数学物理学报, 2023, 43(4): 1062-1072

含混合项的拟线性Schrödinger方程的正规化基态解

归坤明,, 陶虹杉,, 杨俊,*

华南理工大学数学学院 广州 510640

Normalized Ground States for the Quasi-linear Schrödinger Equation with Combined Nonlinearities

Gui Kunming,, Tao Hongshan,, Yang Jun,*

School of Mathematics, South China University of Technology, Guangzhou 510640

通讯作者: *杨俊, E-mail: yangjun@scut.edu.cn

收稿日期: 2022-06-17   修回日期: 2023-01-12  

基金资助: 广东省自然科学基金(2018A0303130196)

Received: 2022-06-17   Revised: 2023-01-12  

Fund supported: NSF of Guangdong Province(2018A0303130196)

作者简介 About authors

归坤明，E-mail:guikunming@qq.com;

陶虹杉，sora1924@outlook.com

摘要

该文研究了一类含混合项拟线性Schrödinger方程正规化基态解的存在性. 推广了文献[1-2]中的结果, 与他们研究的情形相比, 该文中方程对应能量泛函的结构更加复杂.

关键词： 正规化解; 拟线性Schrödinger方程; 摄动法

Abstract

In this paper, we mainly investigate the existence of normalized ground states for the Schrödinger equation with combined nonlinearities. Our results extend those reported in [1-2]. Compared with the case they studied, the structure of the energy function correspongding to the equation in this paper is more complex.

Keywords： Normalized solutions; Quasi-linear Schrödinger equation; Perturbation method

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本文引用格式

归坤明, 陶虹杉, 杨俊. 含混合项的拟线性Schrödinger方程的正规化基态解[J]. 数学物理学报, 2023, 43(4): 1062-1072

Gui Kunming, Tao Hongshan, Yang Jun. Normalized Ground States for the Quasi-linear Schrödinger Equation with Combined Nonlinearities[J]. Acta Mathematica Scientia, 2023, 43(4): 1062-1072

1 引言

拟线性Schrödinger方程

$i \phi_{t}+\Delta \phi+\phi \Delta\left(|\phi|^{2}\right)+f\left(|\phi|^{2}\right) \phi=0, \quad(t, x) \in(0,+\infty) \times \mathbb{R}^{N}$

可以用来描述等离子体物理、非线性光学和流体力学中的一些现象, 参见文献[3⇓-5]等. 这里$\phi$ 表示波函数, $N\geqslant 1$.

考虑方程(1.1)形如$\phi_\lambda(t,x)=u_\lambda(x)e^{-i\lambda t}$的驻波解, 其中$\lambda \in \mathbb{R} $为固定的参数. 代入方程(1.1)式可得一类含变分结构的拟线性方程

$\begin{equation} -\Delta u -u\Delta(u^2)=\lambda u+g(u),uad x\in \mathbb{R} ^N,\end{equation}$

其中$g(u)=f(u^2)u$.

对于给定的$\lambda$, 关于方程(1.2)非平凡解的存在性, 目前已有大量的结果, 如文献[6,7]. 本文关注的是方程(1.2)正规化解的存在性, 即解$u$ 满足(1.2)且有

$\begin{equation} \int_{\mathbb{R} ^N}\left|u\right|^2=a,\end{equation}$

其中$a>0$是固定的常数.

关于方程(1.2)正规化解的研究, 近期吸引了很多学者的关注. 当$g(u)=|u|^{p-2}u$时, 文献[8]考虑了在限制(1.3)式下, 极小解的存在和不存在性. 准确地讲, 作者证明了, 当$2<p<2+\frac{4}{N}$时, 限制极小$m(a)<0,\ \forall a>0$; 当$2+\frac{4}{N}\leqslant p<4+\frac{4}{N}$时, 存在一个常数$C(p,N)>0$, 使得当$0<a<C(p,N)$时, $m(a)=0$且不可达; 当$a>C(p,N)$ 时, $m(a)<0$ 且可达. 借助于限制上的山路定理和摄动方法[9], 文献[10]进一步考虑了$2+\frac{4}{N}\leqslant p<4+\frac{4}{N}$ 时, 多重解的存在性. 近期, 在文献[1]中, 同样采取摄动方法, 得到了$L^2$临界, 即$p=4+\frac{4}{N}$ 时, 基态解的存在性和不存在性, 以及$L^2$ 超临界, 即$4+\frac{4}{N}<p<\frac{N+2}{N-2}$时, 解的存在性. 另外,文献[11]研究了方程(1.2)含有位势项$V(x)u$的情况. 当$2<p<4+\frac{4}{N}$时, 他们得到了基态解在$L^2$临界时的存在性,以及$q$趋于$L^2$临界时的渐近行为.

本文考虑方程(1.2)的一种混合情形, 即非线性项包含$L^2$ 临界和次临界指数的情形

$\begin{equation} -\Delta u -u\Delta(u^2)=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,uad x\in \mathbb{R} ^N,\end{equation} $

其中$\lambda,\mu\in \mathbb{R},\ 2<q<p=\overline{p}\triangleq 4+\frac{4}{N},\ N\geq 3$. 这种情况是文献[2]关于NLS方程结果在拟线性Schrödinger方程上的推广, 也是文献[1]中拟线性Schrödinger方程非线性项的推广. 需要指出的是, 此时方程(1.4)对应能量泛函的结构更复杂, 故要进行更精细的估计.

定义 1.1 称$\tilde{u}$是方程(1.4)式在$S_a$上的基态解, 若它是方程(1.4)在$S_a$ 上有最小能量的解

${\rm d}E_\mu|_{S_a}(\tilde{u})=0,uad E_\mu(\tilde{u})=\mathop{\inf}\limits_{u\in S_a}\{E_\mu(u)|{\rm d}E_\mu|_{S_a}(u)=0\},$

其中泛函为

$\begin{equation} E_\mu(u)=\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\frac{\mu}{q}\int_{\mathbb{R} ^N}\left|u\right|^q-\frac{1}{p}\int_{\mathbb{R} ^N}\left|u\right|^p\end{equation}$

限制在$L^2$球

$u\in S_a\triangleq\left\{u\in X\left|\int_{\mathbb{R} ^N}\left|u\right|^2=a\right.\right\}$

上, 这里

$X\triangleq\left\{u\in H^1(\mathbb{R} ^N)\left|\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2<+\infty\right.\right\}.$

并记

$m(a,\mu)\triangleq\mathop{\inf}\limits_{u\in S_a}E_\mu(u).$

同时令$a^*=\left|Q_{\overline{p}}\right|_1$为临界质量(具体定义见下节).

