Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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On the Oscillation of a Class of Conformable Schrodinger Equations 

 
1 N. Sasikala, 2 V. Sadhasivam 

1 Post Graduate and Research Department of Mathematics, Thiruvalluvar Government  

Arts College , Rasipuram - 637 401, Namakkal Dt. Tamil Nadu, India,  

E-mail :krishivyattish@gmail.com. 

2Associate Professor & Head, Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts 

College , Rasipuram - 637 401, Namakkal Dt. Tamil Nadu, India. 

E-mail ovsadha@gmail.com. 

 

Article History: 

Received: 20-04-2024 

Revised: 12-06-2024 

Accepted: 22-06-2024 

Abstract 

In this article, we have derived a new oscillation criteria for a class of conformable 

Schrodinger equations. Based on the generalized  Riccati technique, the results were 

obtained here. Also we have extended the Hartman-Winter oscillation criteria to 

conformable Schrodinger equation. 

Keywords:. Oscillation, Conformable Schrodinger equation, Elliptic partial 

differential equation. 

AMS Subject Classification: 34A08, 34A34, 34K11, 35B05, 35R20. 

 

1. INTRODUCTION 

The area of research that has grown the fastest in recent years is differential equations in fractional 

calculus. Although there are various fractional derivative notions, including Riemann-Liouville and 

Caputo fractional derivatives, which are based on singular integrals and non-locality, they are 

commonly utilized. In 2014, Khalil et al. [13] developed the conformable fractional derivative, 

which was based on a limit concept similar to that of integer order derivatives. 

The Conformable derivative of Khalil was quickly made general by Katugampola fractional 

derivative or alpha fractional derivative [11, 12]. It has wide application in biophysics, quantum 

mechanics, wave theory and polymers and it is a crucial tool for simulating a variety of physical 

phenomena, including electromagnetic waves and viscoelastic systems [9, 14]. Numerous studies 

have been done in the literature on the oscillation of conformable fractional differential equations [2, 

4, 8, 17]. In mathematically oriented sciences like physics and engineering, conformable partial 

differential equations are widely encountered [19, 20]. For instance, they form the basis of current 

scientific understanding of diffusion, electrostatics, materials, dynamical theory, hydrodynamics, 

electrodynamics, viscoelasticity and quantum mechanics. Moreover, fractional partial differential 

equations have gained popularity in recent years as a tool for mathematical modelling. The 

oscillation of conformable partial differential equations has been researched by numerous authors, 

see [5, 6]. 

The concept of elliptic equation has undergone an important growth over the last two centuries. 

Together with electro statistics heat and mass diffusion, hydrodynamics and many other applications 

it has become one of the most richly enhanced field of mathematics. Numerous authors have been 



Communications on Applied Nonlinear Analysis 

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motivated by the oscillation theory of elliptic equations in recent years [1, 7]. Over the past few 

years, there has been a lot of attention paid to the issue of oscillation and non-oscillation of 

differential equation solutions [10, 18]. 

A linear partial differential equation called the Schrodinger equation controls how a quantum 

mechanical system behaves in terms of its wave function. The Schrodinger equation is the 

cornerstone of quantum mechanics, the study of microscopic events. The Schrodinger equation, 

created in 1926 by the Austrian scientist Erwin Schrodinger, is as essential to understanding quantum 

mechanics as Newton's laws of motion are to understanding large-scale classical mechanics 

occurrences. Later, Noussair [21] used the n-dimensional Emden-Fowler method to generate 

solutions to the nonlinear Schrodinger equation in the outer domain Bn. E.Muller- Pfeiffer [22] also 

obtained oscillation criteria for the Schrodinger equation in sobolev space and generalized the 

derivative of order 2 summable on every compact subdomain of G. By using the Emden-Fowler 

equation, Hirashi Onose [23] has explored some conclusions on the sublinear Schrodinger equation. 

Swanson [24] used a modified sublinear hypothesis to build the article in an emden-Fowler type 

sublinear equation. Zhang [25] has introduced unique standards for the absence of positive results by 

using the Perturbed Schrodinger equation. 

