西南大学学报 (自然科学版)  2020, Vol. 42 Issue (2): 48-54.  DOI: 10.13718/j.cnki.xdzk.2020.02.008
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  • Heisenberg型群上的广义Picone恒等式及其应用    [PDF全文]
    王胜军1, 窦井波2    
    1. 青海师范大学 数学与统计学院, 西宁 810008;
    2. 陕西师范大学 数学与信息科学院, 西安 710119
    摘要:利用Heisenberg型群上p-退化椭圆算子的广义Picone恒等式给出了Hardy不等式、Sturmiam比较原理、Liouville型定理和主特征值的单调性结论.讨论了具有奇异项的拟线性方程的弱解问题.
    关键词Heisenberg型群    广义Picone恒等式    Hardy不等式    Sturmiam比较原理    Liouville型定理    

    在欧式空间上,经典的Picone恒等式为

    $ {\left| {\nabla u} \right|^2} + \frac{{{u^2}}}{{{v^2}}}{\left| {\nabla v} \right|^2} - 2\frac{u}{v}\nabla u\nabla v = {\left| {\nabla u} \right|^2} - \nabla \left( {\frac{{{u^2}}}{v}} \right)\nabla v \ge 0 $ (1)

    其中u≥0,v>0,同时uv是可微函数[1-3].文献[4]将(1)式推广到p-Laplace算子上,接着,文献[5]又将(1)式作了进一步的推广,得到了较为一般的Picone恒等式

    $ \begin{array}{l} {\left| {\nabla u} \right|^2} + \frac{{{{\left| {\nabla u} \right|}^2}}}{{f'\left( v \right)}} + {\left( {\frac{{u\sqrt {{f^\prime }(v)} \nabla v}}{{f(v)}} - \frac{{\nabla u}}{{\sqrt {{f^\prime }(v)} }}} \right)^2} = \\ {\left| {\nabla u} \right|^2} - \nabla \left( {\frac{{{u^2}}}{{f\left( v \right)}}} \right)\nabla v \ge 0 \end{array} $ (2)

    最近,文献[6]将(2)式推广到p-Laplace算子上,给出了更加广义的Picone恒等式

    $ \begin{array}{l} {\left| {\nabla u} \right|^p} - \frac{{p{u^{p - 1}}\nabla u{{\left| {\nabla v} \right|}^{p - 2}}\nabla v}}{{f(v)}} + \frac{{{u^p}{f^\prime }(v){{\left| {\nabla v} \right|}^p}}}{{{{[f(v)]}^2}}} = \\ {\left| {\nabla u} \right|^p} - \nabla \left( {\frac{{{u^2}}}{{f\left( v \right)}}} \right){\left| {\nabla v} \right|^{p - 2}}\nabla v \ge 0 \end{array} $ (3)

    关于Baouendi-Grushi p-退化椭圆算子,文献[7]给出了类似(3)式的结果.本文将(3)式推广到Heisenberg型群的p-退化椭圆算子上,得到了一类广义Picone恒等式,其结果包含了(3)式的情形.作为应用,在第三部分,利用本文得到的广义Picone恒等式证明了Hardy不等式、Sturmiam比较原理、主特征值的单调性结论和Liouville型定理,避免了正则性的讨论.最后,讨论了具有奇异项的拟线性方程的弱解问题.

    1 预备知识

    文献[8]提出了Heisenberg型群,它是Heisenberg群的推广,是一类与亚椭圆问题相联系的Carnot型群.作为满足Hömander条件的一般向量场的重要模型,Heisenberg型群被更多学者广泛研究,并得到许多重要结果[9-12].关于Heisenberg型群,在这里作一个简要的叙述,详细内容可参考本文中提到的参考文献.

    G是具有李代数$\mathscr{G}$ =V1V2的一个2步Carnot群,且$\mathscr{G}$被赋予內积〈·,·〉,定义映射JV2→End(V1):

    $ \left\langle {J\left( {{\xi _2}} \right)\xi _1^\prime ,\xi _1^{\prime \prime }} \right\rangle = \left\langle {{\xi _2},\left[ {\xi _1^\prime ,\xi _1^{\prime \prime }} \right]} \right\rangle \;\;\;\;\xi _1^\prime ,\xi _1^{\prime \prime } \in {V_1},{\xi _2} \in {V_2} $

    若对任意ξ2V2,|ξ2|=1,映射J(ξ2):V1V1是正交的,则称G是一个Heisenberg型群,简称H型群.