下面是本文的主要结论.

定理 1.1 当$a<a^*$时,有

(i)当指标处于以下范围之一

$ \begin{eqnarray*} \left\{ \begin{array}{lll} \mu>0, 2<q<2+\frac{4}{N},\\[3mm] \mu>\mu_1, q=2+\frac{4}{N},\\[3mm] \mu>\mu_2, 2+\frac{4}{N}<q<4+\frac{4}{N} \end{array}\right. \end{eqnarray*}$

时, 有$-\infty<m(a,\mu)<0$, 此时下确界可达. 其中$\mu_1,\mu_2>0$为常数, 其值见(3.2),(3.3)式. 记达到函数为$\tilde{u}$, 则$\exists \lambda_a <0$, 使$(\lambda_a,\tilde{u})$是方程(1.4)的基态解.

(ii)当$\mu<0$时, 有$m(a,\mu)=0$, 此时方程(1.4)无解.

定理 1.2 当$a=a^*$时,有

(i) 当$\mu>0,\ 2<q<2+\frac{4}{N}$ 时, 有$-\infty<m(a,\mu)<0$, 此时下确界可达. 记达到函数为$\tilde{u}$, 则$\exists \lambda_a <0$, 使$(\lambda_a,\tilde{u})$ 是方程(1.4)的基态解.

(ii) 当指标处于以下范围之一

$ \begin{eqnarray*} \left\{ \begin{array}{lll} \mu>\mu_3,& & q=2+\frac{4}{N},\\[3mm] \mu>0,& & 2+\frac{4}{N}<q<4+\frac{4}{N} \end{array}\right. \end{eqnarray*}$

时, 有$m(a,\mu)=-\infty$, 其中$\mu_3>0$为常数, 其值见(5.1)式.

(iii) 当$\mu<0$时, 有$m(a,\mu)=0$, 此时方程(1.4)无解.

定理 1.3 当$a>a^*$时, 有$m(a,\mu)=-\infty$.

研究该问题的首要困难在于: 能量泛函中如下的积分项$\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2$ 在$H^1(\mathbb{R} ^N)$上不可微, 受文献[9,10]的启发, 本文采取添加扰动项的办法解决该问题.另一方面, 相比于不含拟线性项的问题[2], 本文中指标$p,q$取值范围更大. 而泛函下确界的性质, 以及原问题的可解性, 与$p,q,\mu$的取值密切相关, 因此需要细致地划分多种情形.

注 1.1 本文用$\left| \cdot \right|_p$表示$L^p(\mathbb{R} ^N)$空间中的通常范数,即$\left|u\right|_p \triangleq \left(\int_{\mathbb{R} ^N}\left|u\right|^p\right)^{\frac{1}{p}}$. $C, C_1, C_2,\cdots$通常指的是一些变化的常数, 对主要结论没有影响.

2 预备

为了证明主要结论, 需要用到下列的Gagliardo-Nirenberg不等式.

引理 2.1 (文献[Gagliardo Nirenberg不等式]) 设$2<p<2(2^*)$, $u\in L^1(\mathbb{R} ^N)$且$\nabla u\in L^2(\mathbb{R} ^N)$, 则有

$\begin{equation} \int_{\mathbb{R} ^N}\left|u\right|^{\frac{p}{2}}\leqslant \frac{C(p,N)}{\left|Q_p\right|^{\frac{p-2}{N+2}}_1} \left(\int_{\mathbb{R} ^N}\left|u\right|\right)^{\eta_p} \left(\int_{\mathbb{R} ^N}\left|\nabla u\right|^2\right)^{\frac{p\gamma_p}{N+2}},\end{equation}$

其中

$\gamma_p=\frac{N(p-2)}{2p},uad \eta_p=\frac{4N-p(N-2)}{2(N+2)}$

和

$C(p,N)=\frac{p(N+2)}{[4N-p(N-2)]^{\frac{4-N(p-4)}{2(N+2)}}[2N(p-2)]^{\frac{p\gamma_p}{N+2}}},$

$Q_p$为下列问题的唯一非负径向对称解

$\begin{eqnarray*}\left\{ \begin{array}{lll} -\Delta u+1=\lambda u^{\frac{p}{2}-1},& & x\in B_R, \ \ R>0,\ \ \mbox{supp} u\subset \subset B_R,\\ u=\frac{\partial u}{\partial n}=0,& & x\in \partial B_R. \end{array}\right.\end{eqnarray*}$

取$u\in X$, 在不等式(2.1)中用$u^2$替换$u$, 可得

$\begin{equation} \frac{1}{p}\int_{\mathbb{R} ^N}\left|u\right|^p\leqslant K(p,N)\left(\int_{\mathbb{R} ^N}\left|u\right|^2\right)^{\eta_p}\left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\right)^{\frac{p\gamma_p}{N+2}},\end{equation}$

其中$K(p,N)=\frac{C(p,N)}{p\left|Q_p\right|^{\frac{p-2}{N+2}}_1}\cdot4^{\frac{p\gamma_p}{N+2}}$.

当$p=\overline{p}$时, (2.2)式化为

$\begin{equation}\label{cGN} \frac{1}{\overline{p}}\int_{\mathbb{R} ^N}\left|u\right|^{\overline{p}}\leqslant \left(\frac{a}{\left|Q_{\overline{p}}\right|_1}\right)^{\frac{2}{N}}\left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\right).\end{equation}$

于是定义$a^*=\left|Q_{\overline{p}}\right|_1$为临界质量. 且由文献[11,引理2.1]知, $Q_p$与临界质量还具有如下关系

$\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{\overline{p}}{2}}=(N+1)a^*,uad \int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}\right|^2=Na^*.$

下面以$\mu=0$的情况为例, 观察临界质量如何影响泛函结构. 由(1.5)和(2.2)式得

$\begin{eqnarray*} E_0(u)\geqslant \frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-K(p,N)a^{\eta_p} \left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\right)^{\frac{p\gamma_p}{N+2}},\end{eqnarray*}$

注意到

$\begin{eqnarray*} \frac{p\gamma_p}{N+2}\left\{ \begin{array}{lll} <1, 2 <p<\overline{p},\\ =1, p =\overline{p},\\ >1, \overline{p} <p<2(2^*). \end{array}\right.\end{eqnarray*}$

显然, 当$2<p<\overline{p}$时, $E_0$在$S_a$上有下界; 当$p=\overline{p},\ a\leqslant a^*$时,有

$1-K(\overline{p},N)a^{\lambda_{\overline{p}}}=1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\geqslant 0.$

故$E_0$在$S_a$上也有下界.

