega $ 中几乎处处收敛于 $ u $, 则

$ \underset{n \to \infty}{\lim}\int_\Omega {u_n}^2 \log {u_n}^2\mathrm{d}x=\int_\Omega {u}^2 \log {u}^2\mathrm{d}x. $

3 解的存在性

引理 3.1 对于任意一个 $ \rho_1,\rho_2>0 $, $ I|_M $ 不是强制的.

证利用引理 2.3, Hölder 不等式和 Gagliardo-Nirenberg 不等式, 可得

(3.1)$\begin{aligned}&\int_\Omega \Bigg(\mu_1 u_1^2(\log u_1^2-1)+\frac {4}{p+q}\left| u_1\right|^p \left| u_2\right|^q+\mu_2 u_2^2(\log u_2^2-1)+\frac {2\varepsilon}{m+n}\left| u_1\right|^m \left| u_2\right|^n\Bigg)\mathrm{d}x\\<&\frac{N-2}{2e}S(N)(\mu_1 |\nabla u_1|^{2^*}+\mu_2 |\nabla u_2|^{2^*})+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\frac {4}{p+q}C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}\\&|\nabla u_1|^{\frac{2ap}{p+q}}|\nabla u_2|^{\frac{2aq}{p+q}}+\varepsilon S(N)\rho_1^{\frac{2r^{'}m}{m+n}}\rho_2^{\frac{2r^{'}n}{m+n}}|\nabla u_1|^{\frac{2a^{'}m}{m+n}}|\nabla u_2|^{\frac{2a^{'}n}{m+n}}.\end{aligned}$

其中 $ a^\prime:=\frac{N(m+n-2)}{4}=\frac{2^*}{2}, r^\prime:=\frac{m+n}{4}-\frac{N(m+n-2)}{8}=0. $

对于任意的 $ (u_1,u_2)\in M $, 当 $ \|(u_1,u_2)\|_{H_0^1(\Omega)}\rightarrow+\infty $ 时可得

$\begin{align*} I(u_1,u_2)>&\frac{1}{2} \int_\Omega(\left|\nabla u_1\right|^2+\left|\nabla u_2\right|^2)\mathrm{d}x-\frac{1}{2}\Bigg(\frac{N-2}{2e}S(N)(\mu_1 |\nabla u_1|^{2^*}+\mu_2 |\nabla u_2|^{2^*})\\ &+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\frac {4}{p+q}C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}|\nabla u_1|^{\frac{2ap}{p+q}}|\nabla u_2|^{\frac{2aq}{p+q}}\\ &+\varepsilon S(N)|\nabla u_1|^m|\nabla u_2|^n\Bigg)\rightarrow -\infty. \end{align*}$

该引理得证.

设 $ X=\left|\nabla u_1\right|_2, Y=\left|\nabla u_2\right|_2 $, 考虑函数$ \Phi:\mathbb{R}_+^2\rightarrow\mathbb{R} $,

(3.2) $\begin{aligned}\Phi (X,Y)=&\frac{1}{2}\Bigg( (X^2+Y^2)-\Bigg(\frac{N-2}{2e}S(N)(\mu_1 X^{2^*}+\mu_2 Y^{2^*})+\frac{|\Omega|}{e}(\mu_1+\mu_2)\\&+\frac {4}{p+q}C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}X^{\frac{2ap}{p+q}}Y^{\frac{2aq}{p+q}}+\varepsilon S(N)X^m Y^n\Bigg)\Bigg).\end{aligned}$

定义如下集合

$\begin{align*} \mathcal{U}_\alpha=\{(x,y)\in \mathbb{R}_+^2:x^2+y^2=(\rho_1+\rho_2)\alpha \}\quad \mathcal{V}_\alpha=\mathcal{U}_\alpha \cap \{x\geq \sqrt{\rho_1 \lambda_1(\Omega)},y \geq \sqrt{\rho_2 \lambda_2(\Omega)}\}, \end{align*}$

可得

(3.3)$\begin{aligned}\hat{c}_\alpha=\underset{\mathcal{U}_\alpha}{\inf}\,\, I \geq& \underset{ \mathcal{V}_\alpha}{\min}\,\,\Phi(X,Y).\end{aligned}$

引理 3.2 任意的 $ \alpha>\max\{\lambda_1(\Omega),\frac{1}{\rho_1+\rho_2}\} $, $ \hat{c}_\alpha>\frac{1}{2}((\rho_1+\rho_2)\alpha-\Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^a \alpha^a) $. 其中

(3.4)$\begin{aligned}\Lambda=\Lambda(\rho_1,\rho_2):=&\underset{t\in[0,\frac{\pi}{2}]}{\max}\Bigg\{\frac{N-2}{2e}S(N)((\rho_1+\rho_2)\alpha)^{\frac{2^*}{2}-a}(\mu_1 \cos^{2^*}t+\mu_2 \sin^{2^*}t)\\&+\frac{4}{p+q} C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}\cos^{\frac{2ap}{p+q}}t \sin^{\frac{2aq}{p+q}}t+\frac{|\Omega|}{e}(\mu_1+\mu_2)\\&+\varepsilon S(N)(\rho_1+\rho_2)\alpha^{\frac{2^*}{2}-a}\cos^{m}t \sin^{n}t \Bigg\}.\end{aligned}$

证取 $ X=\sqrt{(\rho_1+\rho_2)\alpha}\cos t $, $ Y=\sqrt{(\rho_1+\rho_2)\alpha}\sin t $. 代入 (3.3) 式可得

$\begin{align*} &\underset{t\in[0,\frac{\pi}{2}]}{\max}\bigg\{\frac{N-2}{2e}S(N)(\mu_1 (\sqrt{(\rho_1+\rho_2)\alpha}\cos t)^{2^*}+\mu_2 (\sqrt{(\rho_1+\rho_2)\alpha}\sin t)^{2^*})\\ &+\frac{4}{p+q} C(N,p+q)\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}(\sqrt{(\rho_1+\rho_2)\alpha}\cos t)^{\frac{2ap}{p+q}} (\sqrt{(\rho_1+\rho_2)\alpha}\sin t)^{\frac{2aq}{p+q}}\\ &+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\varepsilon S(N)(\sqrt{(\rho_1+\rho_2)\alpha}\cos t)^{m} (\sqrt{(\rho_1+\rho_2)\alpha}\sin t)^{n} \bigg\}\\ =&\Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^a \alpha^a. \end{align*}$

由 $ \mathcal{V}_\alpha \subset \mathcal{U}_\alpha=\{(\cos t,\sin t)\sqrt {(\rho_1+\rho_2)\alpha}, t\in[0,\frac{\pi}{2}]\} $, 则有对于任意的 $ \alpha>\max\{\lambda_1(\Omega),\frac{1}{\rho_1+\rho_2}\} $,

$\hat{c}_{\alpha}>\frac{1}{2}\left(\left(\rho_{1}+\rho_{2}\right) \alpha-\Lambda\left(\rho_{1}, \rho_{2}\right)\left(\rho_{1}+\rho_{2}\right)^{a} \alpha^{a}\right).$

该引理得证.

