A Liouville type theorem to an extension problem relating to the Heisenberg group

We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.


1.
Introduction. We mention that there have been many works for the symmetry property about the solutions of the Laplace equations with the Neumann boundary in a half space, see [1,17,32] and the references therein, and then Lou and Zhu [31] established classifications of nonnegative solutions to the Neumann problem on the upper half space R n + of R n    ∆u = 0, u ≥ 0 in R n + , ∂u ∂t = u p , on ∂R n + , where R n + = {(x , t)|x = (x 1 , x 2 , · · · , x n−1 ) ∈ R n−1 , t > 0} (n ≥ 2) and p > 1, by providing some technical lemmas and using the method of moving planes.
Chen and Zhu in [15] considered the extension problem    div(y 1−α ∇w) = 0, (x, y) ∈ R n × (0, ∞), − lim y→0 y 1−α ∂w ∂y = w p (x, 0), x ∈ R n for the fractional Laplace equation where 0 < α < 2 and p > 1, and obtained classifications of nonnegative solutions to the extension problem by extending the method in [31] to the fractional Laplacian case. Here the fractional Laplacian (−∆) α/2 is a nonlocal pseudodifferential operator in R n defined by where 0 < α < 2, C n,α is a constant and u belongs to the Schwartz space. The nonlocality of (−∆) α/2 makes it difficult to investigate. To overcome this difficulty, Caffarelli and Silvestre in [9] introduced the extension method to reduce the nonlocal problem to a local one in higher dimensions. The fractional Laplacian has attracted much attention recently. It has extensive applications in mathematical physics, mathematical finances and so on. Especially, it appears in turbulence and water wave, anomalous dynamics, chemical reactions in liquids, population dynamics, geophysical fluid dynamics, and American options in finance. It also connects to conformal geometry. The Heisenberg group H n plays the same role as R n in conformal geometry, as shown by Folland and Stein in [20], hence, the fractional CR covariant operator on H n is also received close attention.
Motivated by [9], Frank, Gonzalez, Monticelli and Tan [21] proved that the equation on the Heisenberg group H n P α/2 u = −u p corresponding to the fractional CR covariant operator P α/2 on H n can be extended to the Dirichlet Neumann problem onĤ n where c α is a constant, 0 < α < 2, u : H n → R, U :Ĥ n + → R, λ > 0, p > 1. In this paper, we establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem (1.1). The function U ∈ C 2 (Ĥ n + ) ∩ C 1 (Ĥ n + ) is called the cylindrical solution to (1.1), if U (x, y, t, λ) = U (r 0 , t, λ) = U (|(x, y)| , t, λ), for (x, y, t, λ) ∈Ĥ n + , r 0 = (|x| Our main result is the following For convenience, we use the notation The idea of proofs for theorems comes from [15] and [31]. Noting that the structures of H n and R n are different, we use the H-reflection on H n introduced by Birindelli and Prajapat [4] (see (4.4) below). Since the subLaplacian ∆ H n is invariance about the H-reflection, we can move planes on H n to reach our aim. The more content for the method of moving planes the readers may refer to [12,13,33].
The paper is organized as follows. Some well known results for H n and ∆ H n are collected in Section 2. The generalized CR inversion of cylindrical solutions is given in Section 3. In section 4, we prove Theorem 1.2 by considering two cases: (1) the supercritical case and (2) the subcritical and critical case, and use the method of moving planes. Theorem 1.1 is obtained from Theorem 1.2.

