Liouville type theorem in the Heisenberg group with general nonlinearity
The author studies the nonexistence of positive solutions of a semilinear elliptic equation in the Heisenberg group with general nonlinearity. The main method used is the moving plane one which goes back to Alexandrov and was first applied to the study of nonlinear differential equations by \textit{J. B. Serrin} [Contrib. Nonlin. Functional Analysis, proc. Sympos. Univ. Wisconsin 1971, 565--601 (1971; Zbl 0271.35004)]. The moving plane method has been widely used in many different problems, both in bounded and unbounded domains. Although the Heisenberg Laplacian satisfies the maximum principle, it is not invariant by the usual reflection which respect to a hyperplane. In order to overcome this difficulty, the author introduces a new notion of reflection and only studies the cylindrical solution. This is different from the Euclidean case. The author also conjectures that any nonnegative solution of the considered semilinear equation in the Heisenberg group is cylindrical symmetric.
scientific article; zbMATH DE number 6143253