本文考虑拟线性椭圆系统

(1.1) $\begin{equation}\label{eq:1.1} \left\{\begin{array}{ll} \Delta_{p_1}u_1+\zeta_1 (|x|)|\nabla u_1|^{p_1-1}=\eta_1(|x|)f_1 (u_1,\cdots, u_m),\\ ~~\vdots\\ \Delta_{p_m}u_m+\zeta_m(|x|)|\nabla u_m|^{p_m-1}=\eta_m(|x|)f_m(u_1,\cdots, u_m), \end{array}\right.\;\ x \in {\mathbb{R}} ^N, \end{equation}$

其中$i=1, \cdots, m$, $p_i \ge 2$, $\Delta_{p_i} u_i=\mbox{div}(|\nabla u_i|^{p_i-2} \nabla u_i)$, $\zeta_i$和$\eta_i$是正连续函数, $f_i$是非负连续函数且关于每个分量是非减的.

拟线性算子$\Delta_p$ ($p \neq 2$)常用于描述许多物理现象的数学模型.在非牛顿流中流体的切向$\overrightarrow{\nu}$和速度$\nabla u$是$\overrightarrow{\nu}=r(x)|\nabla u|^{p-2} \nabla u$的这种相关形式.而$p=2$ ($p <2$, $p>2$)的情形在一些反应扩散方程和扭转蠕变问题分别表示流体是牛顿(伪塑性,膨胀)的. Diaz^[1]和Lions^[2]描述了拟线性系统的物理背景,且给出了含$\Delta_p$算子的自由边界问题的数学处理方法.近年来拟线性椭圆系统解的性质引起许多专家关注,如文献[3-10].对于$m=2$和$p_1=p_2=2$情形,文献[11-13]在适当条件下获得了爆破解的存在性.而对于$p_1, p_2, \cdots, p_m \neq 2$关于非径向解的研究有很多困难且非常有意义.

定义1.1  如果系统(1.1)的解$(u_1, \cdots, u_m)$对每个分量都有$u_i(x) \to \infty$($|x| \to \infty$),则称其为爆破解.

对于径向解情形,张新光等人^[9]考虑了系统

$\begin{eqnarray*} \mbox{div}({\cal L}_i(|\nabla u_i|)\nabla u_i)+h_i(|x|) {\cal L}_i(|\nabla u_i|)|\nabla u_i|=a_i(|x|)g_i(u_1,\cdots,u_m), \;\ x \in{\mathbb{R}}^N, \end{eqnarray*}$

当$a_i$, $h_i$${\cal L}_i$和$g_i$满足适当条件时,证明其存在无穷多个正径向爆破解的充要条件是

$\begin{eqnarray*} \int_0^{\infty} L^{-1}_i \left (H_i(r)^{-1}\int_0^r H_i(s) a_i(s){\rm d}s \right){\rm d}r =\infty, \quad i=1, \cdots, m, \end{eqnarray*}$

其中$H_i(r)=r^{N-1}{\rm e}^{\int_0^r h_i(s){\rm d}s}$和$L_i(s)=s{\cal L}_i(s)$.而Covei^[4]研究了类似于系统(1.1)的正径向解的非存在性,其条件是对任意的$i \in \{1, 2, \cdots, m\}$, $f_i$在$+\infty$的邻域内是有界的.

对于非径向解情形, Cingolani等人^[14]考虑了临界椭圆方程具有有限能量的正解

(1.2) $\begin{equation}\label{eq:1.2} -\Delta u+\ell a(x)u=u^{(N+2)/(N-2)}, \; \ x\in {\mathbb{R}}^N, \end{equation}$

其中$\ell>0$, $N> 4$和$a(x)$是一个不恒为$0$的非负连续函数.他们应用局部Pohozaev恒等式证明了如果$a(x)$在零点$x_0$的平坦阶是$\rho \in [2, N-4)$且$N \geq 7$,则当$\ell \rightarrow+\infty$时,方程(1.2)不存在具有爆破和集中现象的解$u_{\ell}$. 2012年, Hamydy等人^[15]给出了系统

$ \begin{eqnarray*} \left\{\begin{array}{ll} \Delta_pu+\lambda(|x|)|\nabla u|^{p-1}=\phi(|x|)f(v), \\ \Delta_qv+\mu(|x|)|\nabla v|^{q-1}=\psi(|x|)g(u), \end{array}\right.\;\ x \in{\mathbb{R}} ^N \end{eqnarray*}$

爆破解存在和非存在的充分条件.

