Abstract
This paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zero Dirichlet boundary condition in bounded domain by using the subsuper solutions method.
1. Introduction
In this paper, we consider the following system of differential equations:where is a bounded smooth domain with boundary B: are continuous functions, and and are positive parameters, where and The peculiarity of this type of problem, and by far the most important, is that it is not local. This is based on the presence of the operatorwhich contains an integral on all the fields and implies that the equation is not a specific identity. It is clear that these problems contribute to the transition from academia to application. Indeed, very popular for its physical motivations, problem (1) is none other than a stationary version of the following model which regulates the behavior of elastic whose ends are fixed and which is subjected to nonlinear vibrations:where T is a positive constant and and are given functions. In such problems, u expresses the displacement, the extreme force, the initial tension, and relates to the intrinsic properties of the wire material (such as Young’s modulus). For more details, see [1], as well as their references. Basically, this is a generalization to larger dimensions of the model originally proposed in one dimension by Kirchhoff [2] in (1883):where is the initial tension, represents Young’s modulus of the material of the wire, and L its length. The latter is known to be an extension of the equation of D’Alembert waves. Indeed, Kirchhoff took into account the changes caused by transverse oscillations along the length of the wire. With their implications in other disciplines, and given the breadth of their fields of application, nonlocal problems will be used to model several physical phenomena, and they also intervene in biological systems or describe a process dependent on its average, such as particle density population. Moreover, With this significant impact strengthening the field of applications, this type of problem has caught the interest of mathematicians and a lot of work on the existence of solutions has emerged, particularly after the coup de force provided by the famous Lions article [3], where the latter has adopted an approach based on functional analysis. Nevertheless, in most of these articles, the benefit method is purely topological. It is only in the last decades that this approach has been removed from variational methods when Alves and his colleagues [4] obtained for the first time the results of their existence through these methods. Since then, a very fruitful development has given rise to many works based on this advantageous axis (see [1, 3, 5]).
In recent years, problems relating to Kirchhoff operators have been studied in several papers (we refer to [6]), where the authors used different methods to obtain solutions (1) in the case of single equation (see [6]). The concept of weak sub- and supersolutions was first formulated by Hess and Deuel in [7, 8] to obtain existence results for weak solutions of semilinear elliptic Dirichlet problems and was subsequently continued by several authors (see, e.g., [9–18]).
In our recent paper [19], we have discussed the existence of weak positive solution for the following Kirchhoff elliptic systems:
Motivated by the ideas of [20], which the authors considered a system (1) in the case , more precisely, under suitable conditions on we will prove that the problem which is defined in (1) admits a positive solution In current paper, motivated by previous works in ([19, 20]), we discuss the existence of weak positive solution for sublinear Kirchhoff elliptic systems in bounded domains by using subsupersolutions method combined with comparison principle (see Lemma 2.1 in [4]).
The outline of the paper is as follows. In the second section, we give some assumptions and definitions related to problem (1). In Section 3, we prove our main result.
2. Assumptions and Definitions
Let us assume the following assumption.
Assume that are two continuous and increasing functions and there exists such that
Suppose that and
Now, in order to discuss our main result of problem (1), we need the following two definitions.
Definition 1. Let ; is said to be a weak solution of (1) if it satisfiesfor all
Definition 2. A pair of nonnegative functions in is called a weak subsolution and supersolution of (1) if they satisfy on :for all
Lemma 1 [4]. Assume that is a continuous and nonincreasing function satisfyingwhere is a positive constant and assume that are two nonnegative functions such thatand then in
3. Main Result
In this section, we shall state and prove the main result of this paper.
Theorem 1. Suppose that - hold and M is a nonincreasing function satisfying (9). Then, problem (1) has a large positive weak solution for each positive parameters and
Proof of Theorem 1. Let σ be the first eigenvalue of with Dirichlet boundary conditions and the corresponding eigenfunction with satisfying in and on
Since we can take k such thatWe shall verify that is a subsolution of problem (1), where is small and specified later.
A simple calculation:Similarly,Let be such thatand on where
We have from (14) thatOn the other hand, in letNote thatWe have from (11) thatThus, we choose such thatwhere Furthermore,These relations and Young inequality show thatHence, from (15)–(23), it follows thatThen, by (24) and (25), is a subsolution of (1).
Next, we shall construct a supersolution of problem (1). Let ω be the solution of the following problem:Letwhere e is given by (26) and are large positive real numbers to be chosen later. We shall verify that is a supersolution of problem (1). Let with in Then, we obtain from (26) and the condition thatLet Since , these imply that there exist positive large constants such thatThus,From (26) and (30), we can deduce that the couple is a subsolution of problem (1) with and for large.
In order to obtain a weak solution of problem (1) we shall use the arguments by Azouz and Bensedik [19]. For this purpose, we define a sequence as follows: and is the unique solution of the systemProblem (32) is linear in the sense that if is given, the right hand sides of (32) are independent of
Set Then, since and .
We deduce from a result in [4] that system (32) has a unique solution
By using (32) and the fact that is a supersolution of (1), we haveand by Lemma 1, and Also, since , and the monotonicity of and τ one hasfrom which, rding to Lemma 1, for we writeand then Similarly, and becauseRepeating this argument, we get a bounded monotone sequence satisfyingUsing the continuity of the functions and t and the definition of the sequences there exist constants independent of n such thatFrom (39), multiplying the first equation of (32) by and integrating using the Holder inequality and Sobolev embedding, we can show thatorwhere is a constant independent of . Similarly, there exists independent of n such thatFrom (42) and (43), we infer that has a subsequence which weakly converges in to a limit with the properties and . Being monotone and also using a standard regularity argument, converges itself to Now, letting in (32), we deduce that is a positive solution of system (1). The proof of theorem is now completed.
4. Conclusion
In this work, we study the existence of weak positive solutions for a sublinear Kirchhoff elliptic systems in bounded domains by using the subsuper solutions method (SSM) combined with comparison principle which have been widely applied in many work (see for example [4, 19, 21–25]).Validity of the comparison principle and of the SSM for local and nonlocal problems as the stationary Kirchhoff Equation was an important subject in the last few years (see, for example, [26] and [23]. 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 notice that in [4], Alves and Correa developed a new SSM for problem (1) to deal with the increasing M case. The 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 [4] for a problem involving a nonlocal term with a Lebesgue norm, instead of the Sobolev norm appearing in (1). In our next study, we will try to apply an alternative approach using the variational principle which has been presented in [27–29].
Data Availability
The data used to support the findings of the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally to this article. They have all read and approved the final manuscript.