Abstract

In this paper, by using the mountain pass theorem and the concentration compactness principle, we prove the existence of a positive solution for a -Kirchhoff-type problem with critical Sobolev exponent.

1. Introduction and Main Result

In this article, we study the existence of a positive solution for the following -Kirchhoff-type problem: where is a bounded domain, , , , , , is the usual norm in given by , and is the critical Sobolev exponent corresponding to the noncompact embedding of into . This problem contains an integral over , and it is no longer a pointwise identity; therefore, it is often called nonlocal problem. It is also called nondegenerate if and , while it is named degenerate if and .

In the past several decades, much attention has been paid to the Kirchhoff-type problem which is closely related to the stationary analog of the following equation: proposed by Kirchhoff in [1] as an extension of the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the strings during the vibrations, where , , , , and are constants. Kirchhoff’s model takes into account the changes in length of the strings produced by transverse vibrations. These problems also serve to model other physical phenomena as biological systems where describes a process which depends on the average of itself (for example, population density). The presence of the nonlocal term makes the theoretical study of these problems so difficult; then, they have attracted the attention of many researchers in particular after the work of Lions [2], where a functional analysis approach was proposed to attack them.

In the last few years, great attention has been paid to the study of Kirchhoff problems involving critical nonlinearities. These problems create many difficulties in applying variational methods because of the lack of the compactness of the Sobolev embedding. It is worth mentioning that the first work on the Kirchhoff-type problem with critical Sobolev exponent is Alves et al. in [3]. After that, many researchers dedicated to the study of several kinds of elliptic Kirchhoff equations with critical exponent of Sobolev in bounded domain or in the whole space ; some interesting studies can be found in [4–9] and the references therein. More precisely, Naimen in [8] generalized the results of [10] to the semilinear Kirchhoff problem: where is a bounded domain, , and . Under several conditions on and , he proved the existence and nonexistence of solutions. For larger dimensional case, Figueiredo in [5] considers the case if is sufficiently large. Matallah et al. in [7] studied the existence and nonexistence of solutions for the following -Kirchhoff problem: where is a continuous function satisfying some extra assumptions and is a continuous function satisfying some conditions. Benaissa and Matallah in [4] discussed the problem where satisfies some conditions. Very recently, Benchira et al. in [11] have generalized the results of [12] to the nonlocal problem (1) with , , , , if , and if ( is the best Sobolev constant for the imbedding .

Inspired by the above works, especially by [8, 11], we are devoted to studying the existence of positive solutions for problem (1) for all positive. In our problem, a typical difficulty occurs in proving the existence of solutions because of the lack of the compactness of the Sobolev embedding . Furthermore, in view of the corresponding energy, the interaction between the Kirchhoff-type perturbation and the critical nonlinearity is crucial.

The main result of this paper is the following.

Theorem 1. Assume that , , , and . Then, problem (1) has a positive solution for all .

Remark 2. If then In the case where , it is difficult to show that a Palais-Smale sequence of the corresponding energy is bounded; in this case, the authors in [7, 9] used the truncation method to show the existence of solution under the condition “ sufficiently large.” Our objective in this paper is the existence of solution for all .

Let us simply give a sketch of the Proof of Theorem 1. The main tool is variational methods; more precisely, by using the mountain pass theorem [13], we obtain a critical point of the corresponding energy. The main difficulties appear in the fact that problem (1) contains the critical Sobolev exponent; then, the functional energy does not satisfy the Palais-Smale condition in all range; to overcome the lack of compactness, we need to determine a good level of the Palais-Smale condition, and we must verify that the critical value is contained in the range of this level. This is the key point to obtain the existence of a mountain pass solution.

This paper is composed of two sections in addition to the introduction. In Section 2, we give some preliminary results which we will use later. Section 3 is devoted to the proof of main result.

2. Preliminary Results

In this paper, we use the following notations: denotes strong (resp., weak) convergence, denotes as , is the ball centred at and of radius , , and denote various positive constants. We define the best Sobolev constant for the imbedding as

Recall that the infimum is attained in by the functions of the form

Moreover, satisfies

Let be a positive constant and set such that for and for and Set and take so that Then, we have the well-known estimates as :

(See [14, 15]).