下面说明其它情形时, $E_0(u)$在$S_a$上无下界. 为此, 对于$\forall s\in \mathbb{R},\ u\in S_a$, 引入保$L^2$ 范数的变换

$(s\star u)(x)\triangleq e^{\frac{Ns}{2}}u(e^sx)uad \mbox{a.e. } x\in \mathbb{R} ^N,$

直接计算得

$E_0(s\star u)=\frac{e^{2s}}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+e^{(N+2)s}\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\frac{e^{p\gamma_p s}}{p}\int_{\mathbb{R} ^N}\left|u\right|^p,$

显然, 当$p>\overline{p}$时, $\lim\limits_{s\rightarrow +\infty}E_0(s\star u)=-\infty$;

当$p=\overline{p},\ a>a^*$时, 由(2.3)式等号可达知, 存在$\overline{u}\in S_a$使

$\frac{1}{\overline{p}}\int_{\mathbb{R} ^N}\left|\overline{u}\right|^{\overline{p}}>\int_{\mathbb{R} ^N}\left|\overline{u}\right|^2\left|\nabla \overline{u}\right|^2,$

则$\lim\limits_{s\rightarrow +\infty}E_0(s\star \overline{u})=-\infty$.

3 基态解的存在性

在引言中提到, 拟线性项对应的能量泛函可微性的不足是寻找问题(1.4)基态解的首要阻碍. 为此, 本节采取添加扰动项的方式解决该问题. 且由于证明过程十分相似, 本节将一并说明定理1.1和1.2(i).

令

$E_\mu^\delta(u)=\frac{\delta}{4}\int_{\mathbb{R} ^N}\left|\nabla u\right|^4+E_\mu(u),uad \tilde{m}(a,\mu)\triangleq \mathop{\inf}\limits_{u\in \tilde{S}_a}E_\mu^\delta(u),$

其中$0<\delta \leqslant 1$.

$u\in \tilde{S}_a\triangleq \left\{u\in \tilde{X}\left|\int_{\mathbb{R} ^N}\left|u\right|^2=a\right.\right\},$

其中$\tilde{X}\triangleq W^{1,4}(\mathbb{R} ^N)\cap W^{1,2}(\mathbb{R} ^N)$是自反的Banach空间, 具有如下范数

$\left|\left|u\right|\right|_{\tilde{X}}=\left|\left|u\right|\right|_{W^{1,2}}+\left|\left|u\right|\right|_{W^{1,4}}.$

同时由文献[9]知, $E_\mu^\delta \in C^1(\tilde{X})$.

引理 3.1 在定理$1.1$和$1.2$的(i)所述的指标范围内, 有$-\infty<\tilde{m}(a,\mu)<0$.

证 当$a<a^*$时, 利用(2.2)式和Cauchy不等式, 注意到$\frac{q\gamma_q}{N+2}<1$, 则对$\forall u \in \tilde{S_a}$ 有

$\begin{eqnarray*} E_\mu^\delta(u)&\geqslant& \frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+\left[1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right]\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\mu K(q,N)a^{\eta_q}\left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\right)^{\frac{q\gamma_q}{N+2}}\\ & \geqslant& \left[1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right]\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\mu K(q,N)a^{\eta_q}\left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\right)^{\frac{q\gamma_q}{N+2}}\\ &\geqslant& -C. \end{eqnarray*}$

当$a=a^*,\ 2<q<2+\frac{4}{N}$时, 利用Sobolev不等式

$\begin{equation} \int_{\mathbb{R} ^N}\left|u\right|^q\leqslant C\left(\int_{\mathbb{R} ^N}\left|u\right|^2\right)^{\frac{q(1-\gamma_q)}{2}}\left(\int_{\mathbb{R} ^N}\left|\nabla u\right|^2\right)^{\frac{q\gamma_q}{2}}=\tilde{C}\left(\int_{\mathbb{R} ^N}\left|\nabla u\right|^2\right)^{\frac{q\gamma_q}{2}}. \end{equation}$

注意到$\frac{q\gamma_q}{2}<1$, 则对$\forall u \in \tilde{S_a}$ 有

$\begin{eqnarray*} E_\mu^\delta(u)&\geqslant& \frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2-\frac{\mu}{q}\int_{\mathbb{R} ^N}\left|u\right|^q\\& \geqslant& \frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2-\frac{\mu C}{q}\left(\int_{\mathbb{R} ^N}\left|\nabla u\right|^2\right)^{\frac{q\gamma_q}{2}}\\ & \geqslant& -C. \end{eqnarray*}$

上述$C>0$与$u$无关, 故总有$\tilde{m}(a,\mu)>-\infty$.

另一方面, 对于$a\leqslant a^*,\ 2<q<2+\frac{4}{N}$, 直接计算得

$ \begin{eqnarray*} E_\mu^\delta(s\star u)&=& \frac{\delta e^{(N+4)s}}{4}\int_{\mathbb{R} ^N}\left|\nabla u\right|^4+\frac{e^{2s}}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2-\frac{\mu e^{q\gamma_qs}}{q}\int_{\mathbb{R} ^N}\left|u\right|^q\\& & +e^{(N+2)s}\left(\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\frac{1}{\overline{p}}\int_{\mathbb{R} ^N}\left|u\right|^{\overline{p}}\right), \end{eqnarray*}$

注意到$q\gamma_q<2$, 将$u$固定, 将$s$视为自变量, 上式即可简记为

$\begin{eqnarray*} E_\mu^\delta(s\star u)&=& C_1 e^{(N+4)s}+C_2 e^{2s}+C_3 e^{(N+2)s}-C_4 e^{q\gamma_q s}\\& =& Ce^{q\gamma_q s}\left(C_1 e^{(N+4-q\gamma_q)s}+C_2 e^{(2-q\gamma_q)s}+C_3 e^{(N+2-q\gamma_q)s}-1\right), \end{eqnarray*}$

注意到

$e^{(N+4-q\gamma_q)s}+e^{(2-q\gamma_q)s}+e^{(N+2-q\gamma_q)s}\mathop{\longrightarrow}\limits^{s\rightarrow -\infty}0,$

则当$s\ll -1$充分小时, 有$E_\mu^\delta(s\star u)<0$.

