Existence of Solutions for p-Kirchhoff Problem of Brézis-Nirenberg Type with Singular Terms

Abstract

In this paper, we prove the existence of positive solution for a p-Kirchhoff problem of Brézis-Nirenberg type with singular terms, nonlocal term, and the Caffarelli-Kohn-Nirenberg exponent by using variational methods, concentration compactness, and maximum principle.

1. Introduction and Main Result

In this paper, we consider the following p-Kirchhoff problem of Brézis-Nirenberg type with singular terms where is a bounded smooth domain in , , , is the critical Caffarelli-Kohn-Nirenberg exponent corresponding to the noncompact embedding of into where is the closure of with respect to the norm and denotes the usual weighted space with the weight .

Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the term which implies that the equation in (1) is no longer a pointwise identity. In the case and , it is analogous to the stationary version of equations that arise in the study of string or membrane vibrations, namely, where denotes the displacement and is the external force. Equations of this type were first proposed by Kirchhoff in 1883 [1] to describe the transversal oscillations of a stretched string.

These problems serve also to model other physical phenomena as biological systems where describes a process which depends on the average of itself (for example, population density).

In recent years, Kirchhoff-type problems received much attention, mainly after the famous article of Lions [2]; they have been studied in many papers by using variational methods, see [3–9] and the references therein.

The problem (1) without nonlocal term and without singular terms has been treated by Brézis and Nirenberg [10] for . Subsequently, an increasing number of researchers have paid attention to semilinear or quasilinear elliptic equations with critical exponent of Sobolev or Caffarelli-Kohn-Nirenberg, for example, see [11, 12] and the references therein.

In [7], Naimen generalized the results of [10] to the nonlocal problem (1) with and without singular terms Kang in [1] generalized the results of [10] to a quasilinear problem with singular terms and without the nonlocal term .

Thus, it is natural for us to consider the quasilinear Brézis-Nirenberg problem in [10] with nonlocal term and singular weights, The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of into prevent us from using the variational methods in a standard way. So, motivated by all the works mentioned above, we prove existence results of our problem for large range of and under some little, as possible, conditions on . We show that the existence of solutions depends on the parameter and the position of with respect to . Here, we need more delicate estimates.

To the best of our knowledge, many of the results are new for , and even in the case .

Our technique is based on variational methods and concentration compactness argument [2].

Note that if , and , the problem (1) has positive radial ground state solution . Moreover, for any the extremal functions is a minimizer for and satisfies

On the other hand, for , and we defined the first eigenvalue of the problem as and it is positive, (see for instance [1]).

The main result is concluded as the following theorem.

Theorem 1. Let , and with Then, the problem (1) has a positive solution in the following cases: (1) and (2) and

Remark 2. In the case where and is a star-shaped domain with respect to the origin, we can easily verify that the problem (1) has no nontrivial solution by using a Pohozaev-type identity.

This paper is organized as follows. In Section 2, we study the variational framework and give some preliminary results. In Section 3, we show the existence result and we will prove Theorem 1.

2. Variational Framework and Preliminary Results

The starting point of the variational approach to problem (1) is the following Caffarelli-Kohn-Nirenberg inequality in [13] which is also called the Hardy-Sobolev inequality. Assume that and and then, for some positive constant . In the case where , we have and we have the following Hardy inequality:

Definition 3. We say that is a weak solution of equation if for any

Next, we define the energy functional associated to problem (1), for all

Notice that the functional is well defined in and belongs to and a critical point of is a weak solution of problem (1).

Definition 4. 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

Lemma 5. Assume , and Let and be a sequence for . Then, for some with

Proof. We have That is, for any

Then, as , it follows that

As and , we obtain that is bounded in . Up to a subsequence if necessary, there exists a function such that in in in for all and a.e on Then, and thus . This completes the proof of Lemma 5.

The following lemma is very important for giving the local Palais-Smale condition.

