In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to a p(z)–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:

(1)

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

In recent years, several researchers have been interested in equations or systems involving the p(z)–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 p(z)–Laplacian systems:

(2)

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:

(3)

by using the variational techniques. Chen and Wu in [14] examined the semilinear version of

with more general parametric functions f_α, g_β, and convex-concave critical nonlinearity. Mercuri and Willem in [15] proved a representation theorem for Palais-Smale sequences involving the p–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 p–Laplacian. We refer to Chen and Squassina [19], Pawan and Sreenadh [20], and Biswas and Tiwari [21] for fractional p(z)–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 Δ_p(z) 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 ‖w‖_p(z) and ρ_q(z)(|Dw|), 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 B_vol(α, y) property. Then, there exists a constant K such that if 0 < α + β < K, 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

with

We define the Lebesgue space with variable exponent

as the set of all measurable function

such that

(4)

endowed with the Luxembourg norm. And the associated Sobolev space is given by

(5)

with the norm

(6)

And

Lemma 2 (see [12].)Let such that ν₁ ≥ 0, ν₁ ≠ 0. Let be a measurable function such that ν₁(z)ν₂(z) ≥ 1 a.e. in Then, for every

(7)

2.1. Sobolev Spaces on Manifolds

Definition 3 (see [22].)Let and be an atlas of

(8)

where

, g_ij are the components of the Riemannian metric g in the chart and dz is the Lebesgue measure of

Definition 4 (see [22].)The Sobolev space is the completion of with respect to the norm , where

(9)

with

is the norm of the k-th covariant derivative of w. 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 C¹. The length of ξ is

(10)

Let , we define the distance between u and v by

(11)

Definition 6 (see [22].)Log-Hölder continuity: let; we say that t is log-Hölder continuous, if there exists c > 0 such that

(12)

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

(13)

like bilinear forms. Then,

Definition 8 (see [22].)We say that has property B_vol(α, y), if the Ricci tensor of g noted by Rc(g) verifies for some α, and for all there exists some y > 0 such that where B₁(z) are the balls of radius 1 centered at some point z in terms of the volume of smaller concentric balls.

To compare the functionals ‖·‖_q(·) and ρ_q(·)(·), one has the relation

(14)

Proposition 9 (see [23].)Hölder’s inequality: for all and , we have

(15)

where r_q is a positive constant depending on q⁻ and q⁺.

Definition 10 (see [22], [26].)We define the Sobolev space on by

(16)

endowed by the norm

(17)

and we define

as the closure of

Theorem 11 (see [22], [23].)Let with as a compact Riemannian manifold.

(i)
If

(18)

Then, we have

(19)

(ii)
If

(20)

Then, we have

(21)

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 ‖(w, y)‖^p(z) = ‖Dw(z)‖^p(z) + ‖Dy(z)‖^p(z). In what follows, is the space of C^∞ 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 (w, y) ∈ J is a weak solution of the system if (w, y) ∈ J one has

(22)

for all

The functions

are assumed to satisfy the following assumption:

(23)

and the following condition holds

(24)

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

defined as

(25)

By a direct calculation, we have

and

(26)

for any

Consider the Nehari manifold

(27)

Then,

if and only if

(28)

which implies that

(29)

The Nehari manifold

is closely linked to the behavior of the function of the form

for s > 0 defined by

(30)

Lemma 14. Let (w, y) ∈ J\{(0, 0)}, then if and only if ζ′_{(w, y)}(s) = 0.

Proof. The result is a consequence of the fact that

(31)

From this lemma, we have that the elements in correspond to stationary points of the maps ζ_{(w, y)}.

Hence, we note that

(32)

(33)

By Lemma 14,

if and only if ζ′_{(w, y)}(1) = 0. Hence, according to (32), we have for

that

(34)

Thus, it is natural to split

into three corresponding to local minima, local maxima, and points of inflexion of ζ_{(w, y)}, i.e.,

(35)

Lemma 15. Let (w, y) ∈ J, then we have

(i)

(36)

(ii)

(37)

for some constants c₁, c₂ > 0.

Proof. (i) Using Theorem 11 (i), and Lemma 2, we get

(38)

Hence,

(39)

(ii) By Young’s inequality, Lemma 2, and Theorem 11 (ii), we have that

(40)

Hence,

(41)

Lemma 16. For each (α, β) ∈ ℝ²\{(0, 0)}, there exists a constant K₁ > 0 such that for any 0 < α + β < K₁, we have

Proof. Suppose otherwise, that for all (α, β) ∈ ℝ²\{(0, 0)}. Let such that ‖(w, y)‖ > 1. Then, by Lemma 15, (34), and the definition of we have

(42)

that is,

(43)

Then,

(44)

Analogously,

(45)

Then,

(46)

thus,

(47)

According to (44) and (47), we deduce that

(48)

which is a contradiction. Hence, we can conclude that for any 0 < α + β < K₁, we have

Lemma 17. If (w, y) is a minimizing of on such that Then, (w, y) is a critical point of

Proof. Let (w, y) be a local minimizing of in any subset of Then, in any case, (w, y) is a minimizer of under the constraint

(49)

Since the constraint is nondegenerate in (w, y), then by the theory of Lagrange multipliers, there exists σ ∈ ℝ such that

(50)

Thus,

(51)

Since ζ′′_{(w, y)}(1) ≠ 0 and we obtain that σ = 0, which completes the proof.