注 3.3

由上可知 $ \Lambda(\rho_1,\rho_2) $ 取决于 $ \rho_1 $, $ \rho_2 $, $ \mu_1 $, $ \mu_2 $ 等. 对任意的 $ (\tau_1,\tau_2) $, 存在 $ t\in[0,\frac{\pi}{2}] $, 使得

$ |\nabla \tau_1|_2=(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{1}{2}} \cos t, |\nabla \tau_2|_2=(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{1}{2}} \sin t. $

将 $ \cos t=\frac{|\nabla \tau_1|_2}{(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{1}{2}}}, \sin t=\frac{|\nabla \tau_2|_2}{(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{1}{2}}} $, 代入 (3.4) 式得

$\begin{align*} \Lambda(\rho_1,\rho_2) =&\underset{t\in[0,\frac{\pi}{2}]}{\max}\bigg \{\frac{1}{(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{2^*}{2}}}\frac{N-2}{2e}S(N)((\rho_1+\rho_2)\alpha)^{\frac{2^*}{2}-a}(\mu_1 |\nabla \tau_1|_2 ^{2^*}\\ &+\mu_2 |\nabla \tau_2|_2 ^{2^*})+\frac{1}{(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{a}}(C(N,p+q)\frac{4}{p+q}\rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}|\nabla \tau_1|_2 ^{\frac{2ap}{p+q}}\\ &|\nabla \tau_2|_2 ^{\frac{2aq}{p+q}})+\frac{|\Omega|}{e}(\mu_1+\mu_2) +\frac{1}{(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{2^*}{2}}} \varepsilon S(N)|\nabla \tau_1|_2 ^{m}|\nabla \tau_2|_2 ^{n}\bigg\}, \end{align*}$

则当 $ \alpha>\max\{\lambda_1(\Omega),\frac{1}{\rho_1+\rho_2}\} $ 时,

(1) 当 $ 0<|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2 \leq 1 $ 时, 可得

(3.5)$\begin{aligned}(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{a}\Lambda(\rho_1,\rho_2)>&\frac{N-2}{2e}S(N)(\mu_1 |\nabla \tau_1|_2 ^{2^*}+\mu_2 |\nabla \tau_2|_2 ^{2^*})\\&+C(N,p+q)\frac{4}{p+q} \rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}|\nabla \tau_1|_2 ^{\frac{2ap}{p+q}}|\nabla \tau_2|_2 ^{\frac{2aq}{p+q}}\\&+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\varepsilon S(N)|\nabla \tau_1|_2 ^{m}|\nabla \tau_2|_2 ^{n};\end{aligned}$

(2) 当 $ |\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2 \geq K>1 $ 时, 可得

(3.6)$\begin{aligned}(|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2)^{\frac{2^*}{2}}\Lambda(\rho_1,\rho_2)>&\frac{N-2}{2e}S(N)(\mu_1 |\nabla \tau_1|_2 ^{2^*}+\mu_2 |\nabla \tau_2|_2 ^{2^*})\\&+C(N,p+q)\frac{4}{p+q} \rho_1^{\frac{2rp}{p+q}}\rho_2^{\frac{2rq}{p+q}}|\nabla \tau_1|_2 ^{\frac{2ap}{p+q}}|\nabla \tau_2|_2 ^{\frac{2aq}{p+q}}\\&+\frac{|\Omega|}{e}(\mu_1+\mu_2)+\varepsilon S(N)|\nabla \tau_1|_2 ^{m}|\nabla \tau_2|_2 ^{n}.\end{aligned}$

引理 3.4 设 $ \Omega\subset \mathbb{R}^N (N\geq3) $ 是一个光滑有界区域, $ p+q<2^* $, $ m+n=2^* $. 令 $ \alpha \geq \max\{\lambda_1(\Omega),\frac{1}{\rho_1+\rho_2}\} $, $ \rho_1,\rho_2>0 $ 满足

(3.7)$\begin{aligned}(\rho_1+\rho_2)(\alpha-\lambda_1(\Omega))<\min\Bigg\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\Bigg\}.\end{aligned}$

令 $ (u_{1,n},u_{2,n})_n $ 满足

(3.8)$\begin{aligned}\begin{cases}\{|u_{i,n}|_2^2=\rho_i+o(1),i=1,2,\\|\nabla u_{1,n}|_2^2+|\nabla u_{2,n}|_2^2\leq \alpha (\rho_1+\rho_2)+o(1), \,\,\,\,\,\,(n\rightarrow \infty),\\c_\alpha\leq I(u_{1,n},u_{2,n})\leq c_\alpha+o(1),\end{cases}\end{aligned}$

则 $ (u_{1,n},u_{2,n}) \rightarrow (\bar{u_1},\bar{u_2})\in \mathcal{H}(\Omega) $. 特别地, $ c_\alpha $ 可达.

证由假设可知存在 $ (\bar{u_1},\bar{u_2}) \in H_0^1(\Omega, \mathbb{R}^2) $, 使得

$\begin{align*} \begin{cases} |\bar{u_i}|_2^2=\rho_i,i=1,2,\\ u_{i,n}\rightharpoonup \bar{u_i}\in H_0^1(\Omega),i=1,2,\\ |\nabla \bar{u_i}|_2^2\leq \underset {n\rightarrow \infty}{\lim} \inf|\nabla u_{i,n}|,i=1,2, \end{cases} \end{align*}$

此时可得 $ I(\bar{u_1},\bar{u_2})\geq c_\alpha $. 令 $ \tau _{i,n}=u_{i,n}-\bar{u_i},i=1,2 $, 则可知证明 $ (u_{i,n},u_{2,n}) $ 的强收敛等价于证明存在序列 $ (\tau _{1,n_k},\tau _{2,n_k}) $, 使得 $ |\nabla \tau _{1,n_k}|_2^2+|\nabla \tau _{2,n_k}|_2^2 \rightarrow 0. $

下用反证法证明 $ |\nabla \tau _{1,n_k}|_2^2+|\nabla \tau _{2,n_k}|_2^2 \rightarrow 0 $. 假设 $ |\nabla \tau_{1,n}|_2 ^2+|\nabla \tau_{2,n}|_2 ^2 \geq K>0, $

(3.9)$\begin{aligned}I(u_{1,n},u_{2,n})=&I(\tau _{1,n}+\bar{u_1},\tau _{2,n}+\bar{u_2})\\=&\frac{1}{2}(|\nabla (\bar{u_1}+\tau _{1,n})|_2^2+|\nabla (\bar{u_2}+\tau _{2,n})|_2^2)+\frac{1}{2}\int_{\Omega}(\mu_1(\bar{u_1}+\tau _{1,n})^2+\mu_2(\bar{u_2}+\tau _{2,n})^2)\mathrm{d}x\\&-\frac{1}{2}\int_{\Omega} \Bigg(\mu_1(\bar{u_1}+\tau _{1,n})^2 \log(\bar{u_1}+\tau _{1,n})^2+\frac{4}{p+q}|(\bar{u_1}+\tau _{1,n})|^p|(\bar{u_2}+\tau _{2,n})|^q \\&+\mu_2(\bar{u_2}+\tau _{2,n})^2 \log(\bar{u_2}+\tau _{2,n})^2+\frac{2\varepsilon}{m+n}|(\bar{u_1}+\tau _{1,n})|^m|(\bar{u_1}+\tau _{1,n})|^n\Bigg)\mathrm{d}x.\end{aligned}$

