经典的Picone恒等式为

$|\nabla u|^{2}+\frac{u^{2}}{v^{2}}|\nabla v|^{2}-2\frac{u}{v}\nabla u\cdot\nabla v=|\nabla u|^{2}-\nabla\left(\frac{u^{2}}{v}\right)\cdot\nabla v\geq0,$

(1.1)

其中$u\geq0, v>0$, 同时$u, v$是可微函数.以后, Allegreto-Huang在文献[1]中将(1.1)式推广到$p$-Laplace算子上.接着, Tyagi在文献[2]中又将(1.1)式做进一步的推广, 得到了较为一般的Picone恒等式

$|\nabla u|^{2}-\frac{|\nabla u|^{2}}{f'(v)}+\left(\frac{u\sqrt{f'(v)}\nabla v}{f(v)}-\frac{\nabla u}{\sqrt{f'(v)}}\right)^{2}=|\nabla u|^{2}-\nabla(\frac{u^{2}}{f(v)})\cdot\nabla v\geq 0.$

(1.2)

其中$u, v$是可微函数, 且$u\geq0, v>0$; 当$y\neq 0$时, $f(y)\neq 0, f'(y)\geq 1$; 当$y=0$时, $f(0)=0$.最近, Kaushik在文献[3]中将(1.2)式推广到$p$-Laplace算子上, 给出了更加广义的Picone恒等式

$|\nabla u|^{p}-\frac{pu^{p-1}\nabla u\cdot|\nabla v|^{p-2}\nabla v}{f(v)}+\frac{u^{p}f'(v)|\nabla v|^{p}}{[f(v)]^{2}} =|\nabla u|^{p}-\nabla\left(\frac{u^{p}}{f(v)}\right)\cdot|\nabla v|^{p-2}\nabla v\geq 0.$

(1.3)

本文将(1.3)式推广到Heisenberg-Greiner $p$ -退化椭圆算子上, 得到了一类广义Picone恒等式, 这个结果包含了(1.3)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、Sturmium比较原理和主特征值的单调性结论.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.

关于Heisenberg-Greiner $p$ -退化椭圆算子, 在这里作一个简要的叙述, 详细内容可参考文献[4-5]及其中的参考文献. Heisenberg-Greiner $p$ -退化椭圆算子形为

${\Delta _{L,p}}u = {\rm{di}}{{\rm{v}}_L}(|{\nabla _L}u{|^{p - 2}}{\nabla _L}u),$

(1.4)

其中$p>1, \ X_{j}=\frac{\partial }{\partial x_{j}}+2ky_{j}|z|^{2k-2}\frac{\partial }{\partial t}, \ Y_{j}=\frac{\partial }{\partial y_{j}}-2kx_{j}|z|^{2k-2}\frac{\partial }{\partial t}, $ $ z_{j}=x_{j}+\sqrt{-1} y_{j}\in {\Bbb C}, $ $ j=1, 2, \cdots, n, \ t\in {\Bbb R}, $ $ \nabla_L=(X_1, \cdots, X_n, Y_1, \cdots, Y_n), \ \div_{L}(u_{1}, \cdots, u_{2n}) =\sum\limits_{j=1}^{n}(X_{j}u_{j}+Y_{j}u_{n+j}), $ $ k\geq1$.注意到, 当$p=2, k=1$时, $\triangle_{L, p}$就成为Heisenberg群$H^{n}$上的次Laplace算子$\triangle_{H^{n}}$ (见文献[6]).当$p=2, k=2, 3, \cdots, $时, $\triangle_{L, p}$就成为Greiner算子^[7]

${\Delta _L} = \sum\limits_{j = 1}^n {\left( {X_j^2 + Y_j^2} \right)} .$

(1.5)

相应于(1.4)式的一个自然伸缩为

$\delta _{\tau}(z, t)=(\tau z, \tau^{2k}t), \tau>0, (z, t)=(x, y, t)\in{\Bbb R}^{2n+1}.$

(1.6)

由(1.6)式诱导的一个拟距离为

$d(z, t)=[|z|^{4k}+t^{2}]^\frac{1}{4k}.$

(1.7)

算子$\triangle_{L, p}$及$\triangle_{L}$均为一类具有高奇性的平方和退化椭圆算子, 被更多的学者所关注, 并得到了许多重要的成果(见文献[4-5]及其中的参考文献).当$k>1$时, (1.1)式中向量场${X_{j}, Y_{j}}, j=1, 2, \cdots, n$不满足Hörmander有限秩条件, 从而$\triangle_{L}$的亚椭圆性无法由此导出, 增加了研究的难度(见文献[8]及其中的参考文献).