    Heisenberg型群p-退化椭圆算子形为

    $ {\Delta _p}u = {\rm{div}}{_L}\left( {{{\left| {{\nabla _L}u} \right|}^{p - 2}}{\nabla _L}u} \right) $ (4)

    其中p>1.

    $ {X_j} = \frac{\partial }{{\partial {x_j}}} + \frac{1}{2}\sum\limits_{i = 1}^n {\left\langle {\left[ {\xi ,{X_j}} \right],{Y_i}} \right\rangle } \frac{\partial }{{\partial {y_i}}}\;\;\;\;j = 1, \cdots ,m $

    V1的一组标准正交基,其中

    $ \xi = {\xi _1} + {\xi _2} \in \mathscr{G} = {V_1} \oplus {V_2} $
    $ \begin{array}{*{20}{c}} {x = \left( {{x_1}, \cdots ,{x_m}} \right) \in {\mathbb{R}^m}}&{y = \left( {{y_1}, \cdots ,{y_k}} \right) \in {\mathbb{R}^n}} \end{array} $
    $ \begin{array}{*{20}{c}} {{\nabla _L} = \left( {{X_1}, \cdots ,{X_m}} \right)}&{{\rm{div}}{_L}\left( {{u_1}, \cdots ,{u_m}} \right) = \sum\limits_{j = 1}^m {{X_j}} {u_j}} \end{array} $

    相应于(4)式的非迷向伸缩为

    $ \begin{array}{*{20}{c}} {{\delta _\tau }\left( {x,y} \right) = \left( {\tau x,{\tau ^2}y} \right)}&{\tau > 0,\left( {x,y} \right) \in G} \end{array} $ (5)

    与此伸缩相应的G的齐次维数是Q=m+2n.

    ξ=(xy),$\widetilde{\xi}=(\widetilde{x}, \widetilde{y}) \in G$,在H型群G上得到一个拟距离为

    $ {\text{d}}(\xi ) = {\left( {|x(\xi ){|^4} + 16|y(\xi ){|^2}} \right)^{\frac{1}{4}}} $

    定义中心在ξ,半径为R的拟开球为

    $ {B_R}(\xi ) = \{ \xi \in G|{\text{d}}(\xi ) < R\} $

    ΩGΩ是开子集,C0k(Ω)表示Ck(Ω)中具有紧支集的函数构成的集合,

    $ {D^{1,p}}(\mathit{\Omega }) = \left\{ {u:\mathit{\Omega } \to \mathbb{R}||{\nabla _L}u| \in {L^p}(\mathit{\Omega })} \right\} $

    D01,p(Ω)(1<p<∞)是C0(Ω)在范数

    $ {\left\| u \right\|_{D_0^{1,p}}} = {\left( {\int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^p}} {\text{d}}\xi } \right)^{\frac{1}{p}}} $

    下的完备化.

    2 Heisenberg型群p -退化椭圆算子的一类广义Picone恒等式

    下文中,总是假设g满足下列条件:g:(0,∞)→(0,∞)是局部Lipchitz函数,且在(0,∞)上

    $ {g^\prime }(v) \geqslant (p - 1){[g(v)]^{\frac{{p - 2}}{{p - 1}}}} $ (6)

    几乎处处成立.

    定理1 (广义Picone恒等式)若1<p<∞,ΩGuvΩ上的可微函数,在Ω上几乎处处v>0,且g满足(6)式,设

    $ L(u,v) = {\left| {{\nabla _L}u} \right|^p} - p\frac{{|u{|^{p - 2}}u}}{{g(v)}}{\nabla _L}u \cdot {\nabla _L}v{\left| {{\nabla _L}v} \right|^{p - 2}} + \frac{{{g^\prime }(v)|u{|^p}}}{{{{[g(v)]}^2}}}{\left| {{\nabla _L}v} \right|^p} $
    $ R(u,v) = {\left| {{\nabla _L}u} \right|^p} - {\nabla _L}\left( {\frac{{|u{|^p}}}{{g(v)}}} \right){\left| {{\nabla _L}v} \right|^{p - 2}}{\nabla _L}v $

    $ L(u,v) = R(u,v) \geqslant 0 $

    而且在Ω上,L(uv)=0几乎处处成立的充要条件是$\nabla_{L}\left(\frac{u}{v}\right)=0$几乎处处成立.