鉴于已有成果,自然会产生一个问题:能否证明系统(1.1)不存在非径向爆破解?

本文利用新建立的比较原理对上面问题给出了一个肯定回答.记$\mathbb{I}:=\{1, 2, \cdots, m\}$.对所有的$r \ge 0$和$i \in \mathbb{I}$,定义

$ \begin{eqnarray*} A_i(r):= r^{N-1} {\rm e}^{\int_0^r \zeta_i(s){\rm d}s}\;\; \mbox{和}\;\; B_i(r) : =\left (A_i(r)^{-1}\int_0^r A_i(s) \eta_i(s){\rm d}s \right )^{\frac{1}{p_i-1}}. \end{eqnarray*}$

如果$j$ (或$i)=m$,则记$j+1\; (\mbox{或}\; i+1):=1$.假设存在某个指标$i \in \mathbb{I}$,使得

(f1)存在常数${\cal K}$和连续函数$\widetilde{f}_i$,使得对任意的$j \in \mathbb{I}$,当$t_j$足够大时,有$f_i(t_1, \cdots, t_m) \le {\cal K} \widetilde{f}_i(t_{i+1})$成立且$\widetilde{f}_i^{\ \frac{2-p_i}{p_i-1}} \widetilde{f}_i'$在$+\infty$处有界.

(f2)存在常数$K$,有$f_{i+1}(t_1, \cdots, t_m) \le Kt_{i+2}^{s_{i+1}}$,其中$0 < s_{i+1} < p_{i+1}-1$.

(f3) $f_j(t_1, \cdots, t_m) \le Kt_{j+1}^{s_j}$,其中$0 < s_j \le p_j-1$和$j \in \mathbb{I} \backslash \{i, i+1\}$.

定理1.1  如果存在某个指标$i$,使得下面条件之一成立

(1) (f1)-(f3)成立和

(1.3) $\begin{equation}\label{eq:1.3}\int_0^{+\infty}B_j(t){\rm d}t <+\infty, \quad \forall j\in \mathbb{I}.\end{equation} $

(2) $f_i$有界且$ \int_0^{\infty} B_i(t){\rm d}t <\infty$.

则系统(1.1)不存在连续非径向爆破解.

注1.1  (ⅰ)相比于文献[15,定理1.2],本文建立了一类椭圆系统非径向爆破解的非存在性,而系统的方程个数是任意有限个,且非线性项$f_i$与所有未知变量$u_1, \cdots, u_m$相关.

(ⅱ)相比于文献[4,定理1.1],本文利用新建立的比较原理证明系统(1.1)非径向爆破解的非存在性.

(ⅲ)采用类似于文献[5-6, 8-9]的方法可以发现,如果条件(f1)-(f3)成立且

$ \begin{eqnarray*} \int_0^{\infty} \left (A_j(t)^{-1} \int_0^t A_j(s)\eta_j(s){\rm d}s \right )^{\frac{1}{p_j-1}} {\rm d}t =+\infty, \quad \forall j \in \mathbb{I}. \end{eqnarray*}$

则系统(1.1)有无穷多个正径向爆破解.

(ⅳ)条件(f2)和(f3)的假设只要求在$+\infty$处的邻域内成立.

推论1.1  设条件(1.3)成立,且存在某个$i \in \mathbb{I}$使得

(f1') $f_i(t_1, \cdots, t_m) \le Kt_{i+1}^{s_i}$;

(f2') $f_j(t_1, \cdots, t_m) \le Kt_{j+1}^{s_j}$,

其中$0 <s_i <p_i-1$, $0 <s_j \le p_j-1$和$j\in \mathbb{I} \backslash \{i\}$,则系统(1.1)只有有界连续解.

全文用${\cal C}$表示与$f_i$和$u_i$无关的正常数. $B_r(x)$记为$\mathbb{R} ^N$中在$x$点以$r$为半径的球.记$B_r=B_r(0)$.设$\Omega \subset \mathbb{R} ^N$是一个开有界集.