Lemma 18. For every (α, β) ∈ ℝ²\{(0, 0)} such that 0 < α + β < K₁. The functional is bounded and coercive on

Proof. For any according to (23), (24), (29), and Lemma 15, we get

(52)

As p⁻ > q⁺, then as ‖(w, y)‖⟶∞. It follows that is coercive and bounded below on for 0 < α + β < K₁.

Lemma 19.

(i)
If , then
(ii)
If , then

Proof. (i) Since we have ζ′′_{(w, y)}(1) > 0. Then, using (23) and (34), we get

(53)

then,

(54)

Hence,

(55)

(ii) Since

we have ζ′′_{(w, y)}(1) < 0. Thus, according to (23) and (34), we obtain that

(56)

then,

(57)

Hence,

(58)

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

(59)

Lemma 21. The following facts hold:

(i)
If α + β < K₁, then
(ii)
If α + β < K₂, then we have for some

Proof. (i) Let ; by (34), we have

(60)

then,

(61)

Hence, by (61) and (29), we have

(62)

According to (24) and (23), we get Therefore, from the definition of σ_α,β and it follows that(73)(ii) Let ; by (47), we have

(63)

and by Lemma 15 (ii), we get

(64)

then,

(65)

Hence,

(66)

According to (29), (66), and Lemma 15 (i), we deduce that

(67)

Thus, if we choose

(68)

we deduce that

for some positive constant c₀ depending on

, and β.

Lemma 22. For each (w, y) ∈ J\{(0, 0)}, there exists a constant K₃ > 0 such that for all α + β < K₃, we have the following:

(i)
If then there exists a unique (s⁻w, s⁻y) > 0 such that and
(ii)
If then there exist s_max > 0 and unique numbers 0 < s⁺ < s_max < s⁻, such as and

(69)

Proof. Before tackling our proof, we define s_max as follows:

(70)

for every

Hence, we have that ζ_{(w, y)}(s) is increasing for s ∈ [0, s_max] and decreasing for s ∈ (s_max, +∞) and achieves its maximum. We set ; by Lemma 19, we have that

(71)

with

(72)

(i)
For 0 < s < 1 which is sufficiently small, we have

(73)

and for s > 1 which is sufficiently large, we get

(74)

Since ζ_{(w, y)}(s) achieves its maximum, then by Lemma 14, On the other hand, if then there is a unique s⁻ > s_max such that and since

(75)

We obtain that

For s > 1, we get by (74) and (34) that

(76)

and for 0 < s < 1, we deduce again by (73) and (34) that

(77)

Thus, s⁻ is unique, which achieves the proof.

(ii)
If we have

(78)

Therefore, there are unique s⁺ and s⁻ such that 0 < s⁺ < s_max < s⁻,

(79)

thus, by (i), we have

and

Hence,

(80)

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 α + β < K = min{K₁, k₂}, the functional has a minimizer (w₀, y₀) in which satisfies the following assumptions:

(i)
(ii)
is a solution of

Proof. (i) Thanks to Lemma 18, is bounded below on which in particular is bounded below in . Then, there exists a minimizing sequence such that

(81)

Since, is coercive, is bounded on J. Hence, we suppose that, without loss generality, on J, and by the compact embedding (Theorem 11), we have

(82)

Now, we shall demonstrate that and in as n⟶+∞. Otherwise, let or in as n⟶+∞. Then, we have

(83)

using (82), we obtain that

(84)

since,

we get

(85)

That is,

(86)

By (82) and (83), we have

(87)

Since, p⁺ > q⁻ for ‖(w, y)‖ > 1, we deduce that

(88)

which is a contradiction with Lemma 21. Hence,

(89)

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,

(90)

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

Lemma 25. If α + β < K = min {K₁, K₂}, then has a minimizer in such that

(i)
(ii)
is a solution of

Proof. (i) As is bounded below on and so on Then, there exists a minimizing sequence such that

(91)

As is coercive, is bounded in J, and thus, there exists such that up to a subsequence and according to Theorem 11, we obtain

(92)

According to (92) and Lemma 24, we deduce that

(93)

On the other hand, if then there exists a constant s > 0 such that , and according to (92) and (93), we have

(94)

Considering (32) and (94), we get

(95)

For n large enough, Since for all n ∈ ℕ, we have and for every n ∈ ℕ. By Lemma 22, we get for s > 0, then from (95), we must have s < 1. Since and by Lemma 22, we conclude that 1 is the global maximum for Therefore, from Lemma 23, it follows that

(96)

It contradicts that Hence, strongly in J as n⟶+∞ and Using the fact that

(97)

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

(98)

Moreover, hence, we can assume w^± ≥ 0, y^± ≥ 0. From Lemma 17, (w^±, y^±) are two critical points of and, thus, are nonnegative nontrivial solutions of system

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.

Open Research

Data Availability

No data were used to support this study.

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On p(z)–Laplacian System Involving Critical Nonlinearities

Abstract

1. Introduction

2. Notations and Basic Properties

2.1. Sobolev Spaces on Manifolds

3. Proof of the Main Results

3.1. Nehari Manifold Analysis for

3.2. Existence of Nonnegative Solutions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Information

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On p(z)–Laplacian System Involving Critical Nonlinearities

Abstract

1. Introduction

2. Notations and Basic Properties

2.1. Sobolev Spaces on Manifolds

3. Proof of the Main Results

3.1. Nehari Manifold Analysis for

3.2. Existence of Nonnegative Solutions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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