在欧氏空间中, Hardy不等式最初由Hardy在文[1]中建立; Landou在文[2]中给出了该不等式的最佳常数; Azorero和Alonso在文[3] 中获得了高维情形下的Hardy不等式; Skrzypczak在文[4]-[6]中, 通过研究Lipschitz函数的包含偏微分不等方程非负弱解的Caccioppoli估计, 得到了区域上的Hardy型不等式. 近几年欧氏空间上关于Hardy不等式的研究还可参见[7]和[8]. Hardy不等式通常被用于嵌入定理, Gadliardo-Nirenberg插值不等式和实插值理论等(见[9]-[15]).

本文将研究Heisenberg群$ \mathbb{H}^n $上由加权次椭圆$ p $-Laplace不等方程导出的Hardy型不等式及应用. 先介绍$ \mathbb{H}^n $上已有的一些结果. 2001年, Niu, Zhang和Wang在文[16]中建立了Picone恒等式, 并利用Picone恒等式得到Hardy不等式, 这个结果包括了Garofalo和Lanconelli对$ p=2 $时建立的Hardy不等式; 此外, Wang和Dou在Heisenberg型群上也利用广义Picone恒等式得到了Hardy不等式, 见[23]. Niu及其合作者在$ \mathbb{H}^n $上有关Hardy不等式的文章还有[17]-[21]; 2004年, D'Ambrosio在文[22]中证明了有界域$ \Omega \subseteq {\mathbb{H}^n} $上的Hardy不等式.

我们知道, $ \mathbb{H}^n $上现有工作通常需对区域加限制条件. 本文试图在不对区域加限制条件的情况下, 建立$ \mathbb{H}^n $中区域(包括$ \mathbb{H}^n $)的Hardy型不等式, 且该不等式与加权次椭圆$ p $-Laplace不等方程的非负弱解有关.

考虑Heisenberg群$ \mathbb{H}^n $上加权次椭圆$ p $-Laplace不等方程:

$ \begin{equation} - {\Delta _{H,p,{\upsilon _1}}}u \ge f\left( u \right){\upsilon _2}\left( \xi \right),\;\;\xi \in \Omega , \end{equation} $

(1.1)

其中$ \Omega \subseteq {\mathbb{H}^n} $为任意开集, 算子$ {\Delta _{H,p,{\upsilon _1}}}u: = \hbox{div}_H\left( {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right) $是带权$ {\upsilon _1}\left( \cdot \right):\Omega \to \left[ {0,\infty } \right) $的次椭圆$ p $-Laplace算子, $ 1 < p < \infty $; $ u:\Omega \to \left[ {0,\infty } \right) $; 式(1.1)右端的权函数$ {\upsilon _2}\left( \cdot \right) $为定义在$ \Omega $上的可测函数; $ f:\left[ {0,\infty } \right) \to \left[ {0,\infty } \right) $为给定的连续函数. 本文的方法受到[4]-[6]中方法的启发, 我们将那里的方法应用于Heisenberg群.

本文主要结果如下.

定理1.1 (Hardy型不等式)  若假设$ \left( {{H_0}} \right){\rm{ - }}\left( {{H_5}} \right) $(见第二节)成立, $ 1<p<\infty $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $(见第二节)是(1.1)的非负弱解(见第二节), 则对任意在$ \Omega $中有紧支集的Lipschitz函数$ \zeta \in H{\cal L}_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, 成立不等式

$ \begin{equation} \int_\Omega {{{\left| \zeta \right|}^p}{\mu _1}\left( {d\xi } \right)} \le \int_\Omega {{{\left| {{\nabla _H}\zeta } \right|}^p}{\mu _2}\left( {d\xi } \right)} , \end{equation} $

(1.2)

其中

$ {\mu _1}\left( {d\xi } \right) = \left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( u \right){\chi _{\left\{ {u > 0} \right\}}}d\xi, $

$ {\mu _2}\left( {d\xi } \right) = {\left( {\frac{{p - 1}}{{c - \delta }}} \right)^{p - 1}}{\upsilon _1}\left( \xi \right){\cal H}\left( u \right){\chi _{\left\{ {u > 0,\left| {{\nabla _H}u} \right| \ne 0} \right\}}}d\xi. $

本文组织如下: 在第二节我们介绍Heisenberg群及其上的带权Beppo Levi空间和Sobolev空间, 给出本文用到的假设条件和引理. 在第三节我们应用(1.1)导出了非负Lipschitz函数$ \varphi $的包含(1.1)非负弱解$ u $的Caccioppoli不等式. 在第四节利用Caccioppoli不等式, 证明定理1.1, 并利用它导出一个精确Hardy-Poincaré不等式.

我们首先介绍Heisenberg群$ \mathbb{H}^n $, 更多的细节见专著[24]. $ \mathbb{H}^n $是对$ \mathbb{R}^{2n+1} $赋予群运算

$ \xi \circ \xi '{\rm{ = }}\left( {x + x',y + y',t + t' + 2\sum\limits_{i = 1}^n {\left( {{x_i}{{y'}_i} - {{x'}_i}{y_i}} \right)} } \right) $

所得的群, 其中

$ \xi = \left( {z,t} \right) = \left( {x,y,t} \right) = \left( {{x_1},{x_2}, \cdots ,{x_n},{y_1},{y_2}, \cdots {y_n},t} \right) \in {\mathbb{R}^{2n + 1}},\;n \ge 1; $

$ \xi ' = \left( {z',t'} \right) = \left( {x',y',t'} \right) = \left( {{{x'}_1},{{x'}_2}, \cdots ,{{x'}_n},{{y'}_1},{{y'}_2}, \cdots {{y'}_n},t'} \right) \in {\mathbb{R}^{2n + 1}},\;n \ge 1. $

左不变向量场为

$ {X_i} = \frac{\partial }{{\partial {x_i}}} + 2{y_i}\frac{\partial }{{\partial t}},\;\;{Y_i} = \frac{\partial }{{\partial {y_i}}} - 2{x_i}\frac{\partial }{{\partial t}},\;\;i = 1,2, \cdots ,n. $

记水平梯度为

$ {\nabla _H} = \left( {{X_1},{X_2}, \cdots ,{X_n},{Y_1},{Y_2}, \cdots ,{Y_n}} \right). $

散度为

$ \hbox{div}_H\left( {{\upsilon _1},{\upsilon _2}, \cdots {\upsilon _{2n}}} \right) = \sum\limits_{i = 1}^n {\left( {{X_i}{\upsilon _i} + {Y_i}{\upsilon _{n + i}}} \right)} . $

Heisenberg群上的齐次维数为$ Q = 2n + 2 $, Haar测度等价于$ \mathbb{R}^{2n+1} $中的Lebesgue测度.

