# Interval Oscillation Theorems for the Weightedp-Sub-Laplacian Equation in the Heisenberg Group

Authors: Duan Wu; Pengcheng Niu
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we derive a Riccati-type inequality in the Heisenberg groupHn. Based on it, some oscillation criteria are established for the weighted p-sub-Laplacian equations in Hn. Our results generalize the oscillation theorems for p-sub-Laplacian equations in Rn to ones in Hn.


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## Body

## 1. Introduction

In this paper, we consider the nonlinear degenerate elliptic equation in the Heisenberg groupHn:
(1)∇H·∇Hup−2Az,t∇Hu+cz,tup−2u=0,where p>1, z,t∈Ω, Ω is an outer region in Hn, ∇H denotes the Heisenberg gradient (see (19)), and Az,t and cz,t 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,(2)rtx′t′+qtxt=0.A classical result of the oscillation is the famous Fite-Wintner theorem which states that iflimt⟶∞qt=∞, 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 limt⟶∞qt<∞.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:
(3)∇·Ax∇y+qxfy=0.Usami [4] established Fite-Wintner-type theorem to the quasilinear elliptic equation in divergence form:
(4)∇·∇up−2∇u+pxup−2u=0.Xu [5] and Zhuang and Wu [6] studied, respectively, the oscillation problem for the weighted elliptic equation:
(5)∇·∇up−2Ax∇u+pxup−2u=0.For more results about differential equations, one can refer to [7–9] and references therein.It knows that thep-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 Hn. One of the difficulties is that there does not exist a good divergence formula in Hn as in Rn. In this paper, we overcome this difficulty.Before stating our main results, we introduce some notations and notions. For positive constantsa1,a2, we denote
(6)Ga1,a2=z,t∈Hn:a1<z,tH≤a2,Ga1,a2=z,t∈Hn:a1≤z,tH<a2,Ga1,a2=z,t∈Hn:a1≤z,tH≤a2,Ga1,+∞=z,t∈Hn:z,tH≥a1,wherez,tH denotes the norm in Hn (see (22)). A domain Ω is called the outer region in Hn if there exists a positive constant a0 such that Ga0,+∞⊂Ω. Let us restrict our attention to the nontrivial solution uz,t of (1), that is, to the solution uz,t satisfying
(7)supuz,t:z,t∈Ω>0.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.Definition 1.
Let(8)D0=r,s∈R2:r>s≥a0,D=r,s∈R2:r≥s≥a0.
We call thatF belongs to the function class Ψ, if F∈CD,R and there exist f1,f2∈CD,R and ϕ∈C1a0,+∞,R+ such that
F1Fr,r=0 for r≥a0 and Fr,s>0 for all r,s∈D0
F2∂/∂rFr,s+ϕ′r/ϕrFr,s=f1r,sFr,sp−1/p
F3∂/∂sFr,s+ϕ′s/ϕsFr,s=f2r,sFr,sp−1/pIn this paper, we always assume that the following conditions are satisfied.C1 The coefficient matrix Az,t=aijz,t2n×2n is a real symmetric positive definite matrix function (i.e., it is the ellipticity condition in R2n) with aij∈Cloc2Ω,R, the smallest (necessarily positive) eigenvalue of Az,t is denoted by λminz,t, and there exists a function ρr∈CR+,R+ such that
(9)ρr≤minz,tH=rλminz,tAz,tp/p−1,r>a0,where Az,t=∑i,j=12naij2z,t1/2.C2cz,t∈Cloc2Ω,R.For convenience, denote(10)θr=ϕr∫SHo,rcz,tdH,where SHo,r denotes the sphere in Hn with the center o=0,0 and the radius r and dH denotes the 2n-dimensional Hausdorff measure in R2n+1 (see [11]):
(11)gr=1ρr℘α℘r2n+1ϕr1/p−1,where ℘=2n+2 is the homogeneous dimension of Hn and ℘α℘ (α℘ is a constant) denotes the area of unit sphere SHo,1 in Hn.One of the main results is the following.Theorem 2.
Assume that for anyT≥a0, there exist T≤a<c<b and F∈Ψ such that
(12)1Fc,a∫acFs,aθsds+1Fb,c∫cbFb,sθsds>1pp1Fc,a∫ac4s2+1p/2gp−1sf1s,apds+1pp1Fb,c∫cb4s2+1p/2gp−1sf2b,spds.Then, (1) is oscillatory.Denote(13)Rr=∫a0r14s2+1p/2p−1gsds,forr≥a0,and take
(14)Fr,s=Rr−Rsλ,forr≥s≥a0,ϕr=1,in (10), where λ>max1,p−1 is a constant. The following is a Kamenev-type oscillation criterion.Theorem 3.
AssumeRr⟶∞r⟶∞. If for each l≥a0, there exist λ>max1,p−1 such that
(15)limsupr⟶∞1Rλ−p+1r∫lrRs−Rlλθsds>λpppλ−p+1,(16)limsupr⟶∞1Rλ−p+1r∫lrRr−Rsλθsds>λpppλ−p+1.Then, (1) is oscillatory.The paper is organized as follows. In Section2, 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.