由于在 $ H_0^1(\Omega) $ 中, $ u_{i,n}\rightharpoonup \bar{u_i} $, 由引理 2.4 可得

(3.10)$\begin{aligned}&|\nabla (\bar{u_i}+\tau _{i,n})|_2^2=|\nabla \bar{u_i}|_2^2+|\nabla \tau _{i,n}|_2^2+o(1),\,\,\,\,\, n\rightarrow\infty,\\&|(\bar{u_i}+\tau _{i,n})|_2^2=|\bar{u_i}|_2^2+|\tau _{i,n}|_2^2+o(1),\,\,\,\,\, n\rightarrow\infty.\end{aligned}$

由 (3.9) 和 (3.10) 式可知

$ I(\tau _{1,n},\tau _{2,n})\leq o(1), n\rightarrow\infty. $

下面分两种情况进行讨论

(1) 当 $ 0<|\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2 \leq 1 $ 时,

$\begin{align*} |\nabla \tau_{1,n}|_2 ^2+|\nabla \tau_{2,n}|_2 ^2 \leq &\frac{N-2}{2e}S(N)(\mu_1 |\nabla \tau_{1,n}|_2 ^{2^*}+\mu_2 |\nabla \tau_{2,n}|_2 ^{2^*})+C(N,p+q)\frac{4}{p+q} \rho_1^{\frac{2rp}{p+q}}\\ &\rho_2^{\frac{2rq}{p+q}} |\nabla \tau_{1,n}|_2 ^{\frac{2ap}{p+q}}|\nabla \tau_{2,n}|_2 ^{\frac{2aq}{p+q}}+\varepsilon S(N)|\nabla \tau_{1,n}|_2 ^{m}|\nabla \tau_{2,n}|_2 ^{n}+o(1)\\ \leq &\Lambda(\rho_1,\rho_2)(|\nabla \tau_{1,n}|_2 ^2+|\nabla \tau_{2,n}|_2 ^2)^a+o(1), \end{align*}$

即 $ (\rho_1+\rho_2)(\alpha-\lambda_1(\Omega))\geq\frac{1}{\Lambda^{\frac{1}{a-1}}} $

与条件矛盾, 此种情况不成立;

(2) 当 $ |\nabla \tau_1|_2 ^2+|\nabla \tau_2|_2 ^2 \geq K> 1 $ 时,

$\begin{align*} |\nabla \tau_{1,n}|_2 ^2+|\nabla \tau_{2,n}|_2 ^2 \leq &\frac{N-2}{2e}S(N)(\mu_1 |\nabla \tau_{1,n}|_2 ^{2^*}+\mu_2 |\nabla \tau_{2,n}|_2 ^{2^*})+C(N,p+q)\frac{4}{p+q} \rho_1^{\frac{2rp}{p+q}}\\ &\rho_2^{\frac{2rq}{p+q}} |\nabla \tau_{1,n}|_2 ^{\frac{2ap}{p+q}}|\nabla \tau_{2,n}|_2 ^{\frac{2aq}{p+q}}+\varepsilon S(N)|\nabla \tau_{1,n}|_2 ^{m}|\nabla \tau_{2,n}|_2 ^{n}+o(1)\\ \leq& \Lambda(\rho_1,\rho_2)(|\nabla \tau_{1,n}|_2 ^2+|\nabla \tau_{2,n}|_2 ^2)^{\frac{2^*}{2}}+o(1), \end{align*}$

即 $ (\rho_1+\rho_2)(\alpha-\lambda_1(\Omega))\geq\frac{1}{\Lambda^{\frac{N-2}{2}}} $ 与条件矛盾, 此种情况不成立. 综上, $ |\nabla \tau _{1,n_k}|_2^2+|\nabla \tau _{2,n_k}|_2^2 \rightarrow 0 $ 得证.

定理 1.1 的证明 由引理 3.4, 得证.

引理 3.5 假设 $ \rho_1,\rho_2>0 $, 存在 $ \alpha_1,\alpha_2 $ 满足 $ \max\{\frac{1}{\rho_1+\rho_2}, \lambda_1(\Omega)\}\leq \alpha_1<\alpha_2 $, $ \hat{c_{\alpha_1}}<\hat{c_{\alpha_2}} $, 且

$\alpha_2<\lambda_1(\Omega)+\frac{1}{\rho_1+\rho_2}\min\Bigg\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\Bigg\},$

则 $ c_{\alpha_2}<\hat{c_{\alpha_2}} $, 且 $ c_{\alpha_2} $ 可作为 (1.1) 式正解可达.

证由引理 3.4 可知, 集合 $ B_{\alpha_2} $ 中存在 $ (\bar{u_1},\bar{u_2}) $, 使得 $ c_{\alpha_2} $ 可达, 且有%即 $ (\bar{u_1},\bar{u_2}) $ 在$ B_{\alpha_2}\setminus U_{\alpha_2} $ 内取到.

$c_{\alpha_{2}}=\min \left\{\hat{c_{\alpha}}: \max \left\{\frac{1}{\rho_{1}+\rho_{2}}, \lambda_{1}(\Omega)\right\} \leq \alpha \leq \alpha_{2}\right\} \leq \hat{c_{\alpha_{1}}}<\hat{c_{\alpha_{2}}},$

该引理得证.

引理 3.6 由 $ U_\alpha $ 定义可知

$ (\sqrt {\rho_1}\varphi_1,\sqrt {\rho_2}\varphi_1)\in U_{\lambda_1(\Omega)},\ \bigg(\sqrt {\rho_1} \frac{\varphi_2^+}{|\varphi_2^+|_2},\sqrt {\rho_2} \frac{\varphi_2^-}{|\varphi_2^-|_2}\bigg)\in U_{\lambda_2(\Omega)}. $

若 $ {\varphi_1}^2\geq \frac {e}{\min\{\rho_1,\rho_2\}} $ 或 $ {\varphi_1}^2 < \frac {e}{\min\{\rho_1,\rho_2\}} $, 且

$\begin{align*} &\mu_1\rho_1(\log(\sqrt {\rho_1}\varphi_1)^2-1)+\frac{4}{p+q} \rho_1^{\frac{p}{2}}\rho_2^{\frac{q}{2}}|\varphi_1|^{p-2}|\varphi_1|^{q}\\ &+\mu_2\rho_2(\log(\sqrt {\rho_2}\varphi_1)^2-1)-\varepsilon \rho_1^{\frac{m}{2}}\rho_2^{\frac{n}{2}}|\varphi_1|^{m-2}|\varphi_1|^{n}\geq 0, \end{align*}$

可得

$ \hat{c_{\lambda_1(\Omega)}}\leq\frac{\rho_1+\rho_2}{2}\lambda_1(\Omega). $

若 $ {\varphi_2}^2\geq \frac {e}{\min\{\rho_1,\rho_2\}} $ 或 $ {\varphi_2}^2 < \frac {e}{\min\{\rho_1,\rho_2\}} $, 且

$\begin{align*} &\mu_1\rho_1(\log(\sqrt {\rho_1}\varphi_2)^2-1)+\frac{4}{p+q} \rho_1^{\frac{p}{2}}\rho_2^{\frac{q}{2}}|\varphi_2|^{p-2}|\varphi_2|^{q}\\ &+\mu_2\rho_2(\log(\sqrt {\rho_2}\varphi_2)^2-1)-\varepsilon \rho_1^{\frac{m}{2}}\rho_2^{\frac{n}{2}}|\varphi_2|^{m-2}|\varphi_2|^{n}\geq 0, \end{align*}$