定义2.1  设$ \Omega \subseteq {\mathbb{H}^n} $为开集, $ {\cal M}\left( \Omega \right) $为定义在$ \Omega $上的所有Borel可测实函数构成的集合. 记

$ {\cal W}\left( \Omega \right) = \left\{ {\upsilon \in {\cal M}\left( \Omega \right):0 < \upsilon \left( \xi \right) < \infty ,a.e.\xi \in \Omega } \right\}. $

若$ \upsilon \in {\cal W}\left( \Omega \right) $, 满足$ {\upsilon ^{ - \frac{1}{{p - 1}}}} \in L_{loc}^1\left( \Omega \right) $, 其中$ p>1 $, 则称$ \upsilon $满足$ {B_p}\left( \Omega \right) $条件, 简记为$ \upsilon \in {B_p}\left( \Omega \right) $.

若对开集$ \Omega \subseteq {\mathbb{H}^n} $和函数$ u $有

$ \int_{\Omega } {{{\left| u \right|}^p}\upsilon d\xi } < \infty , $

则称$ u \in L_{\upsilon }^p\left( \Omega \right) $. 相应地, $ L_{\upsilon ,loc}^p\left( \Omega \right) $定义为

$ L_{\upsilon ,loc}^p\left( \Omega \right) = \left\{ {u|\int_{\Omega '} {{{\left| u \right|}^p}\upsilon d\xi } < \infty ,\;\forall \Omega ' \subset \subset \Omega } \right\}. $

命题2.1^[6]  设$ \Omega \subseteq {\mathbb{H}^n} $为开集, $ p > 1 $, $ \upsilon \in {B_p}\left( \Omega \right) $, 则

$ L_{\upsilon ,loc}^p\left( \Omega \right) \subseteq L_{loc}^1\left( \Omega \right), $

且当在$ L_{\upsilon ,loc}^p\left( \Omega \right) $中$ {u_n}\rightarrow u $时, 在$ {L_{loc}^1\left( \Omega \right)} $中有$ {u_n}\rightarrow u $.

假设$ \upsilon \in {B_p}\left( \Omega \right) $, $ D'\left( \Omega \right) $为广义函数组成的集合, 记$ \mathbb{H}^n $上带权Beppo Levi空间为

$ H{\cal L}_\upsilon ^{1,p}\left( \Omega \right) = \left\{ {u \in D'\left( \Omega \right):{\nabla _H}u \in L_\upsilon ^p\left( \Omega \right)} \right\}; $

$ \mathbb{H}^n $上局部带权Beppo Levi空间为

$ H{\cal L}_{\upsilon ,loc}^{1,p}\left( \Omega \right): = \left\{ {u \in D'\left( {\Omega '} \right):{\nabla _H}u \in L_{\upsilon ,loc}^p\left( {\Omega '} \right),\;\forall \Omega ' \subset \subset \Omega } \right\}. $

设$ {\upsilon _1}\left( \cdot \right) \in {\cal W}\left( \Omega \right) $, $ {\upsilon _2}\left( \cdot \right) \in {B_p}\left( \Omega \right) $. 记$ \mathbb{H}^n $上带权Sobolev空间为

$ HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right) = L_{{\upsilon _1}}^p\left( \Omega \right) \cap H{\cal L}_{{\upsilon _2}}^{1,p}\left( \Omega \right), $

即

$ \begin{equation} HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right): = \left\{ {u \in L_{{\upsilon _1}}^p\left( \Omega \right) \cap D'\left( \Omega \right):{\nabla _H}u \in L_{{\upsilon _2}}^p\left( \Omega \right)} \right\}, \end{equation} $

(2.1)

其上范数为

$ {\left\| u \right\|_{HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right)}}: = {\left\| u \right\|_{L_{{\upsilon _1}}^p\left( \Omega \right)}} + {\left\| {{\nabla _H}u} \right\|_{L_{{\upsilon _2}}^p\left( \Omega \right)}}. $

设$ \Omega \subseteq {\mathbb{H}^n} $为开集, $ {\upsilon _1}\left( \cdot \right) \in {\cal W}\left( \Omega \right) $, $ {\upsilon _2}\left( \cdot \right) \in {B_p}\left( \Omega \right) $, $ p > 1 $, 则$ HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right) $为$ C^\infty \left( \Omega \right) $在范数$ {\left\| u \right\|_{HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right)}} $下的完备化空间, $ HW_{\left( {{\upsilon _1},{\upsilon _2}} \right),0}^{1,p}\left( \Omega \right) $为$ C_0^\infty \left( \Omega \right) $在范数$ {\left\| u \right\|_{HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right)}} $下的完备化空间. 当$ {\upsilon _1} = {\upsilon _2} $时, 记$ HW_{\left( {{\upsilon _1},{\upsilon _2}} \right)}^{1,p}\left( \Omega \right) $为$ HW_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, $ HW_{\left( {{\upsilon _1},{\upsilon _2}} \right),0}^{1,p}\left( \Omega \right) $为$ HW_{{\upsilon _1},0}^{1,p}\left( \Omega \right) $. 若$ u \in H{\cal L}_{\upsilon ,loc}^{1,p}\left( \Omega \right) $, $ p>1 $, $ \upsilon \left( \cdot \right) \in {B_p}\left( \Omega \right) \cap L_{loc}^1\left( \Omega \right) $, 则对任意$ \Omega ' \subset \subset \Omega $, 有$ \upsilon {\left| {{\nabla _H}u} \right|^{p - 1}} \in L_{loc}^1\left( \Omega \right) $. 进一步有$ \upsilon {\left| {{\nabla _H}u} \right|^{p - 2}}{\nabla _H}u \in L_{loc}^1\left( \Omega ,\mathbb{R}^{2n}\right) $.