## 2. Preliminaries

The Heisenberg groupHn is R2n+1 (or Cn×R) endowed with the group law ∘ defined by
(17)ξ¯∘ξ=x+x¯,y+y¯,t+t¯+2∑i=1nxiy¯i−yix¯i,where ξ=x1,x2,⋯,xn,y1,y2,⋯,yn,t≔x,y,t≔z,t∈R2n×R,ξ¯=z¯,t¯. The group Hn plays the important roles as Rn in conformal geometry, geometry of several complexes, and harmonic analysis (e.g., see Folland and Stein in [12]).The left invariant vector fields onHn are of the form
(18)Xj=∂∂xj+2yj∂∂t,Yj=∂∂yj−2xj∂∂t,j=1,2,⋯,n,T=∂∂t.The familyX1,⋯,Xn,Y1,⋯,Yn satisfies Hörmander’s rank condition (see [13]). The Heisenberg gradient of a smooth function u is defined by
(19)∇Hu=X1u,⋯,Xnu,Y1u,⋯,Ynu.The divergence of a smooth vector value functionF=F1,⋯,F2n on Hn is defined by
(20)∇H·F=X1F1+⋯+XnFn+Y1Fn+1+⋯+YnF2n.ForF=F1,⋯,F2n+1, the usual divergence divF on R2n+1 is
(21)divF=∂F1∂x1+⋯+∂Fn∂xn+∂Fn+1∂y1+⋯+∂F2n∂yn+∂F2n+1∂t.The normξH for ξ∈Hn is
(22)ξH=x2+y22+t21/4.With the norm, the distance between two pointsξ and η in Hn is defined by
(23)dHξ,η=η−1∘ξH,where η−1 denotes the inverse of η with respect to ∘, that is, η−1=−η.The sphere of radiusr>0 centered at the origin o=0,0 of Hn is the set:
(24)SHo,r=ξ∈Hn:dHξ,o=r,and the open ball of radius r>0 centered at o is the set:
(25)BHo,r=ξ∈Hn:dHξ,o<r.From [11], we know that the area of SHo,r is
(26)SHo,r=℘α℘r2n+1,where α℘ is the volume of BHo,1 [14]. For simplicity, we will denote BHo,r and SHo,r by Br and Sr, respectively.Now, we first introduce two well-known lemmas.Lemma 4 (refer to [15]).
Ifα and β are nonnegative constants and q>1, then
(27)αq−qαβq−1+q−1βq≥0.Lemma 5 (the divergence formula inHn [12]).
LetΩ0 be a bounded domain in Hn with C1 boundary ∂Ω0 and ν denote the unit outward normal to ∂Ω0. For any C1Ω0 vector field V=V1,⋯,V2n, we have
(28)∫Ω0∇H·Vdzdt=∫∂Ω0MV·νdH,where
(29)M=In00In2y−2x2n+1×2n.In is the identity matrix of Rn.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 anyC1Br vector field V=V1,⋯,V2n and p>1, we have
(30)MV·νL1Sr2≤℘α℘r2n+12/p4r2+1VLp/p−1Sr2.