In this paper, we are concerned with conformable elliptic equations is of the type 

Δ𝑥
𝛼𝑢 + 𝑝(𝑥)𝑢 = 0, ⬚Δ𝑥

𝛼𝑢 = ∑  𝑛
𝑖=1

∂2𝛼𝑢

∂𝑥𝑖
2𝛼                                           (1.1) 

where 𝛼 ∈ (0,1), 𝑥 = (𝑥1, 𝑥2, … 𝑥𝑛), Δ𝑥
𝛼 is the conformable nabla operator and 𝑝(𝑥) : ℝ𝑛 → ℝ is 

potential function and each compact subset of Ω. 

Define the set 

Ω(𝑎) = {𝑥 ∈ ℝ𝑛: 𝑎 ≤ 𝑟},

Ω(𝑎, 𝑏) = {𝑥 ∈ ℝ𝑛: 𝑎 ≤ 𝑟 ≤ 𝑏},

𝑆(𝑎) = {𝑥 ∈ ℝ𝑛: 𝑟 = 𝑎},

 

𝑢(𝑥): Ω → ℝ is almost always absolutely continuous in 𝛼-fractional derivative of compact subsets 

that fulfills equation (1.1) on Ω is almost everywhere. 

A constrained area 𝐺 ⊂ Ω is a nodal domain for (1.1) if there exists a nontrivial function u ∈

𝐶2(𝐺; ℝ) ∩ 𝐶(𝐺‾; ℝ) such that 1.1 is equal to zero and u = 0 on ∂𝐺. If for each r > 0 equation (1.1) 

has a nodal domain contained and enclosed in Ω𝑟 = Ω ∩ {𝑥 ∈ ℝ𝑛: |𝑥| > 𝑟}, then equation (1.1) is 

called nodally oscillatory. If the function f(x) has zero outside of arbitrary ball in ℝ𝑛 that is centered 

in the origin, is said to be oscillatory; if not, it is said non-oscillatory. We get the function P(t) form 

the Hartman-Winter Theorem 

𝑃(𝑟) =
1

𝑟𝛼 ∫  
𝑟

1
∫  

Ω(1,𝑟)
𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥𝑑𝛼𝑟                                                 (1.2) 

We distinguish two cases, 

(i) There is a finite limit 

lim
𝑟→∞

 𝑃(𝑟) = 𝑃0                                                                                           (1.3) 



Communications on Applied Nonlinear Analysis 

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(ii) The (1.3) condition is fails to hold and lim inf𝑟→∞  𝑃(𝑟) > −∞. 

To my knowledge, aware, there is no literature exists on the oscillation of the conformable elliptic 

equation. Inspired by Robert Marik [16] and Lomtatidze [3, 15] we investigating the following 

conformable elliptic equation of the form 

𝑀(𝑟) = 𝑟 (𝛼𝑃𝑜 − ∫  
Ω(1,𝑟)

  𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥)                      (1.4),

𝑁(𝑟) =
1

𝑟
∫  

Ω(1,𝑟)

  𝑟3−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥.                                            (1.5)

 

2. PRELIMINARIES 

In order to make our method clear, We provide certain fundamental definitions, properties and 

lemmas 

Definition 2.1. Given 𝑢: [0, ∞) → ℝ. Conformable fractional derivative of 𝑢 of order 𝛼 is given by 

𝑇𝛼(𝑢)(𝑥) = lim
𝜖→0

 
𝑢(𝑥 + 𝜖𝑥1−𝛼) − 𝑢(𝑥)

𝜖
⬚∀𝑥 > 0, ⬚𝛼 ∈ (0,1). 

If 𝑢 can be 𝛼-differentiable in some (0, 𝑎), 𝑎 > 0 and lim𝑥→0+  𝑢𝛼(𝑥) exists, then we define 

𝑢𝛼(0) = lim
𝑥→0+

 𝑢𝛼(𝑥). 

Definition 2.2. 𝐼𝛼
𝑎(𝑢)(𝑥) = 𝐼1

𝛼(𝑥𝛼−1)(𝑢) = ∫
𝑎

𝑥
 
𝑢(𝑥)

𝑥1−𝛼 𝑑𝑥, where the integral is the standard Riemann 

improper integral and 𝛼 ∈ (0,1). 

Properties 2.1. Let 𝛼 ∈ (0,1] and at some point 𝑥 > 0 . 𝑢and 𝑣 will eventually be 𝛼 differentiable. 