对于$a<a^*,\ q\geqslant 2+\frac{4}{N}$, 取

$w=\left(\frac{a}{a^*}\right)^{\frac{1}{2}}Q_{\overline{p}}^{\frac{1}{2}},uad w_t=t^{\frac{N}{2}}w(tx),$

显然有$w,w_t\in \tilde{S_a}$, 代入$E_\mu^\delta$计算得

$ \begin{eqnarray*} E_\mu^\delta(w_t)&=& \frac{\delta t^{N+4}}{4}\left(\frac{a}{a^*}\right)^2\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^4+\frac{t^2}{2}\frac{a}{a^*}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2+\frac{Na^*t^{N+2}}{4}\left(\frac{a}{a^*}\right)^2\left\{1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right\}\\ && - \frac{\mu t^{q\gamma_q}}{q}\left(\frac{a}{a^*}\right)^{\frac{q}{2}}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{q}{2}}\\& =& t^2\left(\frac{a}{a^*}\right)^2\left[\frac{\delta t^{N+2}}{4}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^4+\frac{1}{2}\frac{a^*}{a}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2+\frac{Na^*t^N}{4}\left\{1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right\} \right. \\ && - \left. \frac{\mu t^{q\gamma_q-2}}{q}\left(\frac{a}{a^*}\right)^{\frac{q}{2}-2}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{q}{2}}\right]\\ & \triangleq& t^2\left(\frac{a}{a^*}\right)^2\varphi(t). \end{eqnarray*}$

当$q=2+\frac{4}{N}$时, 有

$ \begin{eqnarray*} E_\mu^\delta(w_t)&=& \frac{\delta t^{N+4}}{4}\left(\frac{a}{a^*}\right)^2\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^4+\frac{Na^*t^{N+2}}{4}\left(\frac{a}{a^*}\right)^2\left\{1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right\}\\&& + \frac{at^2}{a^*}\left(\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2-\frac{\mu}{2+\frac{4}{N}}\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{1+\frac{2}{N}}\right).\\ \end{eqnarray*}$

令

$\begin{equation}\label{mu1} \mu_1=\left(1+\frac{2}{N}\right)\left(\frac{a^*}{a}\right)^{\frac{2}{N}}\frac{\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2}{\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{1+\frac{2}{N}}}, \end{equation}$

则$\mu>\mu_1$时,有

$\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2-\frac{\mu}{2+\frac{4}{N}}\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{1+\frac{2}{N}}<0.$

同理得, 当$t\ll -1$充分小时, 有$E_\mu^\delta(w_t)<0$.

当$q>2+\frac{4}{N}$时, 考察$\varphi'(t)=0$, 即

$\begin{eqnarray*}& & \frac{(N+2)\delta t^{N+4-q\gamma_q}}{4}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^4+\frac{N^2a^*t^{N+2-q\gamma_q}}{4}\left\{1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right\}\\ & =& \frac{\mu(q\gamma_q-2)}{q}\left(\frac{a}{a^*}\right)^{\frac{q}{2}-2}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{q}{2}}, \end{eqnarray*}$

注意到$N+4-q\gamma_q>N+2-q\gamma_q>0>2-q\gamma_q$, 则上式左侧关于$t\geqslant 0$ 从$0$单增到$+\infty$, 而右侧关于$t$是正常数. 故存在唯一$\overline{t}>0$, 使$\varphi'(\overline{t})=0$, 且$\overline{t}$是$\varphi$的极小值点.

令$\varphi(\overline{t})<0$得

$\begin{equation}\mu>\mu_2=\frac{q\overline{t}^{2-q\gamma_q}}{4}\cdot \left(\frac{a}{a^*}\right)^{\frac{4-q}{2}}\cdot\frac{\delta\overline{t}^{N+2}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^4+Na^*\overline{t}^N\left\{1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right\}+2\frac{a^*}{a}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2}{\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{q}{2}}}, \end{equation}$

故$\mu>\mu_2$时, $E_\mu^\delta(w_{\overline{t}})=\overline{t}^2\left(\frac{a}{a^*}\right)^2\varphi(\overline{t})<0$.

上述过程证明了总有$\tilde{m}(a,\mu)<0$. 综上, 证毕.

注 3.1 若记$u^*$为$u$的\rm Schwarz对称化,则由文献[8,引理4.3]知

$F(u^*)\leqslant F(u)=\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2,$

且还有

$\int_{\mathbb{R} ^N}\left|u^*\right|^p=\int_{\mathbb{R} ^N}\left|u\right|^p,uad \forall p\in [\overline{p}].$

因此在之后的证明中, 若出现$E_\mu^\delta$的极小化序列, 则始终默认其是\rm Schwarz 对称的.

引理 3.2 任意给定$0<\delta \leqslant 1$, 设$\{u_n\}\subset \tilde{S}_a$ 满足 $E_\mu ^\delta(u_n)\rightarrow \tilde{m}(a,\mu)$, 则在定理$1.1$和$1.2$的(i)所述的指标范围内, $\{u_n\}\subset \tilde{X}$有界.