令$C_{0}^{k}({\Bbb R}^{2n+1})$表示$C^{k}({\Bbb R}^{2n+1})$中具有紧支集的函数构成的集合, $D^{1, p}({\Bbb R}^{2n+1})$ $(1＜p＜\infty)$是$C_{0}^{\infty}({\Bbb R}^{2n+1})$在范数$\|u\|_{D^{1, p}}=(\int_{R^{2n+1}}|\nabla_{L}u|^{p}{\rm d}\xi)^{\frac{1}{p}}$ ($\xi=(x, y, t)\in{\Bbb R}^{2n+1}$)下的完备化.

下文中, 总是假设$g$满足下列条件$g:(0, \infty)\rightarrow(0, \infty)$是局部Lipchitz函数, 且在$(0, \infty)$上

$g'(x)\geq(p-1)[g(x)]^{\frac{p-2}{p-1}}$

(2.1)

几乎处处成立.

定理2.1(广义Picone恒等式)   若$1＜p＜\infty, \ \Omega \subset{\Bbb R}^{2n+1}, \ u, v\in\Omega $, 在$\Omega $上几乎处处$v>0$, 且$g$满足(2.1)式, 定义

$L(u, v)=|\nabla_{L}u|^{p}-p\frac{|u|^{p-2}u}{g(v)}\nabla_{L}u\cdot\nabla_{L}v|\nabla_{L}v|^{p-2}+\frac{g'(v)|u|^{p}}{[g(v)]^{2}}|\nabla_{L}v|^{p}, \\ R(u, v)=|\nabla_{L}u|^{p}-\nabla_{L}(\frac{|u|^{p}}{g(v)})|\nabla_{L}v|^{p-2}\nabla_{L}v,$

则$L(u, v)=R(u, v)\geq0$.而且在$\Omega $上$L(u, v)=0$几乎处处成立的充要条件是在$\Omega $上$\nabla_{L}(\frac{u}{v})=0$几乎处处成立.

证  经计算

$\nabla_{L}(\frac{|u|^{p}}{g(v)})=\frac{pg(v)|u|^{p-2}u\nabla_{L}u-g'(v)|u|^{p}\nabla_{L}v}{[g(v)^{2}]}\\ \quad \quad \quad =p\frac{|u|^{p-2}u\nabla_{L}u}{g(v)}-\frac{g'(v)|u|^{p}\nabla_{L}v}{[g(v)]^{2}}, \\ R(u, v)=|\nabla_{L}u|^{p}-\nabla_{L}(\frac{|u|^{p}}{g(v)})|\nabla_{L}v|^{p-2}\nabla_{L}v\\ \quad \quad \quad =|\nabla_{L}u|^{p}-p\frac{|u|^{p-2}u}{g(v)}\nabla_{L}u\cdot\nabla_{L}v|\nabla_{L}v|^{p-2}+\frac{g'(v)|u|^{p}}{[g(v)]^{2}}|\nabla_{L}v|^{p},$

得到$L(u, v)=R(u, v)$.又因为

$L(u, v)=|\nabla_{L}u|^{p}-p\frac{|u|^{p-2}u}{g(v)}\nabla_{L}u\cdot\nabla_{L}v|\nabla_{L}v|^{p-2}+\frac{g'(v)|u|^{p}}{[g(v)]^{2}}|\nabla_{L}v|^{p}\\ \quad \quad \quad \ge |\nabla_{L}u|^{p}-p\frac{|u|^{p-1}}{g(v)}|\nabla_{L}v|^{p-1}|\nabla_{L}u|+\frac{g'(v)|u|^{p}}{[g(v)]^{2}}|\nabla_{L}v|^{p}\\ \quad \quad \quad \ge |\nabla_{L}u|^{p}-|\nabla_{L}u|^{p}-(p-1)\frac{|u|^{p}|\nabla_{L}v|^{p}}{[g(v)]^{\frac{p}{p-1}}}+\frac{g'(v)|u|^{p}}{[g(v)]^{2}}|\nabla_{L}v|^{p}$

并且$g(x)$满足(2.1)式, 所以$L(u, v)=R(u, v)\geq0$.