Then, 

(1) 𝑇𝛼(𝑎1𝑢 + 𝑎2𝑣) = 𝑎1𝑇𝛼(𝑢) + 𝑎2𝑇𝛼(𝑣), ⬚∀𝑎1, 𝑎2 ∈ ℝ 

(2) 𝑇𝛼(𝑢𝑣) = 𝑢𝑇𝛼(𝑣) + 𝑣𝑇𝛼(𝑢) 

(3) 𝑇𝛼(𝑥𝑝) = 𝑝𝑥𝑝−𝛼, ⬚∀𝑝 ∈ ℝ 

(4) 𝑇𝛼(𝑎) = 0, ⬚𝑢(𝑥) = 𝑎 for every constant functions. 

(5) 𝑇𝛼 (
𝑢

𝑣
) =

𝑣𝑇𝛼(𝑢)−𝑢𝑇𝛼(𝑣)

𝑣2  

(6) If 𝑢 is differential, then 𝑇𝛼(𝑢(𝑥)) = 𝑥1−𝛼 𝑑𝑢(𝑥)

𝑑𝑥
. 

Proof . Refer [13] 

Definition 2.3. Let 𝑢 be a function with 𝑚 variable 𝑥1, … . . . , 𝑥𝑚, and the conformable partial 

derivative of 𝑢 of order 0 < 𝛼 ≤ 1 in 𝑥𝑖 is defined as follows 

∂𝛼

∂𝑥𝑖
𝛼 𝑢(𝑥1, … … , 𝑥𝑚) = lim

𝜖→0
 
𝑢(𝑥1, . . , 𝑥𝑖−1, 𝑥𝑖 + 𝜖𝑥𝑖

1−𝛼 … . , 𝑥𝑚) − 𝑢(𝑥1, … … , 𝑥𝑚)

𝜖
 

Lemma 2.1. If r⃗ = xi⃗⃗⃗ ⃗ + yj⃗⃗⃗ ⃗ + zk⃗⃗ ⃗⃗  and 𝑟 = |r⃗| then grad𝛼  𝑓(𝑟) = 𝑟1−𝛼grad 𝑓(𝑟). 



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Proof. If 𝑓 is differentiable then by using the properties of 2.1 we get 

grad𝛼  𝑓(𝑟) = 𝑟1−𝛼grad 𝑓(𝑟). 

Hence the proof is complete. 

First we introduce the Riccati method. There exists a Ω𝑟 = {𝑥 ∈ ℝ𝑛: ∥ 𝑥 ∥≥ 𝑟} and a solution 𝑢 of 

(1.1) that is non negative on Ω𝑟. Let �⃗⃗⃗⃗� =
grad𝛼u

u
 be the vector function representing the solution to 

the Riccati equation defined on the set Ω𝑟. 

div𝛼  �⃗⃗⃗⃗� + 𝑝(𝑥)+∥ 𝑊 ∥2= 0                                                              (2.1) 

The operator 𝑑𝑖𝑣𝛼  is typical divergent operator, i.e. for �⃗⃗⃗⃗� = (𝑊1, … . , 𝑊𝑛) where the common 

Euclidean norm in ℝ𝑛 is represented by ∥.∥. 

Lemma 2.2. Let equation (1.1) be non-oscillatory, i.e, (1.1) has a positive solution on Ω𝑎 for some 

𝑎 ≤ 1. The below statements are equivalent: 

i) Its 

∫  
Ω(𝑎,∞)

𝑟1−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝛼𝑥 < ∞                                                        (2.2) 

ii) There's a finite limit 

lim
𝑟→∞

 𝑃(𝑟) = 𝑃0                                                                                    (2.3) 

iii) Its holds 

lim inf
𝑟→∞

 𝑃(𝑟) > −∞                                                                             (2.4) 

Proof. Let equation (1.1) be non-oscillatory. There is a number a ∈ ℝ+and a solution u of (1.1) that 

is non negative on Ω𝑎. Let W⃗⃗⃗⃗ =
grad𝛼 𝑢

𝑢
 be the vector function representing the solution of Riccati 

equation defined on Ω𝑎 and using the Gauss divergence theorem and the identity 

∫  
𝑆(𝑟)

 𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆 − ∫  
𝑆(𝑎)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆

+ ∫  
Ω(𝑎,𝑟)

  𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥 + ∫  
Ω(𝑎,𝑟)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥

− ∫  
Ω(𝑎,𝑟)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥.      (2.5)

 

i) ⇒ ii) If (2.2) holds. then the Cauchy inequality gives 

∫  
Ω(𝑎,𝑟)

  𝑟−𝑛+𝜆 ∥ 𝑊 ∥ 𝑑𝑥≤ (∫  
Ω(𝑎,𝑟)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥)

1

2

(∫  
Ω(𝑎,𝑟)

  𝑟−𝛼−𝑛+𝜆𝑑𝑥)

1

2

= (∫  
Ω(𝑎,𝑟)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥)

1

2

(𝜔𝑛 ∫  
𝑟

𝑎

  𝑟𝛼+𝜆−3𝑑𝑟)

1

2

.

 



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here, 𝜔𝑛 is represented as the measure of the sphere in ℝ𝑛 and 𝜔𝑛 =
2Π

𝑛
2

Γ
n
2

, 

∫  
Ω(𝑎,∞)

𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥 < ∞                                                    (2.6) 

not diverges. Evaluate of (2.5) and (2.6) gives 

�̂� − ∫  
Ω(1,𝑟)

  𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥 = ∫  
𝑆(𝑟)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆

+ ∫  
Ω(𝑟,∞)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥 − ∫  
Ω(𝑟,∞)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥,
     (2.7) 

where 

�̂� = ∫  
𝑆(𝑎)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆 + ∫  
Ω(𝑎,∞)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥

+ ∫  
Ω(1,𝑎)

  𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥 − ∫  
Ω(𝑎,∞)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥,
 

is a finite number. We will show that 

�̂� = 𝛼𝑃0.                                                                                          (2.8) 

Then it follows that �̂� actually does not always depend on the choice of the number for 𝑎. Using 

(2.7) and the inequality 

|𝑏 + 𝑐 + 𝑑|2 ≤ 4|𝑏|2 + 4|𝑐|2 + 4|𝑑|2. 

Taking integration from 𝑎 → 𝑅 and multiply by 
1

𝑅𝛼 on both side 

1

𝑅𝛼
∫  

𝑅

𝑎

  |�̂� − ∫  
Ω(1,𝑟)

  𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|

2

𝑑𝛼𝑟 =
4

𝑅𝛼
∫  

𝑅

𝑎

  |∫  
𝑠(𝑟)

 𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆|

2

𝑑𝛼𝑟   (2.9)

+
4(𝛼 − 𝑛 + 𝜆)2

𝑅𝛼
∫  

𝑅

𝑎

  |∫  
Ω(𝑟,∞)

  𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥|

2

𝑑𝛼𝑟  (2.10)

+
4

𝑅𝛼
∫  

𝑅

𝑎

  |∫  
Ω(𝑟,∞)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥|

2

𝑑𝛼𝑟 = 0.  (2.11)

 

The terms (2.10) and (2.11) tends to zero for 𝑟 → ∞, according to the L'Hospital rule, (2.2) and (2.6). 

If follows from the Cauchy inequality 

1

𝑅𝛼
∫  

𝑅

𝑎

  |∫  
𝑆(𝑟)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆|

2

𝑑𝛼𝑟

≤
1

𝑅𝛼
∫  

𝑅

𝑎

 (∫  
𝑆(𝑟)

  𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑆) (𝜔𝑛 ∫  
𝑆(𝑟)

  𝑟𝛼+𝜆−1𝑑𝑆) 𝑑𝛼𝑟.

 

and the term (2.9) tends to zero by using (2.2). Hence 

1

𝑅𝛼 ∫  
𝑅

𝑎
|�̂� − ∫  

Ω(1,𝑟)
  𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|

2

𝑑𝛼𝑟 → 0⬚ for ⬚𝑅 → ∞                                      (2.12) 

 



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Therefore 

|
1

𝑅𝛼
∫

𝑎

𝑅
  (�̂� − ∫

Ω(1,𝑟)
 𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥) 𝑑𝛼𝑟| ≤ (

1

𝑅𝛼
∫

𝑎

𝑅
  |�̂� − ∫

Ω(1,𝑟)
 𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|

2

𝑑𝛼𝑟)

1

2

 and from 

(2.12) it follows that (2.3), 

�̂� = 𝛼𝑃0⬚ holds.  

ii) ⇒ iii) is trivial. iii) ⇒ i) If the (2.4) holds and equation (2.2) does not hold. 

