Abstract

This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, where , , , , , , and satisfy with is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights , near their minima and the dimension . Moreover, we use the well known Pohozaev identity for prove the nonexistence result.

1. Introduction and Main Results

Let be a bounded smooth domain in and consider the following weighted quasilinear elliptic system:where , and and are given positive weights defined on such that and , is the critical Sobolev exponent of the noncompact embedding into , satisfy , and the parameters and satisfy the following assumption:where and denotes the first eigenvalue of in .

Note that and are called strongly-coupled terms and , are called weakly coupled terms.

The problem is important in many fields of sciences; it arises in biological applications (e.g., population dynamics) or physical applications (e.g., models of a nuclear reactor) and have drawn a lot of attention (see [1–4] and references therein).

Our system is posed in the framework of the Sobolev space , endowed with the following norm:

The energy functional of is defined on by the following equation:

It is clear that andwhere , and denote the Fréchet derivative of at .

A pair of functions is said to be a weak solution of if and on satisfy for all . Therefore, the weak solutions of are the critical points of .

Before stating our main results, let us recall a brief history.

For the scalar case, that is, when , , and , then the system reduces to the single elliptic equation as follows:

In the special case , much interest has grown on this critical problem, starting from the celebrated paper by Brézis and Nirenberg [5] for the semilinear equation . They established existing results in dimension when is a ball, namely, they ensured the existence of a positive constant such that problem admits a positive solution for all , where is the first eigenvalue of the operator . In higher dimensions, , they proved the existence of a positive solution for all and no positive solution for or and is a star-shaped domain. After that, many authors generalized the results of reference [5] for the quasilinear case, for example, see [6–9] and the references therein.

In the case where is not constant, Hadiji and Yazidi [10] extended the results of [5] to the weighted problem with and such thatwhere , , , , and when . They showed that the existence of solutions depends not only on parameter but also on the behavior of near its minima. More recently, Benhamida and Yazidi [11] have generalized the results of reference [10] for the quasilinear case .

Concerning the vectorial case and without weights, a lot of papers have appeared in recent years dealing with system involving Laplacian or p-Laplacian operator, see for instance [1, 12–15] and the references therein. On the other hand, it should be mentioned that when , Bouchekif and Hamzaoui [13] studied the following weighted system:where are real parameters and denotes the critical Sobolev’s exponent of the embedding into . They proved the existence of at least one positive solution under suitable assumptions on the data.

A natural interesting question is whether the results concerning the solutions of with in [13] remain true for . By using [11, 13], we gave some positive answers. To the best of our knowledge, the results are new in the case when . Note that this quasilinear problem creates many difficulties in applying variational methods in the fact that contains the critical exponent and weights , then the functional does not satisfy the Palais–Smale condition in all the range. To overcome the lack of compactness, we need to determine a good level of the Palais–Smale condition. On the other hand, it is very difficult to prove that the critical value is contained in the range of this level, so we need more delicate estimates where and play an essential role.

Now, we introduced some notations and hypotheses.

We assume the existence of in such that, in a neighborhood of , the weighted and behave similar toandwhere , , , , , and are positive constants and tends to 0 as goes to .

The parameters and will play an essential role in the study of our system. In fact, if , the case and is treated by a classical procedure. For the other cases, we restricted ourself to the case where and satisfy the following additional conditions:andwhere

Letwhere .

Now, we are in a position to state the results of our paper.

For the nonexistence results, we have the following theorem.

Theorem 1. Assume that and is a starshaped domain with respect to . Then, has no nontrivial solution.
For the existence results, we have the following theorems.

Theorem 2. Suppose that , holds and satisfy equations (9) and (10), respectively. Then, there exist constants such that has a positive solution, under one of the following hypotheses: . . . .

Theorem 3. Suppose that , holds and and satisfy equations (11) and (12), respectively. Then, there exist constants such that has a positive solution, under one of the following hypotheses: . . . . .

This paper is organized as follows: In Section 2, we collected some preliminaries results that will be used throughout the work. In Section 3, we proved Theorem 1 (nonexistence result). In Section 4, we proved Theorems 2 and 3 (existence results) by using the mountain pass theorem.

2. Some Preliminary Results

Throughout this paper, we shall denote by and , for the various positive constants. The diameter of will be denoted by , we use and to denote the strong and weak convergence in the related function spaces, respectively, and and , represents the ball of radius centered at .

We define the following equation:where

First, we recall the following Hardy’s inequality, see for example [16].

Lemma 1. Let such that , we have the following equation:

Moreover, the constant is optimal and not achieved.

We note that direct calculations imply that if , Lemma 1 applies even if we replace with .