      经计算,有

    $ {\nabla _L}\left( {\frac{{|u{|^p}}}{{g(v)}}} \right) = p\frac{{|u{|^{p - 2}}u{\nabla _L}u}}{{g(v)}} - \frac{{{g^\prime }(v)|u{|^p}{\nabla _L}v}}{{{{[g(v)]}^2}}} $
    $ R(u,v) = {\left| {{\nabla _L}u} \right|^p} - p\frac{{|u{|^{p - 2}}u}}{{g(v)}}{\nabla _L}u \cdot {\nabla _L}v{\left| {{\nabla _L}v} \right|^{p - 2}} + \frac{{{g^\prime }(v)|u{|^p}}}{{{{[g(v)]}^2}}}{\left| {{\nabla _L}v} \right|^p} $

    得到

    $ L(u,v) = R(u,v) $

    因为

    $ L(u,v) \geqslant {\left| {{\nabla _L}u} \right|^p} - {\left| {{\nabla _L}u} \right|^p} - (p - 1)\frac{{|u{|^p}{{\left| {{\nabla _L}v} \right|}^p}}}{{{{[g(v)]}^{\frac{{p - 2}}{{p - 1}}}}}} + \frac{{{g^\prime }(v)|u{|^p}}}{{{{[g(v)]}^2}}}{\left| {{\nabla _L}v} \right|^p} $

    并且g(v)满足(6)式,所以

    $ L(u,v) = R(u,v) \geqslant 0 $

    当下面3个等式同时成立时,L(uv)=R(uv):

    $ {g^\prime }(v) = (p - 1){[g(v)]^{\frac{{p - 2}}{{p - 1}}}} $ (7)
    $ \frac{{{{\left| u \right|}^{p - 2}}u}}{{g(v)}}{\nabla _L}u \cdot {\nabla _L}v{\left| {{\nabla _L}v} \right|^{p - 2}} = \frac{{|u{|^{p - 1}}}}{{g(v)}}{\left| {{\nabla _L}v} \right|^{p - 1}}\left| {{\nabla _L}u} \right| $ (8)
    $ \left| {{\nabla _L}u} \right| = \frac{{\left| {u{\nabla _L}v} \right|}}{{g{{(v)}^{\frac{1}{{p - 1}}}}}} $ (9)

    $ \omega = \left\{ {\xi \in \mathit{\Omega }|\left. {\frac{{\left| {u{\nabla _L}v} \right|}}{{g{{(v)}^{\frac{1}{{p - 1}}}}}} = 0} \right)} \right. $

    ξω时,由(9)式得到

    $ \frac{{\left| {u{\nabla _L}v} \right|}}{{g{{(v)}^{\frac{1}{{p - 1}}}}}} = \left| {{\nabla _L}u} \right| = 0 $ (10)

    在(7)式中,取g(v)=vp-1,结合(10)式得

    $ \frac{u}{v}{\nabla _L}v = {\nabla _L}u = 0 $ (11)

    ξωc时,设

    $ \tilde \omega = \frac{{\left| {{\nabla _L}u} \right|{{[g(v)]}^{\frac{1}{{p - 1}}}}}}{{\left| {u{\nabla _L}v} \right|}} $

    L(uv)=0,得

    $ {{\tilde \omega }^p} - p\tilde \omega + p - 1 = 0 $

    从而$\tilde \omega$=1,即

    $ \frac{{\left| {{\nabla _L}u} \right|{{[g(v)]}^{\frac{1}{{p - 1}}}}}}{{\left| {u{\nabla _L}v} \right|}} = 1 $

    g(v)=vp-1,得

    $ {\nabla _L}u \cdot \left( {{\nabla _L}u - {\nabla _L}v\frac{u}{v}} \right) = 0 $ (12)

    综合(11)式和(12)式得到:在Ω上,L(uv)=0几乎处处成立的充要条件是$\nabla_{{L}}\left(\frac{u}{v}\right)=0$几乎处处成立.

    注1   在定理1中,当Ω=G时,结论仍然成立.

    注2   在定理1中,取g(v)=vp-1u≥0,得到文献[13]中的Picone恒等式.