定义2.2   给定权$ {\upsilon _1}\left( \cdot \right) \in {B_p}\left( \Omega \right) \cap L_{loc}^1\left( \Omega \right) $, $ {\upsilon _2}\left( \cdot \right) $可测, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $非负, 且不恒为0, $ f:\left[ {0,\infty } \right) \to \left[ {0,\infty } \right) $为连续函数, $ f(u){\upsilon _2} \in L_{loc}^1\left( \Omega \right) $. 若对每一个非负有紧支集, 满足

$ \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)h\left( \xi \right)d\xi } > - \infty $

的函数$ h \in H{\cal L}_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, 有

$ \begin{equation} \langle { - {\Delta _{H,p,{\upsilon _1}}}u,h} \rangle \ge \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)h\left( \xi \right)d\xi } , \end{equation} $

(2.2)

则说$ u $是(1.1)的弱解.

现在我们给出假设:

$ \left( {{H_0}} \right) $ $ {\upsilon _1} \in L_{loc}^1\left( \Omega \right) \cap {B_p}\left( \Omega \right) $, $ {\upsilon _2}\left( \cdot \right) $可测;

$ \left( {{H_1}} \right) $设$ \Xi $在$ \left( {0,\infty } \right) $中的任意闭区间上为Lipschitz函数, 且连续函数对$ \left( {\Xi ,\omega } \right):\left( {0,\infty } \right) \times \left( {0,\infty } \right) \to \left( {0,\infty } \right) \times \left( {0,\infty } \right) $满足如下相容性条件：

1) 不等式

$ \begin{equation} \omega \left( t \right)\Xi '\left( t \right) \le - c\Xi \left( t \right) \end{equation} $

(2.3)

在$ \left( {0,\infty } \right) $中几乎处处成立, 其中$ c \in \mathbb{R} $, 且$ c $与$ t $无关, $ \Xi $是单调的(不必严格单调).

2) 函数

$ \begin{equation} t \mapsto {\cal H}\left( t \right): = \Xi \left( t \right){\omega ^{p - 1}}\left( t \right)\;\hbox{和}\;t \mapsto \Xi \left( t \right) \cdot {\omega ^{ - 1}}\left( t \right) \end{equation} $

(2.4)

均在0的某邻域内非增或有界.

$ \left( {{H_2}} \right) $设$ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $非负, $ \Omega \subseteq {\mathbb{H}^n} $为开集, $ \left( {{H_0}} \right) $成立, $ f:\left[ {0,\infty } \right) \to \left[ {0,\infty } \right) $连续, 对每一个非负有紧支集的函数$ h \in H{\cal L}_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, 有

$ \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)h\left( \xi \right)d\xi } > - \infty , \;f\left( u \right){\upsilon _2} \in L_{loc}^1\left( \Omega \right). $

更进一步地, 我们考虑集合$ \mathfrak{A} $, $ \mathfrak{A} $由满足如下不等式的$ \delta \in \mathbb{R} $构成:

$ \begin{equation} f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{\left| {{\nabla _H}u} \right|^p} \ge 0,\;\;a.e. \;\;\xi \in \Omega \cap \left\{ {u > 0} \right\}. \end{equation} $

(2.5)

记

$ \begin{equation} {\delta _0} = \inf \mathfrak{A} . \end{equation} $

(2.6)

因为$ \inf \emptyset = + \infty $, 所以$ \mathfrak{A} $既不能是空集, 也不能无下界.

$ \left( {{H_3}} \right) $在假设$ \left( {{H_1}} \right) $和$ \left( {{H_2}} \right) $下, 参数$ \delta $满足$ {\delta _0} \le \delta < c $, 其中$ {\delta _0} $和$ c $分别由式(2.6)和(2.3)给出.

$ \left( {{H_4}} \right) $在假设$ \left( {{H_1}} \right) $和$ \left( {{H_2}} \right) $下, 对任意的$ R > 0 $有

$ {\upsilon _2}^ + \left( \xi \right)f\left( u \right)\Xi \left( u \right){\chi _{\left\{ {0 < u \le R} \right\}}} \in L_{loc}^1\left( \Omega \right). $

$ \left( {{H_5}} \right) $在假设$ \left( {{H_1}} \right) $和$ \left( {{H_2}} \right) $下, 对任意子集$ \Omega ' \subset \subset \Omega $, 有

$ \Xi \left( R \right)\int_{\Omega ' \cap \left\{ {u \ge \frac{R}{2}} \right\}} {{{\left| {{\nabla _H}u\left( \xi \right)} \right|}^{p - 1}}{\upsilon _1}\left( \xi \right)d\xi } \rightarrow 0, R\rightarrow \infty, $

$ \Xi \left( R \right)\int_{\Omega ' \cap \left\{ {u \ge \frac{R}{2}} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)d\xi } \rightarrow 0, R\rightarrow \infty. $

注2.1  对$ \Xi \left( t \right) = {t^{1 - p}} $, $ \omega \left( t \right) = t $, $ f\left( u \right) = {u^{p - 1}},\;1 < p < \infty $, 上述假设$ \left( {{H_0}} \right) $-$ \left( {{H_5}} \right) $满足.

引理2.1^[6]  若假设$ \left( {{H_0}} \right){\rm{ - }}\left( {{H_2}} \right) $成立, $ 1 < p < \infty $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $是(1.1)的非负弱解, $ {\upsilon _2} $在$ \Omega $中a.e.大于0, 则由式(2.6)给定的$ {\delta _0} $存在且有限的充要条件为$ u $ a.e.在$ \Omega $中不为常数.

本节主要目的是证明非负Lipschitz函数$ \varphi $的包含(1.1)非负弱解$ u $的Caccioppoli不等式.

定理3.1(Caccioppoli不等式)  若假设$ \left( {{H_0}} \right){\rm{ - }}\left( {{H_5}} \right) $成立, $ 1 < p < \infty $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $是(1.1)的非负弱解, 则对任意在$ \Omega $中有紧支集且使得

$ \int_{supp\varphi \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}{\upsilon _1}\left( \xi \right)d\xi } < \infty $

成立的非负Lipschitz函数$ \varphi $, 有

$ \begin{align} &\int_{\Omega \cap \left\{ {u > 0} \right\}} {\left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( u \right)\varphi d\xi }\\ \le& M\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u > 0} \right\} \cap supp\varphi } {{\upsilon _1}\left( \xi \right){\cal H}\left( u \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi }, \end{align} $

(3.1)

其中$ M: = \frac{1}{{{p^p}}}{\left( {\frac{{p - 1}}{{c - \delta }}} \right)^{p - 1}} $.

