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  quations 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.