当下面三个等式同时成立时, $L(u, v)=R(u, v)=0$.

$g'(x)=(p-1)[g(x)]^{\frac{p-2}{p-1}}.$

(2.2)

$\frac{|u|^{p-2}u}{g(v)}\nabla_{L}u\cdot\nabla_{L}v|\nabla_{L}v|^{p-2}=\frac{|u|^{p-1}}{g(v)}|\nabla_{L}v|^{p-1}|\nabla_{L}u|.$

(2.3)

$|\nabla_{L}u|=\frac{|u\nabla_{L}v|}{g(v)^{\frac{1}{p-1}}}.$

(2.4)

令

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

当$\xi\in\omega$时, 由(2.4)式得到

$\frac{|u\nabla_{L}v|}{g(v)^{\frac{1}{p-1}}}=|\nabla_{L}u|=0.$

(2.5)

解(2.2)式得$g(x)=x^{p-1}$, 结合(2.5)式得

$\frac{u}{v}\nabla_{L}v=\nabla_{L}u=0.$

(2.6)

当$\xi\in\omega^{c}$ ($\omega^{c}$ =$\Omega /\omega$)时, 设

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

由$L(u, v)=0$, 得

${\varpi ^p} - p\varpi + p - 1 = 0.$

从而$\varpi =1$.即

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

上式中, 取$g(x)=x^{p-1}$, 得

$\nabla_{L}u\cdot(\nabla_{L}u-\nabla_{L}v\frac{u}{v})=0.$

(2.7)

综合(2.6)和(2.7)式得到, 在$\Omega $上$L(u, v)=0$几乎处处成立的充要条件是在$\Omega $上$\nabla_{L}(\frac{u}{v})=0$几乎处处成立.

注2.1  在定理2.1中, 当$\Omega ={\Bbb R}^{2n+1}$时, 结论仍然成立.

作为应用, 本节首先讨论Hardy不等式.证明Hardy不等式时, 需要下面一个关键性的引理.

引理3.1   若$v\in C^{1}$, 在$\Omega $上$v>0$, 并且满足

$ - {\Delta _{L,p}}v \ge \lambda hg(v),$

其中$h$是非负连续函数, 则对于$u\in C_{0}^{\infty}, \ u\geq0$有

$\int_{{\Bbb R}^{2n+1}}|\nabla_{L}u|^{p} \geq\lambda\int_{{\Bbb R}^{2n+1}}h|u|^{p},$

(3.1)

其中$g$满足(2.1)式.

证   取$\phi\in C_{0}^{\infty}({\Bbb R}^{2n+1})$, $\phi>0$.由定理2.1得

$\begin{eqnarray*} 0&\leq&\int_{{\Bbb R}^{2n+1}}L(\phi,v)=\int_{{\Bbb R}^{2n+1}}R(\phi,v)\\ &=&\int_{{\Bbb R}^{2n+1}}|\nabla_{L}\phi|^{p}-\int_{{\Bbb R}^{2n+1}}\nabla_{L}(\frac{\phi^{p}}{g(v)})|\nabla_{L}v|^{p-2}\nabla_{L}v\\ &=&\int_{{\Bbb R}^{2n+1}}|\nabla_{L}\phi|^{p}+\int_{{\Bbb R}^{2n+1}}\frac{\phi^{p}}{g(v)}\triangle_{L,p}v\\ &\leq&\int_{{\Bbb R}^{2n+1}}|\nabla_{L}\phi|^{p}-\lambda\int_{{\Bbb R}^{2n+1}}h\phi^{p}. \end{eqnarray*}$

令$\phi\rightarrow u$, 就得到(3.1)式.