为了证明定理3.1, 我们需要四个引理.

引理3.1   设$ u,\;\varphi $如定理3.1中假设所示, 固定$ 0 < r < R $, 记

$ \begin{equation} {u_{r,R}}\left( \xi \right) = \min \left( {u\left( \xi \right) + r,R} \right),\;\;\Psi \left( \xi \right) = \Xi \left( {{u_{r,R}}\left( \xi \right)} \right)\varphi \left( \xi \right), \end{equation} $

(3.2)

则

$ {u_{r,R}} \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right),\;\;\Psi \in H{\cal L}_{{\upsilon _1},0}^{1,p}\left( \Omega \right). $

证  证明分如下两步进行.

1). 先证$ {u_{r,R}} \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $. 事实上, 由式(3.2)得, $ r \le {u_{r,R}} \le R $. 又因为$ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $, 即对任意的$ \Omega ' \subset \subset \Omega $有

$ u \in D'\left( \Omega \right),\;\;{\nabla _H}u \in L_{{\upsilon _1}}^p\left( {\Omega '} \right), $

所以

$ {u_{r,R}} \in D'\left( \Omega \right),\;\;\left| {{\nabla _H}{u_{r,R}}} \right| \le \left| {{\nabla _H}u} \right| \in L_{{\upsilon _1}}^p\left( {\Omega '} \right). $

因此

$ {u_{r,R}} \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right). $

2). 再证$ \Psi \in H{\cal L}_{{\upsilon _1},0}^{1,p}\left( \Omega \right) $. 实际上, 因为$ r \le {u_{r,R}} \le R $, $ {u_{r,R}} \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $, $ \Xi $属于$ C\left( {\left( {0,\infty } \right)} \right) $, 且在任意闭区间上为Lipschitz函数, 所以对a.e. $ \xi $有

$ \begin{align} {\nabla _H}\Psi \left( \xi \right)& = \Xi '\left( {{u_{r,R}}\left( \xi \right)} \right){\nabla _H}{u_{r,R}}\left( \xi \right) \cdot \varphi \left( \xi \right) + \Xi \left( {{u_{r,R}}\left( \xi \right)} \right){\nabla _H}\varphi \left( \xi \right)\\ & = \Xi '\left( {u\left( \xi \right) + r} \right){\nabla _H}u\left( \xi \right) \cdot {\chi _{\left\{ {u\left( \xi \right) \le R - r} \right\}}} \cdot \varphi \left( \xi \right) + \Xi \left( {{u_{r,R}}\left( \xi \right)} \right){\nabla _H}\varphi \left( \xi \right), \end{align} $

(3.3)

又因为在$ \Omega $中$ \varphi $是非负的有紧支集的Lipschitz函数, 所以(3.3)中右端两项均属于$ L_{{\upsilon _1}}^p\left( \Omega \right) $, 因此

$ \Psi \in H{\cal L}_{{\upsilon _1},0}^{1,p}\left( \Omega \right). $

引理3.2^[21]   设$ u \in HW_{loc}^{1,1}\left( \Omega \right) $, 其中$ u $在每一点$ \xi \in \Omega $的值定义为

$ u\left( \xi \right): = \mathop {\lim \sup }\limits_{\rho \to 0} \frac{1}{{\left| {B\left( {\xi ,\rho } \right) \cap \Omega } \right|}}\int_{B\left( {\xi ,\rho } \right) \cap \Omega } {u\left( \eta \right)d\eta } . $

给定$ t \in \mathbb{R} $, 则有

$ \begin{equation} \left\{ {\xi \in \Omega :u\left( \xi \right) = t} \right\} \subseteq \left\{ {\xi \in \Omega :{\nabla _H}u\left( \xi \right) = 0} \right\} \cup N, \end{equation} $

(3.4)

其中$ N $的Haar测度为零.

引理3.3(局部估计)  若假设$ \left( {{H_0}} \right){\rm{ - }}\left( {{H_3}} \right) $成立, $ 1 < p < \infty $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $是(1.1)的非负弱解, 则对任意在$ \Omega $中有紧支集, 且使得

$ \int_{supp\varphi \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}{\upsilon _1}\left( \xi \right)d\xi } < \infty $

成立的非负Lipschitz函数$ \varphi $和$ 0 < r < R $, 成立不等式

$ \begin{align} & \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {\left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( {u + r} \right)\varphi d\xi } \\ \le& M\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 < u \le R - r} \right\} \cap supp\varphi } {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } + C\left( {r,R} \right), \end{align} $

(3.5)

其中

$ M = \frac{1}{{{p^p}}}{\left( {\frac{{p - 1}}{{c - \delta }}} \right)^{p - 1}}, $

$ C\left( {r,R} \right): = \Xi \left( R \right)\left[ {\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u > R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 1}}\left| {{\nabla _H}\varphi } \right|d\xi } - \int_{\Omega \cap \left\{ {u > R - r} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\varphi d\xi } } \right]. $

证  由引理3.1知, $ \Psi \in H{\cal L}_{{\upsilon _1},0}^{1,p}\left( \Omega \right), $取$ h = \Psi $为不等式(2.2)中的试验函数, 则有

$ \begin{align} I: =& \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)\Psi \left( \xi \right)d\xi } = \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)\Xi \left( {{u_{r,R}}\left( \xi \right)} \right)\varphi d\xi } \\ =& \int_{\Omega \cap \left\{ {0 \le u \le R - r} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } + \Xi \left( R \right)\int_{\Omega \cap \left\{ {u > R - r} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\varphi d\xi } \\ = :&\int_{\Omega \cap \left\{ {0 \le u \le R - r} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } + {C_1}\left( {r,R} \right). \end{align} $

(3.6)

