Existence of Positive Solutions for a Class of (p(x), q(x))-Laplacian Elliptic Systems with Multiplication of Two Separate Functions

Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed Boudiaf El Mnaouar, Bir El’Djir, Oran 31000, Algeria Department of Mathematics, College of Sciences and Arts, Qassim University, Al-Rass, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria Department of Mathematics, College of Sciences, Qassim University, Buraydah, Saudi Arabia

In contrary to ODEs, there is no general result such as the Picard-Lindelöf theorem for PDEs to settle the existence and uniqueness of solutions. Malgrange-Ehrenpreis theorem states that linear partial di erential equations with constant coe cients always have at least one solution; another powerful and general result in case of polynomial coe cients is the Cauchy-Kovalevskaya theorem ensuring the existence and uniqueness of a locally analytic solution for PDEs with coe cients that are analytic in the unknown function and its derivatives; otherwise, the existence of solutions is not guaranteed at all for nonanalytic coecients even if they have derivatives of all orders (see [16]). Given the rich variety of PDEs, there is no general theory of solvability. Instead, research focuses on particular PDEs that are important for applications. It would be desirable when solving a PDE to prove the existence and uniqueness of a regular solution that depends on the initial data given in the problem, but perhaps we are asking too much. A solution with enough smoothness is called a classical solution, but in most cases as for conservation laws, we cannot achieve that much and allow generalized or weak solutions. e point is this: looking for weak solutions allows us to investigate a larger class of candidates, so it is more reasonable to consider as separate the existence and the regularity problems. For various PDEs, this is the best that can be done, and naturally nonlinear equations are more di cult than linear ones. Overall, we know too much about linear PDEs and in best cases, we can express their solutions but too little about nonlinear equations. For linear PDEs, various methods and techniques can be used for separation of variables, method of characteristics, integral transform, change of variables, superposition principle, or even finding a fundamental solution and taking a convolution product to obtain the solution. Variational theory is the most accessible and useful of the methods for nonlinear PDEs, but there are other nonvariational techniques of use for nonlinear elliptic and parabolic PDEs such as monotonicity and fixed point methods, semigroup theory, and sub-supersolutions method that played an important role in the study of nonlinear boundary value problems for a long time. Scorza-Dragoni's work in [17] was one of the earliest papers using a pair of ordered solutions of differential inequalities to establish the existence of solution to a given boundary value problem for a nonlinear second-order ordinary differential equation; his work was followed later by Nagumo in [18,19] which inspired much work on both ordinary and PDEs during the decade of the sixties. Knobloch in [20] introduced the sub-supersolution method to the study of periodic boundary value problems for nonlinear second-order ordinary differential equations using Cesari's method; similar problems and techniques were studied in [21,22] and still the sub-supersolutions and supersolutions are assumed to be smooth solutions of differential inequalities. en, the SSM were also used to study Dirichlet and Neumann boundary value problems for semilinear elliptic problems in [23,24], and even for nonlinear boundary value problems in [25][26][27] and also for systems of nonlinear ordinary differential equations in [28][29][30].
In the last few years in [51,[58][59][60], the regularity and existence of solutions for differential equations with nonstandard p(x)-growth conditions have been studied and p-Laplacian elliptic systems with p(x) � q(x) � p (a constant) have been archived. In this work, we study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions by using sub-supersolution method.

Preliminaries, Assumptions, and
Statement of the Problem 2.1. Plate Problems and Its History. In this paper, we consider the system of differential equations: where Ω ⊂ R N is a bounded smooth domain with C 2 boundary zΩ and 1 < p(x), q(x) ∈ C 1 (Ω) are functions with and a, b: Ω ⟶ R + are continuous functions, while f, g, h, and τ are monotone functions in R + such that satisfying some natural growth condition at u � ∞. We point out that the extension from p-Laplace operator to p(x)-Laplace operator is not trivial, since the p (x)-Laplacian has a more complicated structure then the p-Laplace operator, such as it is nonhomogeneous. Moreover, many results and methods for p-Laplacians are not valid for the p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient is zero in general, and only under some special conditions, λ p(x) is positive (see [53]). Maybe the first eigenvalue and the first eigenfunction of the p(x)-Laplacian do not exist, but the fact that the first eigenvalue λ p is positive and the existence of the first eigenfunction are very important in the study of p-Laplacian problem. ere are more difficulties in discussing the existence of solutions of variable exponent problems. In [59], the authors considered the existence of positive weak solutions for the following p-Laplacian problem: where the first eigenfunction has been used to construct the subsolution of p-Laplacian problem. Under the condition that for all M > 0, the authors gave the existence of positive solutions for problem (5) provided that λ is large enough.

