On <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

El Mokhtar, Mohammed El Mokhtar ould

doi:https://doi.org/10.1155/2022/5391028

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5391028 | https://doi.org/10.1155/2022/5391028

On -Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

Mohammed El Mokhtar ould El Mokhtar¹

Academic Editor: Andrea Scapellato

Received27 Jun 2022

Accepted24 Aug 2022

Published05 Sept 2022

Abstract

In this paper, we deal with -Laplace equations with singular nonlinearities and critical Sobolev exponent. By using the Nehari manifold, Mountain Pass theorem, and Maximum principle theorem, we prove the existence of at least four distinct nontrivial solutions.

1. Introduction

Let a bounded smooth domain in and consider the following -Laplace equations with singular nonlinearities where() , , and with is the critical Sobolev exponent, , and is the -Laplace operator which is degenerate if and singular if .

In recent years, researchers have been interested in studying problems of the type: where is a bounded smooth domain in , , and with is the critical Sobolev exponent and is a suitable function containing singularities on (see [1–4] and references therein). For and after the work of Brézis and Nirenberg [5], Problem (2) has studied by many authors (see, e.g.,[6–18]). Problem (2) becomes the well-known Brézis and Nirenberg problem and is studied extensively in [19]. Ding and Tang in [20] studied the existence of positive solutions with and satisfying (AR) condition in the case . Very recently, M. E. O. El Mokhtar et al. [21] considered Problem (1) with

The term represented by the function with is the key to this famous work because we will allow us to combine the perturbation with the variational methods to overcome shortcomings in the form of singularity. He is well known in the scientific literature that the problems dealt with in applied mathematics have their origins in different fields we will cite as example: heterogeneous chemical catalysis, kinetic chemical catalysis, heat induction or electrical induction, non-Newtonian fluid theory, and viscous fluid theory (see, e.g., [22–26]).

We encounter Problem (1) in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mapping, in the generalized reaction-diffusion theory, in the turbulent flow of a gas in a porous medium, and in the non-Newtonian fluid theory (see [27–30]).

Before giving our main results, we state here some definitions, notations, and known results.

We denote by with respect to the norms

We consider the following approximation equation: for any . The energy functional of (4) is defined by

for all , where

We know that is a function on

A point is a weak solution of Equation (1) if it satisfies

Here .,. denotes the product in the duality , .

Let

From [3], is achieved.

Let and be positive numbers such that

and

where,

with

The main results are concluded as the following theorems.

Theorem 1. Assume that , , and verifying then, the system (1) has at least one positive solution.

Theorem 2. In addition to the assumptions of the Theorem 1, there exists such that if satisfying , then (1) has at least two positive solutions.

Theorem 3. Under the assumptions of Theorem 2 then, there exists a positive real such that if satisfies , then (1) has at least four nontrivial solutions.

This paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last section, we prove Theorem 3.

2. Preliminaries

Definition 4 (see [31]). Let be a Banach space and . (i) is a Palais-Smale sequence at level (in short ) in for where tends to as goes at infinity (ii)We say that satisfies the condition if any sequence in for has a convergent subsequence

2.1. Nehari Manifold [32, 33]

It is well known that is of class in and the solutions of (1) are the critical points of which is not bounded below on . Consider the following Nehari manifold:

Note that contains every nontrivial solution of the problem (1). Thus, if and only if and

In our work, we research the critical points as the minimizers of the energy functional associated to the problem (1) on the constraint defined by the Nehari manifold.

In order to obtain the first positive solution, we give the following important lemmas.

Lemma 5. is coercive and bounded from below on .

Proof. Let such that . If , then by (14) and the Hölder inequality, we obtain where, Therefore, we obtain that for
Thus, is coercive and bounded from below on .

Define

Then, for

Splitting in three parts, we set

We have the following results.

Lemma 6. Suppose that is a local minimizer for on.If, then is a critical point of .

Proof. If is a local minimizer for on , then is a solution of the optimization problem: Hence, there exists a Lagrange multipliers such that Thus, But , since . Hence, . This completes the proof.

Lemma 7. There exists a positive number such that for all verifying we have .

Proof. Let us reason by contradiction.
Suppose that for all such that . Then, by (20) and for , we have Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain From (26) and (27), we obtain , which contradicts our hypothesis.

As then . Define

For the sequel, we need the following Lemma.

Lemma 8.
(i) For all such that , one has
(ii) For all such that there exists such hat

Proof. (i)Let . By (20), we haveand so since and . Then, we conclude that . (ii)Let . By(20), we getBy Sobolev embedding theorem, we obtain This implies By the proof of Lemma 5, we have Thus, for all such that we have with As in [34] we have the following result.

Proposition 9.
(i) For all such that , there exists a sequence in
(ii) For all such that , there exists a sequence in and for each .
Define

Lemma 10. Suppose that . For each , there exists unique and such that , and

Proof. With minor modifications, we refer the reader to [34].