## 3. A Riccati-Type Inequality

In this section, we establish a Riccati-type inequality and then prove two lemmas.Lemma 7 (type inequality).
Suppose thatuz,t is the nontrivial solution of (1) with uz,t>0 for z,t∈Gc,b. Let
(31)v=vz,t=loguz,t,(32)Wz,t=∇Hvp−2Az,t∇Hv,z,t∈Gc,b,(33)Zr=ϕr∫SHo,rMWz,t·νdH,z,t∈Gc,b.
Then,(34)Z′r≤ϕ′rϕrZr−θr−p−1grZrp/p−14r2+1p/21−p.Proof.
It easily knows that(35)Xiv=∂v∂xi+2yi∂v∂t=1u∂u∂xi+2yi1u∂u∂t=1uXiu,and similarly,
(36)Yiv=1uYiu.Hence,
(37)∇Hv=1u∇Hu.
Combining (1) and (32), we have
(38)∇H·Wz,t=∇H·∇Hvp−2Az,t∇Hv=∇H·1up−1∇Hup−2Az,t∇Hu=1up−1∇H·∇Hup−2Az,t∇Hu+∇Hup−2Az,t∇Hu∇H1up−1=1up−1−cz,tup−2u−p−11up∇HuT∇Hup−2Az,t∇Hu=−cz,t−p−1∇Hvp−2∇HvTAz,t∇Hv,that is,
(39)∇H·Wz,t+cz,t+p−1∇Hvp−2∇HvTAz,t∇Hv=0.
By integrating (39) over SHo,rr≥c, it follows
(40)∫SHo,r∇H·Wz,tdH+∫SHo,rcz,tdH+p−1∫SHo,r∇Hvp−2∇HvTAz,t∇HvdH=0.
Using (28) and (10), it implies
(41)Z′r=ϕ′r∫SHo,rMWz,t·νdH+ϕr∫∂SHo,rMWz,t·νdσ=ϕ′rϕrZr+ϕr∫SHo,r∇H·Wz,tdH=ϕ′rϕrZr−θr−p−1ϕr∫SHo,r∇Hvp−2∇HvTAz,t∇HvdH.
ForWz,t in (32), we have
(42)Wz,t=∇Hvp−2Az,t∇Hv≤∇Hvp−1Az,t.It yields
(43)∇Hv≥Wz,tAz,t1/p−1.
In view ofC1, we have
(44)∇Hvp−2∇HvTAz,t∇Hv≥∇Hvp−2λminz,t∇Hv2=λminz,t∇Hvp≥λminz,tWz,tAz,tp/p−1≥ρz,tHWz,tp/p−1.
By (30) and (33), we have
(45)Zrϕr=∫SHo,rMWz,t·νdH≤℘α℘r2n+11/p4r2+11/2∫SHo,rWp/p−1dHp−1/p.
By combining (41), (44), (45), and (11), it yields
(46)Z′r≤ϕ′rϕrZr−θr−p−1ϕrρrZrϕrp/p−1℘α℘r2n+11/1−p4r2+1p/21−p=ϕ′rϕrZr−θr−p−1grZrp/p−14r2+1p/21−p.
The proof is complete.Using Lemma7, we have the following.Lemma 8.
Suppose thatuz,t is the nontrivial solution of (1) with uz,t>0 for z,t∈Gc,b and F∈Ψ. Let Wz,t and Zr be the same as Lemma 7; then,
(47)1Fb,c∫cbFb,sθsds≤Zc+1ppFb,c∫cb4s2+1p/2gp−1sf2b,spds.Proof.
Changingr to s in (34), multiplying (34) by Fr,s, and integrating from c to r, we have in view of F1 and F3 that
(48)∫crFr,sθsds≤−∫crZ′sFr,sds+∫crϕ′sϕsFr,sZsds−p−1∫crFr,sgsZsp/p−14s2+1p/21−pds≤Fr,cZc+∫crf2r,sFr,sp−1/pZsds−p−1∫crFr,sgsZsp/p−14s2+1p/21−pds.
Take(49)α=p−1Fr,sp−1/pZs4r2+11/2gp−1/ps,β=p−1p−1/p4r2+1p−1/2gp−12/psf2r,sp−1pp−1,q=pp−1.Then, by (27), we get
(50)f2r,sFr,sp−1/pZs−p−1Fr,s4s2+1p/2p−1gsZsp/p−1≤4s2+1p/2gp−1sf2r,sppp.
By combining (48), it shows
(51)∫crFr,sθsds≤Fr,cZc+1pp∫cr4s2+1p/2gp−1sf2r,spds.
By lettingr⟶b− and dividing both sides by Fb,c, it follows (47).Lemma 9.
Suppose thatuz,t is the nontrivial solution of (1) with uz,t>0 for z,t∈Ga,c and F∈Ψ. Let Wz,t and Zr be similarly with Lemma 7 for z,t∈Ga,c; then,
(52)1Fc,a∫acFs,aθsds≤−Zc+1ppFc,a∫ac4s2+1p/2gp−1sf1s,apds.
Its proof is similar to that of Lemma8, so we omit it here.