可得$ \hat{c_{\lambda_2(\Omega)}}\leq\frac{\rho_1+\rho_2}{2}\lambda_2(\Omega). $

证

$\begin{align*} \hat{c_{\lambda_1(\Omega)}} \leq&I(\sqrt{\rho_1}\varphi_1,\sqrt{\rho_2}\varphi_1)\\ \leq &\frac{\rho_1+\rho_2}{2}\lambda_1(\Omega)-\frac{1}{2}\varphi_1^2\int_\Omega\Bigg(\mu_1\rho_1(\log(\sqrt {\rho_1}\varphi_1)^2-1)\\ &+\frac{4}{p+q} \rho_1^{\frac{p}{2}}\rho_2^{\frac{q}{2}}|\varphi_1|^{p+q-2}+\mu_2\rho_2(\log(\sqrt {\rho_2}\varphi_1)^2-1)-\varepsilon \rho_1^{\frac{m}{2}}\rho_2^{\frac{n}{2}}|\varphi_1|^{m+n-2}\Bigg)\mathrm{d}x\\ \leq&\frac{\rho_1+\rho_2}{2}\lambda_1(\Omega). \end{align*}$

同理也可得到

$\begin{align*} \hat{c_{\lambda_2(\Omega)}}\leq I\Bigg(\sqrt {\rho_1} \frac{\varphi_2^+}{|\varphi_2^+|_2},\sqrt {\rho_2} \frac{\varphi_2^-}{|\varphi_2^-|_2}\Bigg) \leq\frac{\rho_1+\rho_2}{2}\lambda_2(\Omega). \end{align*}$

该引理得证.

引理 3.7 令 $ \rho_1,\rho_2>0 $ 使得

$ \Lambda(\rho_1,\rho_2)\cdot(\rho_1+\rho_2)^{a-1} \leq \frac{{(a-1)}^{a-1}}{a^a}\lambda_j(\Omega)^{-(a-1)},\,\,\,j=1,2, $

其中 $ \Lambda>\frac{1}{a} $. 定义 $ \bar{\alpha}=\frac{a}{a-1}\lambda_j(\Omega) $, 则 $ c_{\bar{\alpha}}=I(\bar{u_1},\bar{u_2}) $ 可达到, $ (\bar{u_1},\bar{u_2})\in B_{\bar{\alpha}}\setminus U_{\bar{\alpha}} $ 为 $ I|_M $ 的局部极小值. 相应地, $ (\rho_1, \rho_2)\in A $, 其中 $ A $ 如 (1.5) 式定义.

证由 $ \bar{\alpha}=\frac{a}{a-1}\lambda_j(\Omega) $ 及 $ \Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^{a-1} \leq \frac{{(a-1)}^{a-1}}{a^a}\lambda_j(\Omega)^{-(a-1)}, j=1,2 $. 可得

$ \Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^{a-1} \leq \frac{\bar{\alpha}-\lambda_j(\Omega)}{\bar{\alpha}^a}. $

由引理 3.2 和引理 3.6 可得

$ \hat{c_{\lambda_j(\Omega)}}\leq \frac{\rho_1+\rho_2}{2}\lambda_j(\Omega)\leq \frac{1}{2}(\bar{\alpha}(\rho_1+\rho_2)-\Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^{a}\bar{\alpha}^a)\leq \hat{c_{\overline{\alpha}}}. $

由 $ 2+\frac{2}{N}<p+q<2^* $, 得 $ 0<\frac{1}{a^{\frac{1}{a-1}}}<0, \frac{1}{a-1}>\frac{N-2}{2}>0 $.

(1) 当 $ \Lambda>1 $ 时, $ \min\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\}=\frac{1}{\Lambda^{\frac{1}{a-1}}} $.

$\begin{align*} \bar{\alpha}\leq \frac{1}{\Lambda^{\frac{1}{a-1}}}\cdot\frac{1}{\rho_1+\rho_2}\cdot \frac{1}{a^{\frac{1}{a-1}}} &<\frac{1}{\Lambda^{\frac{1}{a-1}}}\cdot\frac{1}{\rho_1+\rho_2}\\ <\lambda_1(\Omega)+\frac{1}{\rho_1+\rho_2}\min\Bigg\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\Bigg\}. \end{align*}$

(2) 当 $ 0<\Lambda<1 $ 时, $ \min\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\}=\frac{1}{\Lambda^{\frac{N-2}{2}}} $.

$\begin{align*} \bar{\alpha}\leq \frac{1}{\Lambda^{\frac{1}{a-1}}}\cdot\frac{1}{\rho_1+\rho_2}\cdot \frac{1}{a^{\frac{1}{a-1}}} <\lambda_1(\Omega)+\frac{1}{\rho_1+\rho_2}\min\Bigg\{\frac{1}{\Lambda^{\frac{1}{a-1}}},\frac{1}{\Lambda^{\frac{N-2}{2}}}\Bigg\}. \end{align*}$

根据引理 3.5, 该引理得证.

定义

$\begin{aligned}G\left(u_{1}, u_{2}\right) & :=\int_{\Omega}\left(\mu_{1} u_{1}^{2}\left(\log u_{1}^{2}-1\right)+\frac{4}{p+q}\left|u_{1}\right|^{p}\left|u_{2}\right|^{q}+\mu_{2} u_{2}^{2}\left(\log u_{2}^{2}-1\right)+\frac{2 \varepsilon}{m+n}\left|u_{1}\right|^{m}\left|u_{2}\right|^{n}\right) \mathrm{d} x \\M_{\alpha}\left(\rho_{1}, \rho_{2}\right) & :=\sup _{U_{\alpha}\left(\rho_{1}, \rho_{2}\right)} G .\end{aligned}$

可得

$ \hat{c_\alpha}(\rho_1,\rho_2)=\frac{1}{2}(\alpha(\rho_1+\rho_2)-M_\alpha(\rho_1,\rho_2)). $

根据 (1.5) 式中 $ A $ 的定义可知, 对于任意 $ (\rho_1,\rho_2)\in A\setminus \{(0,0)\} $, 存在 $ \alpha $ 满足 $ \alpha \geq \max\{\frac{1}{\rho_1+\rho_2},\lambda_1 (\Omega)\} $ 且 $ (\bar{u}_1,\bar{u}_2)\in B_\alpha $, 使得 $ (\bar{u}_1,\bar{u}_2) $ 为 (1.1) 式的解, 满足 $ I(\bar{u}_1,\bar{u}_2)=c_\alpha<\hat{c_\alpha} $.

可得

$ \int_\Omega (\left|\nabla \bar{u_1}\right|^2+\left|\nabla \bar{u_2}\right|^2)\mathrm{d}x\leq(\rho_1+\rho_2)\alpha. $

定义

$ \bar{\alpha}:=\frac{1}{\rho_1+\rho_2}\int_\Omega (\left|\nabla \bar{u_1}\right|^2+\left|\nabla \bar{u_2}\right|^2)\mathrm{d}x<\alpha, $

因此 $ (\bar{u}_1,\bar{u}_2)\in U_{\bar{\alpha}}(\rho_1,\rho_2), \hat{c_{\overline{\alpha}}}=c_\alpha. $

引理 3.8 若 $ s>0 $, 则有 $ (s\bar{u}_1,s\bar{u}_2)\in U_{\bar{\alpha}}(s^2\rho_1,s^2\rho_2) $, 且满足

$ \hat{c_{\overline{\alpha}}}(s^2\rho_1,s^2\rho_2)=\frac{s^2}{2}\bar{\alpha}(\rho_1+\rho_2)-\frac{1}{2}G(s\bar{u}_1,s\bar{u}_2). $

证由 $ (\bar{u}_1,\bar{u}_2)\in U_{\bar{\alpha}}(\rho_1,\rho_2) $, 因此$ (s\bar{u}_1,s\bar{u}_2)\in U_{\bar{\alpha}}(s^2\rho_1,s^2\rho_2) $. 可得

$\begin{align*} \hat{c_{\overline{\alpha}}}(s^2\rho_1,s^2\rho_2) =\frac{s^2}{2}\bar{\alpha}(\rho_1+\rho_2)-\frac{1}{2}G(s\bar{u}_1,s\bar{u}_2). \end{align*}$

该引理得证.

