Properties of solutions to fractional p-subLaplace equations on the Heisenberg group

The aim of this paper is to study properties of solutions to the fractional p-subLaplace equations on the Heisenberg group. Based on the maximum principles and the generalization of the direct method of moving planes, we obtain the symmetry and monotonicity of the solutions on the whole group and the Liouville property of solutions on a half space.


Introduction
be the fractional p-subLaplacian on the Heisenberg group H n , where 0 < s < 1, Q = 2n + 2, C Q,s is a positive constant, and PV is the Cauchy principal value. In this paper we study the properties of cylindrical solutions to the fractional p-subLaplace equation where 2 ≤ p < ∞.
Recall that the fractional Laplacian in R n is a nonlocal pseudodifferential operator defined by (-) α u(x) = C n,α lim ε→0 R n \B ε (x) u(x)u(y) |x -y| n+2α dy, (1.3) where 0 < α < 1, C n,α is a constant, and u belongs to the Schwartz space. Since the nonlocal property of the operator (-) α brings new difficulties to investigate, Caffarelli and Silvestre in [4] developed the extension method which can reduce the nonlocal problem relating to (-) α to a local one in higher dimensions. This method has been applied to deal with equations involving the fractional Laplacian, and fruitful results have been obtained, see [3] and the references therein. Chen et al. [7] developed a direct method of moving planes to handle the problem involving (-) α for 0 < α < 1, and this direct method has been used successfully to study symmetry, monotonicity, and nonexistence for many fractional Laplace equations, see [6,7] and the references therein. Recently, Chen and Li [6] considered the fractional p-Laplacian and obtained the radial symmetry and monotonicity of solutions to the equations involving operator (1.4).
To the elliptic equation in R n , Li and Ni [19] proved that the positive solutions to (1.5) are radially symmetric with the assumptions that the limit of u is zero at the infinity and g ≤ 0 if u is sufficiently small. Under the same conditions, the authors in [6] extended the result in [19] to the fractional p-Laplace equation and got the radial symmetry and monotonicity of the solutions. They also pointed out that the fractional p-Laplacian becomes p-Laplacian as α → 1 and, furthermore, it reduces to -when p = 2. There are many interesting results about subLaplace and p-subLaplace equations on the Heisenberg group (see [13,15,17,18] and [10,11,20,21,[25][26][27]). There have been several different definitions of the fractional power subLaplacian in H n (see [12,14,22] etc.). The definition of fractional power subLaplacian given by Roncal and Thangavelu in [22] is indeed a generalization of the definition given by Cowling and Haagerup in [9] about the heat semigroup. The fractional power subLaplace equations can also be studied by generalizing the extension method in [4] to H n , although the fractional power subLaplacian (-H ) s (0 < s < 1) does not have the concrete integral expression, for example, see [14] and [8] for s = 1 2 . There are also some results of the fractional power subLaplacian which are the extension of [8], see [23,24]. Note that the expression of fractional power subLaplacian on H n (see [22]) is the special form of fractional p-subLaplacian (1.1). By extending the method of moving planes in [5][6][7] to H n , in this paper, we study the properties of the solutions to (1.2) on H n and H n + = {ξ ∈ H n | t > 0}. Our main results are the following.  9) and suppose that u satisfies (1.8) and is lower semicontinuous onH n + . If f (0) = 0, f (a) is nonpositive and locally bounded for a sufficiently small, then u ≡ 0.
Observe that Theorem 1.1 is the extension of symmetry and monotonicity of solutions to the fractional p-Laplace equation on R n in [6] to the Heisenberg group, and Theorem 1.2 is the Liouville property on a half space in H n . When f (a) = -a + a q (q > 1), our results still hold.
The authors in [22] The paper is organized as follows. Section 2 collects some well-known results on H n , and we show that (1.1) is well defined for u ∈ L sp (H n ) ∩ C 1,1 loc (H n ). In Sect. 3, we establish three maximum principles. Theorem 1.1 and Theorem 1.2 are proved in Sect. 4.