Lemma 2. (1)Assume that and , and there exists such that , then .(2)Suppose that and satisfy equations (9) and (10), respectively, and and a.e. , we have(i)If and and and , then .(ii)If and or and , then respectively.(iii)If , , , and satisfy the conditions (11) and (12), respectively, then

Remark 1. If and , , we obtain the following estimate:

Proof. (1)Let such that on , on and on , where . Let for . We have the following equation: Using the change of variable , we obtain the following equation: Letting , then by the Dominated Convergence theorem we deduce the desired result.(2)First, we proof 2.i). Since in a neighborhood of , then by equations (9) and (10), we can write the following equations: and where such thatFrom equation (25), we obtain the existence of , such thatDefining , thenand from equations (23) and (24), we deduce the following equation:Using the change of variable , and integrating by parts, we obtain the following equation:Hence, by using equation (26), we obtain the following equation:where .
Therefore, for and , we reach that . This concludes the proof of .
To prove , first we start by the case and .
Let for is large enough, where is the positive eigenfunction corresponding to the first eigenvalue of the operator in .
We have the following equation:Using equations (23) and (24), we obtain the following equation:By a simple change of variable and integrating by parts, we obtain the following equation by equation (28):where . Letting , we obtain the following equation:thusSimilarly, we deduce in the case and , thatNow, we proof 2.iii). Since and satisfy equations (11) and (12), respectively, for all we have the following equation:By applying Lemma 1 for , we obtained the following equation:Thus,The proof is complete.
Inspired by [1], we obtain the following result.

Lemma 3. We have the following equation:with

Proof. Consider a minimizing sequence for . Let be chosen later. Taking and in quotient (15), we obtain the following equation:Observe thatLet and define the following function:The minimum of the function is achieved at the point with minimum valueChoosing and in equation (42) such that , we obtain the following equation:hence,To complete the proof, let be a minimizing sequence for and define with the following equation:Then,By Young’s inequality, it follows thatBy equation (49), we have the following equation:Consequently,we know thatthenThus,We know that is achieved if and only if by the following function:where is a normalization constant and is a small positive constant; for more details, see [17, 18].
Set , where is a fixed function such that and in some neighborhood of . We have, from [8], thatwhere , and are positive constants.

Lemma 4. Assume that hold and one of the hypotheses is satisfied. Then,

Proof. Let such thatThen, by and equations (15) and (59), we have the following equation:We know by [11] and withwhere is the area of and by the same way we define .
Using equations (61) and (62), we distinct the following cases:(1)If , and , then(2)If , and , we have the following equation:(3)If , and , we have the following equation:(4)If , and , we have the following equation:(5)If , , and , we have the following equation:(6)If , , and , we obtain the following equation:(7)If , , and , we have the following equation:(8)If , and , it result in the following equation:(9)If , and , we have the following equation:Using the fact that,and from Lemma 3, we obtain the following equation:Let such thatthen, we have The conclusion follows from the previous inequalities.

3. Nonexistence Result

The main goal of this section is the nonexistence result. So we use Pohozaev identity to prove Theorem 1.

Proof of Theorem 1. Let be the solution of . Multiplying the first equation in the system by on both sides and integrating by parts, we obtain the following equation:where denotes the outward normal to .
Similarly, we obtain for the second equation of as follows:Combining equations (77) and (78), we write the following equation:On the other hand, multiplying the two equations in the system by and , respectively, integrating by parts, and by summing the obtained results, we obtain the following equation:Combining equations (79) and (80), we obtain the following equation:If is a star-shaped domain about , we have the following equation:By , we reach thatwhich is a contradiction.

4. Existence Results

We first verify that satisfies the geometric conditions of the mountain pass theorem.

Lemma 5. Assume that is satisfied, then(i)There exist and such that for all with .(ii)There exists , with such that .

Proof. (i)From Hölder’s inequality, Sobolev embedding and it follows that where is a positive constant. Then, there exists such that , for small enough.(ii)We have as , for any ; thus, there exists with such that .Next, we prove an important lemma which ensures the local compactness of a Palais–Smale sequence for .

Lemma 6. If , then satisfies condition.

Proof. Suppose that satisfieswith as , then is bounded in . Going if necessary to a subsequence, we can assume that as It follows that is a weak solution of the system, i.e.,We setApplying the following relations as in Brézis-Lieb Lemma [19]:we obtain the following equations:andSince , thenWe may therefore assume thatFrom the definition of , we obtain the following equation:Thus, . Assume that , then .
Passing to the limit in equation (90), we obtain the following equation:and hence, , for .
On the other hand, we have the following equation:which yields a contradiction. Thus, strongly in .

Lemma 7. Let satisfy equation (59) and as in Lemma 4 with . Then,

Proof. We have the following equation:Denoting by the function in the right-hand side of the last inequality. A forward computation assures thatis the maximum point for . So,By Lemmas 3 and 4 and for small , we obtain the following equation:and thus, equation (97) holds.
Now, we can prove Theorems 2 and 3.

Proofs of Theorem 2 and 3. By Lemma 5, there exists a Palais–Smale sequence a sequence in withLemmas 6 and 7 imply that verifies the condition . Using the mountain pass theorem in [20] whenever and the Ghoussoub–Preiss version in [21] whenever , respectively, we obtain a nontrivial critical point of .
ConsiderRepeating the above process for , we obtain a non-negative solution to the problem . From and by using the maximum principle, we conclude that and .

Data Availability

The functional analysis data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1399) during the academic year 1444AH/2023AD and Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number (1399) during the academic year 1444AH/2023AD. This work was supported by Qassim University, represented by the Deanship of Scientific Research and Algerian Ministry of Higher Education and Scientific Research.