引理2.1^[16]  设$h(x, t, \eta)$关于$x\in \Omega$是可测的,关于$t$是连续的,且存在$\eta$使得在$\Omega \times \mathbb{R} \times \mathbb{R} ^N$上满足$|h(x, t, \eta)| \le \iota (1+|\eta|)^{p-1}$.设$u \in W^{1, p}(\Omega)\cap L^{\infty}(\Omega)$是$\Delta_p u=h(x, u, \nabla u)$的一个弱解.则对任意的$y\in \Omega$和在$\Omega$中的任意球$B_R(y)$($R \in (0, 1)$),存在一个$\alpha>0$和依赖于$N, p, R, \iota$和$\|u\|_{\infty, \Omega}$的常数${\cal C}$,使得对任意的$x, x'\in B_R(y)$,有

$ \begin{eqnarray*} |\nabla u(x)-\nabla u(x')| < {\cal C} |x-x'|^{\alpha}\;\ \mbox{和}\;\ |\nabla u(x)| <{\cal C}. \end{eqnarray*} $

采用引理2.1和文献[17,引理3.2]中类似的证明方法可以建立下面结果:

引理2.2  设$u \in W^{1, p}(\Omega)$是方程$\Delta_p u+\zeta(x) |\nabla u|^{p-1}=F(x, u)$在$\Omega$中的一个解,其中$\zeta$和$F: \Omega\times \mathbb{R} \to [0, \infty)$是连续函数.如果$u\in L_{loc}^{\infty}(\Omega)$,则$|\nabla u| \in L_{loc}^{\infty}(\Omega)$.

受文献[15,引理2.1-2.2]的思想启发可以建立系统(1.1)的一个新的比较原理.

引理2.3  设$(u_1, \cdots, u_m) \in W^{1, p_1}(\Omega) \times \cdots \times W^{1, p_m}(\Omega)$是系统(1.1)的一个解,且对任意的$j \in \mathbb{I}$,有$|\nabla u_j| \in L_{loc}^{\infty}(\Omega)$.如果存在常数$\beta>0$和某个指标$i \in \mathbb{I}$,使得

(ⅰ)在$\partial \Omega$上有$\beta u_i-w_i <0$;

(ⅱ) $\left |\{x \in \Omega: \beta u_i-w_i>0 \} \right | \neq 0$;

(ⅲ) $\beta > \sup\limits_{\Omega} \left ( \frac{f_i(w_1, \cdots, w_m)}{f_i(u_1, \cdots, u_m)} \right )^{\frac{1}{p_i-1}}$.

对任意$j \in \mathbb{I}$, $w_j \in W^{1, p_j}(\Omega)$且$|\nabla w_j| \in L_{loc}^{\infty}(\Omega)$.则$(w_1, \cdots, w_m)$不是系统(1.1)的解.

证  假设矛盾,即$(w_1, \cdots, w_m)$是系统(1.1)的解.

不失一般性,假设条件(i)-(iii)对指标$i=1$成立.取$\delta : =\sup\limits_{\Omega} (\beta u_1-w_1)>0$.记

$\begin{eqnarray*} \Omega_n : =\{x\in \Omega :\, \beta u_1(x)-w_1(x) >\delta_n\}\; \mbox{和}\; \Omega_n' : =\{x\in \Omega_n :\, \nabla \beta u_1 \neq \nabla w_1\}, \end{eqnarray*}$

其中$\delta_n= \left\{\begin{array}{ll} \delta-\frac{1}{n}, & \mbox{如果}\;\ \delta <+\infty, \\ n, & \mbox{如果}\;\ \delta =+\infty. \end{array}\right.$因为$|\nabla u_1|\in L_{loc}^{\infty} (\Omega)$和$|\nabla w_1|\in L_{loc}^{\infty} (\Omega)$,选择

$\begin{eqnarray*} \tau=\left (1+\frac{\inf\limits_{\Omega_n'} \eta_1(|x|)\left [\beta^{p_1-1}f_1(u_1, \cdots, u_m)-f_1(w_1, \cdots, w_m) \right ]}{\|\zeta_1\|_{\infty, \Omega}\left (\|\nabla \beta u_1\|_{\infty, \Omega_n'}^{p_1-1}+2\|\nabla w_1\|_{\infty, \Omega_n'}^{p_1-1}+1 \right )} \right )^{\frac{1}{p_1-1}}. \end{eqnarray*}$

记

$\begin{eqnarray*} \Omega_{n, \tau}^1: & =&\{x \in \Omega_n':\, |\nabla \beta u_1| \ge \tau |\nabla w_1| \}, \\ \Omega_{n, \tau}^2: & =&\{x \in \Omega_n':\, |\nabla w_1| \ge \tau |\nabla \beta u_1| \}, \\ \Omega_{n, \tau}^3: & =&\left \{ x\in \Omega_n' :\, \tau |\nabla \beta u_1|>|\nabla w_1|>\frac{1}{\tau}|\nabla \beta u_1| \right \}. \end{eqnarray*}$