Denoting 

𝜒(𝑟): = ∫  
𝑟

𝑎

∫  
Ω(𝑎,𝑟)

𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥𝑑𝛼𝑟 

this function satisfies 

lim
𝑟→∞

 
𝜒(𝑟)

𝑟
→ ∞⬚ for ⬚𝑟 → ∞                                                                                         (2.13) 

∫  
Ω(𝑎,∞)

𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑥 = ∞. 

From (2.5) we integrate from 𝑎 → R and multiply by 
1

𝑅𝑎 we get 

∣
1

𝑅𝛼
𝜒(𝑅) −

1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
Ω(𝑎,𝑟)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟

+
1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
𝑠(𝑟)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟| = | − 𝑃(𝑅) +
1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
𝑠(𝑎)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟 ∣

 

If iii) holds, lim𝑟→∞  inf𝑃(𝑟) > −∞ hold. 

Less than 
1

4𝑅𝛼 𝜒(𝑅) and the right-hand side is bounded from above if (2.4) is valid. Hence 

3𝜒(𝑅)

4𝑅𝛼
≤ |

1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
Ω(𝑎,𝑟)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟| +

|
1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
𝑠(𝑟)

  𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟|.      (2.14)

 

By the Cauchy inequality, 

1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
𝑠(𝑟)

 𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟 ≤ (𝑇𝛼𝜒(𝑅))
1

2 (
𝜔𝑛𝑅𝜆+2𝛼

(𝜆 + 𝛼)(𝜆 + 2𝛼)
)

1

2

1

𝑅𝛼
∫  

𝑅

𝑎

 ∫  
Ω(𝑎,𝑟)

  (𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟

≤ (𝛼 − 𝑛 + 𝜆)(𝜒(𝑅))
1

2 (
𝜔𝑛𝑅𝜆

𝜆(𝜆 − 𝛼)
)

1

2

.

 

 



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From (2.13) it follows that 

𝜒(𝑅)

42(𝛼−𝑛+𝜆)2𝑅𝜆 ≤
𝜔𝑛

𝜆(𝜆−𝛼)
                                                                               (2.15) 

for 𝑅 large enough, therefore 

1

𝑅𝛼 ∫  
𝑅

𝑎
∫  

Ω(𝑎,𝑟)
(𝛼 − 𝑛 + 𝜆)𝑟−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟 =

𝜒(𝑅)

4
                            (2.16) 

Combining the above computations above for R large enough, we obtain 

𝜒2(𝑅)

4
≤

𝜔𝑛

(𝜆 + 𝛼)(𝜆 + 2𝛼)
𝑇𝛼𝜒(𝑅)𝑅𝜆+2𝛼 

and from here we get 

4𝜔𝑛

𝑇𝛼𝜒(𝑅)

𝜒2(𝑅)
≥

(𝜆 + 𝛼)(𝜆 + 2𝛼)

𝑅𝜆+2𝛼
 for 𝑅 large enough.  

Integration from 𝑟1 → ∞ gives a divergent integral on the right-hand side and convergent integral on 

the left-hand side. This contradicts our proof. 

Introducing this function 𝜌(𝑟) is defined 

𝜌(𝑟) = ∫  
𝑆(𝑟)

𝑟𝛼−𝑛+𝜆⟨𝑊, 𝑒𝑖⟩𝑑𝑆                                                                       (2.17) 

Lemma: 2.3. Let (2.3) holds. Let the equation (1.1) have a non-oscillatory solution. Then, 

𝑀(𝑟) − (
(𝛼 − 𝑛 + 𝜆)

𝛼
+ 1) 𝑔 +

𝑟1−(𝛼+𝜆)𝑔2

𝜔𝑛(𝜆 + 𝛼)
≤ 0 

and 

𝑟−1(𝑟𝑁(𝑟) − 𝜏𝜖𝑁(𝜏𝜖) − 𝜏𝜖
2𝜌(𝜏𝜖)) − 𝐺 (

(𝛼 − 𝑛 + 𝜆)𝑟(1−𝛼)

2 − 𝛼
+

(2𝑟𝛼−1)

𝛼
− 1)

+
𝑟1−(𝛼+𝜆)𝐺2

𝜔𝑛(2 − 𝛼 − 𝜆)
≤ 0.