    3 H型群上的广义Picone恒等式的应用

    作为应用,本节首先讨论Hardy不等式.为此需要下面关键性的引理:

    引理1   若vC1(G),在Ωv>0,并且满足

    $ - {\Delta _p}v \geqslant \lambda hg(v) $

    其中λ>0,h是非负连续函数,g满足(6)式,则对于uC0(G),u≥0,有

    $ \int_G {{{\left| {{\nabla _1}u} \right|}^p}} {\text{d}}\xi \geqslant \lambda \int_G h |u{|^p}{\text{d}}\xi $ (13)

      取ϕC0(G),ϕ >0.由注1得

    $ \begin{array}{*{20}{c}} {0 \leqslant \int_G L (\phi ,v){\text{d}}\xi \leqslant \int_G R (\phi ,v){\text{d}}\xi \leqslant } \\ {\int_G {{{\left| {{\nabla _L}\phi } \right|}^p}} {\text{d}}\xi - \lambda \int_G h {\phi ^p}{\text{d}}\xi } \end{array} $

    ϕu,就得到(13)式.

    利用引理1,取g(v)=vp-1,容易得到文献[14]中的下列Hardy不等式:

    定理2 (Hardy不等式)   设1<pQuC0(G\{0}),有

    $ \int_G {{{\left| {{\nabla _L}u} \right|}^p}} {\text{d}}\xi \geqslant {\left( {\frac{{Q - p}}{p}} \right)^p}\int_G {{{\left( {\frac{{|x|}}{{\text{d}}}} \right)}^{p\alpha }}} \frac{{|u{|^p}}}{{{{\text{d}}^p}}}{\text{d}}\xi $

    成立,其中Q=m+2n是相应于(5)式的齐次维数.

    定理3 (Sturmiam比较原理)   设f1f2是两个权函数,f1f2,且g满足

    $ {g^\prime }(v) \geqslant (p - 1)\left[ {g{{(v)}^{\frac{{p - 2}}{{p - 1}}}}} \right] $

    u是方程

    $ \left\{ {\begin{array}{*{20}{l}} { - {\Delta _p}u = {f_1}(\xi )|u{|^{p - 2}}u}&{\xi \in \mathit{\Omega }} \\ {u = 0}&{\xi \in \partial \mathit{\Omega }} \end{array}} \right. $

    的正解,则方程

    $ \left\{ {\begin{array}{*{20}{l}} { - {\Delta _p}v = {f_2}(\xi )g(v)}&{\xi \in \mathit{\Omega }}\\ {v = 0}&{\xi \in \partial \mathit{\Omega }} \end{array}} \right. $ (14)

    的任意非平凡解一定改变符号.

      假设v>0是方程(14)的解,由广义Picone恒等式,有

    $ \begin{array}{l} 0 \le \int_\mathit{\Omega } L (u,v){\rm{d}}\xi = \int_\mathit{\Omega } R (u,v){\rm{d}}\xi = \\ \;\;\;\;\;\int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^p}} {\rm{d}}\xi - {\nabla _L}\left( {\frac{{{u^p}}}{{g(v)}}} \right){\left| {{\nabla _L}v} \right|^{p - 2}}{\nabla _L}v = \\ \;\;\;\;\;\int_\mathit{\Omega } {\left( {{f_1}(\xi ){u^p} - {f_2}(\xi )|u{|^p}} \right){\rm{d}}\xi } = \\ \;\;\;\;\;\int_\mathit{\Omega } {\left( {{f_1} - {f_2}} \right){u^p}{\rm{d}}\xi } < 0 \end{array} $

    矛盾.因此假设错误,即vΩ上改变符号.

    对于下列不确定特征值问题

    $ \left\{ {\begin{array}{*{20}{l}} { - {\Delta _p}u = \lambda h(\xi )g(u)}&{\xi \in \mathit{\Omega }}\\ {u = 0}&{\xi \in \partial \mathit{\Omega }} \end{array}} \right. $ (15)

    其中h(ξ)是不确定权函数,利用定理1给出的主特征值的严格单调性结论,g(u)满足(6)式.

    定理4 (主特征值的单调性结论)   设λ1+(Ω)>0是问题(15)的主特征值,若Ω1Ω2Ω1Ω2λ1+(Ω1)与λ1+(Ω2)都存在,则λ1+(Ω1)>λ1+(Ω2).