XINJING WANG, PENGCHENG NIU AND XUEWEI CUI
Denote by |ξ| H n the distance from ξ to the zero (see [20]): The distance between two points in H n is defined by The open ball of radius R > 0 centered at ξ is the set By the dilations δ κ of the group, ξ → |ξ| H n is homogeneous of degree one and where |·| denotes the Lebesgue measure, Q = 2n + 2 is the homogeneous dimension of H n . Existence and uniqueness of solutions to the extension problem of the operator P α/2 were given in [21]. More information about the fractional CR covariant operator P α/2 (0 < α < 2) can be referred to [21]. We will consider cylindrical solutions to the extension problem (1.1) for the equation P α/2 = −u p on the extended spacê where z = (x 1 , · · · , x n , y 1 , · · · , y n , t, λ) := (x, y, t, λ) ∈ R n × R n × R n × R,z = (x,ȳ,t,λ), λ ∈ R. Denote by |z|Ĥ n the distance from z ∈Ĥ n to the origin: The distance between z andz onĤ n is of the form where z −1 is the inverse of z with respect to•. When λ =λ = 0, we have dĤ n (z,z) = d H n (ξ,ξ), for z = (x, y, t, 0),z = (x,ȳ,t, 0), ξ = (x, y, t),ξ = (x,ȳ,t). The open ball of radius R centered atz is the set For any z 0 ∈ H n × {0}, we denote For any z = (x, y, t, λ) ∈Ĥ n , denote Proof. Clearly, By (2.7), we arrive at Lemma 2.2 ([7, 10]). Let V be a bounded domain inĤ n , Z a smooth vector field on V , and a(z) a smooth nonnegative function.
Then u ≥ 0 in V .

Lemma 2.3 ([33]
). For a domain V inĤ n , let P 0 ∈ ∂V satisfy the interior Heisenberg ball condition (see [10], where ν is the outer unit normal to ∂V at P 0 . If c 1 (z) = 0, then the above conclusion is also valid when we drop the assumption U (P 0 ) = 0.
3. Generalized CR inversion of cylindrical solutions. We recall the generalized CR inversion onĤ n + , see [10]. For any (x, y, t, λ) ∈Ĥ n + , let then the generalized CR inversion of function U is given by The main result of this section is the following Proof. Note that v is also a cylindrical function. We first compute And by (3.1), Therefore, we have 1 λ and c = 0. Hence, Next, in view of the generalized CR inversion, the boundary condition in (

4.
Proof of the main results. In this section, we prove Theorem 1.2 at first. To do so, let us consider the property of solutions to (1.3) in the supercritical case. The subcritical and critical case can be lifted to a suitable higher dimension space which allows us to use the results for the supercritical case. This will enable us to prove Theorem 1.1.
Proof of Theorem 1.2. We will treat two case: (1) the supercritical case p > Q+α Q−α and (2) the subcritical and critical case 1 < p ≤ Q+α Q−α , respectively, and give the expressions of the cylindrical solutions to (1.3).
Case 1: p > Q+α Q−α . Since the cylindrical solution U (z) to (1.3) does not own the decay at infinity, we apply Lemma 3.1 to see where β = p(Q−α)−(Q+α). The first purpose is to get some symmetry properties of v by using the method of moving planes; then we show that v is independent of t and U (z) is also independent of t. The expression of U (z) can be obtained from the result in [15].
Similarly to the proofs of Propositions 4.2 and 4.3, we can also move the plane from the right to the left. Then it has claimed that v(|(x, y)| , t, λ) is symmetric with respect to t = 0. The above process is based on that the origin ofĤ n + is the center of the generalized CR inversion. By letting any point onĤ n + be the center of the generalized CR inversion and repeating the previous proof, we imply that v(|(x, y)| , t, λ) is symmetric with respect to any point on the t axis and then is independent of t, hence U (z) is also independent of t. Now we can go back to the case in [15] and have by using the result in [15] that where b > 0, a = b p . Case 2. 1 < p ≤ Q+α Q−α . To handle this case, we will lift the dimension of the spaceĤ n + and reduce Case 2 (including critical and subcritical cases) to Case 1 (the supercritical case). Concretely, we choose a positive integer m so large that p > Q+2m+α Q+2m−α and set w(x 1 , · · · , x n , x n+1 , · · · , x n+m , y 1 , · · · , y n , y n+1 , · · · , y n+m , t, λ) =U (x 1 , · · · , x n , y 1 , · · · , y n , t, λ).  By the choice of m, it sees that p is a supercritical exponent. Applying the result to Case 1, we know that w is independent of (x 1 , · · · , x n+m , y 1 , · · · , y n+m , t) and finish the proof of Theorem 1.2.