关键词: Heisenberg 群, Liouville 型定理, 半线性方程, 向量场法

Abstract: 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.

Key words: Heisenberg   group, Liouville type theorem, semi-linear equation, vector field method