Generalised Picone's identity and some Qualitative properties of p-sub-Laplacian on Heisenberg groups

In this article, we derive a generalised nonlinear Picone's identity for p sub-Laplacian on the Heisenberg group. As an application of Picone's identity, we prove a Hardy type inequality and Picone's inequality. We also establish some qualitative results involving the system of nonlinear equations involving p-sub-Laplacian.


Introduction
It is well known that Picone type identities play an important role in the study of qualitative properties of elliptic partial dierential equations. The classical Picone's identity [25] is as follows: If u ≥ 0 and v > 0 are suciently smooth functions, then For some of the applications of this identity, we refer to [1,2,3,22] and the references cited therein. W. Allegretto and Y.X. Huang [4] obtained Picone's identity for p-Laplace equations. Their identity is as follows: J. Tyagi [26] generalised (1) and proved the following nonlinear Picone type identity: where f (y) = 0, ∀ 0 = y ∈ R and α > 0 is such that f (y) ≥ 1 α , ∀ 0 = y ∈ R. K. Bal [5] established a nonlinear Picone's identity for p-Laplace operators. They showed that for all y. T. Feng [14] further generalised Picone's identity for p-Laplace equations as follows: where v > 0, u ≥ 0, g(u) and f (v) satisfy that for p > 1, q > 1, 1 For some interesting Picone type identities and related results in Euclidean domains, we refer to [6,11,12,15,18,19,28].
Research works available for Picone type identities in Heisenberg group are not as exhaustive as it is in the case of Euclidean domain. Niu et al. [24] obtained Picone's identity for p-sub-Laplacian in bounded domains of Heisenberg group. Their identity is as follows: For some further results involving Picone's identity and its applications on the Heisenberg groups, we refer to [16,17,20,21,27] and references therein. For a nonlinear Picone identity for biharmonic operator on the Heisenberg group, see [13].
Motivated by the above research works, aim of this article is to prove a nonlinear analogue of Picone's identity for p-sub-Laplacian on the Heisenberg group. Our main result is stated below: Let us denote Remark 1.1. If we choose f (s) = s p and g(s) = s p−1 , then our result reduces to (4).
The article is organized as follows: In Section 2, we recall some brief results on the Heisenberg group. Section 3 deals with the proof of Theorem 1.1. In section 4, we discuss some applications of the Theorem 1.1.

Preliminaries
In this section, we present some denitions related to the Heisenberg group. The Heisenberg group , is a non-commutative group equipped with the product where x 1 , y 1 , x 2 , y 2 ∈ R n , t 1 , t 2 ∈ R and ·, · is the usual scalar product in R n . With this operation H n is a Lie group and the Lie algebra of H n is generated by the left-invariant vector elds The norm on H n is given by The distance between ξ = (z, t) and ξ = (z , t ) on H n is dened as follows: The Heisenberg gradient is dened as and hence the Heisenberg Laplacian is dened as The p-sub-Laplacian is dened as Denition 2.1 (S 1,p (Ω) and S 1,p 0 (Ω) Space). For an open subset Ω ⊆ H n and 1 < p < ∞, we dene S 1,p (Ω) = {u : Ω → R such that u, |∇ H n u| ∈ L p (Ω)}.
The space S 1,p (Ω) is equipped with the norm By S 1,p 0 (Ω), we denote the closure of C ∞ 0 (Ω) with respect to the norm For further details on Heisenberg group, see [7,9].

Proof of Theorem 1.1
It is easy to see that On using (9), we obtain Next, we show that L(u, v) ≥ 0. Let q be conjugate of p, i.e., 1 p + 1 q = 1. Then Now, we will show that T i ≥ 0, i = 1, 2, 3. Let us recall Young's inequality where 1 p + 1 q = 1. Equality in (10) holds if and only if a p = b q . On choosing a = |∇ H n u| and b = f (u)|∇ H n v| p−1 pg(v) in (10), we obtain This shows that T 1 ≥ 0.
Finally, we need to show that if L(u, v) = 0 then (6), (7) and (8) are satised. If L(u, v) = 0, then and From (11) and equality case of (10), we obtain which gives (7). It is easy to see that (12) implies ( where h ∈ L ∞ (Ω) is a nonnegative weight function. Let 0 ≤ u ∈ S 1,p 0 (Ω) and f (u) ∈ S 1,p 0 (Ω). Further, if f and g satisfy conditions of Theorem 1.1, we have Proof. Let K be a compact subset of Ω and 0 ≤ φ ∈ C ∞ 0 (Ω). By Theorem 1.1, As φ tends to u, we obtain (14). Theorem 4.2. Suppose that h 1 (x) and h 2 (x) are continuous functions such that h 1 (x) < h 2 (x) on Ω ⊂ R n . If f and g satisfy conditions of Theorem 1.1 and there exists u ∈ C 2 (Ω) such that in Ω, Then any nontrivial changes sign.
Proof. Assume that v does not change sign, then which is a contradiction. This completes the proof.  v) in Ω, in Ω, Proof. For any φ 1 , φ 2 ∈ S 1,p 0 (Ω), On On using (20) and (21), we get On applying Theorem 1.1, we get |∇ H n u| = f (u) in Ω.
Next, we prove a generalised Picone type inequality in the spirit of [10].
Let Ω be a bounded domain in H n and f, g satisfy the conditions in Theorem 1.1. Let 0 ≤ u ∈ S 1,p 0 (Ω), and 0 ≤ v ∈ S 1,p 0 (Ω) be such that −∆ H n v ≥ 0 is a bounded Radon measure. We further assume that v ≡ 0 in Ω and v = 0 on ∂Ω. Then Proof. Since v ≥ 0 and v = 0 on ∂Ω, therefore by strong maximum principle [8] either v > 0 or v ≡ 0 in Ω.
Since v ≡ 0 in Ω, v > 0 in Ω. Let v m (ξ) = v(ξ) + 1 m , then −∆ H n v m = −∆ H n v and v m → v in S 1,p (Ω) and almost everywhere. Now, we consider 0 ≤ u ∈ S 1,p 0 (Ω), then there exists a sequence {u n } in C ∞ 0 (Ω) such that u n ≥ 0 for each n and u n → u in S 1,p 0 (Ω). By using Theorem 1.1, we obtain Fatou's lemma and Lebesgue dominated convergence theorem implies that as n, m → ∞, we obtain This completes the proof.