On the oscillation of certain class of conformable Emden-Fowler type elliptic partial differential equations

1.   Introduction

An essential area of applied theory of differential equations that deals with the study of oscillatory processes in the social and technological sciences is the theory of oscillations, see ^{[10,18,19,20,21,22,23,25,34,39,41]}.

Fractional differential equations have popularity and gained importance during the past few years. Several researchers trying to develop the earlier work with definitions of fractional derivatives like Riemann-Liouville and Caputo derivative, see ^{[17,24,28,32,35]}. Khalil et al. ^[27] introduced a new fractional derivative called the conformable derivative. Oscillation of theory conformable were done by several authors ^{[1,3,8,16,26,29,38]}. The conformable fractional derivative seeks to extend the common derivative while satisfying the properties of nature and provides a fresh approach.

The Emden-Fowler equations have been considered one of the important classical objects in the theory of differential equations. This type of equation has a variety of interesting physical applications occurring in astrophysics and atomic physics. The oscillation for the Emden-Fowler equation has interest over the last 50 years, many results appeared in the oscillatory behavior of the Emden-Fowler differential equations, see ^{[5,9,15,40,42]} and references cited therein. The references ^[4,13,14] lists numerous studies have been conducted on Emden-Fowler generalization when the gradient term is utilized.

Elliptic partial differential equations have various uses in physics as well as practically all branches of mathematics, including harmonic analysis, geometry, and Lie theory. The Laplacian equation and Poisson equation serve as the fundamental illustration of an elliptic PDE. The theory of elliptic PDE have application in electrostatics, heat and mass diffusion and hydrodynamics, see ^{[2,6,11,12,30,33,36,43,44]}.

Throughout the past few decades, the issue of oscillation and nonoscillation of elliptic partial differential equation solutions has drawn a lot of attention. To the present time, there exists almost no literature in conformable Emden-Fowler type elliptic partial differential equations is of the form

where $\alpha, \beta \in (0, 1)$, $\Delta_x^{\alpha}$ is the conformable nabla operator, $D^\alpha (u) = (u_{x_1}, u_{x_2}, ..., u_{x_n})$ for the conformable gradient of order $\alpha$ of $u$ with respect to the spatial variable $x$, $\parallel.\parallel$ is the usual Euclidean norm on $\mathbb{R}^n$. $c(x): \mathbb{R}^n \rightarrow \mathbb{R}$ is potential function $c \in$ $L_{loc}^1(\Omega)$, with $\Omega:\{x\in\mathbb{R}^n:\parallel x \parallel\geq 1 \}$.

By a solution of (1.1), we mean a function $u(x): \mathbb{R}^n \rightarrow \mathbb{R}$ which is absolutely continuous with first $\alpha$-fractional derivative in every compact subset of $\Omega$ and satisfies the Eq (1.1) a.e. in $\Omega$.

A bounded domain $G \subset$ $\Omega$ is said to be a nodal domain for (1.1) if there exists a nontrival function $u \in C^{2\alpha}(G; \mathbb{R}) \cap C(\overline{G}; \mathbb{R})$ such that $u$ is a solution of (1.1) with $u = 0$ on $\partial G$. Equation (1.1) is said to be nodally oscillatory in $\Omega$ if for any $r > 0$ Eq (1.1) has a nodal domain contained in $\Omega_r = \Omega \cap \{x \in \mathbb{R}^n:|x| > r\}$. Some different approach in the oscillation theory form the nodal domains. A bounded domain is said to be the nodal domain of a nontrivial solution and $u|_{\partial\Omega} = 0$.

By the Hartman-Winter Theorem, we get the function $C(r)$

where

The motivation for this work comes from the papers ^[7,31,37]. In this paper, we studied the $\alpha$- fractional partial differential equations for Emden-Fowler type elliptic equations (1.1) using the Riccati technique by considering the two cases

$\rm i)$ there exists a finite limit $\lim_{r\rightarrow \infty} C(r) = C_0$;

$\rm ii)$ the case $i)$ fails to hold and $\liminf_{r\rightarrow \infty}C(r) > -\infty.$

We introduce several fundamental definitions, properties, and lemmas that are helpful throughout the rest of this study in the following part, Section 2. We established the main findings in Section 3. We provide an example to emphasize the key findings in Section 4.

2.   Preliminaries

This part introduces several fundamental terms, characteristics, and lemmas that will be helpful throughout the rest of the paper.

Definition 2.1. ^[27] Given $f : [0, \infty) \rightarrow \mathbb{R}$. Then, the conformable fractional derivative of $f$ of order $\alpha$ is defined by

for every $t > 0$, $\alpha \in (0, 1)$. If $f$ is $\alpha$-differentiable in some $(0, a)$, $a > 0$ and $\lim_{t\to o^+} f^\alpha(t)$ exists, then we define

Definition 2.2. ^[27] Let $a \geq 0$ and $t \geq a$. Also, let $f$ be a functiondefined on $(a, t]$ and $\alpha \in (0, 1)$. Then, the $\alpha$ - fractional integral of f is given by

where the integral is the usual Riemann improper integral and $\alpha\in(0, 1)$.