Preliminaries
The Heisenberg group H n is the Euclidean space R 2n+1 (n ≥ 1) endowed with the group law •:ξ The left invariant vector fields corresponding to H n are It is easy to check that X i and Y j satisfy The Heisenberg gradient of a function u is defined by and the subLaplacian H on H n is The family {X 1 , . . . , X n , Y 1 , . . . , Y n } satisfies Hörmander's rank condition (see [16]) which implies that H is hypoelliptic and the maximum principle holds for solutions to the equation involving H (see [2]). The integer Q = 2n + 2 is called the homogeneous dimension of H n . Denote by |ξ | H the distance from ξ to the zero (see [13]) Authors in [22] used the norm |(z, w)| = ( n i=1 (x i 2 + y i 2 ) 2 + 16t 2 ) 1 4 for (x, y, t) := (z, w) ∈ H n , which is equivalent to (2.5). The distance between two points of H n is defined by It is well known that ξ → |ξ | H is homogeneous of degree one with respect to δ κ and where | · | denotes the Lebesgue measure.
where ε is sufficiently small. Noting that u ∈ C 1,1 loc (H n ), Q+sp-p < Q, and Q+sp-2p+2 < Q, we know that I 1 and I 2 are finite; I 3 is clearly convergent when |ξ | H → ∞; and I 4 is finite from u ∈ L sp (H n ). Hence, (1.1) is well defined.

Maximum principles
In this section, we prove three maximum principles which will be used in the process of moving planes. These maximum principles are on a bounded domain in H n , on a bounded domain in the left domain of some hyperplane, and on a narrow region.

Lemma 3.1
Let Ω be a bounded domain in H n . Assume u ∈ L sp (H n ) ∩ C 1,1 loc (H n ) is lower semicontinuous onΩ and satisfies These conclusions also hold on the unbounded region Ω if we further assume that Proof Suppose that (3.2) is not true, then by the lower semicontinuity of u onΩ there exists ξ 0 ∈Ω such that From (3.1), we know that ξ 0 is a point in Ω, and which contradicts (3.1). This implies (3.2). If there exists some point ξ 0 ∈ Ω such that u(ξ 0 ) = 0, then Using u(ξ ) ≥ 0, we have u(ξ ) = 0 almost everywhere in H n . For an unbounded region Ω, the condition lim |ξ | H →∞ u(ξ ) ≥ 0 implies that the negative minimum ξ 0 of u cannot be reached at infinity. Then the condition of lower semicontinuity ensures that the proof can go on as above. The proof is ended.
Set w λ (ξ ) = u λ (ξ )u(ξ ). These conclusions also hold for the unbounded region Ω if we further assume that

Lemma 3.2 Let Ω be a bounded domain in Σ λ . Assume that the cylindrical function u ∈ L sp (H n ) ∩ C 1,1 loc (H n ) is lower semicontinuous onΩ and satisfies
Proof Suppose that (3.4) is incorrect. By the lower semicontinuity of w λ onΩ, there exists ξ 0 ∈Ω such that For simplicity, we denote G(a) = |a| p-2 a, a ≥ 0.
From the antisymmetry of w λ , w λ (ξ ) = 0 almost everywhere in H n .

Lemma 3.3
Let Ω be a bounded narrow domain in Σ λ and locate in {ξ | λl < t < λ} for small l. Assume that the cylindrical function u ∈ L sp (H n )∩C 1,1 loc (H n ) is lower semicontinuous onΩ. If c(x) is bounded from below in Ω and u satisfies Furthermore, if w λ = 0 at some point in Ω, then w λ (ξ ) = 0 almost everywhere in H n .
These conclusions also hold for the unbounded region Ω if we further assume that Proof By the proof of Lemma 3.2, we have Obviously, Similar to (3.6), we know Combining (3.10), (3.11), and I 1 < 0, we have Noting that ξ 0 is a negative minimum of w λ , we infer ∇w λ 0 (ξ 0 ) = 0, and so Now (3.12) and (3.13) contradict (3.8), and then (3.9) is proved.

Proof of the main results
Following the idea in [6], we first use Lemma 3.1, Lemma 3.2, and Lemma 3.3 to prove Theorem 1.1.
The above result provides the starting point of moving planes. Let us move the plane T λ to the right as long as (4.1) holds to its limiting position We will show that λ 0 = 0 (4.5) i.e., In fact, suppose that (4.6) is false, we have by Lemma 3.2 that From the definition of λ 0 , there exist a sequence λ k → λ 0 and a point ξ k ∈ Σ λ k such that If |ξ k | H is sufficiently large, then u(ξ k ) is small and so ς λ k (ξ k ) is also small, this implies f (ς λ k (ξ k )) ≤ 0 (because f (a) ≤ 0 for the sufficiently small a). It follows But this contradicts the fact that ξ k is a negative minimum of w λ k (see Lemma 3.2). Hence, {ξ k } is bounded, i.e., the sequence {ξ k } is bounded.
Next, we give the proof of Theorem 1.2.