另一方面, 从不等式(2.2)推出

$ \begin{align} I =& \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)\Psi d\xi } \le \langle { - \hbox{div}_H\left( {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right),\Psi } \rangle \\ =& \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u \cdot {\nabla _H}\Psi d\xi } \\ =& \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u \cdot {\nabla _H}{u_{r,R}}\Xi '\left( {{u_{r,R}}} \right)\varphi d\xi } \\ &+ \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u \cdot {\nabla _H}\varphi \Xi \left( {{u_{r,R}}} \right)d\xi } \\ = &\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\Xi '\left( {u + r} \right)\varphi d\xi } \\ & + \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}\Xi \left( {u + r} \right){\nabla _H}u \cdot {\nabla _H}\varphi d\xi } \\ &+ \Xi \left( R \right)\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u > R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u \cdot {\nabla _H}\varphi d\xi } \\ = :&a\left( {r,R} \right) + b\left( {r,R} \right) + {C_2}\left( {r,R} \right). \end{align} $

(3.7)

对$ 0 \le u \le R-r $, 以上所有积分有限. 由假设$ \left( {{{\rm H}_1}} \right) $得

$ \begin{equation} a\left( {r,R} \right) \le - c\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi d\xi } = : - c{a_1}\left( {r,R} \right). \end{equation} $

(3.8)

注意到

$ \begin{align*} b\left( {r,R} \right)& \le \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 1}}\left| {{\nabla _H}\varphi } \right|\Xi \left( {u + r} \right)d\xi } \\ & = \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {\left( {\frac{{\left| {{\nabla _H}\varphi } \right|}}{\varphi }\omega \left( {u + r} \right)} \right) \cdot {{\left| {{\nabla _H}u} \right|}^{p - 1}}\left( {{\upsilon _1}\left( \xi \right)\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi } \right)d\xi } , \end{align*} $

对任意的$ \varepsilon > 0 $, $ p > 1 $, 由Young不等式得

$ \begin{align} &b\left( {r,R} \right) \le \frac{{p - 1}}{p}\varepsilon \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{{\left| {{\nabla _H}u} \right|}^p} \cdot \left( {{\upsilon _1}\left( \xi \right)\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi } \right)d\xi } \\ & + \frac{1}{{p{\varepsilon ^{p - 1}}}}\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{{\left( {\frac{{\left| {{\nabla _H}\varphi } \right|}}{\varphi }\omega \left( {u + r} \right)} \right)}^p} \cdot \left( {{\upsilon _1}\left( \xi \right)\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi } \right)d\xi } \\ =& \frac{{p - 1}}{p}\varepsilon {a_1}\left( {r,R} \right) + \frac{1}{{p{\varepsilon ^{p - 1}}}}\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 \le u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left( {\frac{{\left| {{\nabla _H}\varphi } \right|}}{\varphi }} \right)}^p}\Xi \left( {u + r} \right){\omega ^{p - 1}}\left( {u + r} \right)\varphi d\xi } \\ = :&\frac{{p - 1}}{p}\varepsilon {a_1}\left( {r,R} \right) + \frac{1}{{p{\varepsilon ^{p - 1}}}}d\left( {r,R} \right). \end{align} $

(3.9)

将(3.8)和(3.9)代入(3.7)得

$ \begin{align} I & \le a\left( {r,R} \right) + b\left( {r,R} \right) + {C_2}\left( {r,R} \right) \\ & \le - c{a_1}\left( {r,R} \right) + \frac{{p - 1}}{p}\varepsilon {a_1}\left( {r,R} \right) + \frac{1}{{p{\varepsilon ^{p - 1}}}}d\left( {r,R} \right) + {C_2}\left( {r,R} \right) \\ & {\rm{ = }}\left( {\frac{{p - 1}}{p}\varepsilon - c} \right){a_1}\left( {r,R} \right) + \frac{1}{{p{\varepsilon ^{p - 1}}}}d\left( {r,R} \right) + {C_2}\left( {r,R} \right). \end{align} $

(3.10)

进而由$ {C_1}\left( {r,R} \right) $, $ {a_1}\left( {r,R} \right) $和$ d\left( {r,R} \right) $有限, 再将(3.6)和(3.10)结合推出

$ \begin{align*} & \int_{\Omega \cap \left\{ {0 \le u \le R - r} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } + \left( {c - \frac{{p - 1}}{p}\varepsilon } \right){a_1}\left( {r,R} \right) \\ \le& \frac{1}{{p{\varepsilon ^{p - 1}}}}d\left( {r,R} \right) + \left( {{C_2}\left( {r,R} \right) - {C_1}\left( {r,R} \right)} \right). \end{align*} $

记

$ C\left( {r,R} \right) = {C_2}\left( {r,R} \right) - {C_1}\left( {r,R} \right). $

取$ \varepsilon {\rm{ = }}\left( {c - \delta } \right)\frac{p}{{p - 1}} $, 则有式(3.5)成立.

利用引理3.2和引理3.3可证得

引理3.4   若假设$ \left( {{H_0}} \right){\rm{ - }}\left( {{H_4}} \right) $成立, $ 1 < p < \infty $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $是(1.1)的非负弱解, 则对任意在$ \Omega $中有紧支集, 且使得

$ \int_{supp\varphi \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}{\upsilon _1}\left( \xi \right)d\xi } < \infty $

成立的非负Lipschitz函数$ \varphi $和$ R > 0 $, 有不等式

$ \begin{align} & \int_{\Omega \cap \left\{ {0 < u < R} \right\}} {\left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( u \right)\varphi d\xi } \\ \le& M\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,0 < u < R} \right\} \cap supp\varphi } {{\upsilon _1}\left( \xi \right){\cal H}\left( u \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } + C\left( R \right), \end{align} $

(3.11)

其中

$ M = \frac{1}{{{p^p}}}{\left( {\frac{{p - 1}}{{c - \delta }}} \right)^{p - 1}}, $

$ C\left( R \right): = \Xi \left( R \right)\left[ {\int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u \ge \frac{R}{2}} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 1}}\left| {{\nabla _H}\varphi } \right|d\xi } - \int_{\Omega \cap \left\{ {u \ge \frac{R}{2}} \right\}} {f\left( u \right){\upsilon _2}\left( \xi \right)\varphi d\xi } } \right]. $

证  在(3.5)两边令$ r \to {0^ + } $便可证得(3.11), 但其中有一些细节需仔细处理, 这里给出证明过程. 我们分为以下四步:

i) 先证如下不等式成立:

$ \begin{equation} \mathop {\lim \inf }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {{g_r}\left( \xi \right)\varphi d\xi } \ge \int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {{g_0}\left( \xi \right)\varphi d\xi } , \end{equation} $

(3.12)

其中

$ {g_r}\left( \xi \right): = \left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( {u + r} \right), $

$ {g_0}\left( \xi \right): = \left( {f\left( u \right){\upsilon _2}\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( u \right). $

事实上, 由假设$ \left( {{{\rm H}_1}} \right) $知, $ \Xi $与$ \omega $只能有四种情形: a) $ \Xi $非增, $ \frac{\Xi }{\omega } $非增; b) $ \Xi $增, $ \frac{\Xi }{\omega } $非增; c) $ \Xi $非增, $ \frac{\Xi }{\omega } $在0的邻域内有界; d) $ \Xi $增, $ \frac{\Xi }{\omega } $在0的邻域内有界.

