Heisenberg群上一类半线性方程的Liouville型定理  

A Liouville type theorem of semi-linear equations on the Heisenberg group

作　　者：张书陶[1] 赵琼[1] 韩亚洲[1] 

机构地区：[1]中国计量学院理学院,杭州310018

出　　处：《上海大学学报（自然科学版）》2015年第3期319-330,共12页Journal of Shanghai University:Natural Science Edition

基　　金：国家自然科学基金资助项目(11201443);浙江省自然科学基金资助项目(Y6110118)

摘　　要：结合向量场法的思想,研究了Heisenberg群上的一类半线性方程,并给出不存在非平凡正解的Liouville型定理.首先,利用Heisenberg群上左不变向量场的对称性构造一类实泛函,并通过恒等变形获得一些恒等式;然后,利用试验函数的性质,结合Heisenberg群上的极坐标公式、Young不等式等技巧以精确估计,进而证明任一非负解均恒为零.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.

关 键 词：向量场法 LIOUVILLE型定理 半线性方程 HEISENBERG群 

分 类 号：O175.2[理学—数学]