证 当$a<a^*$时, 由(2.2)式得

$\begin{eqnarray*} E_\mu^\delta(u_n)&\geqslant& \frac{\delta}{4}\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4 +\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2+\left[1-\left(\frac{a}{a^*}\right)^{\frac{2}{N}}\right]\int_{\mathbb{R} ^N}\left|u_n\right|^2\left|\nabla u_n\right|^2\\ & &-\mu K(q,N)a^{\eta_q}\left(\int_{\mathbb{R} ^N}\left|u_n\right|^2\left|\nabla u_n\right|^2\right)^{\frac{q\gamma_q}{N+2}}, \end{eqnarray*}$

注意到$\frac{q\gamma_q}{N+2}<1$, 由$E_\mu ^\delta(u_n)\rightarrow \tilde{m}(a,\mu)$知

$\int_{\mathbb{R} ^N}\left|u_n\right|^2\left|\nabla u_n\right|^2<C,uad \int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4<C,uad \int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2<C,$

即$\{u_n\}\subset \tilde{X}$有界.

当$a=a^*,\ q<2+\frac{4}{N}$时, 由(3.1)式得

$E_\mu^\delta(u_n)\geqslant \frac{\delta}{4}\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4 +\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2-\frac{\mu C}{q}\left(\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2\right)^{\frac{q\gamma_q}{2}},$

注意到$\frac{q\gamma_q}{2}<1$, 由$E_\mu ^\delta(u_n)\rightarrow \tilde{m}(a,\mu)$和(3.1)式得

$\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4<C,uad \int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2<C,uad \int_{\mathbb{R} ^N}\left|u_n\right|^q<C,$

注意到$q<\overline{p}<4^*=\frac{4N}{N-4}$, 由Hölder不等式得

$\left|u_n\right|_{\overline{p}}\leqslant \left|u_n\right|_q^{1-\alpha}\left|u_n\right|_{4^*}^\alpha<C.$

将上述四项有界性直接带入$E_\mu^\delta(u_n)$, 立得$\int_{\mathbb{R} ^N}\left|u_n\right|^2\left|\nabla u_n\right|^2<C$, 即$\{u_n\}\subset \tilde{X}$有界.证毕.

引理 3.3 设$a_1,a_2>0$满足$a_1+a_2=a\leqslant a^*$. 则$\tilde{m}(a,\mu)<\tilde{m}(a_1,\mu)+\tilde{m}(a_2,\mu).$

证 任取$0<c<a^*,\theta>1$使$\theta c\leqslant a^*$. 设$\{u_n\}\subset \tilde{S_c}$ 满足$E_\mu ^\delta(u_n)\rightarrow \tilde{m}(c,\mu)$, 直接计算得

$\begin{eqnarray*} \tilde{m}(\theta c,\mu)&\leqslant& E_\mu ^\delta(u_n(\theta ^{-\frac{1}{N}}x))\\ &=&\frac{\delta \theta ^{1-\frac{4}{N}}}{4}\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4+\theta ^{1-\frac{2}{N}}\int_{\mathbb{R} ^N}\left(\frac{1}{2}+\left|u_n\right|^2\right)\left|\nabla u_n\right|^2-\frac{\theta}{\overline{p}}\int_{\mathbb{R} ^N}\left|u_n\right|^{\overline{p}}-\frac{\mu\theta}{q}\int_{\mathbb{R} ^N}\left|u_n\right|^q\\ &\leqslant& \theta E_\mu ^\delta(u_n), \end{eqnarray*}$

令$n\rightarrow +\infty$, 得$\tilde{m}(\theta c,\mu)\leqslant \theta \tilde{m}(c,\mu)$. 注意到等号成立当且仅当

$\lim\limits_{n\rightarrow +\infty}\left(\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4+\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^2+\int_{\mathbb{R} ^N}\left|u_n\right|^2\left|\nabla u_n\right|^2\right)=0,$

显然这不可能. 故有

$\tilde{m}(\theta c,\mu)<\theta \tilde{m}(c,\mu).$

取$\theta=\frac{a}{a_1}>1,\ c=a_1<a^*$, 得$\tilde{m}(a_1,\mu)>\frac{a_1}{a}\tilde{m}(a,\mu)$. 同理, $\tilde{m}(a_2,\mu)>\frac{a_2}{a}\tilde{m}(a,\mu)$. 两式相加即得结论, 证毕.

命题 3.1 设$\{u_n\}\subset \tilde{S}_a$满足$E_\mu ^\delta(u_n)\rightarrow \tilde{m}(a,\mu)$. 则在平移意义下, $\{u_n\}$ 存在子列在$\tilde{X}$中相对紧, 即, $\exists \{u_{n_k}\}\subset \{u_n\},\ \{y_k\}\subset \mathbb{R},\ {u}_\delta\in \tilde{S}_a$, 使$u_{n_k}(x+y_k)\rightarrow{u}_\delta(x)$在$\tilde{X}$中.

证 利用集中紧原理, 以下三条结论有且仅有一条成立

(i)消失性: $\forall R>0,$有

$\lim\limits_{n\rightarrow +\infty}\mathop{\sup}\limits_{y\in \mathbb{R} ^N}\int_{B_R(y)}\left|u_n\right|^2=0.$

(ii)二分性: $\exists a_1\in (0,a),\ \{u_n^1\},\ \{u_n^2\}\subset \tilde{X}$有界, 使

$\begin{eqnarray*}\left\{ \begin{array}{lll} \left|u_n-(u_n^1+u_n^2)\right|_r\rightarrow 0,uad \forall r\in (2,2(2^*)),uad \left|u_n^1\right|_2\rightarrow a_1,uad \left|u_n^2\right|_2\rightarrow a-a_1,\\ \mbox{dist}(\mbox{supp} u_n^1,\mbox{supp}u_n^2)\rightarrow +\infty,\\[2mm] \liminf\limits_{n\rightarrow +\infty}\int_{\mathbb{R} ^N}\{\left|u_n\right|^2\left|\nabla u_n\right|^2-(\left|u_n^1\right|^2\left|\nabla u_n^1\right|^2+\left|u_n^2\right|^2\left|\nabla u_n^2\right|^2)\}\geqslant 0,\\[3mm] \liminf\limits_{n\rightarrow +\infty}\int_{\mathbb{R} ^N}\{\left|\nabla u_n\right|^2-(\left|\nabla u_n^1\right|^2+\left|\nabla u_n^2\right|^2)\}\geqslant 0,\\[3mm]\liminf\limits_{n\rightarrow +\infty}\int_{\mathbb{R} ^N}\{\left|\nabla u_n\right|^4-(\left|\nabla u_n^1\right|^4+\left|\nabla u_n^2\right|^4)\}\geqslant 0. \end{array}\right. \end{eqnarray*}$

(iii)紧性: $\exists \{y_k\}\subset\mathbb{R} ^N$, 使$\forall \epsilon >0,\ \exists R>0$, 有

$\int_{B_R(y_k)}\left|u_n\right|^2\geqslant a-\epsilon.$

若(i)成立, 则由文献[12,引理I.1]知: $u_n\rightarrow 0$于$L^r(\mathbb{R} ^N),\forall r\in (2,2(2^*))$中. 则$\liminf\limits_{n\rightarrow +\infty}E_\mu ^\delta(u_n)\geqslant 0$, 与引理$3.1$矛盾!