      设u1u2分别是相应于λ1+(Ω1),λ1+(Ω2)的正的特征函数,对于φC0(Ω1),利用定理1,得到

    $ \begin{array}{l} 0 \le \int_{{\mathit{\Omega }_1}} R \left( {\varphi ,{u_2}} \right){\rm{d}}\xi = \int_{{\mathit{\Omega }_1}} {{{\left| {{\nabla _L}\varphi } \right|}^p}} {\rm{d}}\xi - \int_{{\mathit{\Omega }_1}} {{\nabla _L}} \left( {\frac{{{\varphi ^p}}}{{g\left( {{u_2}} \right)}}} \right){\left| {{\nabla _L}{u_2}} \right|^{p - 2}}{\nabla _L}{u_2}{\rm{d}}\xi = \\ \;\;\;\;\;\int_{{\mathit{\Omega }_1}} {{{\left| {{\nabla _L}\varphi } \right|}^p}} {\rm{d}}\xi + \int_{{\mathit{\Omega }_1}} {\frac{{{\varphi ^p}}}{{g\left( {{u_2}} \right)}}{\Delta _{L,p}}{u_2}{\rm{d}}\xi } = \\ \;\;\;\;\;\int_{{\mathit{\Omega }_1}} {{{\left| {{\nabla _L}\varphi } \right|}^p}} {\rm{d}}\xi - \lambda _1^ + \left( {{\mathit{\Omega }_2}} \right)\int_{{\mathit{\Omega }_1}} h {\varphi ^p}{\rm{d}}\xi \end{array} $

    D01,p(Ω1)中,令φu1,有

    $ 0 \le \int_{{\mathit{\Omega }_1}} R \left( {{u_1},{u_2}} \right){\rm{d}}\xi = \left( {\lambda _1^ + \left( {{\mathit{\Omega }_1}} \right) - \lambda _1^ + \left( {{\mathit{\Omega }_2}} \right)} \right)\int_{{\mathit{\Omega }_1}} h u_1^p{\rm{d}}\xi $ (16)

    已知

    $ \int_{{\mathit{\Omega }_1}} {{{\left| {{\nabla _L}{u_1}} \right|}^p}} {\rm{d}}\xi = \int_{{\mathit{\Omega }_1}} - \left( {{\Delta _p}{u_1}} \right){u_1}{\rm{d}}\xi = \int_{{\mathit{\Omega }_1}} {\lambda _1^ + } hg\left( {{u_1}} \right){u_1}{\rm{d}}\xi $

    g(u1)=u1p-1,得到

    $ \int_{{\mathit{\Omega }_1}} {{{\left| {{\nabla _L}{u_1}} \right|}^p}} {\rm{d}}\xi = \int_{{\mathit{\Omega }_1}} {\lambda _1^ + } hu_1^p{\rm{d}}\xi $

    从而

    $ \int_{{\mathit{\Omega }_1}} h u_1^p{\rm{d}}\xi \ge 0 $

    结合(16)式得到

    $ \lambda _1^ + \left( {{\mathit{\Omega }_1}} \right) - \lambda _1^ + \left( {{\mathit{\Omega }_2}} \right) \ge 0 $

    由已知条件知道

    $ \lambda _1^ + \left( {{\mathit{\Omega }_1}} \right) \ne \lambda _1^ + \left( {{\mathit{\Omega }_2}} \right) $

    因此

    $ \lambda _1^ + \left( {{\mathit{\Omega }_1}} \right) > \lambda _1^ + \left( {{\mathit{\Omega }_2}} \right) $

    定理5 (Liouville型结果)   若c0>0,p>1,且g满足$g^{\prime}(y) \geqslant(p-1)\left[g(y)^{\frac{p-2}{p-1}}\right]$],则

    $ - {\Delta _p}v \ge {c_0}g\left( v \right) $ (17)

    DLoc1,p(G)中没有正解.