引理 3.9

令 $ s\in(0,1) $, $ |t| $ 足够小,

$\begin{align*} (U_1(t),U_2(t)):=\Bigg(s\sqrt {\rho_1} \frac{\bar{u}_1+t\tau_1}{|\bar{u}_1+t\tau_1|_2}, s\sqrt {\rho_2} \frac{\bar{u}_2+t\tau_2}{|\bar{u}_2+t\tau_2|_2}\Bigg), \end{align*}$

其中 $ (\tau_1,\tau_2)\in H_0^1(\Omega,\mathbb{R}^2) $ 使得

$ \int_\Omega \bar{u}_1\tau_1\mathrm{d}x=\int_\Omega \bar{u}_2\tau_2\mathrm{d}x=0,\int _\Omega (\nabla \bar{u}_1\nabla \tau_1+\nabla \bar{u}_2\nabla \tau_2)\mathrm{d}x<0, $

且

$\begin{align*} \int_\Omega\Bigg( \frac{2p}{p+q}s^{p+q-1}|\bar{u}_1|^{p-1}|\bar{u}_2|^{q}\tau_1+ \frac{2q}{p+q}s^{p+q-1}|\bar{u}_2|^{q-1}|\bar{u}_1|^{p}\tau_2+\frac{m\varepsilon}{m+n}s^{m+n-1}|\bar{u}_1|^{m-1}\\ |\bar{u}_2|^{n}\tau_1+\frac{n\varepsilon}{m+n}s^{m+n-1}|\bar{u}_1|^{m}|\bar{u}_2|^{n-1}\tau_2+\mu_1s\bar{u}_1 \tau_1\log\bar{u}_1^2 s^2+\mu_2s\bar{u}_2 \tau_2\log\bar{u}_2^2 s^2\Bigg)\mathrm{d}x>0. \end{align*}$

则对任意 $ t $, $ (U_1(t),U_2(t))\in M_{(s^2\rho_1,s^2\rho_2)} $ 成立, 且

$ \frac {\mathrm{d}}{\mathrm{d}t}\|(U_1(t),U_2(t))\|_{H_0^1(\Omega)}^2\Bigg|_{t=0}<0,\,\,\,\,\frac {\mathrm{d}}{\mathrm{d}t}I(U_1(t),U_2(t))\Bigg|_{t=0}<0. $

证由于

$ \int_\Omega\Bigg(s\sqrt {\rho_i}\frac{\bar{u}_i+t\tau_i}{|\bar{u}_i+t\tau_i|_2}\Bigg)^2\mathrm{d}x=s^2\rho_i,\,\,\,i=1,2. $

因此 $ (U_1(t),U_2(t))\in M_{(s^2\rho_1,s^2\rho_2)} $ 对任意 $ t $ 成立. 由 $ \frac {\mathrm{d}}{\mathrm{d}t}U_1(t)=s\sqrt {\rho_1}\frac{\tau_1}{|\bar{u}_1+t\tau_1|_2} $ 可得

$\frac {\mathrm{d}}{\mathrm{d}t}(U_1(t),U_2(t))\Bigg|_{t=0}=(s\tau_1,s\tau_2),$

$\begin{align*} \frac {\mathrm{d}}{\mathrm{d}t}\|(U_1(t),U_2(t))\|_{H_0^1(\Omega)}^2\Bigg|_{t=0} =\int_\Omega s^2(\nabla\bar{u}_1\nabla\tau_1+\nabla\bar{u}_2\nabla\tau_2)\mathrm{d}x<0. \end{align*}$

由条件可知

$\begin{align*} &\frac {\mathrm{d}}{\mathrm{d}t}G(U_1(t),U_2(t))\Bigg|_{t=0}\\ =&2s\int_\Omega\Bigg( \frac{2p}{p+q}s^{p+q-1}|\bar{u}_1|^{p-1}|\bar{u}_2|^{q}\tau_1+ \frac{2q}{p+q}s^{p+q-1}|\bar{u}_2|^{q-1}|\bar{u}_1|^{p}\tau_2+\frac{m\varepsilon}{m+n}s^{m+n-1}|\bar{u}_1|^{m-1}\\ &|\bar{u}_2|^{n}\tau_1 +\frac{n\varepsilon}{m+n}s^{m+n-1}|\bar{u}_1|^{m}|\bar{u}_2|^{n-1}\tau_2+\mu_1s\bar{u}_1 \tau_1\log\bar{u}_1^2 s^2+\mu_2s\bar{u}_2 \tau_2\log\bar{u}_2^2 s^2\Bigg)\mathrm{d}x>0. \end{align*}$

则可得

$\begin{align*} \frac {\mathrm{d}}{\mathrm{d}t}I(U_1(t),U_2(t))\Bigg|_{t=0} =&s^2\int _\Omega (\nabla \bar{u}_1\nabla \tau_1+\nabla \bar{u}_2\nabla \tau_2)\mathrm{d}x-s\int_\Omega\Bigg( \frac{2p}{p+q}s^{p+q-1}|\bar{u}_1|^{p-1}|\bar{u}_2|^{q}\tau_1 +\frac{2q}{p+q}\\ &s^{p+q-1}|\bar{u}_2|^{q-1}|\bar{u}_1|^{p}\tau_2+\frac{m\varepsilon}{m+n}s^{m+n-1}|\bar{u}_1|^{m-1}|\bar{u}_2|^{n}\tau_1 +\frac{n\varepsilon}{m+n} s^{m+n-1}\\ &|\bar{u}_1|^{m}|\bar{u}_2|^{n-1}\tau_2 +\mu_1s\bar{u}_1 \tau_1\log\bar{u}_1^2 s^2+\mu_2s\bar{u}_2 \tau_2\log\bar{u}_2^2 s^2\Bigg)\mathrm{d}x<0. \end{align*}$

该引理得证.

下面引入星形集的定义

若 $ \mathbb{R}^N $ 中的一个子集 $ S $ 满足 $ \forall x\in S, \mbox{线段}\ \{\lambda x_0+(1-\lambda)x|\lambda \in [0,1]\}\subseteq S, $ 则称 $ S $ 是一个星形集.

引理 3.10 集合 $ A $ 相对于 (0,0) 是星形集.

证由 $ (\bar{u}_1,\bar{u}_2)\in U_{\bar{\alpha}}(\rho_1,\rho_2) $, 得$ (U_1(0),U_2(0))=(s\bar{u}_1,s\bar{u}_2)\in U_{\bar{\alpha}}(s^2\rho_1,s^2\rho_2) $. 存在极小的正常数 $ \varepsilon,\tau $ 使得 $ (U_1(\tau),U_2(\tau))\in U_{\bar{\alpha}-\varepsilon}(s^2\rho_1,s^2\rho_2) $.