选择函数序列$\psi_n= \left (\beta u_1-w_1-\delta_n \right )^+= \max \{\beta u_1-w_1-\delta_n, 0\} \in W_0^{1, p_1}(\Omega)$.用$\psi_n$乘以下面两个方程并在$\Omega$中积分

$\Delta_{p_1}u_1+\zeta_1 (|x|)|\nabla u_1|^{p_1-1}=\eta_1(|x|)f_1 (u_1, \cdots, u_m), $

$ \Delta_{p_1}w_1+\zeta_1 (|x|)|\nabla w_1|^{p_1-1}=\eta_1(|x|)f_1 (w_1, \cdots, w_m), $

可得

(2.1) $\begin{eqnarray}\label{eq:2.2}&&\int_{\Omega_n} \left (\beta^{p_1-1}|\nabla u_1|^{p_1-2}\nabla u_1-|\nabla w_1|^{p_1-2}\nabla w_1 \right )\nabla \psi_n(x){\rm d}x \nonumber \\&= & \int_{\Omega_n} \zeta_1(|x|) \left (\beta^{p_1-1}|\nabla u_1|^{p_1-1}-|\nabla w_1|^{p_1-1} \right )\psi_n(x){\rm d}x \nonumber \\&& -\int_{\Omega_n} \eta_1(|x|)\Big [ \beta^{p_1-1}f_1(u_1(x), \cdots, u_m(x)) -f_1(w_1(x), \cdots, w_m(x)) \Big ]\psi_n(x){\rm d}x.\end{eqnarray}$

鉴于文献[16,引理1]可知,存在$\lambda_0>0$使得

(2.2) $\begin{equation}\label{eq:2.3}\lambda_0 \int_{\Omega_n} |\nabla \psi_n|^{p_1} {\rm d}x \le \int_{\Omega_n} \left (\beta^{p_1-1}|\nabla u_1|^{p_1-2}\nabla u_1-|\nabla w_1|^{p_1-2}\nabla w_1 \right )\nabla \psi_n(x){\rm d}x.\end{equation}$

显然在$\Omega_{n, \tau}^3$中,有

(2.3) $\begin{equation}\label{eq:2.4}|\nabla \beta u_1|^{p_1-1}-|\nabla w_1|^{p_1-1} \le \left (\tau^{p_1-1}-1 \right )|\nabla w_1|^{p_1-1}.\end{equation}$

计算发现,对任意的$\gamma >1$, $p>2$和${\cal X} \ge \gamma$,有

$\begin{eqnarray*}{\cal X}^{p-1}-1 \le \left (\frac{\gamma}{\gamma-1} \right )^{p-2} ({\cal X}-1)^{p-1}+\gamma^{p-2}-1.\end{eqnarray*}$

则$\frac{|\nabla \beta u_1|}{|\nabla w_1|} \ge \tau >1$意味着

$\begin{eqnarray*}\label{eq:2.5}\left ||\nabla \beta u_1|^{p_1-1}- |\nabla w_1|^{p_1-1} \right | \le \left (\frac{\tau}{\tau-1} \right )^{p_1-2} |\nabla \beta u_1-\nabla w_1|^{p_1-1}+\left (\tau^{p_1-2}-1\right )|\nabla w_1|^{p_1-1}.\end{eqnarray*}$

联合(2.3)式可推出

$\begin{eqnarray*}&&\int_{\Omega_n} \zeta_1(|x|) \left (\beta^{p_1-1}|\nabla u_1|^{p_1-1}-|\nabla w_1|^{p_1-1} \right )\psi_n(x){\rm d}x \\&\le & \|\zeta_1\|_{\infty, \Omega} \left (\frac{\tau}{\tau-1} \right )^{p_1-2} \left (\int_{\Omega_{n, \tau}^1} |\nabla \psi_n|^{p_1-1}\psi_n {\rm d}x +\int_{\Omega_{n, \tau}^2} |\nabla \psi_n|^{p_1-1}\psi_n {\rm d}x\right )\\&& +\|\zeta_1\|_{\infty, \Omega} \left (\tau^{p_1-2}-1\right ) \left (\int_{\Omega_{n, \tau}^1 } |\nabla w_1|^{p_1-1}\psi_n {\rm d}x + \int_{\Omega_{n, \tau}^2} |\nabla \beta u_1|^{p_1-1}\psi_n {\rm d}x \right )\\& &+ \int_{\Omega_{n, \tau}^3} \zeta_1(|x|) \left ( |\nabla \beta u_1|^{p_1-1}-|\nabla w_1|^{p_1-1}\right ) \psi_n {\rm d}x\\&\le & \|\zeta_1\|_{\infty, \Omega} \left (\frac{\tau}{\tau-1} \right )^{p_1-2} \int_{\Omega_n'} |\nabla \psi_n|^{p_1-1}\psi_n {\rm d}x\\&& + \|\zeta_1\|_{\infty, \Omega} \int_{\Omega_n'} \left (\tau^{p_1-1}-1 \right ) (|\nabla \beta u_1|^{p_1-1}+2|\nabla w_1|^{p_1-1}) \psi_n{\rm d}x.\end{eqnarray*}$

