Abstract
This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters , , , and . To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.
1. Introduction
The proposed problem (2) is important in many fields of sciences, and it arises in biological applications (e.g., population dynamics) or physical applications (e.g., models of a nuclear reactor) and has drawn a lot of attention; see [1, 2] and references therein.
A natural question that arises in concert applications is to see what happens if these elliptic problems (degenerate or nondegenerate) are affected by certain singular perturbations.
The degeneracy and singularity occur in system (2); thus, standard variational methods are not applied which means that in our work, we research the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters , , , and .
In recent years, much attention has been paid to the existence of nontrivial solutions for problems of the type as follows:
Wang and Zhou [3] have proved that , for and , has at least two distinct solutions when and under some sufficient conditions on . In [4], Bouchekif and Matallah have shown the existence of two nontrivial solutions of when , , and with a positive constant and under some appropriate conditions on functions and .
Many existing results are available for regular and critical problems that arise from potentials; see, for example, [5–7]. However, to our knowledge, there are few results for singular systems (see [8, 9]).
This paper is organized as follows. In Section 2, we give our system and main results. In Section 3, we cite some preliminaries. Conclusion is presented in Section 4.
2. The Mathematical Model and Main Results
This paper deals with the existence and multiplicity of nontrivial solutions to the following proposed problem:where is a bounded regular domain in containing 0 in its interior, , , , , is the critical Caffarelli–Kohn–Nirenberg exponent, , and are positive real such that , is a real parameter, and is a function defined on .
By , we denote the completion of the space with respect to the norm as follows:
Using the Hardy inequality, this norm is equivalent to . More explicitly, we have
The space is endowed with the norm
Since our approach is a variational method, we define the functional on bywhere
A couple is a weak solution of the proposed problem (2) if it satisfieswith
Here, denotes the product in the duality , .
We list here a few integral inequalities. The first one that we need is the Caffarelli–Kohn–Nirenberg inequality [10], which ensures the existence of a positive constant such that
In (10), as , then , and we have the following weighted Hardy inequality [11]:
Let
From [12], is achieved.
Lemma 1. Let be a domain (not necessarily bounded), , and . Then, we have
For simplicity of writing, let us note the quantity by .
Proof. The proof is essentially given in [1] with minor modifications.
We set assumptions on the function which is somewhere positive but which may change sign in (H1) and in (H2) There exists such that As regards, problems containing the weight function change sign; see [13–15] and references therein.
Here, we can address some background works on the critical points; see, for example, [16, 17].
Let be a positive number such thatwherewithThen, we obtain the following results.
Theorem 2. Assume that , , , , , (H1), and real parameter satisfying , then (2) has at least one nontrivial solution.
Theorem 3. In addition to the assumptions of Theorem 2, if (H2)̧ holds and verifies , then (2) has at least two nontrivial solutions.
3. Preliminaries
Definition 4. Let , be a Banach space and .(i) is a Palais–Smale sequence at level (in short ) in for ifwhere tends to 0 as goes at infinity.(ii)We say that satisfies the condition if any sequence in for has a convergent subsequence.
3.1. Nehari Manifold
It is well known that is of class in and the solutions of (2) are the critical points of which is not bounded below on . Consider the following Nehari manifold:
Thus, if and only if
Note that contains every nontrivial solution of problem (2). Moreover, we have the following results.
Lemma 5. is coercive and bounded from below on .
Proof. Let such that . If , then, by (19) and the Hölder inequality, we obtainwhereand we deduce thatfor .
Thus, is coercive and bounded from below on .
DefineThen, for ,Now, we split into the following three parts:and .
We have the following results.
Lemma 6. Suppose that is a local minimizer for on. Then, if , is a critical point of .
Proof. If is a local minimizer for on , then is a solution of the optimization problemHence, there exists a Lagrange multiplier such thatThus,However, , since . Hence, . This completes the proof.
Lemma 7. There exists a positive number such that, for all verifyingwe have .
Proof. Let us reason by contradiction.
Suppose such that . Then, by (24) and for , we haveMoreover, by the Hölder inequality and the Sobolev embedding theorem, we obtainandFrom (31) and (32), we obtain , which contradicts our hypothesis.
Thus, . DefineFor the sequel, we need the following Lemma.
Lemma 8. (i)For all such that , one has .(ii)For all such that , one haswhere
Proof. (i)Let . By (24), we have and so We conclude that .(ii)Let . By (24), we getMoreover, by Sobolev embedding theorem, we haveThis impliesBy (20), we getThus, for all such that , we have .
For each , we write
Lemma 9. Let satisfy . For each with , one has the following:(i)If , then there exists a unique such that and(ii)If , then there exists unique and such that , , and
Proof. With minor modifications, we refer to [18].
Taking the idea of the work of Brown–Zhang [18], we prove the following result.
Proposition 10. (i)For all such that , there exists a sequence in (ii)For all such that , there exists a sequence in
4. Conclusion
In our work, we have searched the critical points as the minimizers of the energy functional associated with the problem on the constraint defined by the Nehari manifold , which are solutions to our problem. Under some sufficient conditions on coefficients of the proposed problem (2) such that , , , , , , , , and , we split into two disjoint subsets and ; thus, we obtain that (2) has two nontrivial solutions and . Since , this implies that and are distinct.
Appendix
A. Proof of Theorem 2
Drawing on the works of [18, 19], we establish the existence of a local minimum for on .
Proposition 11. For all such that , the functional has a minimizer and it satisfies(i)(ii) is a nontrivial solution of (2)
Proof. If , then by Proposition 10, there exists a sequence in ; thus, it is bounded by Lemma 5. Then, there exists , and we can extract a subsequence which will be denoted by such thatand we haveThus, by (45), is a weak nontrivial solution of (2). Now, we show that converges to strongly in . . Suppose this is not true, then by the lower semicontinuity of the norm, either or , we obtainWe get a contradiction. Therefore, converges to strongly in . Moreover, we have . If not, then by Lemma 9, there are two numbers and , uniquely defined so that and . In particular, we have . Sincethere exists such that . By Lemma 9, we getwhich is a contradiction.
B. Proof of Theorem 3
Next, we establish the existence of a local minimum for on . For this, we require the following lemma.
Lemma 12. For all such that , the functional has a minimizer in , and it satisfies(i)(ii) is a nontrivial solution of (2) in
Proof. If , then by Proposition 10, there exists a , sequence in ; thus, it is bounded by Lemma 5. Then, there exists , and we can extract a subsequence which will be denoted by such thatThis impliesMoreover, by (24), we obtainthus, by (31) and (52), there exists a positive numbersuch thatThis implies thatNow, we prove that converges to strongly in . Suppose this is not true, then either or . By Lemma 9, there is a unique such that . Sincewe haveand this is a contradiction. Hence,Thus,By (54) and Lemma 6, we may assume that is a nontrivial solution of (2).
Now, we complete the proof of Theorem 3. By Proposition 11 and Lemma 12, we obtain that (2) has two nontrivial solutions and . Since , this implies that and are distinct.
Finally, for every , problem (2) has a solution such that . Thus, there exist with as . Then, we get .
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 (4424) during the academic year 1445AH/2024AD.