 

are solvable. 

Proof. Let �⃗⃗⃗⃗� represent the solution of (2.1), which is based on 𝑊𝑎 for each 𝑎 ∈ ℝ. 

Cauchy inequality gives us 

𝜌2(𝑟) = 𝜔𝑛𝑟𝜆+𝛼−1 ∫  
𝑆(𝑟)

𝑟𝛼−𝑛+𝜆 ∥ 𝑊 ∥2 𝑑𝑆                                           (2.18) 

Introducing the Notation 

𝑔 = liminf𝑟𝜌(𝑟)⬚𝐺 = limsup𝑟𝜌(𝑟) 

Obviously, to any 0 < 𝜖 < 𝑚𝑖𝑛{𝑔, 1 − 𝐺} there exists 𝜏𝜖 > 𝑟0 and 𝑟𝜖 > 𝜏𝜖 such that 

𝑔−∈< 𝑟𝜌(𝑟) < 𝐺 + 𝜖                                                                         (2.19) 

 



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From (2.7), (2.8) we easily find that 

𝑟𝜌(𝑟) = 𝑀(𝑟) − (𝛼 − 𝑛 + 𝜆)𝑟 ∫  
∞

𝑟
𝜌(𝑠)𝑠−𝛼𝑑𝑠 +

𝑟

𝜔𝑛
∫  

∞

𝑟
𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠  (2.20) 

Taking the new account and the above argument, we get 

𝑔 − 𝜖 ≥ 𝑀(𝑟) −
(𝛼 − 𝑛 + 𝜆)

𝛼
(𝑔 − 𝜖) +

𝑟1−(𝛼+𝜆)(𝑔 − 𝜖)2

𝜔𝑛(𝜆 + 𝛼)
 

Taking differentiation, Multiply by 𝑟2 and integrating from 𝜏 → R we get 

𝑅𝜌(𝑅) =𝑅−1(𝜏2𝜌(𝜏) − 𝑅𝑁(𝑅) + 𝜏2𝑁(𝜏)) + 𝑅−1 ∫  
𝑅

𝜏

 2𝑠𝛼𝜌(𝑠)𝑑𝑠 +

𝑅−1 ∫  
𝑅

𝜏

  (𝛼 − 𝑛 + 𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠 − 𝑅−1 ∫  
𝑅

𝜏

 
𝑠(3−𝜆−𝛼)𝜌2(𝑠)

𝜔𝑛
𝑑𝑠

 

here substituting 𝑅 = 𝑟 and 𝜏 = 𝜏𝜖 

𝑟𝜌(𝑟) =𝑟−1(𝜏𝜖
2𝜌(𝜏𝜖) − 𝑟𝑁(𝑟) + 𝜏𝜖

2𝑁(𝜏𝜖)) + 𝑟−1 ∫  
𝑟

𝜏

 2𝑠𝛼𝜌(𝑠)𝑑𝑠 +

𝑟−1 ∫  
𝑟

𝜏

  (𝛼 − 𝑛 + 𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠 − 𝑟−1 ∫  
𝑟

𝜏

 
𝑠(3−𝜆−𝛼)𝜌2(𝑠)

𝜔𝑛
𝑑𝑠       (2.21)

𝐺 + 𝜖 ≤𝑟−1(𝜏𝜖
2𝜌(𝜏𝜖) − 𝑟𝑁(𝑟) + 𝜏𝜖𝑁(𝜏𝜖)) +

(𝐺 + 𝜖) (
2𝑟(𝛼−1)

𝛼
+

(𝛼 − 𝑛 + 𝜆)𝑟1−𝛼

2 − 𝛼
−

(𝐺 + 𝜖)𝑟𝑡1−(𝜆+𝛼)

𝜔𝑛(2 − 𝛼 − 𝜆)
)

 

Hence 

𝑀(𝑟) − (
(𝛼 − 𝑛 + 𝜆)

𝛼
+ 1) 𝑔 +

𝑟1−(𝛼+𝜆)𝑔2

𝜔𝑛(𝜆 + 𝛼)
≤ 0 

and 

𝑟−1(𝑟𝑁(𝑟) − 𝜏𝜖𝑁(𝜏𝜖) − 𝜏𝜖
2𝜌(𝜏𝜖)) − 𝐺 (

(𝛼 − 𝑛 + 𝜆)𝑟(1−𝛼)

2 − 𝛼
+

(2𝑟𝛼−1)

𝛼
− 1)

+
𝑟1−(𝛼+𝜆)𝐺2

𝜔𝑛(2 − 𝛼 − 𝜆)
≤ 0

 

Hence it is proved. 