因此从(2.1)式和(2.2)式可得

$\begin{eqnarray*}\lambda_0\int_{\Omega_n} |\nabla \psi_n|^{p_1} {\rm d}x&\le& \|\zeta_1\|_{\infty, \Omega} \left (\frac{\tau}{\tau-1} \right )^{p_1-2} \int_{\Omega_n'} |\nabla \psi_n|^{p_1-1}\psi_n {\rm d}x\\&& +\int_{\Omega_n'} \|\zeta_1\|_{\infty, \Omega} \left (\tau^{p_1-1}-1 \right ) (|\nabla \beta u_1|^{p_1-1}+2|\nabla w_1|^{p_1-1}) \psi_n {\rm d}x\\&& - \int_{\Omega_n}\eta_1(|x|) \left [\beta^{p_1-1}f_1(u_1, \cdots, u_m)-f_1(w_1, \cdots, w_m) \right ] \psi_n {\rm d}x.\end{eqnarray*}$

由$\tau$的定义可知

$\begin{eqnarray*}\lambda_0\int_{\Omega_n} |\nabla \psi_n|^{p_1} {\rm d}x \le \|\zeta_1\|_{\infty, \Omega} \left (\frac{\tau}{\tau-1} \right )^{p_1-2} \left (\int_{\Omega_n'} |\nabla \psi_n|^{p_1} {\rm d}x \right )^{\frac{p_1-1}{p_1}} \left (\int_{\Omega_n'} \psi_n^{p_1} {\rm d}x \right )^{\frac{1}{p_1}}.\end{eqnarray*}$

则

(2.4) $\begin{equation}\label{eq:2.6}\left ( \int_{\Omega_n'}|\nabla \psi_n|^{p_1} {\rm d}x \right )^{\frac{1}{p_1}} \le \frac{\|\zeta_1\|_{\infty, \Omega}}{\lambda_0} \left (\frac{\tau}{\tau-1} \right )^{p_1-2} \left (\int_{\Omega_n'} \psi_n^{p_1} {\rm d}x \right )^{\frac{1}{p_1}}.\end{equation}$

选择$\frac{Np_1}{N+p_1} < \kappa < \min \{N, p_1\}$,利用Sobolev嵌入定理可得

$\begin{eqnarray*}\left (\int_{\Omega} \psi_n^{\kappa^*} {\rm d}x \right )^{\frac{1}{\kappa^*}} \le {\cal C} \left ( \int_{\Omega} |\nabla \psi_n|^{\kappa} {\rm d}x \right )^{\frac{1}{\kappa}}, \end{eqnarray*}$

其中$\kappa^*=\frac{N\kappa}{N-\kappa}$.即

$\begin{eqnarray*}\left (\int_{\Omega_n'} \psi_n^{\kappa^*} {\rm d}x \right )^{\frac{1}{\kappa^*}} \le {\cal C} \left ( \int_{\Omega_n'} |\nabla \psi_n|^{\kappa} {\rm d}x \right )^{\frac{1}{\kappa}}\le {\cal C} \left (\int_{\Omega_n'} |\nabla \psi_n|^{p_1} {\rm d}x \right )^{\frac{1}{p_1}} |\Omega_n'|^{\frac{1}{\kappa}-\frac{1}{p_1}}.\end{eqnarray*}$

