LIOUVILLE-TYPE RESULTS FOR ELLIPTIC EQUATIONS WITH ADVECTION AND POTENTIAL TERMS ON THE HEISENBERG GROUP

We investigate nonlinear elliptic equations of the form −∆_Hu(ξ) + A(ξ) · ∇_Hu(ξ) = V (ξ)f(u), ξ ∈ Hⁿ, where Hⁿ = (R²n⁺¹, ◦) is the (2n+1)-dimensional Heisenberg group, ∆_H is the Kohn-Laplacian operator, ∇_H is the Heisenberg gradient, · is the inner product in R²ⁿ, the advection term A : Hⁿ → R²ⁿ is a C¹ vector field satisfying a certain decay condition, the potential function V : Hⁿ → (0, ∞) is continuous, and the nonlinearity f(u) has the form −u^−p, p > 0, u > 0, or e^u. Namely, we establish Liouville-type results for the class of stable solutions to the considered problems. Next, some special cases of the potential function V are discussed.