Properties 2.1. ^[27] Let $\alpha\in(0, 1]$ and $f, g$ be $\alpha$-differentiable at some point $t > 0$. Then,

(1) $T_\alpha(af+bg) = a T_\alpha(f)+b T_\alpha(g).$

(2) $T_\alpha(fg) = f T_\alpha(g) + g T_\alpha(f).$

(3) $T_\alpha(t^p) = pt^{p-\alpha}.$

(4) $T_\alpha(C) = 0,$ $C \in \mathbb{R}.$

(5) $T_\alpha(\frac{f}{g}) = \frac{gT_\alpha(f)-fT_\alpha(g)}{g^2}.$

(6) If $f$ is differential, then $T_\alpha (f(t)) = t^{1-\alpha}\frac{d f(t)}{dt}.$

Definition 2.3. ^[3] Let $f$ be a function with $m$ variable $x_1, ......, x_m$, the conformable partial derivative of $f$ of order $0 < \alpha \leq 1$ in $x_i$ is defined as follows

Definition 2.4. ^[3] Consider the scalar field $f(x)$ and the vector field $F(x)$ that are assumed to possess partial conformable derivative of order $\alpha$ with respect to all the Cartesian coordinates $x_i, i = 1, 2, 3$. We define the conformable gradient of order $\alpha$ of the scalar field $f$ as follows

where $e_i$ is the unit vector in the $i$ direction. The conformable gradient of order $\alpha$ of the vector field$F$ is defined as follows

Definition 2.5. A solution of Eq (1.1) is called oscilatory if it has arbitrarily large zeros in $G$, and is called nonoscillatory otherwise. Equation (1.1) are said to be oscillatory if all their solutions are oscillatory.

Lemma 2.1. If $\overrightarrow{\rm r} = \overrightarrow{\rm x_i}+\overrightarrow{\rm y_j}+\overrightarrow{\rm z_k}$ and $r = |\overrightarrow{\rm r}|$ then $D^\alpha$ f(r) = $r^{1-\alpha}$D(f(r)).

Proof. If $f$ is differentiable, then using the Properties 2.1, we get

The proof of Lemma 2.1 is completed.

Lemma 2.2. If f(t) = At - B$t^{\frac{\beta+1}{\beta}}$, then

where $\beta \in (0, 1)$.

Proof. If $f$ differentiable, then

so that the maximum point of $f$ is realized in

Consequently,

Hence, the proof is completed. First we introduce the Riccati technique. There exists a $\Omega_r = \{x\in\mathbb{R}^{n}:\parallel x \parallel \geq r\}$ and a solution $u$ of (1.1) which is positive on $\Omega_r$. Let vector function ${W} = \left(\frac{|D^\alpha u|^{\beta-1}D^\alpha u}{|u|^{\beta-1}u}\right)$ be the solution of Riccati equation defined on the set $\Omega_r$.

The $i$ - component of gradient is

Taking the summation, we get

where $div_\alpha {W}$ = $\sum_{i = 1}^n\dfrac{\partial^{\alpha}W_i}{\partial x_i^{\alpha}}$.

For the next, we define

Lemma 2.3. Let the Eq (1.1) be nonoscillatory, i.e., (1.1) has a positive solution on $\Omega_a$ for some $a \leq 1$. The following statements are equivalent:

i)

ii) there exists a finite limit

iii) there exists a infinite limit

Proof. Let the Eq (1.1) be nonoscillatory. There exists a number $a \in\mathbb{R}^{+}$ and a solution $u$ of (1.1) which is positive on $\Omega_a$. Let, vector function ${W} = \left(\frac{|D^\alpha u|^{\beta-1} \ \ D^\alpha u}{|u|^{\beta-1} \ \ \ u}\right)$ is the solution of Riccati equation defined on the set $\Omega_a$.

By the Gauss divergence theorem ^[37],

that is,

where $e_i$ is the unit vector in the $i$ direction and $\langle W, e_i\rangle$ is the usual scalar product in $\mathbb{R}$, implies that $W$ satisfies the equality

Therefore, $i)$ $\Rightarrow$ $ii).$ Next, we suppose that (2.2) holds, then the Cauchy inequality gives

where ${\omega_{n}}$ is the measure of the unit sphere in $\mathbb{R}^n$ and ${\omega_{n}} = \frac{2\Pi^\frac{n}{2}}{\Gamma \frac{n}{2}}$. Therefore,

Combining (2.5) and (2.6), we get

where

is a finite number. Next, we will show that

One can prove this results as in the proof of the Lemma 2.1 in ^[33]. This implies $ii) \Rightarrow iii)$ is trivial. To show, $iii) \Rightarrow i)$, we suppose that (2.4) holds and (2.2) does not hold. Let us define the function

This function satisfies

and

Similarly, the rest of the proof follows from the Lemma 2.1 ^[33].

For the next, we define

and

where $\alpha$, $\beta$, $\lambda$ $\in$ $(0, 1)$.