若(ii)成立, 则有

$\tilde{m}(a,\mu)=\lim\limits_{n\rightarrow +\infty}E_\mu ^\delta(u_n)\geqslant \liminf\limits_{n\rightarrow +\infty}(E_\mu ^\delta(u_n^1)+E_\mu ^\delta(u_n^2))\geqslant \tilde{m}(a_1,\mu)+\tilde{m}(a_2,\mu),$

与引理$3.3$矛盾!

因此必有(iii)成立. 令$\tilde{u}_k(x)\triangleq u_{n_k}(x+y_k)$, 则$\tilde{u}_k\rightarrow{u}_\delta$在$L^2(\mathbb{R} ^N)$中且$ {u}_\delta \in \tilde{S}_a$.

由Hölder不等式和引理$3.2$, 易得$\tilde{u}_k\rightarrow{u}_\delta$于$L^r(\mathbb{R} ^N),\forall r\in (2,2(2^*))$中. 再由Fatou引理得

$\tilde{m}(a,\mu)\leqslant E_\mu ^\delta( {u}_\delta)\leqslant \limsup_{n\rightarrow +\infty}E_\mu ^\delta( {u}_k)=\tilde{m}(a,\mu).$

故$\left|\left|\tilde{u_k}\right|\right|_{\tilde{X}}\rightarrow \left|\left|u_\delta\right|\right|_{\tilde{X}}$, 证毕.

在本文中, 文献[10]中的定理4.1可写为如下引理.

引理 3.4 给定$a>0$, 任取$\{\delta_n\}\rightarrow 0$. 设\rm Schwarz对称序列$\{u_n\}\subset \tilde{S_a}$和$\{\lambda_n\}\subset \mathbb{R} $满足

$\left|E_\mu ^{\delta_n}(u_n)\right|\leqslant C,uad (E_\mu ^{\delta_n})'(u_n)-\lambda_n u_n=0,uad \forall n\geqslant 1,$

其中$C>0$与$n$无关.

则$\exists \tilde{u}\in W^{1,2}\cap L^{\infty}(\mathbb{R} ^N)\setminus \{0\},\ \lambda_a \in \mathbb{R} $, 使

$\lambda_n\rightarrow \lambda_a,uad (E_\mu)'(\tilde{u})-\lambda_a\tilde{u}=0.$

特别的, 若$\lambda_a<0$, 则$u_n\rightarrow \tilde{u}$在$W^{1,2}(\mathbb{R} ^N)$ 中; $u_n\nabla u_n\rightarrow \tilde{u}\nabla \tilde{u}$在$L^2(\mathbb{R} ^N)$中; $\delta_n\int_{\mathbb{R} ^N}\left|\nabla u_n\right|^4\rightarrow 0$. 即$\tilde{u}$ 是$E_\mu$在$S_a$上的临界点.

定理1.1和1.2的(i)的证明 由命题$3.1$, 有$E_\mu ^\delta(u_\delta)= \tilde{m}(a,\mu)$, 这里$u_\delta$ 是$E_\mu^\delta$在$\tilde{S_a}$上的临界点, 即$(E_\mu ^\delta|_{\tilde{S_a}})'(u_\delta)=0$. 由文献[引理3], 这等价于$\exists \lambda_\delta \in \mathbb{R} $, 使

$(E_\mu^\delta)'(u_\delta)-\lambda_\delta u_\delta=0.$

任取$\{\delta_n\}\rightarrow 0$, 记$v_n\triangleq u_{\delta_n},\ \lambda_n \triangleq \lambda_{\delta_n}$, 则有

$\left|E_\mu^{\delta_n}(v_n)\right|=\left|\tilde{m}(a,\mu)\right|\leqslant C,$

其中$C>0$与$\delta,N$无关. 以及

$(E_\mu^{\delta_n})'(v_n)-\lambda_n v_n=0,uad \forall n\geqslant 1.$

由引理$3.4$知, $\exists \tilde{u} \in W^{1,2}(\mathbb{R} ^N)\cap L^\infty(\mathbb{R} ^N)\setminus \{0\}$以及$\lambda_a\in \mathbb{R} $, 使

$\lambda_n\rightarrow \lambda_a,uad (E_\mu)'(\tilde{u})-\lambda_a\tilde{u}=0.$

故有Pohozaev恒等式

$\begin{equation} \frac{N-2}{N}\left(\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla \tilde{u}\right|^2+\int_{\mathbb{R} ^N}\left|\tilde{u}\right|^2\left|\nabla \tilde{u}\right|^2\right)-\frac{\lambda}{2}\int_{\mathbb{R} ^N}\left|\tilde{u}\right|^2-\frac{1}{\overline{p}}\int_{\mathbb{R} ^N}\left|\tilde{u}\right|^{\overline{p}}-\frac{\mu}{q}\int_{\mathbb{R} ^N}\left|\tilde{u}\right|^q=0, \end{equation}$

代入$E_\mu(\tilde{u})$化简得

$0\geqslant \limsup_{n\rightarrow +\infty}E_\mu^{\delta_n}(v_n)\geqslant E_\mu(\tilde{u})=\frac{1}{N}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+\frac{2}{N}\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2+\frac{\lambda_a}{2}\int_{\mathbb{R} ^N}\left|u\right|^2,$

显然有$\lambda_a<0$, 故$\tilde{u}$是$E_\mu$在$S_a$上的临界点, 即为方程(1.4)的基态解.

综上, 定理$1.1$和$1.2$的(i)证毕.