以下对$ \delta $分$ \delta \ge 0 $和$ \delta < 0 $讨论, 证明(3.12):

1. 当$ \delta \ge 0 $时, 因为

$ 0 \le \delta < c,\;\;\Xi '\left( t \right) \le - c\frac{{\Xi \left( t \right)}}{{\omega \left( t \right)}}, $

所以$ \Xi $非增. 因此情形a)和c)可能出现.

(1) 先考虑情形a). 有

$ \Xi \left( {u + r} \right) \le \Xi \left( u \right),\;\;\delta \frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}} \le \delta \frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}}. $

记

$ {\tilde g_r}\left( \xi \right): = \left( {f\left( u \right){\upsilon ^ + }_2\left( \xi \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{{\left| {{\nabla _H}u} \right|}^p}} \right)\Xi \left( {u + r} \right), $

则$ {\tilde g_r} \ge 0 $, 且有

$ {g_r}\left( \xi \right){\rm{ = }}{\tilde g_r}\left( \xi \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge {\tilde g_r}\left( \xi \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right). $

而由Levi定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {{{\tilde g}_r}\left( \xi \right)\varphi d\xi } {\rm{ = }}\int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {{{\tilde g}_0}\left( \xi \right)\varphi d\xi } . $

因此, 不等式(3.12)成立.

(2) 再考虑情形c). 因为

$ {\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge {\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right), $

所以

$ {g_r}\left( \xi \right) \ge f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( {u + r} \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( {u + r} \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right). $

由Levi定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {f\left( u \right){\upsilon ^{\rm{ + }}}_2\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } {\rm{ = }}\int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {f\left( u \right){\upsilon ^{\rm{ + }}}_2\left( \xi \right)\Xi \left( u \right)\varphi d\xi } . $

因为$ \frac{\Xi }{\omega } $在0的邻域内有界, 所以它在整个积分区域有界, 从而由Lebesgue控制收敛定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi d\xi } {\rm{ = }}\int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}}\varphi d\xi } . $

因此, 不等式(3.12)成立.

2. 当$ \delta < 0 $时, 此时a), b), c)和d) 均可能出现.

(1) 先考虑情形a). 当$ r > 0 $, $ u > 0 $时, 有

$ \Xi \left( {u + r} \right) \le \Xi \left( u \right),\;\;\delta \frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}} \ge \delta \frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}}, $

$ {\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge {\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right), $

所以

$ \begin{align*} {g_r}\left( \xi \right)& \ge f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( {u + r} \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( u \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right) \\ & = f\left( u \right){\upsilon _2}\left( \xi \right)\Xi \left( u \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( u \right) + f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( {u + r} \right) - \Xi \left( u \right)} \right)\\ &= {g_0}\left( \xi \right) - f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( u \right) - \Xi \left( {u + r} \right)} \right). \end{align*} $

因为

$ f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( u \right) - \Xi \left( {u + r} \right)} \right) \ge 0, $

$ \mathop {\lim }\limits_{r \to {0^ + }} f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( u \right) - \Xi \left( {u + r} \right)} \right) = 0; $

再由假设$ \left( {{{\rm H}_4}} \right) $得

$ f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( u \right) - \Xi \left( {u + r} \right)} \right)\varphi {\chi _{\left\{ {0 < u \le R - r} \right\}}} \le f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( u \right)\varphi {\chi _{\left\{ {0 < u \le R} \right\}}} \in {L^1}\left( \Omega \right), $

所以, 利用Lebesgue控制收敛定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int\limits_{_{\left\{ {0 < u \le R - r} \right\}}} {f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\left( {\Xi \left( u \right) - \Xi \left( {u + r} \right)} \right)\varphi d\xi } = 0. $

因此, 不等式(3.12)成立.

(2) 再考虑情形b). 当$ u > 0 $时, 有

$ \delta \frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}} \ge \delta \frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}},\;\;{\upsilon ^ + }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge {\upsilon ^ + }_2\left( \xi \right)\Xi \left( u \right), $

从而

$ {g_r}\left( \xi \right) \ge f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( u \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( u \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( u \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right). $

因为

$ f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) < f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right), $

所以, 由Lebesgue控制收敛定理和假设$ \left( {{{\rm H}_2}} \right) $得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int\limits_{_{\left\{ {0 < u \le R - r} \right\}}} {f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } = \int\limits_{_{\left\{ {0 < u \le R} \right\}}} {f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right)\varphi d\xi }. $

因此, 不等式(3.12)成立.

(3) 其次考虑情形c). 因为

$ {\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge {\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right), $

所以

$ {g_r}\left( \xi \right) \ge f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( {u + r} \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( {u + r} \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right). $

又因为$ f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( {u + r} \right) \ge 0 $, 所以由Levi定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int\limits_{_{\left\{ {0 < u \le R - r} \right\}}} {f\left( u \right){\upsilon ^{\rm{ + }}}_2\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } = \int\limits_{_{\left\{ {0 < u \le R} \right\}}} {f\left( u \right){\upsilon ^{\rm{ + }}}_2\left( \xi \right)\Xi \left( u \right)\varphi d\xi }. $

注意到

$ \frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}} \le \sup \left\{ {\frac{{\Xi \left( t \right)}}{{\omega \left( t \right)}}:t \in (0,R]} \right\}, $

所以由Lebesgue控制收敛定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi d\xi } {\rm{ = }}\int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}}\varphi d\xi }. $

因此, 不等式(3.12)成立.