类似于Allegreto-Huang在文献[1]的方法, 利用引理3.1, 取$g(v)=v^{p-1}$, 容易得到下列Hardy不等式.

定理3.1 (Hardy不等式)   设$1＜p＜Q, \ u\in C_{0}^{\infty}(R^{2n+1}\setminus\{0\})$, 有

$\int_{{\Bbb R}^{2n+1}}|\nabla_{L}u|^{p}{\rm d}z{\rm d}t \geq\left(\frac{Q-p}{p}\right)^{p}\int_{{\Bbb R}^{2n+1}}\left(\frac{|z|}{d}\right)^{p\alpha }\frac{|u|^{p}}{d^{p}}{\rm d}z{\rm d}t$

成立, 其中$Q=2n+2k$是相应于(1.6)式的齐次维数.

定理3.2 (Sturmium比较原理)   设$f_{1}, \ f_{2}$是两个权函数, $f_{1}＜f_{2}$且$g$满足

$g'(y)\geq(p-1)[g(y)^{\frac{p-2}{p-1}}], $

若$u$是方程

$\left\{\begin{array}{ll} -\triangle_{L, p}u=f_{1}(\xi)|u|^{p-2}u,&\xi\in\Omega , \\ u=0,&\xi\in\partial \Omega \end{array}\right.$

的一个正解, 则方程

$\left\{\begin{array}{ll} -\triangle_{L, p}v=f_{2}(\xi)g(v),&\xi\in\Omega , \\ v=0,&\xi\in\partial \Omega \end{array}\right.$

(3.3)

的任意非平凡解一定改变符号.

证   假设$v>0$是(3.2)式的一个解, 由广义Picone恒等式, 有

$\begin{eqnarray*} 0&\leq&\int_{\Omega}L(u,v)=\int_{\Omega}R(u,v)\\ &=&\int_{\Omega}|\nabla_{L}u|^{p} -\nabla_{L}\left(\frac{u^{p}}{g(v)}\right) |\nabla_{L}v|^{p-2}\nabla_{L}v\\ &=&\int_{\Omega}f_{1}(\xi)u^{p}-f_{2}(\xi)u^{p}\\ &=&\int_{\Omega}(f_{1}-f_{2})u^{p} < 0, \end{eqnarray*}$

这是一个矛盾式.因此假设错误, 即$v$在$\Omega $上改变符号.

对于下列带有不确定权的特征值问题

$\left\{\begin{array}{ll} -\triangle_{L, p}u=\lambda h(\xi)g(u),&\xi\in\Omega , \\ u=0,&\xi\in\partial \Omega , \end{array}\right.$

(3.3)

其中$h(x)$是一个不确定权函数, $g(u)=u^{p-1}$.下面我们利用定理2.1给出主特征值的严格单调性结论.

定理3.3 (主特征值的单调性)   设$\lambda^{+}_{1}(\Omega )>0$是问题(3.3)的主特征值, 若$\Omega _{1}\subset\Omega _{2}$且$\Omega _{1}\neq\Omega _{2}$, $\lambda^{+}_{1}(\Omega _{1})$与$\lambda^{+}_{1}(\Omega _{2})$都存在, 则$\lambda^{+}_{1}(\Omega _{1})>\lambda^{+}_{1}(\Omega _{2})$.

证  设$u_{1}, u_{2}$分别是相应于$\lambda^{+}_{1}(\Omega _{1}), \lambda^{+}_{1}(\Omega _{2})$的特征函数, 其中$u_{2}>0$.对于$\varphi \in C^{\infty}_{0}(\Omega _{1})$, 利用定理2.1, 得到

$\begin{eqnarray*} 0&\leq&\int_{\Omega_{1}}R(\varphi,u_{2})\\ &=&\int_{\Omega_{1}}|\nabla_{L}\varphi|^{p}-\int_{\Omega_{1}}\nabla_{L}\left(\frac{\varphi^{p}}{g(u_{2})}\right)|\nabla_{L}u_{2}|^{p-2}\nabla_{L}u_{2}\\ &=&\int_{\Omega_{1}}|\nabla_{L}\varphi|^{p}+\int_{\Omega_{1}}\frac{\varphi^{p}}{g(u_{2})}\triangle_{L,p}u_{2}\\ &=&\int_{\Omega_{1}}|\nabla_{L}\varphi|^{p}-\lambda^{+}_{1}(\Omega_{2})\int_{\Omega_{1}}h\varphi^{p}. \end{eqnarray*}$