4 问题无解的情形

在引言中提到, 问题(1.4)的可解性既与临界质量有关, 又与扰动系数有关. 本节将说明问题无解的情形, 证明定理$1.1$的(ii)和定理$1.2$的(iii).

若$u$是方程(1.4)的解, 则有(3.4)式成立. 又在(1.4)式左右乘$u$并积分得

$\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+4\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\lambda \int_{\mathbb{R} ^N}\left|u\right|^2-\int_{\mathbb{R} ^N}\left|u\right|^{\overline{p}}-\mu \int_{\mathbb{R} ^N}\left|u\right|^q=0,$

将上式与(3.4)式联立, 消去$\lambda$, 并利用(2.2)式得

$ \begin{eqnarray*} \mu \gamma_q\int_{\mathbb{R} ^N}\left|u\right|^q&=& \int_{\mathbb{R} ^N}\left|\nabla u\right|^2+(N+2)\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\gamma_{\overline{p}}\int_{\mathbb{R} ^N}\left|u\right|^{\overline{p}}\\& \geqslant& \int_{\mathbb{R} ^N}\left|\nabla u\right|^2+(N+2)\Big\{1-(\frac{a}{a^*})^{\frac{2}{N}}\Big\}\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2\\ & \geqslant& 0, \end{eqnarray*}$

但显然$\mu \gamma_q\int_{\mathbb{R} ^N}\left|u\right|^q<0$, 矛盾! 则方程(1.4)无解.

另一方面,利用(2.2)式得, 对$\forall u\in \tilde{S_a}$有

$\begin{eqnarray*} E_\mu(s\star u)&\geqslant& \frac{e^{2s}}{2}\int_{\mathbb{R} ^N}\left|\nabla u\right|^2+e^{(N+2)s} \Big\{1-(\frac{a}{a^*})^{\frac{2}{N}}\Big\}\int_{\mathbb{R} ^N}\left|u\right|^2\left|\nabla u\right|^2-\mu \frac{e^{q\gamma_qs}}{q}\int_{\mathbb{R} ^N}\left|u\right|^q\\& >& 0, \end{eqnarray*}$

且$\lim\limits_{s\rightarrow -\infty}E_\mu(s\star u)=0$, 故$m(a,\mu)=0$.

综上, 定理$1.1$的(ii)和定理1.2的(iii)证毕.

5 泛函无下界的情形

寻找问题(1.4)的基态解, 重点在能量泛函的下确界. 而如果泛函无下界, 则问题的可解性是不明确的. 本节将说明能量泛函无下界的情形, 证明定理$1.2$的(ii)和定理$1.3$.

定理1.2的(ii)的证明 对于$q=2+\frac{4}{N}$, 仿照引理$3.1$的方法. 这时$a=a^*$, 故取

$w=Q_{\overline{p}}^{\frac{1}{2}},uad w_t=t^{\frac{N}{2}}w(tx),$

直接计算得

$E_\mu(w_t)=t^2\left(\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2-\frac{\mu}{2+\frac{4}{N}}\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{\frac{q}{2}}\right).$

令

$\begin{equation} \mu_3=\left(1+\frac{2}{N}\right)\frac{\int_{\mathbb{R} ^N}\left|\nabla Q_{\overline{p}}^{\frac{1}{2}}\right|^2}{\int_{\mathbb{R} ^N}\left|Q_{\overline{p}}\right|^{1+\frac{2}{N}}}, \end{equation}$

则$\mu>\mu_3$时, 有$\lim\limits_{t\rightarrow +\infty}E_\mu(w_t)=-\infty$.

对于$q>2+\frac{4}{N}$, 记(2.2)式的达到函数为$w$, 注意到(2.2)式关于$u$齐次, 则不妨设$w\in S_{a^*}$. 直接计算得

$E_\mu(s\star w)=\frac{e^{2s}}{2}\int_{\mathbb{R} ^N}\left|\nabla w\right|^2-\mu \frac{e^{q\gamma_qs}}{q}\int_{\mathbb{R} ^N}\left|w\right|^q,$

注意到$q\gamma_q>2$, 显然有$\lim\limits_{s\rightarrow +\infty}E_\mu(s\star w)=-\infty$.定理$1.2$的(\romannumeral2)证毕.

定理1.3的证明 由文献[8,定理1.9]知, $m(a,0)=-\infty$. 故$\exists u_0\in S_a$, 使$E_0(u_0)<0$. 直接计算得

$E_\mu(s\star u_0)=e^{(N+2)s}\left(E_0(u_0)-\frac{1}{2}\int_{\mathbb{R} ^N}\left|\nabla u_0\right|^2\right)+\frac{e^{2s}}{2}\int_{\mathbb{R} ^N}\left|\nabla u_0\right|^2-\mu \frac{e^{q\gamma_qs}}{q}\int_{\mathbb{R} ^N}\left|u_0\right|^q,$

注意到$q\gamma_q<N+2$, 显然有$\lim\limits_{s\rightarrow +\infty}E_\mu(s\star u_0)=-\infty$.定理$1.3$证毕.

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Adv Nonlinear Stud, 2018, 18(1): 131-150

DOI:10.1515/ans-2017-6026      URL     [本文引用: 1]

