1. Introduction
A partial differential equation is a function of more than one variable, which is the key distinction between PDEs and ordinary differential equations. Many factors in nature are changing at once and manifest as phenomena. We end up with a PDE since they demand several variables rather than simply one. It has been observed that elliptic equations appear when dealing with physical laws. Therefore, elliptic equations are mostly associated with real-world problems and are important to study. In the literature, several such problems have been presented and solved with different assumptions [
1,
2,
3,
4,
5].
We concentrate our attention on the following system in this paper:
where
is the smooth boundary of the open bounded domain
of
,
is the anisotropic operator and
are real numbers with
are continuous positive functions on
and
:
are continuous maps. The Anisotropic Kirchhoff type system (
1) is a generalized variant of the basic Kirchhoff system. Problems with population dynamics and some physical events are examples of such systems. In general, such a problem can be solved in one of two ways: variational or topological. When employing variational approaches to solve a problem, the requirements on the second item of system (
1) are frequently more restrictive. We will employ a topological method here, specifically the sub-super solution method (combined with Schauder’s fixed point) which is based on the comparison principle.
It is demonstrated in [
6] that the comparison principle fails in this situation, making the conventional sub-super solution approach ineffective for Kirchhoff equations. Therefore, the results of the p-Kirchhoff type system’s existence based on the original application of the sub-super solution approach were unsuccessful.
In this study, we show that under weaker criteria than those suggested in the preceding papers, we can prove the desired result in more general circumstances. In order to do this, we refer back to [
6] in which the researchers proposed a unique sub-super solution strategy for the classical Kirchhoff system, based on the Schauder Fixed-Point Theorem rather than the comparison principle, with
. However, the proof seems to be ambiguous. As a matter of fact, the embedding of
is not compact, and the function examined from
in
is not compact.
Based on the concepts in [
6] and the information in [
7], we demonstrate that the new version of the sub-super solution approach performs admirably on the non-local issue and can be extended to nonlinear non-local systems. As a result, we establish the existence of several never-before-considered nonlinearities
in a positive solution to the
-Kirchhoff problem (
1).
The research work is divided into sections as follows: We demonstrate the equivalence of a
-Kirchhoff system to a non-local
-Laplacian system, moreover, the viability of the novel non-local
-Laplacian sub-super solution approach
-Laplacian system and
-Kirchhoff system. In
Section 3, we demonstrate how the modified sub-super solution technique suggested in [
6] may be applied to the nonlinear problem. In
Section 4, we demonstrate the method’s effectiveness for the issue (1.1) and present the fundamental theory underlying the existence of a positive solution. Finally, we give an explanation of the results obtained.
2. Preliminaries
Here, we indicate the bounded domain by
in
Further, take the real numbers
and take a vector
such that
In the next step, we take the Sobolev space
given by
which is a Banach space equipped with the norm
where
denotes the usual norm of
The closure of
in
with regard to the norm
defines the Banach space, which will be designated as
Think about the harmonic mean
of
,
provided by
Suppose that
and define the symbol
If
then there exists an embedding
which is compact for
and is continuous in the case
see [
7]. As a consequence of this, the norm is
is equivalent to the norm mentioned in the above (
2).
Lemma 1 ([
7])
. Let . Then the problemhas a solution in Lemma 2 ([
7])
. Assume that bounded domain Φ and take fulfillingwhere on implies that Then in Lemma 3 ([
7])
. Take a bounded and admissible such that one can find a continuous embedding and the best constant of the mentioned embedding is indicated by . Then we have thatwhere Numerous ideas about anisotropic has been presented in the literature [
8,
9,
10,
11]. Adapting the ideas of [
7] to the anisotropic setting, we obtain the result below which is significant for the current work.
Lemma 4 ([
7])
. Let and consider the unique solution ofDefine In the case then with and . If where and are positive constants that rely only on and with given in (3).
4. The Sub-Super Approach for Non-Local -Laplacian System
In the literature, the solution of different systems has been investigated via sub-super approach [
13,
14,
15,
16]. Here, we demonstrate how the linear example with
in the sub-super solution approach [
6] may be modified for the
-Laplacian in the non-local situation. Consider the non-local system as
where
and
Definition 1. The pair of solutions is a sub-super solution of (6) if the following holds true a.e in and on for
Inequalities in
and
are in the weak sense for all
where, for
. in
is defined by,
In the upcoming theorem, we will prove the main result of this research work.
Theorem 1. Let us assume the following
is a continuous operator for which maps bounded sets to bounded sets
is continuous for in
Then, the system (6) has a solution if a pair of sub-super solutions of (6) exists in the sense of Definition 1. Proof. First of all take the operators,
introduced in the following manner
Remark that
,
and as
, there exist constants
such that
a.e. in
Then,
. Through the continuity of
in
, we can find a positive constant of continuity
of
Therefore, we have the following inequality:
Here, the Nemytskii operator is denoted by
and is given by
the boundedness of
is obtained through the
-boundedness of
. At the end, through the continuity of
and dominated convergence theorem, we deduce the continuity of
and
Here, we fix
by the Minty–Browder Theorem, then a unique pair solution
exists of
In these circumstance, we introduce the operator
by
where
is the unique solution of (8). □
Claim 1: Take a bounded sequence
in
and
while
S is compact. After that
Using the test function
in (
9) and the inequality (
7), we get
where
are constants which have no rely on
The inequality
provides that
for all
As
for
and
where
rely only on
b and
N; this implies that
for all
with
being a constant that does not depend on
From (
12), it follows that the sequence
is bounded in
Since the embedding
is compact, the boundness of
in
implies that there exists a convergent sub-sequence of
in
The compactness of
S implies that
is bounded in
. It is easy to find a convergent sub-sequence of
in
through compact embedding.
From (
10), we conclude there exists
in a manner that
Remark that in (
7), (
10) and (
13), the constants are independent of the choice of
. Then, for some
we conclude that
. By the Schauder fixed point theorem, in
, one can find a unique
in a manner that
, that is
Finally,
is a solution of problem
if, and only if,
and
, furthermore
and
. We have to establish that
and
. To establish the result, take
. The remaining cases can be supported by the same logic. Assume
As
is a sub-solution then, for
and
we have
and as
is a solution of (
14), we have
Then
In
through a remark, then we have
As a result,
in
due to the monotonicity of the
-Laplacian. Then, we have
in
. In the same way, we also arrive at
in
. We get to the conclusion that
and
. This indicates that
, the solution of (
14), is also a solution of the main problem (
6), since
and
.
5. The Sub-Super Solution for -Kirchhoff
Systems
The solution of Kirchhoff systems is investigated with different assumptions [
17,
18,
19,
20]. As in [
6] for anisotropic
-Laplacian system, the sub-super solution of problem (2.1) is now defined, and the existence theorem for the main system is subsequently presented (
1).
Definition 2. The pair of sub-super solutions is a solution of (5) if the following holds: in and on for
For all purposes, the last two inequalities are regarded in the weak sense for
Setting
and
in (
6), we get the below result.
Theorem 2. Assume the following for : is invertible,
in
is continuous in
In the sense of Definition 2, if a pair of sub-super solutions exists of (5), then one can find a solution of (5) in a manner that Proof. Through the continuity of
, on
, and
is satisfied. In addition to this, due to
and
hypothesis
of Theorem 3.1 is also verified. From these, we can say that (
5) has a solution
. Consequently, the pair
is also has a solution of
Furthermore, recall
for the definition of the operator
□