则可得

$ \hat{c_{\overline{\alpha}-\varepsilon}}(s^2\rho_1,s^2\rho_2)\leq I(U_1(\tau),U_2(\tau))<I(U_1(0),U_2(0))=\hat{c_{\overline{\alpha}}}(s^2\rho_1,s^2\rho_2). $

由引理 3.5 得 $ (s^2\rho_1,s^2\rho_2)\in A $. 该引理得证.

定理 1.2 的证明 由 $ \Lambda $ 的定义可得

$\begin{align*} \Lambda(\rho_1,\rho_2)\leq \frac{N-2}{2e}S(N)((\rho_1+\rho_2)\alpha)^{\frac{2^*}{2}}\max\{ \mu_1,\mu_2 \}+\frac{|\Omega|}{e}(\mu_1+\mu_2). \end{align*}$

由条件可得

$\begin{align*} \Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^{a-1} \leq R(\Omega,N,p+q). \end{align*}$

记 $ R(\Omega,N,p+q)=\frac{{(a-1)}^{a-1}}{a^a}\lambda_j(\Omega)^{-(a-1)} $, 则有

$ \Lambda(\rho_1,\rho_2)(\rho_1+\rho_2)^{a-1}\leq \frac{{(a-1)}^{a-1}}{a^a}\lambda_j(\Omega)^{-(a-1)}. $

根据引理 3.7 得 $ (\rho_1,\rho_2)\in A $, 该定理得证.

4 解的稳定性

定义集合

$ G_{\overline{\alpha}}:=\{(u_1,u_2)\in H_0^1(\Omega;\mathbb{C}^2):(|u_1|,|u_2|)\in B_{\overline{\alpha}},I(u_1,u_2)=c_{\overline{\alpha}}\}. $

引理 4.1 对于 $ (u_1,u_2)\in G_{\overline{\alpha}} $, 存在 $ \theta_1,\theta_2 \in \mathbb{R} $, 使得 $ (u_1,u_2)=({\rm e}^{{\rm i}\omega_1\theta_1}|u_1|,{\rm e}^{{\rm i}\omega_2\theta_2}|u_2|), $且

$\begin{align*} &\inf \{ I(u_1,u_2):(u_1,u_2) \in H_0^1(\Omega,\mathbb{C}^2),(|u_1|,|u_2|)\in B_{\overline{\alpha}}\}=c_{\overline{\alpha}},\\ &\inf \{ I(u_1,u_2):(u_1,u_2) \in H_0^1(\Omega,\mathbb{C}^2),(|u_1|,|u_2|)\in U_{\overline{\alpha}}\}:=\tilde{c_{\overline{\alpha}}}\leq \hat{c_{\overline{\alpha}}}. \end{align*}$

证若 $ (\tau_1,\tau_2)\in U_\alpha $, 则 $ (\tau_1,\tau_2)\in H_0^1(\Omega;\mathbb{C}^2), (|\tau_1|,|\tau_2|)\in U_{\overline{\alpha}} $, 可得 $ \tilde{c_{\overline{\alpha}}}\leq \hat{c_{\overline{\alpha}}} $.

若 $ (u_1,u_2)\in G_{\overline{\alpha}} $, 则 $ (u_1,u_2)\in H_0^1(\Omega;\mathbb{C}^2), I(u_1,u_2)=c_{\overline{\alpha}} $, 由 diamagnetic 不等式及引理 2.5 得

$\begin{align*} \int_\Omega|\nabla|u_i||^2\mathrm{d}x\leq \int_\Omega|\nabla u_i|^2\mathrm{d}x, \,\,|u_i|^2 \log |u_i|^2= u_i^2 \log u_i^2,\,\,\,i=1,2. \end{align*}$

由 $ I(u_1,u_2) $ 定义可得

$\begin{align*} I(|u_1|,|u_2|)\leq I(u_1,u_2),\,\,\,c_{\overline{\alpha}} \leq I(|u_1|,|u_2|)\leq I(u_1,u_2)=c_{\overline{\alpha}} \end{align*}$

则 $ (u_1,u_2) $ 在复数域上与实数域上一致, 前面所证在复数域上仍成立.

该引理得证.

引理 4.2 假设 $ \{ (\varphi_{1,n},\varphi_{2,n})\}\subset H_0^1(\Omega,\mathbb{C}^2) $ 满足

(4.1)$\begin{aligned}\int_\Omega|\varphi_{i,n}|^2\mathrm{d}x \rightarrow \rho_i, i=1,2, I(\varphi_{1,n},\varphi_{2,n})\rightarrow c_{\overline{\alpha}}, n\rightarrow\infty,\end{aligned}$

且当 $ n $ 足够大时, 满足

(4.2) $\begin{aligned}\int_\Omega(|\nabla \varphi_{1,n}|^2+|\nabla \varphi_{2,n}|^2)\mathrm{d}x\leq(\rho_1+\rho_2)\bar{\alpha}+o(1).\end{aligned}$

则存在 $ (u_1,u_2)\in G_{\overline{\alpha}} $, 使得 $ (\varphi_{1,n},\varphi_{2,n})\rightarrow (u_1,u_2) $ 在 $ H_0^1(\Omega,\mathbb{C}^2) $ 中成立.

证由引理 3.4 及引理 4.1 可得该引理得证.

引理 4.3 若 $ c_{\overline{\alpha}}<\tilde{c_{\overline{\alpha}}} $, 则 $ G_{\overline{\alpha}} $ 是轨道稳定的.

证假设 $ \{ (\varphi_{1,n},\varphi_{2,n})\}\subset H_0^1(\Omega,\mathbb{C}^2) $, $ (u_{1,n},u_{2,n})\in G_{\overline{\alpha}} $. 且 $ \overline{\varepsilon}>0 $ 满足

$\begin{align*} &\underset{n\rightarrow\infty}{\lim}\|(\varphi_{1,n},\varphi_{2,n})-(u_{1,n},u_{2,n})\|_ {H_0^1(\Omega,\mathbb{C}^2)}=0,\\ &\underset{t>0}{\sup} \mathrm{dist}_{H_0^1}((\Phi_{1,n}(t,\cdot),\Phi_{2,n}(t,\cdot)), G_{\overline{\alpha}})\geq2\overline{\varepsilon}, \end{align*}$

其中 $ (\Phi_{1,n},\Phi_{2,n}) $ 是 (1.1) 式的解. 则存在 $ \{t_n\} $, 令 $ \Psi_{i,n}:=\Phi_{i,n}(t_n,x) $, 即 $ \mathrm{dist}_{H_0^1}(\Psi_{1,n},\Psi_{2,n}), G_{\overline{\alpha}})\geq\overline{\varepsilon}. $ 引理 4.2 说明 $ G_{\overline{\alpha}} $ 是紧的, 则存在 $ (u_1,u_2)\in G_{\overline{\alpha}} $, 使得 $ (\varphi_{1,n},\varphi_{2,n})\rightarrow (u_1,u_2) $ 在 $ H_0^1(\Omega,\mathbb{C}^2) $ 中成立. 由 Sobolev 嵌入的连续性及引理 2.4 和引理 2.6 可得

$\begin{align*} &\int_\Omega\varphi_{i,n}^2\log\varphi_{i,n}^2\mathrm{d}x=\int_\Omega u_i^2\log u_i^2\mathrm{d}x,\\ &\int_\Omega|\varphi_{1,n}|^p|\varphi_{2,n}|^q\mathrm{d}x=\int_\Omega|u_1|^p|u_2|^q\mathrm{d}x,\,\,\,i=1,2. \end{align*}$

由 $ (u_1,u_2)\in G_{\overline{\alpha}} $, 可知 $ (\varphi_{1,n},\varphi_{2,n}) $ 满足 (4.1) 式, 则 $ \Psi_{i,n}(x) $ 满足 (4.1) 式.