In this paper, we study the quasilinear Schrödinger equation \n \n \n \n \n \n \n -\n \n Δ\n ⁢\n u\n \n \n +\n \n V\n ⁢\n \n (\n x\n )\n \n ⁢\n u\n \n \n -\n \n \n γ\n 2\n \n ⁢\n \n (\n \n Δ\n ⁢\n \n u\n 2\n \n \n )\n \n ⁢\n u\n \n \n =\n \n \n \n |\n u\n |\n \n \n p\n -\n 2\n \n \n ⁢\n u\n \n \n \n \n {-\\Delta u+V(x)u-\\frac{\\gamma}{2}(\\Delta u^{2})u=|u|^{p-2}u}\n \n, \n \n \n \n x\n ∈\n \n ℝ\n N\n \n \n \n \n {x\\in\\mathbb{R}^{N}}\n \n, where \n \n \n \n \n V\n ⁢\n \n (\n x\n )\n \n \n :\n \n \n ℝ\n N\n \n →\n ℝ\n \n \n \n \n {V(x):\\mathbb{R}^{N}\\to\\mathbb{R}}\n \n is a given potential, \n \n \n \n γ\n &gt;\n 0\n \n \n \n {\\gamma&gt;0}\n \n, and either \n \n \n \n p\n ∈\n \n (\n 2\n,\n \n 2\n *\n \n )\n \n \n \n \n {p\\in(2,2^{*})}\n \n, \n \n \n \n \n 2\n *\n \n =\n \n \n 2\n ⁢\n N\n \n \n N\n -\n 2\n \n \n \n \n \n {2^{*}=\\frac{2N}{N-2}}\n \n for \n \n \n \n N\n ≥\n 4\n \n \n \n {N\\geq 4}\n \n or \n \n \n \n p\n ∈\n \n (\n 2\n,\n 4\n )\n \n \n \n \n {p\\in(2,4)}\n \n for \n \n \n \n N\n =\n 3\n \n \n \n {N=3}\n \n. If \n \n \n \n γ\n ∈\n \n (\n 0\n,\n \n γ\n 0\n \n )\n \n \n \n \n {\\gamma\\in(0,\\gamma_{0})}\n \n for some \n \n \n \n \n γ\n 0\n \n &gt;\n 0\n \n \n \n {\\gamma_{0}&gt;0}\n \n, we establish the existence of a positive solution \n \n \n \n u\n γ\n \n \n \n {u_{\\gamma}}\n \n satisfying \n \n \n \n \n \n max\n \n x\n ∈\n \n ℝ\n N\n \n \n \n ⁡\n \n |\n \n \n γ\n μ\n \n ⁢\n \n u\n γ\n \n ⁢\n \n (\n x\n )\n \n \n |\n \n \n →\n 0\n \n \n \n {\\max_{x\\in\\mathbb{R}^{N}}|\\gamma^{\\mu}u_{\\gamma}(x)|\\to 0}\n \n as \n \n \n \n γ\n →\n \n 0\n +\n \n \n \n \n {\\gamma\\to 0^{+}}\n \n for any \n \n \n \n μ\n &gt;\n \n 1\n 2\n \n \n \n \n {\\mu&gt;\\frac{1}{2}}\n \n. Particularly, if \n \n \n \n \n V\n ⁢\n \n (\n x\n )\n \n \n =\n λ\n &gt;\n 0\n \n \n \n {V(x)=\\lambda&gt;0}\n \n, we prove the existence of a positive classical radial solution \n \n \n \n u\n γ\n \n \n \n {u_{\\gamma}}\n \n and up to a subsequence, \n \n \n \n \n u\n γ\n \n →\n \n u\n 0\n \n \n \n \n {u_{\\gamma}\\to u_{0}}\n \n in \n \n \n \n \n \n H\n 2\n \n ⁢\n \n (\n \n ℝ\n N\n \n )\n \n \n ∩\n \n \n C\n 2\n \n ⁢\n \n (\n \n ℝ\n N\n \n )\n \n \n \n \n \n {H^{2}(\\mathbb{R}^{N})\\cap C^{2}(\\mathbb{R}^{N})}\n \n as \n \n \n \n γ\n →\n \n 0\n +\n \n \n \n \n {\\gamma\\to 0^{+}}\n \n, where \n \n \n \n u\n 0\n \n \n \n {u_{0}}\n \n is the ground state of the problem \n \n \n \n \n \n -\n \n Δ\n ⁢\n u\n \n \n +\n \n λ\n ⁢\n u\n \n \n =\n \n \n \n |\n u\n |\n \n \n p\n -\n 2\n \n \n ⁢\n u\n \n \n \n \n {-\\Delta u+\\lambda u=|u|^{p-2}u}\n \n, \n \n \n \n x\n ∈\n \n ℝ\n N\n \n \n \n \n {x\\in\\mathbb{R}^{N}}\n \n.

Colin M, Jeanjean L, Squassina M.

Stability and instability results for standing waves of quasi-linear Schrödingerr equations

Adv Nonlinear Stud, 2010, 23(6): 1353-1385

[本文引用: 3]

Liu X Q, Liu J Q, Wang Z Q.

Quasilinear elliptic equations with critical growth via perturbation method

J Differ Equ, 2013, 254(1): 102-124

DOI:10.1016/j.jde.2012.09.006      URL     [本文引用: 3]

Jeanjean L, Luo T J, Wang Z Q.

Multiple normalized solutions for quasi-linear Schrödingerr equations

J Differ Equ, 2015, 259(8): 3894-3928

DOI:10.1016/j.jde.2015.05.008      URL     [本文引用: 3]

Zeng X Y, Zhang Y M.

Existence and asymptotic behavior for the ground state of quasilinear elliptic equations

Adv Nonlinear Stud, 2018, 18(4): 725-744

DOI:10.1515/ans-2018-0005      URL     [本文引用: 2]

In this paper, we are concerned with the existence and asymptotic behavior of minimizers of a minimization problem related to some quasilinear elliptic equations. Firstly, we prove that there exist minimizers when the exponent q is the critical one \n \n \n \n \n q\n *\n \n =\n \n 2\n +\n \n 4\n N\n \n \n \n \n \n {q^{*}=2+\\frac{4}{N}}\n \n. Then, we prove that all minimizers are compact as q tends to the critical case \n \n \n \n q\n *\n \n \n \n {q^{*}}\n \n when \n \n \n \n a\n &lt;\n \n a\n \n q\n *\n \n \n \n \n \n {a&lt;a_{q^{*}}}\n \n is fixed. Moreover, we find that all the minimizers must blow up as the exponent q tends to the critical case \n \n \n \n q\n *\n \n \n \n {q^{*}}\n \n for any fixed \n \n \n \n a\n &gt;\n \n a\n \n q\n *\n \n \n \n \n \n {a&gt;a_{q^{*}}}\n \n.

Lions P L.

The concentration-compactness principle in the calculus of variations, the locally compact case, part 2

Ann I H Poincare-AN, 1984, 1(4): 223-283

[本文引用: 1]

Berestycki H, Lions P L.

Nonlinear scalar field equations, II existence of infinitely many solutions

Arch Rat Mech AN, 1983, 82: 347-375

DOI:10.1007/BF00250556      URL    

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