On <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 30.8425 16.7875" width="30.8425pt"><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><g transform="matrix(.017,0,0,-0.017,10.177,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,16.115,0)"><path id="g113-123" d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"/></g><g transform="matrix(.017,0,0,-0.017,24.662,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>–Laplacian System Involving Critical Nonlinearities

Aberqi, Ahmed; Bennouna, Jaouad; Benslimane, Omar; Ragusa, Maria Alessandra

doi:https://doi.org/10.1155/2022/6685771

Journal of Function Spaces

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Abstract Introduction Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 6685771 | https://doi.org/10.1155/2022/6685771

On –Laplacian System Involving Critical Nonlinearities

Ahmed Aberqi,¹Jaouad Bennouna,²Omar Benslimane,²and Maria Alessandra Ragusa³

Academic Editor: Giovanni Di Fratta

Received15 Feb 2022

Accepted05 Apr 2022

Published25 Apr 2022

Abstract

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to a –Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari’s method and doing a refined analysis on the associated fiber map and some variational techniques.

1. Introduction

In the present work, we investigate the existence of nonnegative nontrivial solutions to the following system:

Here, is a complete compact Riemannian N-manifold, to be specified later, and satisfying the assumptions (23) and (24) in Section 3. is the Laplacian operator on .

In recent years, several researchers have been interested in equations or systems involving the –Laplacian, not only for their application in several scientific fields, such as fluid filtration in porous media, constrained heating, elastoplasticity, and optimal control, but also for their mathematical importance in the theory of function spaces with variable exponents. For example, in [1], Zhang proved the existence of positive solutions under some conditions of the following class of –Laplacian systems: in bounded open set without assuming the symmetric radial conditions. And by using the subsuper solution technique, Boulaaras et al. in [2] have studied the asymptotic behavior of the system . In [3], Aberqi et al. established the existence of a renormalized solution for a class of nonlinear parabolic systems using the Gagliardo-Nirenberg theorem, with the source term being less regular. In addition, we refer to the work of Marino and Winkert [4] who studied this kind of system with nongrowth conditions, governed by a double-phase operator. For the systems with singular source data, we refer to Saoudi [5] and Papageorgiou et al. [6]. For more results, we refer to [7–11], as well as to [12], and the references therein.

Before explaining the novelty of this paper, we give an overview of the literature on this kind of system in . Adriouch and El Hamidi in [13] proved the existence and multiplicity of solutions to the following system: by using the variational techniques. Chen and Wu in [14] examined the semilinear version of with more general parametric functions , , and convex-concave critical nonlinearity. Mercuri and Willem in [15] proved a representation theorem for Palais-Smale sequences involving the –Laplacian and critical nonlinearities. For a deeper comprehension, see [16–18].

Next, we will mention some papers that deal with the same problem with the fractional –Laplacian. We refer to Chen and Squassina [19], Pawan and Sreenadh [20], and Biswas and Tiwari [21] for fractional –Laplacian. Readers may refer to the references given therein for more background.

Our goal in the present contribution is to study this kind of system with nonstandard convex-concave nonlinearity, in the Sobolev spaces on the complete manifold. We prove the existence of nonnegative nontrivial solutions using the Nehari manifold technique. However, we address the challenges due to the fact that is not homogeneous, also due to the non-Euclidean framework of the system. Moreover, we do not have enough background on this space, such as embedding results, Hölder inequality, and the relation between the and including the pertinent result proven in ([22], Proposition 2.5). This is the first existing result in this field to the best of our knowledge.

The theorem below contains our main result.

Theorem 1. Let satisfy the property. Then, there exists a constant such that if then the system admits at least two nonnegative weak solutions.

The organization of this contribution is given as follows. We start in Section 2 by presenting some definitions and properties of Lebesgue spaces with variable exponents on a bounded set of and on a complete manifold . After that, in Section 3, we give some properties of the Nehari manifold and set up the variational framework of the system Then, we establish the existence of two nonnegative nontrivial solutions to the system .

2. Notations and Basic Properties

This section is devoted to recalling some definitions and properties which will be used in the next sections (see [22–27]).

Consider an open-bounded set of with We define the Lebesgue space with variable exponent as the set of all measurable function such that endowed with the Luxembourg norm. And the associated Sobolev space is given by with the norm

And

Lemma 2 (see [12]). Let such that Let be a measurable function such that a.e. in Then, for every

2.1. Sobolev Spaces on Manifolds

Definition 3 (see [22]). Let and be an atlas of where , are the components of the Riemannian metric in the chart and is the Lebesgue measure of

Definition 4 (see [22]). The Sobolev space is the completion of with respect to the norm , where with . is the norm of the -th covariant derivative of If is a subset of then is the completion of with respect to where denotes the vector space of continuous functions whose support is a compact subset of

Definition 5 (see [22]). Let a curve of class . The length of is Let , we define the distance between and by

Definition 6 (see [22]). Log-Hölder continuity: let; we say that is log-Hölder continuous, if there exists such that

The set of Log-Hölder continuous functions on will be denoted by which is linked to by the proposition below.

Proposition 7 (see [24, 25]). Let be a chart of , such that like bilinear forms. Then,

Definition 8 (see [22]). We say that has property if the Ricci tensor of noted by verifies for some and for all there exists some such that where are the balls of radius 1 centered at some point in terms of the volume of smaller concentric balls.
To compare the functionals and , one has the relation

Proposition 9 (see [23]). Hölder’s inequality: for all and , we have where is a positive constant depending on and

Definition 10 (see [22, 26]). We define the Sobolev space on by endowed by the norm and we define as the closure of in

Theorem 11 (see [22, 23]). Let with as a compact Riemannian manifold. (i)IfThen, we have (ii)IfThen, we have

Proposition 12 (see [24]). We have if is complete.