从(2.4)式可知

$\begin{eqnarray*}\left (\int_{\Omega_n'} \psi_n^{\kappa^*} {\rm d}x \right )^{\frac{1}{\kappa^*}} \le \frac{{\cal C} \|\zeta_1\|_{\infty, \Omega}}{\lambda_0} \left (\frac{\tau}{\tau-1}\right )^{p_1-2} \left (\int_{\Omega_n'} \psi_n^{p_1}{\rm d}x \right )^{\frac{1}{p_1}} |\Omega_n'|^{\frac{1}{\kappa}-\frac{1}{p_1}}.\end{eqnarray*}$

由$\kappa^* >p_1$,应用ölder不等式可得

$\begin{eqnarray*}\left (\int_{\Omega_n'} \psi_n^{\kappa^*} {\rm d}x \right )^{\frac{1}{\kappa^*}} \le \frac{{\cal C} \|\zeta_1\|_{\infty, \Omega}}{\lambda_0} \left (\frac{\tau}{\tau-1}\right )^{p_1-2} \left (\int_{\Omega_n'} |\psi_n|^{\kappa^*} {\rm d}x \right )^{\frac{1}{\kappa^*}} |\Omega_n'|^{\frac{1}{\kappa}-\frac{1}{\kappa^*}}.\end{eqnarray*}$

即$1 \le \frac{{\cal C} \|\zeta_1\|_{\infty, \Omega}}{\lambda_0} \left (\frac{\tau}{\tau-1}\right )^{p_1-2} |\Omega_n'|^{\frac{1}{\kappa}-\frac{1}{\kappa^*}}$.令$n \to \infty$可以发现矛盾.

定理1.1的证明  假设系统存在一个连续爆破解$(w_1, \cdots, w_m)$.

(1)不失一般性,设$i=1$时条件(f1)-(f3)成立.

利用文献[8,定理1]的方法可以证明系统(1.1)有正径向解.因此,我们首先考虑系统(1.1)的径向解$(u_1, \cdots, u_m)$且对任意的$j \in \mathbb{I}$,有$u_j' \ge 0$.计算可得

$ \begin{eqnarray*} \left\{\begin{array}{ll} \left ( (u_1')^{p_1-1} \right )'+\frac{N-1}{r} \left (u_1'\right )^{p_1-1}+\zeta_1(r) \left (u_1'\right )^{p_1-1}=\eta_1(r)f_1(u_1, \cdots, u_m), \\~~\vdots \\ \left ( (u_m')^{p_m-1} \right )'+\frac{N-1}{r} \left (u_m'\right )^{p_m-1}+\zeta_m(r) \left (u_m' \right )^{p_m-1}=\eta_m(r)f_m(u_1, \cdots, u_m). \end{array}\right. \end{eqnarray*}$

用$A_j(r)=r^{N-1} {\rm e}^{\int_0^r \zeta_j(s){\rm d}s}$乘以系统第$j$个方程可得

(3.1) $ \begin{equation}\label{eq:3.1} \left\{\begin{array}{ll} \left ((u_1')^{p_1-1} A_1(r)\right )'=A_1(r)\eta_1(r)f_1(u_1, \cdots, u_m), \\~~\vdots\\ \left ((u_m')^{p_m-1} A_m(r)\right )'=A_m(r)\eta_m(r)f_m(u_1, \cdots, u_m). \end{array}\right. \end{equation} $

对任意的$j \in \mathbb{I}$,设$u_j(0):=\alpha_j$.则对任意的$r>0$,对(3.1)式两边从$0$到$r$积分

$\begin{eqnarray*} \left\{\begin{array}{ll} u_1(r)=\alpha_1+\displaystyle \int_0^r \left (A_1(t)^{-1} \displaystyle \int_0^t A_1(s)\eta_1(s) f_1(u_1, \cdots, u_m){\rm d}s \right )^{\frac{1}{p_1-1}}{\rm d}t, \\ ~~ \vdots\\ u_m(r)=\alpha_m+\displaystyle \int_0^r \left (A_m(t)^{-1} \displaystyle \int_0^t A_m(s)\eta_m(s) f_m(u_1, \cdots, u_m){\rm d}s \right )^{\frac{1}{p_m-1}}{\rm d}t. \end{array}\right. \end{eqnarray*} $

对$j=3, \cdots, m$,取$\alpha_j \le 0$.则条件(f3)和$f_j$的单调性意味着

(3.2) $\begin{eqnarray}\label{eq:3.2} u_j(R) &=& \alpha_j+\displaystyle \int_0^R \left (A_j(t)^{-1} \displaystyle \int_0^t A_j(s)\eta_j(s) f_j(u_1(s), \cdots, u_m(s)){\rm d}s \right )^{\frac{1}{p_j-1}}{\rm d}t \nonumber \\& \le & \left [Ku_{j+1}(R)\right ]^{\frac{s_j}{p_j-1}} \displaystyle \int_0^R B_j(t){\rm d}t. \end{eqnarray}$

