A nonlinear Liouville theorem for fractional equations in the Heisenberg group

Abstract

Abstract. We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the subLaplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre [9], as established in [16]. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space H n × R + .

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…and  3R (x) is the ball in R N+1 with radius 3R and its center at the x;  + 3R =  3R ∩ R N+1 + is the upper half ball; and ′  + 3R is the flat part of  + 3R , which is the ball B 3R in R N . For other results of fractional Laplacian equations, please see some works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and reference therein.…”

Liouville type theorem for nonlinear boundary value problem on Heisenberg group

“…We finally introduce the CR inversion onĤ n + to end this section, which will be used later. As in [6]…”

“…Let Σ µ = {(x, y, t, λ) ∈Ĥ n + | t ≥ µ}, T µ = {(x, y, t, λ) ∈ H n + |t = µ} and p µ = (0, 0, 2µ, 0). As in [6], we define the H-reflection function v µ of v with respect to T µ on Σ µ by…”

“…where c is a positive constant depending only on n. In a sequencing paper [6], Cinti and Tan proved the nonexistence of positive solution for problem equation (1.6). More precisely, they proved problem (1.6) possesses no positive cylindrical solutions provided 0 < p <…”

Properties for Nonlinear Fractional SubLaplace Equations on the Heisenberg Group

“…The definition of fractional power subLaplacian in Roncal and Thangavelu [14] is indeed a generalization of Cowling and Haagerup [15] about the Heat semigroup. The fractional power subLaplace equations can also be studied by generalizing the extension method in [1] to H n , for example, see [13,16,17]. Extending the method of moving planes in [3,4,18] to H n , we establish the Liouville type result of the solution to (1.2) on H n + = {ξ ∈ H n |t > 0}, and the symmetric and monotone of the solution to (1.3) on H n .…”