3. Proof of the Main Results

In this section, we prove our main result, and we note that endowed with norm In what follows, is the space of functions with compact support in .

3.1. Nehari Manifold Analysis for

First, we define the weak solution of system as follows.

Definition 13. We say that is a weak solution of the system if one has for all

The functions are assumed to satisfy the following assumption: and the following condition holds

To prove our main result, we will use Nehari manifold and fibering maps. The fact that is a weak solution is equivalent to being a critical point of the following functional defined as

By a direct calculation, we have and for any

Consider the Nehari manifold

Then, if and only if which implies that

The Nehari manifold is closely linked to the behavior of the function of the form for defined by

Lemma 14. Let then if and only if

Proof. The result is a consequence of the fact that

From this lemma, we have that the elements in correspond to stationary points of the maps

Hence, we note that

By Lemma 14, if and only if Hence, according to (32), we have for that

Thus, it is natural to split into three corresponding to local minima, local maxima, and points of inflexion of , i.e.,

Lemma 15. Let then we have
(i) (ii) for some constants

Proof. Using Theorem 11 (i), and Lemma 2, we get Hence, (ii) By Young’s inequality, Lemma 2, and Theorem 11 (ii), we have that Hence,

Lemma 16. For each there exists a constant such that for any we have

Proof. Suppose otherwise, that for all Let such that . Then, by Lemma 15, (34), and the definition of we have that is, Then, Analogously, Then, thus, According to (44) and (47), we deduce that which is a contradiction. Hence, we can conclude that for any we have

Lemma 17. If is a minimizing of on such that Then, is a critical point of

Proof. Let be a local minimizing of in any subset of Then, in any case, is a minimizer of under the constraint Since the constraint is nondegenerate in then by the theory of Lagrange multipliers, there exists such that Thus, Since and we obtain that , which completes the proof.

Lemma 18. For every such that The functional is bounded and coercive on

Proof. For any according to (23), (24), (29), and Lemma 15, we get As then as It follows that is coercive and bounded below on for

Lemma 19. (i)If , then (ii)If , then

Proof. Since we have Then, using (23) and (34), we get then, Hence, (ii) Since we have Thus, according to (23) and (34), we obtain that then, Hence,

Remark 20. As a consequence of Lemmas 16–18, we have for every with , and is coercive and bounded below on and . We define

Lemma 21. The following facts hold: (i)If then (ii)If then we have for some

Proof. Let ; by (34), we have then, Hence, by (61) and (29), we have According to (24) and (23), we get Therefore, from the definition of and it follows that(73)(ii) Let ; by (47), we have and by Lemma 15 (ii), we get then, Hence, According to (29), (66), and Lemma 15 (i), we deduce that Thus, if we choose we deduce that for some positive constant depending on , and

Lemma 22. For each there exists a constant such that for all we have the following: (i)If then there exists a unique such that and (ii)If then there exist and unique numbers such as and

Proof. Before tackling our proof, we define as follows: for every .
Hence, we have that is increasing for and decreasing for and achieves its maximum. We set ; by Lemma 19, we have that with (i)For which is sufficiently small, we haveand for which is sufficiently large, we get Since achieves its maximum, then by Lemma 14, On the other hand, if then there is a unique such that and since We obtain that
For we get by (74) and (34) that and for we deduce again by (73) and (34) that Thus, is unique, which achieves the proof. (ii)If we haveTherefore, there are unique and such that thus, by (i), we have and Hence,

3.2. Existence of Nonnegative Solutions

This section is devoted to proving the existence of minimizers in , , also to show the existence of two nonnegative solutions of system .

Lemma 23. For , the functional has a minimizer in which satisfies the following assumptions: (i)(ii) is a solution of

Proof. Thanks to Lemma 18, is bounded below on which in particular is bounded below in . Then, there exists a minimizing sequence such that Since, is coercive, is bounded on Hence, we suppose that, without loss generality, on , and by the compact embedding (Theorem 11), we have Now, we shall demonstrate that and in as Otherwise, let or in as Then, we have using (82), we obtain that since, we get That is, By (82) and (83), we have Since, for we deduce that which is a contradiction with Lemma 21. Hence, Consequently, is a minimizer of on
(ii) According to Lemma 17, we deduce that is a solution of

Lemma 24. Let and be any two bounded sequences in . Then,

Proof. Similar to the proof ([21], Theorem 5.2), we will omit it.

Lemma 25. If then has a minimizer in such that (i)(ii) is a solution of

Proof. As is bounded below on and so on Then, there exists a minimizing sequence such that As is coercive, is bounded in , and thus, there exists such that up to a subsequence and according to Theorem 11, we obtain According to (92) and Lemma 24, we deduce that On the other hand, if then there exists a constant such that , and according to (92) and (93), we have Considering (32) and (94), we get For large enough, Since for all we have and for every By Lemma 22, we get for then from (95), we must have Since and by Lemma 22, we conclude that 1 is the global maximum for Therefore, from Lemma 23, it follows that It contradicts that Hence, strongly in as and Using the fact that and Lemma 15, we conclude that
(ii) From Lemma 17, is a solution of

Proof of Theorem 1. From Lemma 23 and Lemma 25, there are and such that Moreover, hence, we can assume From Lemma 17, are two critical points of and, thus, are nonnegative nontrivial solutions of system

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors declare that their contributions are equal.

Acknowledgments

This paper has been supported by PRIN 2017 n.2017AYM8XW 004.

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Copyright © 2022 Ahmed Aberqi et al. 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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