Interval Oscillation Theorems for the Weighted p-Sub-Laplacian Equation in the Heisenberg Group

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Introduction
In this paper, we consider the nonlinear degenerate elliptic equation in the Heisenberg group H n : where p > 1, ðz, tÞ ∈ Ω, Ω is an outer region in H n , ∇ H denotes the Heisenberg gradient (see (19)), and Aðz, tÞ and cðz, tÞ are to be specified later.
In the qualitative theory of nonlinear partial differential equations, one of the important problems is to determine whether or not solutions of the equations are oscillatory. For the second-order linear ordinary differential equation, A classical result of the oscillation is the famous Fite-Wintner theorem which states that if lim t⟶∞ qðtÞ = ∞, then the solutions of (2) are oscillatory (see [1]). In [2], Kamenev studied the oscillatory behavior of the solutions of (2) under the assumption that lim t⟶∞ qðtÞ < ∞.
Soon after, Fite-Wintner's theorem and Kamenev's theorem were extended to various forms of second-order differential equations. In [3], by using the Riccati-type transformation, Noussair and Swanson extended Fite-Wintner's theorem to the equation: Usami [4] established Fite-Wintner-type theorem to the quasilinear elliptic equation in divergence form: Xu [5] and Zhuang and Wu [6] studied, respectively, the oscillation problem for the weighted elliptic equation: For more results about differential equations, one can refer to [7][8][9] and references therein.
It knows that the p-Laplacian equations play a critical role in physical phenomena. We refer the readers to Díaz [10] for detailed references on physical background of the p-Laplacian equations. In this paper, we derive several oscillation criteria for the weighted p-sub-Laplacian equation in H n . One of the difficulties is that there does not exist a good divergence formula in H n as in R n . In this paper, we overcome this difficulty.
Before stating our main results, we introduce some notations and notions. For positive constants a 1 , a 2 , we denote where jðz, tÞj H denotes the norm in H n (see (22)). A domain Ω is called the outer region in H n if there exists a positive constant a 0 such that G½a 0 ,+∞Þ ⊂ Ω. Let us restrict our attention to the nontrivial solution uðz, tÞ of (1), that is, to the solution uðz, tÞ satisfying A nontrivial solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.
Now, we give a definition.
where S H ðo, rÞ denotes the sphere in H n with the center o = ð0, 0Þ and the radius r and dH denotes the 2n-dimensional Hausdorff measure in R 2n+1 (see [11]): where ℘ = 2n + 2 is the homogeneous dimension of H n and ℘α ℘ (α ℘ is a constant) denotes the area of unit sphere S H ðo, 1Þ in H n . One of the main results is the following.
and take in (10), where λ > max ð1, p − 1Þ is a constant. The following is a Kamenev-type oscillation criterion.
The paper is organized as follows. In Section 2, we collect some well-known results for the Heisenberg group and introduce two lemmas. Section 3 is devoted to the proofs of the Riccati-type inequality. The proofs of Theorems 2 and 3 are given in Section 4.

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Advances in Mathematical Physics

Preliminaries
The Heisenberg group H n is R 2n+1 (or C n × R) endowed with the group law ∘ defined by where ξ = ðx 1 , x 2 ,⋯,x n , y 1 , y 2 ,⋯,y n , tÞ ≔ ðx, y, tÞ ≔ ðz, tÞ ∈ R 2n × R, ξ = ð z, tÞ. The group H n plays the important roles as R n in conformal geometry, geometry of several complexes, and harmonic analysis (e.g., see Folland and Stein in [12]). The left invariant vector fields on H n are of the form The family fX 1 , ⋯, X n , Y 1 , ⋯, Y n g satisfies Hörmander's rank condition (see [13]). The Heisenberg gradient of a smooth function u is defined by The divergence of a smooth vector value function F = ðF 1 , ⋯, F 2n Þ on H n is defined by The norm jξj H for ξ ∈ H n is With the norm, the distance between two points ξ and η in H n is defined by where η −1 denotes the inverse of η with respect to ∘, that is, η −1 = −η.
The sphere of radius r > 0 centered at the origin o = ð0, 0Þ of H n is the set: and the open ball of radius r > 0 centered at o is the set: From [11], we know that the area of S H ðo, rÞ is where α ℘ is the volume of B H ðo, 1Þ [14]. For simplicity, we will denote B H ðo, rÞ and S H ðo, rÞ by B r and S r , respectively. Now, we first introduce two well-known lemmas.
Lemma 4 (refer to [15]). If α and β are nonnegative constants and q > 1, then Lemma 5 (the divergence formula in H n [12]). Let Ω 0 be a bounded domain in H n with C 1 boundary ∂Ω 0 and ν denote the unit outward normal to ∂Ω 0 . For any C 1 ðΩ 0 Þ vector field where M = I n is the identity matrix of R n .
The following lemma plays a critical role in proving the Riccati-type inequality. The proof is similar to Theorem 2.3 in [14], and we omit it. Lemma 6. For any C 1 ðB r Þ vector field V = ðV 1 , ⋯, V 2n Þ and p > 1, we have

A Riccati-Type Inequality
In this section, we establish a Riccati-type inequality and then prove two lemmas.

Advances in Mathematical Physics
Using Lemma 7, we have the following.
Lemma 8. Suppose that uðz, tÞ is the nontrivial solution of (1) with uðz, tÞ > 0 for ðz, tÞ ∈ G½c, bÞ and F ∈ Ψ. Let Wðz, tÞ and ZðrÞ be the same as Lemma 7; then, Proof. Changing r to s in (34), multiplying (34) by Fðr, sÞ, and integrating from c to r, we have in view of ðF 1 Þ and Its proof is similar to that of Lemma 8, so we omit it here.

Proofs of the Main Results
The following lemma is useful for proving Theorem 2.
Proof. Suppose that the statement is incorrect; without loss generality, we may assume that there exists a solution uðz, tÞ of (1) such that u z, t ð Þ> 0 for z, t ð Þ∈ G a, b ½ : ð53Þ From Lemmas 8 and 9, it implies that (47) which contradicts to (12). Thus, the claim is true. Now, we give the following.
Proof of Theorem 1. Take the sequence By assumptions, we see that for each i ∈ N, there exist a i , b i , c i ∈ R such that T i ≤ a i < c i < b i and (12) holds. In view to Lemma 10, we conclude that every nontrivial solution u ðz, tÞ of (1) has at least one zero ðz, tÞ in G½a i , b i . By noting jðz, tÞj H ≥ a i ≥ T i , i ∈ N, it follows that every solution has arbitrarily large zeros. Hence, (1) is oscillatory.
As an immediate consequence of Theorem 2, the following result is true.