## 4. Proofs of the Main Results

The following lemma is useful for proving Theorem2.Lemma 10.
If there existc∈a,b and F∈Ψ such that (12) holds, then every nontrivial solution uz,t of (1) has at least one zero in Ga,b.Proof.
Suppose that the statement is incorrect; without loss generality, we may assume that there exists a solutionuz,t of (1) such that
(53)uz,t>0forz,t∈Ga,b.
From Lemmas8 and 9, it implies that (47) and (52) hold. Using them, we have
(54)1Fc,a∫acFs,aθsds+1Fb,c∫cbFb,sθsds≤1pp1Fc,a∫ac4s2+1p/2gp−1sf1s,apds+1pp1Fb,c∫cb4s2+1p/2gp−1sf2b,spds,which contradicts to (12). Thus, the claim is true.Now, we give the following.Proof of Theorem 1.
Take the sequence(55)Ti⊂a0,+∞withlimi⟶∞Ti=∞.
By assumptions, we see that for eachi∈N, there exist ai,bi,ci∈R such that Ti≤ai<ci<bi and (12) holds. In view to Lemma 10, we conclude that every nontrivial solution uz,t of (1) has at least one zero z,t in Gai,bi. By noting z,tH≥ai≥Ti,i∈N, it follows that every solution has arbitrarily large zeros. Hence, (1) is oscillatory.
As an immediate consequence of Theorem2, the following result is true.Corollary 11.
If (12) in Theorem 2 is replaced by
(56)limsupr⟶∞∫lrFs,lθs−4s2+1p/2gp−1sf1s,lpppds>0,(57)limsupr⟶∞∫lrFr,sθs−4s2+1p/2gp−1sf2r,spppds>0,for each sufficient large l≥a0; then, (1) is oscillatory.Proof.
For anyT≥a0, let l=T. We choose a=l in (56); then, there exists c>a such that
(58)∫acFs,aθs−4s2+1p/2gp−1sf1s,apppds>0.
By lettingc=l in (57), there exists b>c such that
(59)∫cbFb,sθs−4s2+1p/2gp−1sf2b,spppds>0.
By combining (58) and (59), it yields (12). The conclusion is proven from Theorem 2.Proof of Theorem 2.
From Definition1 and the definitions of Fr,s and Rr, we get
(60)f1r,s=λRr−Rsλ−p/p14r2+1p/2p−1gr,f2r,s=−λRr−Rsλ−p/p14s2+1p/2p−1gs.Thus, it follows
(61)∫lr4s2+1p/2gp−1sf1s,lpds=∫lrλpRs−Rlλ−p14s2+1p/2p−1gsds=λpλ−p+1Rr−Rlλ−p+1,∫lr4s2+1p/2gp−1sf2r,spds=∫lrλpRr−Rsλ−p14s2+1p/2p−1gsds=λpλ−p+1Rr−Rlλ−p+1.
Notinglimr⟶∞Rr=∞, we see
(62)limr⟶∞1ppRλ−p+1r∫lr4s2+1p/2gp−1sf1s,lpds=λpppλ−p+1,(63)limr⟶∞1ppRλ−p+1r∫lr4s2+1p/2gp−1sf2r,spds=λpppλ−p+1.
By combining (63) and (15), it yields
(64)limsupr⟶∞1Rλ−p+1r∫lrRs−Rlλθs−1pp4s2+1p/2gp−1sf1s,lpds=limsupr⟶∞1Rλ−p+1r∫lrRs−Rlλθsds−λpppλ−p+1>0,which implies (56). Similarly, (57) holds by combining (63) and (16). From Corollary 11, (1) is oscillatory.



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