The Hartman Winter theorem and newly discovered oscillation requirements for conformable elliptic 

equations are covered in the following session. 

3. MAIN RESULTS 

In this section, the following results has been established. The below theorem is an oscillation  

criterion of the Hartman Winter type. 

Theorem 3.1. If−∞ < lim
𝑟→∞

 inf𝑃(𝑟) < lim
𝑟→∞

 sup𝑃(𝑟) ≤ ∞    (3.1) 



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or if lim
𝑟→∞

 𝑃(𝑟) = ∞                                                                           (3.2) 

Then (1.1) is oscillatory 

Proof. Assume 3.1 is true by contradiction and ∃𝑟 there is a non negative solution of 1.1 on Ω𝑟 

exists. It follows that the Riccati equations corresponding solution is defined on 𝑅. Lemma 2.1’ s ( iii 

) ⇒ ( ii ) portion and the first inequality in 3.1 indicates the existence of a finite limit lim𝑟→∞  𝑃(𝑟) 

which is in opposition to 3.1. The same proof applies to 3.2 

Theorem  3.2. Let equation (1.1) have oscillatory solution 𝑢. Then 

𝑀(𝑟) > 𝐵 + 𝜖 +
(𝛼 − 𝑛 + 𝜆)2𝜔𝑛

4(𝛼 − 𝜆)
𝑟𝜆−𝛼+1 

and 

𝑁(𝑟) > 𝑟−1(𝑟𝜖
2𝜌(𝑟𝜖) + 𝑟𝜖

2𝑁(𝑟𝜖)) −

𝜔𝑛𝑟−1

4
(

(𝛼 − 𝑛 + 𝜆)2𝑟𝜆+2−𝛼

𝜆 − 𝛼 + 2
+

4𝑟𝜆+3𝛼−2

𝜆 + 3𝛼 − 2
+

4(𝛼 − 𝑛 + 𝜆)𝑟𝛼+𝜆

𝛼 + 𝜆
) − 𝐴 + 𝜖

 

are oscillatory. 

Moreover, 

lim inf 𝑟𝜌(𝑟) ≥ 𝐴, ⬚liminf𝑟𝜌(𝑟) ≤ 𝐵 

where 𝐴 is the least non-negative root of equation and 𝐵 is the largest root of equation. 

Proof. Assume the contradiction. Let equation (1.1) have the non-oscillatory solution. From lemma 

2.1, (2.5) and (2.7) there exists 𝑟𝜖 > 𝑟0 such that 

𝐴 − 𝜖 < 𝑟𝜌(𝑟) < 𝐵 + 𝜖 for 𝑟 > 𝑟𝜖 

Integrating from 𝑟 → ∞ and taking into account of (2.5) and (2.7) we get that 

𝑟𝜌(𝑟) = 𝑀(𝑟) − (𝛼 − 𝑛 + 𝜆)𝑟 ∫  
∞

𝑟

 𝜌(𝑠)𝑠−𝛼𝑑𝑠 +
𝑟

𝜔𝑛
∫  

∞

𝑟

 𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠

𝑀(𝑟) = 𝑟𝜌(𝑟) − 𝑟 (
𝑟

𝜔𝑛
∫  

∞

𝑟

 𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠 − (𝛼 − 𝑛 + 𝜆) ∫  
∞

𝑟

 𝜌(𝑠)𝑠−𝛼𝑑𝑠)

𝑀(𝑟) < 𝐵 + 𝜖 +
(𝛼 − 𝑛 + 𝜆)2𝜔𝑛

4(𝛼 − 𝜆)
𝑟𝜆−𝛼+1                         (3.3)

 

Similarly, from (2.5) 

𝑟𝑁(𝑟) =𝜏𝜖
2𝜌(𝜏𝜖) + 𝜏𝜖𝑁(𝜏𝜖) − ∫  

𝜏𝜖

𝑟

 (
𝑠3−𝛼−𝜆𝜌2(𝑠)