假设 $ \int_\Omega(|\nabla \Psi_{1,n}|^2+|\nabla\Psi_{2,n}|^2)\mathrm{d}x\geq(\rho_1+\rho_2)\bar{\alpha}+\bar{\varepsilon} $. 对任意足够大的 $ n $, 存在 $ \bar{t_n}\in(0,t_n) $, 使得 $ (\Phi_{1,n}(\bar{t_n},\cdot),\Phi_{1,n}(\bar{t_n},\cdot)) $ 满足 (4.2) 式, 且

$\begin{align*} \int_\Omega(|\nabla \Phi_{1,n}(\bar{t_n},\cdot)|^2+|\nabla \Phi_{2,n}(\bar{t_n},\cdot)|^2)\mathrm{d}x=(\rho_1+\rho_2)\bar{\alpha}+o(1). \end{align*}$

由引理 3.3.2, 存在 $ (\bar{u_1},\bar{u_2})\in G_{\overline{\alpha}} $, 使得

$\begin{align*} \int_\Omega(|\nabla \bar{u_1}|^2+|\nabla \bar{u_2}|^2)\mathrm{d}x=(\rho_1+\rho_2)\bar{\alpha}, \end{align*}$

所以 $ \tilde{c_{\overline{\alpha}}}<\hat{c_{\overline{\alpha}}}=c_{\overline{\alpha}} $ 与题中条件矛盾. 由引理 4.2 可知该引理得证.

引理 4.4 $ c_{\overline{\alpha}}<\widetilde{c_{\overline{\alpha}}} $.

证假设 $ c_{\overline{\alpha}}=\widetilde{c_{\overline{\alpha}}} $, 则存在 $ \varepsilon_n\rightarrow0, (\tau_{1,n},\tau_{2,n})\in H_0^1(\Omega,\mathbb{C}^2) $, 使得

$\begin{align*} &\|(\tau_{1,n},\tau_{2,n})\|_{H_0^1(\Omega,\mathbb{C}^2)}=\int_\Omega(|\nabla \tau_{1,n}|^2+|\nabla \tau_{2,n}|^2)\mathrm{d}x=(\rho_1+\rho_2)\bar{\alpha},\\ &\int_\Omega|\tau_{i,n}|^2\mathrm{d}x = \rho_i,\ c_{\overline{\alpha}}\leq I(\tau_{1,n},\tau_{2,n})\leq c_{\overline{\alpha}}+\varepsilon_n,\,\,i=1,2. \end{align*}$

定义 $ u_{i,n}=|\tau_{i,n}|,\,\,i=1,2 $, 则

$\begin{align*} \|(u_{1,n},u_{2,n})\|_{H_0^1(\Omega,\mathbb{R}^2)}^2 \leq \|(\tau_{1,n},\tau_{2,n})\|_{H_0^1(\Omega,\mathbb{C}^2)}^2=(\rho_1+\rho_2)\bar{\alpha}. \end{align*}$

所以 $ (u_{1,n},u_{2,n}) $ 可以作为 $ c_{\overline{\alpha}} $ 最小化问题的一个可容许的解, 且

(4.3)$\begin{matrix} c_{\overline{\alpha}}\leq I(u_{1,n},u_{2,n})\leq I(\tau_{1,n},\tau_{2,n})\leq c_{\overline{\alpha}}+\varepsilon_n. \end{matrix}$

特别地

(4.4)$\begin{matrix}\frac{1}{2}(\|(\tau_{1,n},\tau_{2,n})\|_{H_0^1(\Omega,\mathbb{C}^2)}^2-\|(u_{1,n},u_{2,n})\|_{H_0^1(\Omega,\mathbb{R}^2)}^2)=I(\tau_{1,n},\tau_{2,n})-I(u_{1,n},u_{2,n}) \leq\varepsilon_n. \end{matrix}$

由引理 4.2 可知, 当 $ n $ 足够大时, $ (\tau_{1,n},\tau_{2,n})\rightarrow (\tau_{1,\infty},\tau_{2,\infty}) $ 且 $ (u_{1,n},u_{2,n})\rightarrow (u_{1,\infty},u_{2,\infty}) $. 由 (4.3), (4.4) 式, Sobolev 嵌入的性质及引理 2.4, 引理 2.6 可推断出

$\begin{align*} &\int_\Omega|u_{1,n}|^p|u_{2,n}|^q\mathrm{d}x=|u_{1,\infty}|^p|u_{2,\infty}|^q,\ I(u_{1,\infty},u_{2,\infty})=c_{\overline{\alpha}}. \end{align*}$

则

$\left(u_{1, \infty}, u_{2, \infty}\right) \in U_{\bar{\alpha}}, \widehat{c_{\bar{\alpha}}} \leq I\left(u_{1, \infty}, u_{2, \infty}\right)=c_{\bar{\alpha}}.$

此时与 $ c_{\overline{\alpha}}<\widehat{c_{\overline{\alpha}}} $ 矛盾, 所以 $c_{\overline{\alpha}}<\widetilde{c_{\overline{\alpha}}}$. 该引理得证.

定理 1.3 的证明

结合引理 4.3 及引理 4.4, 该定理得证.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

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Brezis

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薛艳昉, 韩建新.

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Existence and multiplicity of normalized solutions for a class of quasilinear Schrödinger equations

Advances in Applied Mathematics, 2022, 11 (12): 8485-8503

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Deng

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, et al.