The energy function corresponding to problem (1) is given by

Notice that is well defined in and belongs to . We say that is a weak solution of (1), if for any there holds

Hence, a critical point of functional is a weak solution of problem (1).

Definition 3. Let ; a sequence is called a sequence (Palais-Smale sequence at level ) if

Let . We say that satisfies the Palais-Smale condition at level , if any sequence contains a convergent subsequence in

By [11], we have the following result.

Lemma 4. Let , , , and . For , we consider the function , given by

Then, (1)when , the equation has a unique positive solution and for all (2)when , the equation has a unique positive solution and for all

3. Proof of Main Result

To prove our main result, we use the mountain pass theorem. First, we will verify that the functional exhibits the mountain pass geometry.

Lemma 5. Suppose that , , , and . Then, there exists and positive numbers and such that (a), with (b) and

Proof. (1)Let ; by Sobolev and Young inequalities, we haveLet , from (14), one has As and there exists a sufficiently small positive numbers and such that (2)Let ; as , it holds that as , so we can easily find with , such that . The proof is complete

We define where is taken from Lemma 5.

Now, we prove the following lemma which is important to ensure the local compactness of sequences for .

Let and be defined in Lemma 4 and define with

Lemma 6. Assume that , , , , and is a sequence for with

Then, contains a subsequence converging strongly in .

Proof. As and , we have Then, is bounded in . Hence, by the concentration compactness principle due to Lions (see [6, 16]), there exists a subsequence, still denoted by , such that where is an at most countable index set, are nonnegative numbers, and is the Dirac mass at . Moreover, by the Sobolev inequality, we infer that We now claim that . To this end, by contradiction, suppose that ; then, there exists . For , let be a smooth cut-off function centered at such that , , and . Clearly, is bounded in . It follows from as that On the one hand, by Hölder’s inequality and (6), we have Since and is bounded in , then Moreover, by using Hölder’s inequality, we find So, as is bounded in , we deduce that By (22), (26), (28), and Hölder’s inequality, we obtain that is, Then, by (23), we obtain Now, we assume by contradiction that . Set and ; then by (31), we get It is clear that thanks to . So, from (32) and the definition of in Lemma 4, we get According to Lemma 4, there exist and such that and if with which implies that Moreover, using (23), we conclude that On the other hand, by the fact , one can get which implies that This is a contradiction. Hence, is empty and so On the other hand, we have for any . Set as ; then, from (40) and (41), we deduce that Taking the test function in (43), we get Therefore, the equalities (42) and (44) imply that . Consequently, converges strongly in , which is the desired result.
The energy functional satisfies the Palais-Smale condition at level for any . So, the existence of the solution follows immediately from the following lemma.

Lemma 7. Let , , , and Then,

Proof. We define the functions and such that Then, Note that and when is close to , so is attained for some . Furthermore, from , it follows that By multiplying the equation in (49) by , we obtain Easy computations show that By applying (9), we have for small enough On the other hand, we multiply the equation in (50) by and by recalling (53), we obtain By applying (9), we have for small enough Now, we estimate .
It follows from that that is, Set and As , then by (57), we get which implies from Lemma 4 that with defined in (34). Therefore, , where As is concave, then is convex and so Since , one has So, we deduce that Consequently, by (9) and as , we have Taking small enough, we obtain . Thus, the proof of this lemma is completed.

Now, we can proof the existence of a mountain pass-type solution.

Proof of Theorem 1. Applying Lemma 5, we get that possesses a mountain pass geometry. Then, from the mountain pass theorem [13], there exists a sequences of . According to Lemmas 6 and 7, has a subsequence (still denoted by ) such that in . Hence, is a critical point of and therefore a solution of (1).

Now, we show that . To obtain a contradiction assume that . We have Then, . By the strong maximum principle [17], one has . Theorem 1 can be concluded.

Data Availability

Data from functional analysis and variational methods used to support the results of this study are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge (1) Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number 1125 during the academic year 1443AH/2021AD and (2) Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number 1423 during the academic year 1443AH/2021AD.