Abstract
In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in . Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.
1. Introduction and Main Results
Let be a regular bounded domain with a smooth boundary, , , , , and positive constants. Given a function specified later, we consider the following critical singular elliptic Kirchhoff type problem:
It is pointed out that the appearance of the integral over the domain implies that the equation is no longer a pointwise identity, and so the problem under consideration is nonlocal. This fact provokes some mathematical difficulties which make the study of this class of problems, particularly interesting. We refer to [1] and the references therein, for previous papers dealing with this subject. So, interested reader can find information on its historical development as well as the description of situations that can be modeled realistically, by the nonlocal problems, for example, physical and biological systems where the variable describes a process depending on the average of itself as population density. Then, the study of the solvability of nonlocal problems is motivated by its various applications.
For , problem is related to the stationary analogue of the Kirchhoff model introduced by Kirchhoff himself [2] in 1883 as an extension of the classical D’Alembert wave equation; he take into account the changes in length of the strings produced by transverse vibrations.
Actually, despite the intense development on elliptic Kirchhoff type problems without singular term (i.e., for ), see, for example, [3] and the references therein, results on the multiplicity of solutions are still not very abundant.
In the regular case and without the Kirchhoff term, more precisely when , Tarantello [4] mainly imposed a suitable assumption on and proved the existence of at least two solutions for Benmansour and Bouchekif [5] extended the results obtained by Tarantello to the nonlocal case where is a smooth bounded domain of , and are positive constants, and belongs to satisfying certain condition. They proved the existence of two weak solutions.
The same multiplicity result has been established by Sabri et al. in [6] when they considered under an appropriate condition on , the following problem where is a smooth bounded domain of and and are positive constants.
Elliptic problems with singularity are widely studied in the literature, and there are many results dealing with this kind of problems; we refer interested readers to [7–10]. In particular, Kang and Deng [11] generalized the main result of [4] to the following singular problem: with , , where is the first eigenvalue of the operator Here, the authors imposed the presence of the term which provided them with the main tool to obtain the second solution if .
In this paper, we would like to consider the nonlocal elliptic singular operator and a critical inhomogeneous nonlinearity also containing a Hardy term which is different than the previous works in the literature. Our main purpose is to give a multiplicity result. To our best knowledge, this kind of problems has not been considered before.
This problem is related to the following well-known weighted Hardy inequality [11]:
Let for be the Sobolev weighted space endowed with the norm , which is equivalent to the norm . From the work of Chaudhuri and Ramaswamy [9], we know that
We denote by the usual -norm, and is the norm in , the topological dual of .
Our main result reads as follows.
Theorem 1. Let , , , , and satisfy the condition where Then, the problem admits at least two solutions in
This work is organized as follows: Section 2 is devoted to some preliminary results which we will use later. In Section 3, we give the definition of the Palais-Smale condition and the proof of our main result.
2. Some Preliminary Results
Seeking a weak solution to the problem is equivalent to find a critical point to the energy functional given by
It means that is said to be a weak solution of if it satisfies
Clearly, is not bounded from below on , so we introduce the following appropriate subset of : which we split into three subsets corresponding to local minima, points of inflection, and local maxima of , respectively (for more details, see [12]). where
Let for and Put where
The function attains its maximum at the point where with
The following lemmas play crucial roles in the sequel of this work.
Lemma 2. The functional is coercive and bounded below on
Proof. We know that (since ). Therefore, from the Hardy inequality, we get
in particular, , where .
Thus, is coercive and bounded from below on .
Lemma 3. Under the condition and for all , there exist unique , , and such that , , , , , , and .
Proof. If is verified then, there exist unique , , , and such that and , which implies that and , for all ; and , which implies that and , for all ; and and , which implies that and , for all .
For , we know that the best constant is attained when by the functions
(see [13, 14]). Let be a positive constant and set such that for and for and Set and ; then, by [15], we have the following lemma.
Lemma 4. Let be a solution of ; then, we get
Lemma 5. Let and assume that the condition is verified; then, for small enough, we have where for every .
3. Proof of Theorem 1
3.1. Existence of Solution in
Proposition 6. If holds, then , and there is a critical point of such that and is a local minimizer for .
Proof. The proof is exactly the same as the one given in the proof of Theorem 1.1 in [5]. We omit the details here.
3.2. Existence of Solution in
In this part, we prove the existence of a second solution such that
Proposition 7. If holds, then , and there is a critical point of such that .
Due to the presence of the critical Sobolev exponent, a loss of compactness occurs, so we need the concept of the Palais-Smale condition.
A sequence is said to be a Palais-Smale sequence at level ( in short) for in if
When we say that a functional verifies condition at level , we mean that any sequence for has a convergent subsequence in .
In order to obtain the second solution, we give the following important lemma.
Lemma 8. Let verifying , then satisfies the condition for where
Proof. Let be a sequence with ; then, is a bounded sequence in Thus, by the compact embedding theorem, it has a subsequence still denoted such that weakly in a.e in , in , and in for all
Let It follows that weakly in and from the Brézis-Lieb lemma [16], we obtain
Since we get
By the fact that and we have
So, we get
Assume that with , it follows that
From the definition of , we lead to
As we deduce that
Consequently, we obtain
which is a contradiction. Therefore, ; then, strongly in .
In the search of our second solution, it is natural to show that Set It is obvious that given by belongs to
Lemma 9. Let and assume that the condition is verified for a small enough positive number. Then, for every , there exists such that for every .
Proof. Let us consider the functional defined by
We know that , and for near so , where
From Lemma 4 and as , we deduce that
On the other hand, we have
where
Then, for small enough, there exists also small enough such that for all
Therefore, as , there exists small enough such that we get
As in [4], we remark that under the condition , the manifold disconnects in exactly two components and . More precisely, and where
In particular, .
From a direct computation, we deduce that
Therefore, for sufficiently large, the estimate (38) holds for all
Thus, we derive that
Set
It is obvious that given by belongs to We conclude that
As the range of any intersects , one has from this and applying another time the Ekeland variational principle, we can obtain a minimizing sequence such that
We also deduce that
Consequently, we obtain a subsequence of and such that
This implies that is a critical point for , and
Therefore, according to Proposition 6 and Proposition 7, the proof of Theorem 1 is achieved.
Data Availability
The 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
This work is supported by Qassim University, represented by the Deanship of Scientific Research and Algerian Ministry of Higher Education and Scientific Research.