𝜔𝑛
− 𝜌(𝑠)((𝛼 − 𝑛 + 𝜆)𝑠2−𝛼 + 2𝑠𝛼)) 𝑑𝑠 − 𝑟2𝜌(𝑟)

𝑁(𝑟) <𝑟−1(𝑟𝜖
2𝜌(𝑟𝜖) + 𝑟𝜖

2𝑁(𝑟𝜖)) −

𝜔𝑛𝑟−1

4
(

(𝛼 − 𝑛 + 𝜆)2𝑟𝜆+2−𝛼

𝜆 − 𝛼 + 2
+

4𝑟𝜆+3𝛼−2

𝜆 + 3𝛼 − 2
+

4(𝛼 − 𝑛 + 𝜆)𝑟𝛼+𝜆

𝛼 + 𝜆
) − 𝐴 + 𝜖

 

 



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which contradicts the theorem and it is proved. 

Theorem 3.3. Let equation (1.1) have oscillatory solution 𝑢. Then 

𝑀(𝑟) > 𝐵 + 𝜖 + (𝐴 + 𝜖) (
(𝛼 − 𝑛 + 𝜆)𝑟(1−𝛼)

𝛼
−

(𝐴 + 𝜖)𝑟1−(𝜆+𝛼)

𝜔𝑛(𝜆 + 𝛼)
) 

and 

𝑁(𝑟) > 𝜖 − 𝐴 + 𝑟−1(𝑟𝜖
2𝜌(𝑟𝜖) + 𝑟𝜖𝑁(𝑟𝜖)) + (𝐵 + 𝜖)𝑟−1 ((𝛼 − 𝑛 + 𝜆)

𝑟2−𝛼

(2−𝛼)
+

2𝑟𝛼

𝛼
−

(𝐵+𝜖)𝑟2−𝛼−𝜆

𝜔𝑛(2−𝛼−𝜆)
) are 

oscillatory. 

Proof. Assume the contradiction. Let equation (1.1) have the non-oscillatory solution. By lemma 2.1, 

(2.5) and (2.7) 

𝑀(𝑟) = 𝑟𝜌(𝑟)) + (𝛼 − 𝑛 + 𝜆)𝑟 ∫  
∞

𝑟

 𝜌(𝑠)𝑠−𝛼𝑑𝑠 −
𝑟

𝜔𝑛
∫  

∞

𝑟

 𝜌(𝑠)2𝑠1−(𝜆+𝛼)𝑑𝑠

𝑀(𝑟) < 𝐵 + 𝜖 + (𝐴 + 𝜖) (
(𝛼 − 𝑛 + 𝜆)𝑟(1−𝛼)

𝛼
−

(𝐴 + 𝜖)𝑟1−(𝜆+𝛼)

𝜔𝑛(𝜆 + 𝛼)
)      (3.4)

 

Similarly, we obtain from (2.5) 

−𝑁(𝑟) = 𝑡𝜌(𝑟) − 𝑟−1(𝜏2𝜌(𝜏) + 𝜏𝑁(𝜏)) +

𝑟−1 ∫  
𝑟

𝜏

 (
𝑠(3−𝜆−𝛼)𝜌(𝑠)2

𝜔𝑛
− (𝛼 − 𝑛 + 𝜆)𝑠2−𝛼𝜌(𝑠) − 2𝑠𝛼𝜌(𝑠)) 𝑑𝑠

𝑁(𝑟) < 𝜖 − 𝐴 + 𝑟−1(𝑡𝜖
2𝜌(𝑟𝜖) + 𝑟𝜖𝑁(𝑟𝜖))

⬚ + (𝐵 + 𝜖)𝑟−1 ((𝛼 − 𝑛 + 𝜆)
𝑟2−𝛼

(2 − 𝛼)
+

2𝑟𝛼

𝛼
−

(𝐵 + 𝜖)𝑟2−𝛼−𝜆

𝜔𝑛(2 − 𝛼 − 𝜆)
)    (3.5)

 

which contradicts the theorem and it is proved. 

4. CONCLUSION  

Using the traditional Riccati Substitution, this work presents some oscillation results for the class of 

conformable Schrodinger equations. The result demonstrates that Hartman-Winter criteria may be 

effectively applied in the theorem to derive oscillation criterion. Our newly obtained results in this 

study have improved, extending and adopting a broad perspective of certain known results that are 

already there in the literature. 

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