这里$u_{m+1}:=u_1$.由(1.3)式可假定$ \int_0^R B_j(t){\rm d}t \ge 1$ ($\forall j \in \mathbb{I}$).选择$\alpha_1 =0$和$\alpha_2 \ge 0$,则

(3.3) $\begin{eqnarray}\label{eq:3.3} u_1(R)&= & \int_0^R \left (A_1^{-1}(t) \int_0^t A_1(s)\eta_1(s)f_1(u_1(s), \cdots, u_m(s)){\rm d}s \right )^{\frac{1}{p_1-1}}{\rm d}t \nonumber \\& \le & f_1(u_1(R), \cdots, u_m(R))^{\frac{1}{p_1-1}} \displaystyle \int_0^R B_1(t){\rm d}t. \end{eqnarray}$

联合(3.2)式和(3.3)式可得

$\begin{eqnarray*} u_2(R) &\le & \alpha_2+ f_2(u_1(R), \cdots, u_m(R))^{\frac{1}{p_2-1}} \int_0^R \left (A_2(t)^{-1} \int_0^t A_2(s)\eta_2(s){\rm d}s \right )^{\frac{1}{p_2-1}} {\rm d}t \\ & \le & \alpha_2+K^{m-1}\left (f_1(u_1(R), \cdots, u_m(R))^{\frac{1}{p_1-1}} \right )^{\frac{s_m}{p_m-1}\times \cdots \times \frac{s_2}{p_2-1}} \prod\limits_{j=1}^m \int_0^R B_j(t){\rm d}t. \end{eqnarray*}$

记${\cal M}(R):=\prod\limits_{j=1}^m \int_0^R B_j(t){\rm d}t$.则

(3.4) $\begin{eqnarray}\label{eq:3.5} 0& \le & 1-\frac{\alpha_2}{u_2(R)} \le \frac{K^{m-1} {\cal M}(R) f_1(u_1(R), \cdots, u_m(R))^{\frac{\sigma}{p_1-1}} }{u_2(R)} \nonumber \\ &\le & \frac{{\cal M}(R) K^{m-1}{\cal K}^{\frac{\sigma}{p_1-1}} \widetilde{f}_1(u_2(R))^{\frac{\sigma}{p_1-1}}}{u_2(R)}, \end{eqnarray}$

其中$\sigma: =\frac{s_m}{p_m-1}\times \cdots \times \frac{s_2}{p_2-1}$.显然$0 <\sigma < 1$.利用L'Hôpital法则和条件(f1)可得

$ \begin{eqnarray*} \lim\limits_{t \to \infty} \frac{\widetilde{f}_1(t)^{\frac{1}{p_1-1}}}{t^{\frac{1}{\sigma}}}=\lim\limits_{t\to \infty} \frac{\sigma \widetilde{f}_1(t)^{\frac{2-p_1}{p_1-1}}\widetilde{f}_1'(t)}{(p_1-1) t^{\frac{1}{\sigma}-1}}=0. \end{eqnarray*}$

从而联合(3.4)式意味着

${\mathfrak U}_2:=\mathop {\sup }\limits_{{{\mathbb{R}}^N}} {u_2}(|x|) <\infty . $

由(3.3)式容易推出

$\begin{eqnarray*}u_1(R) &\le & f_1(u_1(R), \cdots, u_m(R))^{\frac{1}{p_1-1}} \int_0^R B_1(t){\rm d}t \\&\le & \widetilde{f}_1(u_2(R))^{\frac{1}{p_1-1}}\int_0^{\infty} B_1(t){\rm d}t \\&& +\left [f_1(u_1(R), \cdots, u_m(R))^{\frac{1}{p_1-1}}-\widetilde{f}_1(u_2(R))^{\frac{1}{p_1-1}} \right ]\int_0^R B_1(t){\rm d}t \\&\le &{\cal M}(\infty)({\cal K}+2) \widetilde{f}_1(u_2(R))^{\frac{1}{p_1-1}}.\end{eqnarray*}$