Complexity
In [48], the existence and nonexistence of positive weak solutions to the following quasilinear elliptic system: has been considered where the first eigenfunction has been used to construct the subsolution of problem (7) and the following results were obtained: (7) has a positive weak solution for each λ > 0. (7) has no nontrivial nonnegative weak solution. For further generalizations of system (7), we refer to [49,50].
As already discussed before, on the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of the p(x)-Laplacian do not exist even if the first eigenfunction of the p(x)-Laplacian exists. Because of the nonhomogeneous property of the p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolutions of p(x)-Laplacian problems. Moreover, in [47,61], the authors studied the existence of solutions for problem (5), where some symmetry conditions are imposed. en, in [46], the existence of positive solutions of the system was investigated: without any symmetry conditions. Motivated by the ideas introduced in [47], the authors proved the existence of a positive solution when λ is large enough and satisfies condition (6) and they did not assume any symmetric condition and did not assume any sign condition on f(0) and g(0). Also the authors proved the existence of positive solutions with multiparameters; in this paper, we extend this given system of differential equations, where we establish the existence of a positive solution for a new class of this system with respect to the symmetry conditions by constructing a positive subsolution and supersolution and p, q ∈ C 1 (Ω) are functions, λ, λ 1 , λ 2 , μ 1 , and μ 2 are positive parameters, and Ω ⊂ R N is a bounded domain and we did not assume any sign condition on f(0), g(0), h(0), and τ(0).

Preliminary Results.
In order to discuss problem (1), we need some theories on W 1,p(x) 0 (Ω) which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces W 1,p(x) 0 (Ω) which will be used later (for details, see [54]).
Define (17) where h(x, u) is continuous on Ω × R and h(x) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [44], we have the following lemma: Lemma 1 (see [45]) (comparison principle). Let then u ≥ v a.e. in Ω.
Here, and hereafter, we will use the notation d(x, zΩ) to denote the distance of x ∈ Ω to denote the distance of Ω.
Since zΩ is C 2 regularly, there exists a constant δ ∈ (0, 1) such that d(x) ∈ C 2 (zΩ 3δ ) and |∇d(x)| � 1. Denote we have the following result Lemma 2 (Lemma 2.1 in [52]). If positive parameter η is large enough and ω is the unique solution of (22), then we have (i) For any θ ∈ (0, 1), there exists a positive constant C 1 such that (ii) and, there exists a positive constant C 2 such that
Step 2. We will construct a supersolution of problem (1); we consider where r > 0 is the positive number that verifies (H3) and β � max x∈Ω z 1 (x). We shall prove that (z 1 , z 2 ) is a supersolution of problem (1).
In the definition of v 1 (x), let We claim that From the definition of v 1 , it is easy to see that Since v 1 − ϕ 1 ∈ C 1 (zΩ δ ), there exists a point x 0 ∈ zΩ δ such that If v 1 (x 0 ) − ϕ 1 (x 0 ) < 0, it is easy to see that 0 < d(x) < δ and then From the definition of v 1 , we have It is a contradiction to us, (50) is valid. Obviously, there exists a positive constant C 3 such that c ≤ C 3 λ.
According to the comparison principle, we have From (50) and (57), when η ≥ λ p + and λ ≥ 1 are sufficiently large, we have, for all x ∈ Ω, According to the comparison principle, when μ is large enough, we have, for all x ∈ Ω, Combining the definition of v 1 (x) and (58), it is easy to see that, for all x ∈ Ω, When μ ≥ 1 and λ is large enough, from Lemma 2, we can see that β is large enough, and then λ q+ b 2 β r is a large enough. Similarly, we have ϕ 2 ≤ z 2 . is completes the proof.

Conclusion
Validity of the comparison principle and of the SSM for local and nonlocal problems as the stationary and evolutionary Kirchhoff Equation was an important subject in the last few years (see, for example, [44,53,58,[62][63][64][65][66]), where the authors showed by giving different counterexamples that the simple assumption M increasing somewhere is enough to make the comparison principle and SSM hold false contradiction and clear up some results in the literature. Moreover, the two conditions that M is nonincreasing and H is increasing turn out to be necessary and sufficient, at least for the validity of the comparison principle. It is worth to note that in [45,67], C. O. Alves and F. J. S. A. Correa developed a new SSM for problem (1) to deal with the increasing M case. e result is obtained by using a kind of Minty-Browder theorem for a suitable pseudomonotone operator, but instead of constructing a subsolution, the authors assumed the existence of a whole family of functions which satisfy a stronger condition than just being subsolutions; the inconvenience is that these stronger conditions restrict the possible right hand sides in (1). Another SSM for nonlocal problems is obtained in [45] for a problem involving a nonlocal term with a Lebesgue norm, instead of the Sobolev norm appearing in (1).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this manuscript.