張書陶,趙 瓊,韓亞洲
(中國計量學院理學院,杭州 310018)
Heisenberg群上一類半線性方程的Liouville型定理
張書陶,趙瓊,韓亞洲
(中國計量學院理學院,杭州310018)
結合向量場法的思想,研究了Heisenberg群上的一類半線性方程,并給出不存在非平凡正解的Liouville型定理.首先,利用Heisenberg群上左不變向量場的對稱性構造一類實泛函,并通過恒等變形獲得一些恒等式;然后,利用試驗函數的性質,結合Heisenberg群上的極坐標公式、Young不等式等技巧以精確估計,進而證明任一非負解均恒為零.
向量場法;Liouville型定理;半線性方程;Heisenberg群
在Euclidean空間,半線性方程
Heisenberg群的概念來源于量子力學、多復變幾何等學科.作為非交換幾何的典型代表,學者們對Heisenberg群上的半線性問題
進行了大量的研究[9-11].為敘述需要,下面首先給出Heisenberg群的一些概念和記號.
Hn上的一個伸縮族為
記Q=2n+2為相應的齊次維數.Hn上的一組左不變向量場為
則Xj,Yj,j=1,2,···,n關于伸縮一次齊次,T關于伸縮二次齊次,且
記
分別為廣義梯度和次Laplace算子.定義距離函數為
定理1C2(Hn)表示由Hn上全體二階連續可微函數組成的集合,令為方程
的非負解,其中h(ξ)為Hn上的非負函數且滿足
Xu[16]引入的向量場采用了復數的形式,具有較強的對稱性,那么對于一些對稱性較弱的算子(如左不變向量場(式(3))的推廣形式Greiner算子),是否可直接推廣呢?本工作結合Xu的思想,采用Heisenberg群上的左不變向量場(式(3)),通過引入一類實泛函從而導出了一些恒等式,并對定理1重新證明.這將為進一步研究以Greiner算子為主部的半線性方程的Liouville性質[12]做充分的準備.
令u≥0滿足方程(4),并記
取??Hn,(?)表示?上全體無窮次可微且具有緊支集的函數全體,?∈C∞0(?),0≤?≤1,考慮非負積分
式中,u=v-k(k/=0),q,k,r待定.為書寫方便,式(8)及以下積分中均省略積分域?和積分微元dxdydt.
由式(7)和(8),可得
將u=v-k代入式(9),并結合左不變向量場的關系,運用分部積分技巧,經過計算可得
另一方面,注意到
結合左不變向量場的關系,運用分部積分可得
式中,
下面對式(15)中的Ⅵ3采用不同的分部積分,分別給出和.一方面,
另一方面,
由式(15)~(17),可得
式中,η待定.
由上節的討論,可得
即有
式中,
首先處理λ5Rhvr+k-pk?q|?Hv|2.注意到
另外,通過分部積分有
由式(20)和(21),可得
把式(22)代入到λ5Rhvr+k-pk?q|?Hv|2中,則式(19)化為
式中,
證明(定理1):選取
如果由式(23)得到v≡0,則可證u≡0.為使式(23)中“=”號的左邊為正,取及 r均非負.由Young不等式
可得
式中,∈為充分小正數,C為某正常數.由?≤1可知,?q-1≤?q-2,從而有
同理可證
因為λ3≥0,λ4≥0,所以0≤η≤(n-1)/(n+2).若r=0,則λ1≤0,將無法控制式(26)和(27)中的正項∈Rvr-2?q|?Hv|4,所以r>0.由λ10=k2r(1-η)可知,λ10/=0,從而有
需λ4>0,從而0<η≤(n-1)/(n+2).由λ9/=0可得
為了控制∈Rvr?q(Tv)2,需要λ3>0,故0<η<(n-1)/(n+2).
通過分部積分,可得
若
則
故在式(33)成立的情況下,
將式(25)~(34)代入式(23),有
選取?={ξ∈Hn:|ξ|≤R,R為任一實數},?∈(?)滿足
再由式(36)和(38),可得
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A Liouville type theorem of semi-linear equations on the Heisenberg group
ZHANG Shu-tao,ZHAO Qiong,HAN Ya-zhou
(College of Sciences,China Jiliang University,Hangzhou 310018,China)
Referring to the method of vector fields,this paper studies a class of semilinear equations on the Heisenberg group and gives a Liouville type theorem,namely,the nonexistence of nontrivial positive solutions.A class of real functional constituted by leftinvariant vector fields on the Heisenberg group is introduced.Some identities are obtained by identical deformation.It is proved that any nonnegative solution is trivial according to the properties of test function and some techniques such as polar coordinates formula on the Heisenberg group and Young inequality.
vector field method;Liouville type theorem;semi-linear equation;Heisenberg group
O 175.2
A
1007-2861(2015)03-0319-12
10.3969/j.issn.1007-2861.2013.07.052
2013-10-29
國家自然科學基金資助項目(11201443);浙江省自然科學基金資助項目(Y6110118)
韓亞洲(1978—),男,副教授,博士,研究方向為橢圓方程的定性理論.E-mail:yazhou.han@gmail.com