3. Proof of Theorem 1

Now, taking as a starting point the work of Tarantello [35], 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 (1)

Proof. If , then by Proposition 9, (i) there exists a sequence in , thus, it bounded by Lemma 5. Then, there exists and we can extract a subsequence which will denoted by such that By (15) and (39), we have Thus, by (39), is a weak nontrivial solution of (1). Now, we show that converges to strongly in . Suppose otherwise. By the lower semi-continuity of the norm, if we have and we obtain We get a contradiction. Therefore, converge to strongly in . Moreover, we have . If not, then by Lemma 10, there are two numbers and , uniquely defined so that and . In particular, we have . Since there exists such that . Then, we get which contradicts the fact that . Since and , then by Lemma 6, we may assume that is a nontrivial nonnegative solution of (1). By the Harnack inequality, we conclude that (see, e.g., [29]).

4. Proof of Theorem 2

Next, we establish the existence of a local minimum for on . For this, we require the following Lemmas.

Lemma 12. Let be sequence for for some with in .
Then, and , with .

Proof. Let be a minimizing sequence for with is the unit sphere. By Ekeland’ s variational principle [12], we may assume . So is a sequence and therefore after passing to a subsequence. Hence and , which implies that , and Therefore, From (15) and considering small enough, we get which implies that with Set for all with Using (46), we obtain that where We have since . Then, we conclude that

Lemma 13. Let; then, the functional satisfies the condition in with , where

Proof. If , then by Proposition 9, (ii) there exists a , sequence in ; thus, it bounded by Lemma 5. Then, there exists and we can extract a subsequence which will denoted by such that Then, is a weak solution of (1). Let ; then, by Brézis-Lieb [36], we obtain Since and by (55) and (56), we deduce that Hence, we may assume that Moreover, by Sobolev inequality, we have Combining (60) and (59), we obtain Either, Then from (58), (59), Lemma 13 and Lemma 12, we obtain which is a contradiction. Therefore, , and we conclude that converges to strongly in .
Thus,

Lemma 14. There exists and such that for all , one has In particular, for all .

Proof. Let satisfies (4). Then, we have We consider the two functions: Then, for all for all , By the continuity of , there exists such that On the other hand we have Then, we obtain Now, taking such that we obtain Set We deduce that for all ; then, there exists such that with satisfying (4) and for all

Lemma 15. For all such that , the functional has a minimizer in and it satisfies is a nontrivial solution of (1) in .

Proof. By Proposition 9 (ii), there exists a sequence for , in for all . From Lemmas 13, 18, and 8(ii), for , satisfies condition and . Then, we get that is bounded in . Therefore, there exist a subsequence of still denoted by and such that converges to strongly in and for all .
Finally, by using the same arguments as in the proof of Proposition 11 for all , we have that is a solution of (1).

Now, we complete the proof of Theorem 2. By Proposition 11 and Lemma 15, we obtain that (1) has two positive solutions and . Since , then, and are distinct.

5. Proof of Theorem 3

Now, we consider the following Nehari submanifold of :

Thus, if and only if

Firstly, we need the following Lemmas.

Lemma 16. Under the hypothesis of theorem 3, there exist such that is nonempty for any .

Proof. Fix and let Clearly and as . Moreover, we have for put ; then, we obtain since with . Thus, we obtain if .
Then, there exists such that . Thus, and is nonempty for any .

Lemma 17. There exist positive real numbers such that for and any verifying

Proof. Let then by (14), (20) and the Holder inequality, it allows us to write Thus, if and choosing with defined in Lemma 16, then we obtain that

Lemma 18. Suppose and when Then, there exist and positive constants such that (i)We have(ii)There exists when , with , such that

Proof. We can suppose that the minima of are realized by and . The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have the following: (i)By (20) and (85), we getBy exploiting the function which achieve its maximum at the point such that if and the fact that, then, we obtain that (ii)Let , then we have for all Letting for large enough, we obtain For large enough, we can ensure .

Let and defined by

Proof of Theorem 19. If then, by the Lemma 5 and Proposition 9 (ii), verifying the Palais-Smale condition in . Moreover, from the Lemmas 6, 17, and 18, there exists such that Thus, is the third solution of our system such that and . Since (1) is odd with respect , we obtain that is also a solution of (1).
Finally, for every problem (4) has solution such that . Thus, there exist with as Then, we get .

6. Conclusion

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem on the constraint defined by the Nehari manifold , which are solutions of our problem. Under some sufficient conditions on coefficients of equation of (1) such that , and , we split in two disjoint subsets and ; thus, we consider the minimization problems on and , respectively. In Sections 3 and 4 we have proved the existence of at least two nontrivial solutions on for all if and .

Data Availability

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

Conflicts of Interest

The author declares that there is no conflicts of interest.

Acknowledgments

The author gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1250) during the academic year 1443AH/2022 AD.

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Copyright © 2022 Mohammed El Mokhtar ould El Mokhtar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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