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

Advanced Nonlinear Studies, 2023, 23 (1): 20220049

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We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: \n \n \n \n \n \n \n \n \n \n −\n Δ\n u\n =\n \n \n \n ∣\n \n u\n \n ∣\n \n \n \n \n \n 2\n \n \n ∗\n \n \n −\n 2\n \n \n u\n +\n λ\n u\n +\n μ\n u\n log\n \n \n u\n \n \n 2\n \n \n \n \n \n x\n ∈\n Ω\n,\n \n \n \n \n u\n =\n 0\n \n \n \n x\n ∈\n ∂\n Ω\n,\n \n \n \n \n \n \n \\left\\{\\phantom{\\rule[-1.25em]{}{0ex}}\\begin{array}{ll}-\\Delta u={| u| }^{{2}^{\\ast }-2}u+\\lambda u+\\mu u\\log {u}^{2}\\hspace{1.0em}& x\\in \\Omega,\\\\ u=0\\hspace{1.0em}& x\\in \\partial \\Omega,\\end{array}\\right.\n \n where \n \n \n \n Ω\n ⊂\n \n \n R\n \n \n N\n \n \n \n \\Omega \\subset {{\\mathbb{R}}}^{N}\n \n is a bounded open domain, \n \n \n \n λ\n,\n μ\n ∈\n R\n \n \\lambda,\\mu \\in {\\mathbb{R}}\n \n, \n \n \n \n N\n ≥\n 3\n \n N\\ge 3\n \n and \n \n \n \n \n \n 2\n \n \n ∗\n \n \n ≔\n \n \n 2\n N\n \n \n N\n −\n 2\n \n \n \n {2}^{\\ast }:= \\frac{2N}{N-2}\n \n is the critical Sobolev exponent for the embedding \n \n \n \n \n \n H\n \n \n 0\n \n \n 1\n \n \n \n (\n \n Ω\n \n )\n \n \n ↪\n \n \n \n L\n \n \n \n \n 2\n \n \n ∗\n \n \n \n \n \n (\n \n Ω\n \n )\n \n \n {H}_{0}^{1}\\left(\\Omega )\\hspace{0.33em}\\hookrightarrow \\hspace{0.33em}{L}^{{2}^{\\ast }}\\left(\\Omega )\n \n. The uncertainty of the sign of \n \n \n \n s\n log\n \n \n s\n \n \n 2\n \n \n \n s\\log {s}^{2}\n \n in \n \n \n \n \n (\n \n 0\n,\n +\n ∞\n \n )\n \n \n \\left(0,+\\infty )\n \n has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided \n \n \n \n λ\n ∈\n R\n,\n μ\n >\n 0\n \n \\lambda \\in {\\mathbb{R}},\\mu \\gt 0\n \n and \n \n \n \n N\n ≥\n 4\n \n N\\ge 4\n \n. While the case of \n \n \n \n μ\n <\n 0\n \n \\mu \\lt 0\n \n is thornier. However, for \n \n \n \n N\n =\n 3\n,\n 4\n \n N=3,4\n \n, \n \n \n \n λ\n ∈\n \n (\n \n −\n ∞\n,\n \n \n λ\n \n \n 1\n \n \n \n (\n \n Ω\n \n )\n \n \n )\n \n \n \\lambda \\in \\left(-\\infty,{\\lambda }_{1}\\left(\\Omega ))\n \n, we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for \n \n \n \n μ\n <\n 0\n \n \\mu \\lt 0\n \n and \n \n \n \n −\n \n \n \n (\n \n N\n −\n 2\n \n )\n \n μ\n \n \n 2\n \n \n +\n \n \n \n (\n \n N\n −\n 2\n \n )\n \n μ\n \n \n 2\n \n \n log\n \n \n −\n \n \n \n (\n \n N\n −\n 2\n \n )\n \n μ\n \n \n 2\n \n \n \n \n +\n λ\n −\n \n \n λ\n \n \n 1\n \n \n \n (\n \n Ω\n \n )\n \n ≥\n 0\n \n -\\frac{\\left(N-2)\\mu }{2}+\\frac{\\left(N-2)\\mu }{2}\\log \\left(-\\frac{\\left(N-2)\\mu }{2}\\right)+\\lambda -{\\lambda }_{1}\\left(\\Omega )\\ge 0\n \n if \n \n \n \n N\n ≥\n 3\n \n N\\ge 3\n \n. Comparing with the results in the study by Brézis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter \n \n \n \n μ\n \n \\mu \n \n on logarithmic perturbation is not zero.

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张恭庆.

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Alfaro

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Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

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Chen

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Journal of Mathematical physics, 2020, 61 : Art 101510

[本文引用: 1]

Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents

1983

... 对数薛定谔方程是一种非线性薛定谔方程, 它是薛定谔方程的一种推广, 在凝聚态物理和量子信息领域中被广泛应用. Brézis 和 Nirenberg 在文献 [1] 中提出了方程 ...

Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains

2015

... 解的存在性取决于 $ \lambda $ 和 $ N $ 的值. Noris, Tavares 和 Verzini 在文献 [2] 中考虑了非线性方程 ...

Normalized solutions for nonlinear Schr?dinger systems on bounded domains

2019

... 在球面域 $ \Omega =B_1 $ 上的解的存在性和轨道稳定性. Tavares 和 Verzini 在文献 [3] 中考虑了方程 (1.2) 在有界域上正规化解的存在性. Squassina 和 Szulkin 在文献 [5] 中研究了带有周期势的对数薛定谔方程 $ -\Delta u+V(x)u=u\log u^2, x \in \mathbb{R}^N $正基态解的存在性. 薛艳昉, 韩建新在文献 [5] 中研究了一类薛定谔方程正解的存在性问题. 景丽, 滕凯民在文献 [6] 中利用扰动方法证明了一类拟线性薛定谔方程基态规范解的存在性并分析了解的收敛性. Deng, He, Pan 和 Zhong 在文献 [7] 中证明了方程 $ -\Delta u+\omega u=\mu u\log u^2+\left|u\right|^{p-2} u $ 在 $ \mathbb{{R}}^N $ 上正解的存在性. ...

... 由文献 [3] 可知, 当 $ (\sqrt {\rho_1}\varphi_1,\sqrt {\rho_2}\varphi_2)\in B_\alpha $, 则 $ B_\alpha \neq \emptyset $. 此外, 令 ...

Multiple solutions to logarithmic Schr?dinger equations with periodic potential

2015

薛定谔方程在 Berestycki-Lions 条件下解的存在性

2022

... ] 中研究了带有周期势的对数薛定谔方程 $ -\Delta u+V(x)u=u\log u^2, x \in \mathbb{R}^N $正基态解的存在性. 薛艳昉, 韩建新在文献 [5] 中研究了一类薛定谔方程正解的存在性问题. 景丽, 滕凯民在文献 [6] 中利用扰动方法证明了一类拟线性薛定谔方程基态规范解的存在性并分析了解的收敛性. Deng, He, Pan 和 Zhong 在文献 [7] 中证明了方程 $ -\Delta u+\omega u=\mu u\log u^2+\left|u\right|^{p-2} u $ 在 $ \mathbb{{R}}^N $ 上正解的存在性. ...

Existence of solutions for the Schr?dinger equation under Berestycki-Lions conditions

2022

一类拟线性薛定谔方程规范化解的存在性和多重性

2022

Existence and multiplicity of normalized solutions for a class of quasilinear Schr?dinger equations

2022

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

2023

1983

... 引理 2.1^[8] ($Hölder不等式$) 设 $ p>1 $, $ q>1 $, 且 $ \frac{1}{p}+\frac{1}{q}=1 $. 若 $ f\in L^p(\mathbb{R}^2) $, $ g\in L^q(\mathbb{R}^2) $, 则有 $ f\cdot g\in L^1(\mathbb{R}^2) $, 且 ...

1983

Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

2017

... 引理 2.2^[9] (Gagliardo-Nirenberg不等式) 对于任意的 $ u\in H^1(\mathbb{R}^N) $,% $ r=\frac{p+q}{4}-\frac{N(p+q-2)}{8} $ ...

Orbital stability of Gausson solutions to logarithmic Schr?dinger equations electron

2016

... 接下来, 分别引入对数项 $ {u}^2 \log {u}^2 $ (见文献 [10]) 和对偶项 $ |u_{1}|^p |u_{2}|^q $ (见文献 [11]) ...

Ground states solution of Nehari-Poho$\check{\rm z}$aev type for periodic quasilinear Schr?dinger system

2020

... 接下来, 分别引入对数项 $ {u}^2 \log {u}^2 $ (见文献 [10]) 和对偶项 $ |u_{1}|^p |u_{2}|^q $ (见文献 [11]) ...

〈

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