在$D^{1, p}({\Bbb R}^{2n+1})$中, 令$\varphi \rightarrow u_{1}$, 有

$0\leq\int_{\Omega _{1}}R(u_{1}, u_{2}) =(\lambda^{+}_{1}(\Omega _{1})-\lambda^{+}_{1}(\Omega _{2}))\int_{\Omega _{1}}hu_{1}^{p}.$

(3.4)

由

$\int_{\Omega _{1}}|\nabla_{L}u_{1}|^{p} =\int_{\Omega _{1}}-(\triangle_{L, p}u_{1})u_{1} =\int_{\Omega _{1}}\lambda^{+}_{1} h u_{1}^{p},$

得到$\int_{\Omega _{1}} h u_{1}^{p}\geq 0.$结合(3.4)式, 有$\lambda^{+}_{1}(\Omega _{1})-\lambda^{+}_{1}(\Omega _{2})\geq 0$.若$\lambda^{+}_{1}(\Omega _{1})=\lambda^{+}_{1}(\Omega _{2})$, 则由(3.4)式知道$R(u_{1}, u_{2})=0$, 从而通过定理2.1, 有$u_{1}=ku_{2}$.而已知$\Omega _{1}\subset\Omega _{2}$且$\Omega _{1}\neq\Omega _{2}$, 所以$u_{1}\neq ku_{2}$.进而$\lambda^{+}_{1}(\Omega _{1})\neq\lambda^{+}_{1}(\Omega _{2})$, 因此$\lambda^{+}_{1}(\Omega _{1})>\lambda^{+}_{1}(\Omega _{2})$.

定理3.4 (具奇异项的拟线性方程组的弱解结论)   若$g$满足$g\prime(y) \geq (p-1)[g(y)^{\frac{p-2}{p-1}}]$, 且$(u, v)$是下列方程组的一组弱解

$\left\{ {\begin{array}{*{20}{l}} { - {\Delta _{L,p}}u(\xi ) = g(v(\xi )),}&{\xi \in \Omega ,}\\ { - {\Delta _{L,p}}v(\xi ) = \frac{{{{[g(v(\xi ))]}^2}}}{{{{[u(\xi )]}^{p - 1}}}},}&{\xi \in \Omega ,}\\ {u(\xi ) > 0,\quad v(\xi ) > 0,}&{\xi \in \Omega ,}\\ {u(\xi ) = 0,\quad v(\xi ) = 0,}&{\xi \in \partial \Omega ,} \end{array}} \right.$

(3.5)

则$u=c_{1}v$, 其中$c_{1}$是常数.

证  设$\phi_{1}, \phi_{2} \in D_{0}^{1, p}(\Omega )$, 有

$\int_\Omega | {\nabla _L}u{|^{p - 2}}{\nabla _L}u \cdot {\nabla _L}{\phi _1} = \int_\Omega g (v){\phi _1}{\rm{d}}\xi ,$

(3.6)

$\int_\Omega | {\nabla _L}v{|^{p - 2}}{\nabla _L}v \cdot {\nabla _L}{\phi _2} = \int_\Omega {\frac{{{{[g(v)]}^2}}}{{{u^{p - 1}}}}} {\phi _2}{\rm{d}}\xi .$

(3.7)

取$\phi_{1} = u, \phi_{2} = \frac{u^{p}}{g(v)}$, 得

$\int_{\Omega }|\nabla_{L}u|^{p}{\rm d}\xi=\int_{\Omega }ug(v){\rm d}\xi=\int_{\Omega }\nabla_{L}\left(\frac{u^{p}}{g(v)}\right)|\nabla_{L}v|^{p-2}\nabla_{L}v{\rm d}\xi.$

从而

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

因此, 由定理2.1中的广义Picone恒等式, 得$\nabla_{L}\left(\frac{u}{v}\right) = 0$, 即$u=c_{1}v$, 其中$c$是常数.