Lemma 2.4. Consider (2.3) holds and the Eq (1.1) have a nonoscillatory solution. Then the equation

and

are solvable.

Proof. Let ${W}$ be the solution of the Riccati Eq (2.1) defined on ${W_a}$ for some $a\in \mathbb{R}$. From Cauchy inequality gives

The equalities (2.7) and (2.8) gives

Differentiate (2.5) with respect to $r$, multiply by $r^2$ and integrating over $\tau$ to $R$, we get

Now, substituting $R = r$ and $\tau = \tau_{\epsilon}$

Let us introduce the notation

Obviously, for any $0 < \epsilon < \min\{l, 1-L\}$ there exists $\tau_\epsilon > r_0$ and $r_\epsilon > \tau_\epsilon$ such that

Due to this fact we have from (2.12) and (2.13) gives

that is,

Therefore,

and

Hence, the Lemma is proved.

Lemma 2.5. Let Eq (1.1) has oscillatory solution $u$. Then all the solutions of Eq (1.1) are oscillatory.

Proof. Assume the contrary. Let $u$ be nonoscillatory solution of Eq (1.1). There exists $r_0 > 0$ such that $u D^\alpha u > 0$. Let us introduce the notation

There exists $r_3 > r_2$ such that

Because of this fact we have from (2.13) that

but this contradicts the Eq (2.15) and hence it is proved.

Lemma 2.6. Let there exists the function $v$ which is locally absolutely continuous together with its first derivative and satisfying the inequalities

almost everywhere. Then Eq (1.1) is nonoscillatory.

3.   Main results

In this paper, the following main results has been established.

Theorem 3.1. (Hartman-Wintner Type Oscillation Criteria) If

or if

then the Eq (1.1) is oscillatory

Proof. Suppose on the contrary that (3.1) holds and there exists number a such that positive solution of (1.1) on $\Omega_a$ exists. Then, Lemma 2.3 would implies that there exists a finite limit $\lim\limits_{r\rightarrow \infty}C(r)$, this contradicts our assumption.

Corollary 3.1. (Leighton-Wintner Type Criteria) If

then Eq (1.1) has no positive solution on $\Omega_a$ for any $a > 1$.

Theorem 3.2. Let Eq (1.1) has oscillatory solution $u$. Then

and

are oscillatory. Moreover,

where $M$ is the least nonnegative root of equation and $N$ is the largest root of equation.

Proof. On the contrary we assume the Eq (1.1) have the nonoscillatory solution. The Eq (2.1) has the solution of $\sigma:[r_0, \infty)\rightarrow (0, \infty)$ satisfying the condition

Clearly, for any $0 < \epsilon < 1-N(0 < \epsilon < 1)$ there exists $r_\epsilon$ such that

where $h(\epsilon) = (N+\epsilon)\left(2r^{\alpha-\beta}+(\alpha-n+\lambda)r^{2-\alpha-\beta}-\frac{\beta r^{2-(\alpha+\beta+\lambda)} }{\omega_{n}}(N+\epsilon)^\frac{1}{\beta}\right)(h(\epsilon) = 1)$. Finally, from (2.12) and (2.13) we get

and

but this contradicts (3.4) and (3.5) and hence the theorem is proved.

Theorem 3.3. Let (2.3) holds and

Then the Eq (1.1) is oscillatory.

Proof. By contradiction, a solution $W$ of the Riccati equation defined on $\Omega_a$ and there exists number $a\geq 1$. We combine the Eqs (2.7) and (2.8) and by using integration by parts, we obtain

that is,

Using Lemma 2.2, we get

Hence,

a contradiction. Hence, the proof is complete.

Corollary 3.2. Assume (2.3) hold, and

Then the Eq (1.1) is oscillatory.

Corollary 3.3. Assume (2.3) hold, and

Then the Eq (1.1) is oscillatory.

Corollary 3.4. Assume (2.3) hold, and each conditions are guarantees the oscillation of the Eq (1.1)

4.   Example

In this section, we give an example to illustrate the main results.

Example 4.1. Consider the conformable partial differential equations in Emden-Fowler type Laplacian equation

In paper [27,Lemma 4], for a linear equation $\beta = 1, \lambda < \beta$ and here $\beta = 1, \lambda = 0$ and $\alpha = 1$. Then

By apply the result of Theorem 3.3, Corollaries 3.2–3.4 in the right hand side, the solution is satisfied.

5.   Conclusions

In this paper, the authors have obtained some new oscillation criteria for certain class of conformable Emden-Fowler type elliptic partial differential equations by using Riccati technique. These newly derived results extend and complements the known results in the existing literature for the integer-order equations. To prove the effectiveness of our result we have illustrate with an example.

Acknowledgments

The authors are thankful to the reviewers valuable comments and suggestions that improved the quality of the paper.

The work of U.F. G. was supported by the government of the Basque Country for the ELKARTEK21/10 KK-2021/00014 and ELKARTEK22/85 research programs, respectively.

Conflict of interest

The authors declare no conflict of interest.