Liouville type results for semilinear elliptic equation and inequality on pseudo-Hermitian manifolds

In this paper, we study the semilinear elliptic equation and inequality on pseudo-Hermitian manifolds. In particular, we first obtain a Liouville theorem for the equation ${\mathrm{\Delta }}_{b}u+F(u)=0$ based on a generalized Jerison-Lee’s formula. Next, we prove the nonexistence of a positive solution to the inequality ${\mathrm{\Delta }}_{b}u+F(u)\le 0$ under the volume estimate.