Liouville Type Theorems of Semilinear Equations with Square Sum of Vector Fields

Abstract

Let X_j ; j = 1, …, k, be first order smooth quasi-homogeneous vector fields on Rn with the property that the dimension of the Lie algebra generated by these vector fields is n at x = 0 and X^∗_j = -X_j, j = 1, …, k. Let L = \sum^k_{i=1} X²_i . In this paper, we study the nonnegative solutions of semilinear equation Lu + f(x, u) = 0 (or ≤ 0 ) in Rn and generalized cone domain, respectively, and prove that the solutions must be vanish under some suitable conditions.