(4) 最后考虑情形d). 当$ u > 0 $时, 有

$ \Xi \left( {u + r} \right) \ge \Xi \left( u \right), $

从而

$ {g_r}\left( \xi \right) \ge f\left( u \right){\upsilon ^ + }_2\left( \xi \right)\Xi \left( u \right) + \delta \frac{{{\upsilon _1}\left( \xi \right)}}{{\omega \left( {u + r} \right)}}{\left| {{\nabla _H}u} \right|^p}\Xi \left( {u + r} \right) + f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right). $

因为$ {\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right) \le {\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right) $, $ \frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}} \le \sup \left\{ {\frac{{\Xi \left( t \right)}}{{\omega \left( t \right)}}:t \in \left( {0,R} \right)} \right\} $, 所以由Lebesgue控制收敛定理得

$ \mathop {\lim }\limits_{r \to {0^ + }} \int\limits_{_{\left\{ {0 < u \le R - r} \right\}}} {f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( {u + r} \right)\varphi d\xi } = \int\limits_{_{\left\{ {0 < u \le R} \right\}}} {f\left( u \right){\upsilon ^ - }_2\left( \xi \right)\Xi \left( u \right)\varphi d\xi } , $

$ \mathop {\lim }\limits_{r \to {0^ + }} \int_{\Omega \cap \left\{ {0 < u \le R - r} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( {u + r} \right)}}{{\omega \left( {u + r} \right)}}\varphi d\xi } {\rm{ = }}\int_{\Omega \cap \left\{ {0 < u \le R} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^p}\frac{{\Xi \left( u \right)}}{{\omega \left( u \right)}}\varphi d\xi } . $

因此, 不等式(3.12)成立.

ii) 其次证如下极限成立: 当$ r \to {0^ + } $时, 有

$ \begin{align} & \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{\upsilon _1}\left( \xi \right)\Xi \left( {u + r} \right){\omega ^{p - 1}}\left( {u + r} \right){\chi _{\left\{ {0 < u + r \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } \\ \to& \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{\upsilon _1}\left( \xi \right)\Xi \left( u \right){\omega ^{p - 1}}\left( u \right){\chi _{\left\{ {0 < u \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi }. \end{align} $

(3.13)

由条件$ \left( {{H_1}} \right) $中ii)知$ {\cal H}\left( t \right) = \Xi \left( t \right){\omega ^{p - 1}}\left( t \right) $在0的邻域内a) 非增或b)有界.

a) 先考虑$ {\cal H} $在0的邻域内非增的情形, 即$ \exists \tau > 0 $, s.t. 对$ \forall t < \tau $, 函数$ {\cal H}\left( t \right) $非增. 不失一般性, 假设$ 2r \le \tau \le R $,

$ {E_\tau }: = \Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u < \frac{\tau }{2}} \right\} \cap supp\varphi, $

$ {F_\tau }: = \Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u \ge \frac{\tau }{2}} \right\} \cap supp\varphi , $

则有

$ \begin{align} & \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\} \cap supp\varphi } {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\chi _{\left\{ {u + r \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } \\ = & \int_{{E_\tau }} {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\chi _{\left\{ {u + r \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } + \int_{{F_\tau }} {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\chi _{\left\{ {u + r \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } . \end{align} $

(3.14)

先考虑$ {E_\tau } $上的积分, 此时有$ u + r < \tau $. 因为$ {\cal H}\left( t \right) $在$ t \in \left( {0,\tau } \right) $时非增, 所以$ {{\cal H}_r}\left( \xi \right): = {\cal H}\left( {u\left( \xi \right) + r} \right) $在$ {E_\tau } $上非增.

又因为$ {\cal H}\left( {u\left( \xi \right) + r} \right) \to {\cal H}\left( {u\left( \xi \right)} \right) $, 所以由Levi定理得

$ \begin{equation} \mathop {\lim }\limits_{r \to {0^ + }} \int_{{E_\tau }} {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\chi _{\left\{ {u + r \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } {\rm{ = }}\int_{{E_\tau }} {{\upsilon _1}\left( \xi \right){\cal H}\left( u \right){\chi _{\left\{ {0 < u \le R} \right\}}}{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } . \end{equation} $

(3.15)

接下来考虑$ {F_\tau } $上的积分. 因为

$ \begin{align*} {\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\left| {{\nabla _H}\varphi } \right|^p}{\varphi ^{1 - p}}{\chi _{\left\{ {u + r \le R} \right\} \cap {F_\tau }}} & \le {\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){\left| {{\nabla _H}\varphi } \right|^p}{\varphi ^{1 - p}}{\chi _{\left\{ {\frac{\tau }{2} \le u + r \le R} \right\}}} \\ & \le \mathop {\sup }\limits_{t \in \left[ {\frac{\tau }{2},R} \right]} {\upsilon _1}\left( \xi \right){\cal H}\left( t \right){\left| {{\nabla _H}\varphi } \right|^p}{\varphi ^{1 - p}}{\chi _{supp\varphi }} \in {L^1}\left( \Omega \right), \end{align*} $

所以, 由Lebesgue控制收敛定理得

$ \begin{equation} \mathop {\lim }\limits_{r \to {0^ + }} \int_{{F_\tau } \cap \left\{ {u + r \le R} \right\}} {{\upsilon _1}\left( \xi \right){\cal H}\left( {u + r} \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } = \int_{{F_\tau } \cap \left\{ {u \le R} \right\}} {{\upsilon _1}\left( \xi \right){\cal H}\left( u \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } , \end{equation} $

(3.16)

结合(3.14)-(3.16)知(3.13)成立.

b) 再考虑$ {\cal H}\left( t \right) $在0的邻域内有界的情形. 事实上, 注意到$ {\cal H}\left( t \right) $在$ \left[ {0,R} \right] $有界, 从而与前面a)在$ {F_\tau } $上的讨论类似, 只需取$ \tau {\rm{ = }}0 $即可. 由引理3.2知

$ \left| {\left\{ {\xi \in \Omega :u\left( \xi \right) = 0,{\nabla _H}u\left( \xi \right) \ne 0} \right\}} \right| = 0. $

因此, (3.13)成立.

iii) 最后对$ r \le \frac{R}{2} $, 显然有$ C\left( {r,R} \right) \le C\left( R \right) $.

iv) 因此, 结合以上三步和引理2.1可知(3.11)成立.