由$\widetilde{f}_1$的连续性和$u_2(R)$的有界性意味着

$\begin{eqnarray*}{\mathfrak U}_1:=|u_1|_{\infty} <\infty.\end{eqnarray*}$

从(3.3)式可得对$r>0$,有$0 <u_1(r) <{\mathfrak U}_1$.当$|x| \to \infty$时,使用$\alpha_2 \le u_2$和$w_1 (x)\to \infty$可知对所有的$R>0$,有

$ \begin{eqnarray*}\sup\limits_{B_R} \left [\frac{f_1(w_1, \cdots, w_m)}{f_1(u_1, \cdots, u_m)} \right ]^{\frac{1}{p_1-1}} & \le &\sup\limits_{B_R} \left [\frac{{\cal K} \widetilde{f}_1(w_2)}{f_1(\alpha_1, \alpha_2, \cdots, \alpha_m)} \right ]^{\frac{1}{p_1-1}}\\[0.1cm]& <&\beta (R):=\inf\limits_{|x|=R} \frac{w_1}{{\mathfrak U}_1}-\varepsilon < \inf\limits_{|x|=R} \frac{w_1}{u_1}, \end{eqnarray*} $

其中$\alpha_2=|w_2|_{\infty, B_R}$和$\varepsilon$是非常小的正数.因为当$R \to \infty$时有$\beta (R) \to \infty$,则存在$R_1>0$,使得$\left |\{x\in B_{R_1}: \beta (R_1)u_1-w_1>0\}\right |\neq 0$.从$w_1$的爆破性与$u_1$的有界性,容易验证引理2.3的条件(ⅰ)成立.

另一方面,对方程$\Delta_{p_1}w_1+\zeta_1(|x|) |\nabla w_1|^{p_1-1}=\eta_1(|x|) f_1(w_1, \cdots, w_m)$利用引理2.2和联合$f_1(w_1, \cdots, w_m)$在$B_R$中的有界性,可得对所有的$R <R_1$,有$|\nabla w_1| \in L^{\infty} (B_R)$.另外对所有的$R <R_1$,有$|\nabla u_1|\in L^{\infty}(B_R)$.引理2.3意味着这是一个矛盾.

(2)假设$f_1$是有界且$ \int_0^{\infty} B_1(t){\rm d}t <\infty$.

取$\alpha_1>0$,由(3.1)式可得

$\begin{eqnarray*}u_1(R)=\alpha_1+ \int_0^R \left (A_1^{-1}(t) \int_0^t A_1(s)\eta_1(s)f_1(u_1(s), \cdots, u_m(s)){\rm d}s \right )^{\frac{1}{p_1-1}}{\rm d}t.\end{eqnarray*}$

即$u_1$有界且对所有的$r>0$,有$u_1>0$.则存在$R_2>0$和$\beta_0>0$,使得对所有的$R \ge R_2$和$\beta>\beta_0$,有$\left | \{x\in B_R: \beta u_1-w_1>0 \} \right |\neq 0$.取$u_2 \ge 1$,则

$ \begin{eqnarray*}\mathop {\sup }\limits_{{{\mathbb{R}}^N}}\frac{f_1(w_1, \cdots, w_m)}{f_1(u_1, \cdots, u_m)} <\infty.\end{eqnarray*} $

选择$\beta >\max \left \{\beta_0, \mathop {\sup }\limits_{{{\mathbb{R}}^N}} \left ( \frac{f_1(w_1, \cdots, w_m)}{f_1(u_1, \cdots, u_m)} \right )^{\frac{1}{p_1-1}} \right \}$.由$w_1(x) \to \infty$ ($|x| \to \infty$)可知存在$R_3 >0$,使得在$\partial B_R$上对所有的$R \ge R_3$,有$\beta u_1-w_1 <0$.设$R_4 >\max \{R_2, R_3\}$,则

$\begin{eqnarray*}\sup\limits_{B_{R_4}} \left ( \frac{f_1(w_1, \cdots, w_m)}{f_1(u_1, \cdots, u_m)} \right )^{\frac{1}{p_1-1}} <\beta <\inf\limits_{|x|=R_4} \frac{w_1}{u_1}.\end{eqnarray*} $

容易证明: $\forall R <R_4$,有$|\nabla w_1| \in L^{\infty} (B_R)$和$|\nabla u_1|\in L^{\infty}(B_R)$.引理2.3意味着矛盾.

推论1.1的证明  假设矛盾,即假设第$i$个解$u_i$是爆破的.从条件(f1')和(f2')容易证明条件(f1)-(f3)成立.则类似定理1.1证明可以得到一个矛盾.