Lemma 6. Let and for For , we consider the function given by Then, (1)If and then the equation has a unique positive solutionand for all (2)If then the equation has a unique positive solution and for all

Proof. (1)For and , we havethat is, the equation has a unique positive solution and for all (2)For we have andThen, for and for . Hence, is concave function and Moreover, we have and thus, from (25) and the concavity of , we can conclude that the equation has a unique positive solution and for all
Now, we prove an important lemma which ensures the local compactness of the Palais-Smale sequence for

For let be defined in Lemma 6 and define and

Lemma 7. Let , and Assume that and or and Then, the functional satisfies condition for all

Proof. Let is a sequence for with By the proof of Lemma 5, we have is a bounded sequence in Hence, by the concentration compactness principle due to Lions [2], there exists a subsequence, still denoted by such that where is an at most countable index set and is the Dirac mass at . Moreover, by the Sobolev-Hardy inequality we infer that We claim that is finite and for any , for , let be a smooth cut-off function centered at such that , and Since is bounded in and as , it holds by Hölder’s inequality Then, . Therefore, by (29), we deduce that Assume by contradiction that there exists such that Set and then by (32) we get It is clear that thanks to So, from (33) and the definition of in Lemma 6 we get

We will discuss it in two cases:

Case 1. and
According to Lemma 6, we have and if with which implies that

Case 2. , and . In this case, from Lemma 6, we get and if with which implies that

Hence, using (29), we deduce

By Young inequality we have we observe that thus, for we get since for and defined in Lemma 7. Contradiction with Then, is empty, which implies that

Now, set as ; then, we have for any Let , and then, from (42) and (43), we deduce that

Taking the test function in (45), we get

Therefore, the equalities (44) and (45) imply that Consequently, converges strongly in which is the desired result

3. Proof of the Main Result

Let be a positive constant and set such that for and for and Set

We have the well-known estimates as : where and are zeroes of the function that satisfy (see [1]).

On the other hand, the function has the unique minimal point and is increasing on . Thus, if , i.e., we have

Therefore, the equalities (48) and (51) imply that

Consequently,

Next, we show that For any , let

We can show easily that since is increasing on and decreasing on and So,

Lemma 8. Let , and . Assume that and , or and Then,

Proof. We define the following functions Note that and when is close to , so that is attained for some Furthermore, from it follows that Therefore, Choose small enough so that by (47) we have for some

Besides, it holds

For we have by (47)

Then, for small enough, the above estimates yield for some (independently of ).

For , , and for small enough we have by (56), which implies that is bounded above for all , that is, there exists a positive real number (independently of ).

Now, we estimate . It follows from

Set and . Then, by (61) the definition of we get which implies from (26) and the proof of Lemma 6 that Therefore, , where As is concave then is convex and so,

By , we have

So, from (64) we deduce that

Consequently, by (47)

which is the desired result.

Now, we can proof the existence of a positive solution.

Proof of Theorem 1. We verify that the functional satisfies the mountain pass geometry.
Note that . Let by Sobolev and Young inequalities, it holds that Let , since and from (67), one has

Then, we need to consider the following cases.

Case and

As there exists a sufficiently small positive numbers and such that

Since as there exists such that and .

Case and

In this case, we have and then, from (68), one has

As there exist such that

On the other hand, using (47) and taking small enough, we get for all . Then, as , it follows from the above inequality, as Thus, choosing sufficiently large such that and

Set where

By the Mountain Pass Theorem, there exists a Palais-Smale sequence at level Using Lemma 5, we have that has a subsequence, still denoted by , such that in as . Hence, from Lemmas 7 and 8, we have in as . Hence, and So, as , we can conclude that is a nonzero solution of (1) with positive energy. Now, we show that . Because which implies that By the strong maximum principle one has This completes the proof of Theorem 1.

4. Conclusion 1

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem. Our results and setting were more general and delicate, it difficult to obtain the solution in the degenerate case. Our technique was based on variational methods and concentration compactness argument and we needed to estimate the energy levels. We have shown the existence result for our problem (1) if , and with and the problem (1) has a positive solution in the following cases: (1) and 2) and

Data Availability

The [DATA TYPE] 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 (1) Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1123) during the academic year 1443AH/2021 AD and (2) Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number (1123) during the academic year 1443AH/2021 AD.