定理3.1的证明  在(3.11)中令$ R \to \infty $可证得(3.1)成立. 事实上, 不失一般性, 假设

$ \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u > 0} \right\}} {{\upsilon _1}\left( \xi \right)\Xi \left( u \right){\omega ^{p - 1}}\left( u \right){{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}d\xi } < \infty. $

否则不等式(3.1)是平凡的. 因为

$ \int_{\Omega \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0,u > 0} \right\}} {{\upsilon _1}\left( \xi \right){{\left| {{\nabla _H}u} \right|}^{p - 1}}\left| {{\nabla _H}\varphi } \right|d\xi } < \infty , $

$ \int_\Omega {f\left( u \right){\upsilon _2}\left( \xi \right)d\xi } < \infty , $

所以由假设$ \left( {H_5} \right) $知$ \mathop {\lim }\limits_{R \to \infty } C\left( R \right) = 0 $. 因此, (3.1)直接从(3.11)推出, 其中应用了Levi定理.

定理1.1的证明  若$ \zeta $是$ \Omega $中有紧支集的非负Lipschitz函数, 则$ \varphi = {\zeta ^p} $是Lipschitz的, 且有$ {\nabla _H}\varphi = p{\zeta ^{p - 1}}{\nabla _H}\zeta $, 所以$ \left| {{\nabla _H}\varphi } \right|{\rm{ = }}p{\zeta ^{p - 1}}\left| {{\nabla _H}\zeta } \right| $, 从而

$ {\left| {{\nabla _H}\varphi } \right|^p}{\varphi ^{1 - p}}{\rm{ = }}{\left( {p{|\zeta| ^{p - 1}}\left| {{\nabla _H}\zeta } \right|} \right)^p}{\left| \zeta \right|^{p\left( {1 - p} \right)}} = {p^p}{\left| {{\nabla _H}\zeta } \right|^p}, $

因此

$ \int\limits_{supp\varphi \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{{\left| {{\nabla _H}\varphi } \right|}^p}{\varphi ^{1 - p}}{\upsilon _1}\left( \xi \right)d\xi } {\rm{ = }}{p^p}\int\limits_{supp\varphi \cap \left\{ {\left| {{\nabla _H}u} \right| \ne 0} \right\}} {{{\left| {{\nabla _H}\zeta } \right|}^p}{\upsilon _1}\left( \xi \right)d\xi } < \infty . $

所以定理3.1的条件满足, 因此, 由式(3.1)可证得式(1.2).

又因为Lipschitz函数的绝对值也为Lipschitz函数, 所以式(1.2)中的$ \zeta $可用$ \left| \zeta \right| $代替, 即可令$ \varphi = {\left| \zeta \right|^p} $, 重复前述过程, 便得定理1.1.

接下来, 我们应用定理1.1推导出一个精确Hardy-Poincaré不等式.

推论4.1(精确Hardy-Poincaré不等式)   设$ 1 < p < \infty $, $ {\upsilon _1},{\upsilon _2} \in {\cal W}\left( \Omega \right) $, $ {\upsilon _1} \in L_{loc}^1\left( \Omega \right) \cap {B_p}\left( \Omega \right), $ $ {\upsilon _2}{u^{p - 1}} \in L_{loc}^1\left( \Omega \right) $, $ u \in H{\cal L}_{{\upsilon _1},loc}^{1,p}\left( \Omega \right) $是(1.1)的非负非平凡弱解, 则对任意在$ \Omega $中有紧支集的Lipschitz函数$ \zeta \in H{\cal L}_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, 成立不等式

$ \begin{equation} \int_\Omega {{{\left| \zeta \right|}^p}{\upsilon _2}\left( \xi \right)d\xi } \le \int_\Omega {{{\left| {{\nabla _H}\zeta } \right|}^p}{\upsilon _1}\left( \xi \right)d\xi } . \end{equation} $

(4.1)

更进一步, 若$ - {\Delta _{H,p,{\upsilon _1}}}{u_0} = {u_0}^{p - 1}{\upsilon _2}\left( \xi \right) \in L_{loc}^1\left( \Omega \right) $存在非平凡、非负的弱解, 则(4.1)是精确的, 即$ \int_\Omega {{{\left| \zeta \right|}^p}{\upsilon _2}\left( \xi \right)d\xi } \le C\int_\Omega {{{\left| {{\nabla _H}\zeta } \right|}^p}{\upsilon _1}\left( \xi \right)d\xi } $中的常数$ C = 1 $是最佳的.

证  取$ \Xi \left( t \right) = {t^{1 - p}} $, $ \omega \left( t \right) = t $, 则有$ \omega \left( t \right)\Xi '\left( t \right){\rm{ = }} - \left( {p - 1} \right)\Xi \left( t \right) $, 所以$ c = p - 1 $. 再取$ f\left( t \right) = {t^{p - 1}} $, $ \delta = 0 $, 则定理1.1的假设满足. 所以应用定理1.1得式(4.1)成立.

接下来证明精确性: 考虑有紧支集的光滑函数序列$ {\left\{ {{\zeta _k}} \right\}_{k \in N}} $, 假设存在满足定理所有假设的$ {u_0} $, 使得在$ HW_{\left( {{\upsilon _2},{\upsilon _1}} \right)}^{1,p}\left( \Omega \right) $中$ {\zeta _k} \to {u_0} $. 因为每个$ {\zeta _k} $有紧支集且属于$ H{\cal L}_{{\upsilon _1}}^{1,p}\left( \Omega \right) $, 所以我们有

$ \langle { - {\Delta _{H,p,{\upsilon _1}}}{u_0},{\zeta _k}} \rangle = \int_\Omega {{{\left| {{\nabla _H}{u_0}} \right|}^{P - 2}}{\nabla _H}{u_0} \cdot {\nabla _H}{\zeta _k}{\upsilon _1}\left( \xi \right)} d\xi = \int_\Omega {{u_0^{p - 1}}{\upsilon _2}\left( \xi \right){\zeta _k}} d\xi . $

令$ k \to \infty $得

$ \int_\Omega {{{\left| {{\nabla _H}{u_0}} \right|}^p}{\upsilon _1}\left( \xi \right)d\xi } = \int_\Omega {{u_0}^p{\upsilon _2}\left( \xi \right)d\xi }. $

这就证明了精确性.

注  若取(1.1)中的$ {\upsilon _1} \equiv 1 $, 则[4]中的一些结果在Heisenberg群上也可建立.