      假设v是(17)式的正解,取R>0,令ϕ1是相应于第一特征值λ1(BR(ξ))的第一特征函数,使得λ1(BR(ξ))<c0.由文献[15]中的极大值原理知道:$\frac{\phi_{1}^{p}}{g(v)}$可以作为测试函数,且$\frac{\phi_{1}^{p}}{g(v)}$D1,p(BR(ξ)).因此

    $ {c_0}\int_{{B_R}(\xi )} {\phi _1^p} {\rm{d}}\xi - \int_{{B_R}(\xi )} {{{\left| {{\nabla _L}{\phi _1}} \right|}^p}} {\rm{d}}\xi \le - \int_{{B_R}(\xi )} R \left( {{\phi _1},v} \right){\rm{d}}\xi \le 0 $

    从而

    $ {c_0} \le \frac{{\int_{{B_R}(\xi )} {{{\left| {{\nabla _L}{\phi _1}} \right|}^p}} {\rm{d}}\xi }}{{\int_{{B_R}(\xi )} {\phi _1^p} {\rm{d}}\xi }} = {\lambda _1}\left( {{B_R}(\xi )} \right) < {c_0} $

    矛盾.因此假设错误,即(17)式在DLoc1,p(G)中没有正解.

    文献[16]讨论了具有奇异项的p-Laplacian方程解的问题,这里利用定理1来讨论这类问题.

    定理6 (具有奇异项的拟线性方程组的弱解结论)   若g满足

    $ {g^\prime }(y) \ge (p - 1)\left[ {g{{(y)}^{\frac{{p - 2}}{{p - 1}}}}} \right] $

    且(uv)是下列方程组的一组弱解:

    $ \left\{ \begin{array}{l} - {\Delta _p}u(\xi ) = g(v(\xi ))\quad \quad \quad \xi \in \mathit{\Omega }\\ - {\Delta _p}v(\xi ) = \frac{{{{[g(v(\xi ))]}^2}}}{{{{[u(\xi )]}^{p - 1}}}}\quad \;\;\;\xi \in \mathit{\Omega }\\ u(\xi ) > 0,v(\xi ) > 0\quad \;\;\;\;\;\;\;\;\;\xi \in \mathit{\Omega }\\ u(\xi ) = 0,v(\xi ) = 0\;\;\;\;\;\;\;\;\;\;\;\;\xi \in \partial \mathit{\Omega } \end{array} \right. $ (18)

    u=c1v,其中c1是常数.

      设ϕ1ϕ2D01,p(Ω),由方程组(18)有

    $ \int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^{p - 2}}} \left| {{\nabla _L}u} \right|{\nabla _L}{\phi _1}{\rm{d}}\xi = \int_\mathit{\Omega } g (v){\phi _1}{\rm{d}}\xi $ (19)
    $ \int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^{p - 2}}} \left| {{\nabla _L}u} \right|{\nabla _L}{\phi _2}{\rm{d}}\xi = \int_\mathit{\Omega } {\frac{{{{[g(v)]}^2}}}{{{u^{p - 1}}}}{\phi _2}{\rm{d}}\xi } $ (20)

    在(19),(20)式中取ϕ1=u$\phi_{2}=\frac{u^{p}}{g(v)}$,得

    $ \int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^p}} {\rm{d}}\xi = \int_\mathit{\Omega } u g(v){\rm{d}}\xi = \int_\mathit{\Omega } {{\nabla _L}} \left( {\frac{{{u^p}}}{{g(v)}}} \right){\left| {{\nabla _L}v} \right|^{p - 2}}{\nabla _L}v{\rm{d}}\xi $

    从而

    $ \int_\mathit{\Omega } R (u,v){\rm{d}}\xi = \int_\mathit{\Omega } {{{\left| {{\nabla _L}u} \right|}^p}} {\rm{d}}\xi - {\nabla _L}\left( {\frac{{{u^p}}}{{g(v)}}} \right){\left| {{\nabla _L}u} \right|^{p - 2}}{\nabla _L}v = 0 $

    因此,由定理1中的R(uv)>0,得

    $ {\nabla _L}\left( {\frac{u}{v}} \right) = 0 $

    u=c1v,其中c是常数.

    参考文献
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    Generalized Picone's Identity and Its Applications for the Heisenberg-Type Group
    WANG Sheng-jun1, DOU Jing-bo2    
    1. School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, China;
    2. School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China
    Abstract: This paper establishes a generalized version of the Picone's identity of p-degenerated elliptic operators for the Heisenberg-type group. As applications, Hardy-type inequality, Sturmian comparison principle, a Liouville-type theorem and the strict monotonicity of the principal eigenvalue are given. The weak solution of the quasi-linear system with singular nonlinearity is also studied.
    Key words: Heisenberg-type group    generalized Picone's identity    Hardy-type inequality    Sturmian comparison principle    Liouville-type theorem    
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