Journal of Function Spaces
Volume 2022, Issue 1 2486542
Research Article
Open Access

On a Class of Schrödinger System Problem in Orlicz–Sobolev Spaces

H. El-Houari, 

Corresponding Author

H. El-Houari

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques Béni Mellal, Beni Mellal, Morocco Laboratoire de Recherche “Mathématiques Appliquées et Calcul Scientifique”, usms.ac.ma

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L. S. Chadli, 

L. S. Chadli

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques Béni Mellal, Beni Mellal, Morocco Laboratoire de Recherche “Mathématiques Appliquées et Calcul Scientifique”, usms.ac.ma

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H. Moussa, 

H. Moussa

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques Béni Mellal, Beni Mellal, Morocco Laboratoire de Recherche “Mathématiques Appliquées et Calcul Scientifique”, usms.ac.ma

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First published: 30 March 2022
https://doi.org/10.1155/2022/2486542
Citations: 2
Academic Editor: Salah Mahmoud Boulaaras

Abstract

Using the mountain pass theorem, we obtain the existence of a nontrivial and nonnegative weak solution of a quasi-linear Schrödinger system driven by the ω(⋅)-Laplacian operator in Orlicz–Sobolev spaces.

1. Introduction

For a long time, the theory of function spaces has been more or less strongly attached with the investigation of partial differential equations, between 1935 and 1938. S. L. Sobolev introduced the classical Sobolev spaces , p ∈ [1, +∞], q ∈ ℕ, as follows:
(1)
where ♌⊂ℝN is an open set and Dα is the weak (or distributional) partial derivative. The study of diverse mathematical problems with homogeneous operators satisfies nonpolynomial growth conditions rather than the classical p-structure using the classical model of monotonic operators which are mainly based on the analysis of Sobolev spaces for finding the weak solutions. However, the problems do not have any equivalence in the traditional Lebesgue and Sobolev spaces. This has led to the employment of a general class of spaces of functions called Orlicz–Sobolev spaces. Those spaces are composed of functions that have weak derivatives and satisfy specific integrability conditions. We can quote here [1–7] and reference therein.
In recent years, much of the work focused on the study of pulse propagation in nonlinear optical fiber. In a single-mode optical fiber, when nonlinear third-order effects are involved, the pulse propagation is qualified by the nonlinear Schrödinger equation. This equation was widely studied by several authors (see [8–11] and reference therein), and it has been proved that the existence and uniqueness of a soliton solution in the ground state, i.e., a positive, radial, symmetrical solution with the lowest energy level among all possible solutions, is a significant advantage. A mono-mode optical fiber is not precisely mono-mode; however, it is dual-mode due to the occurrence of birefringence. Birefringence tends to split a pulse in half in both polarization directions, but nonlinear effects can trap them together to prevent them from splitting. Menyuk in [11] has shown that both polarization components in a birefringent fiber are regulated by a nonlinear coupled Schrödinger system:
(2)
where b > 0 which depends on the fiber anisotropy. In the search for standing wave solutions of (2), i.e., solutions of the form:
(3)
and by rescaling the variables, we obtain that u and v satisfy the following system:
(4)
In this work, we deal with the existence of a weak solution to the following Schrödinger problem system type:
(5)
where defined by
(6)
and ϱi are two continuous functions satisfied the following:
  • (i)

    (A1): ϱi ∈ C(ℝN, ℝ) and

  • (ii)

    (A2): meas({y ∈ ℝN : ϱi(y) ≤ H}) < ∞, for all H > 0, where meas(⋅) denotes the Lebesgue measure in ℝN

Recently, problem (5) has attracted the interest of researchers. It was studied by L. Wang et al, in [12]. We weakened the hypotheses used to solve this problem in [12] by defining a new uniformly convex Banach space , which is injected into the Orlicz–Sobolev space , where Λ : ℝ+⟶ℝ+ is an N-function, i.e., Λ is a convex even function and is represented as follows:
(7)
where ω : ℝ+⟶ℝ+ satisfies
(8)
and is right continuous function and nondecreasing function. We present some examples of an N-functions as follows:
  • (1)

    ω(t) = tp−1, t > 0, 1 < p + 1 < N

  • (2)

    ω(t) = 2p(1 + t2)p−1,  t > 0,1 ≤ p < (N/2)

  • (3)

    ω(t) = (log(1 + |t|)/|t|),  t ∈ ℝ∗

We note by N the set of all N-functions. The relationship between the complementary function of Λ is given by
(9)
The N-function Λ ∈ Δ2, if for a certain constant 2 < k,
(10)

This article proceeds in this way. In the second section, we remind some of the properties on Orlicz and Orlicz–Sobolev spaces. The third section discusses the data assumption specification. In section four, we explain the existence of a result and its proof based on the mountain pass theorem (see [13]).

2. Some Preliminary Results and Hypotheses

In this section, we mention a few basic properties of the Orlicz–Sobolev space. We refer the reader to [14–16] for some proofs of the results of this section.

Let ♌ be an open subset of ℝN, N ≥ 2. The Orlicz space is defined as the set of real-valued measurable functions u on ♌ such that
(11)
Recall that is a Banach space under the norm
(12)
The closure in of all bounded measurable functions with compact support in is denoted by . The Space is reflexive if and only if (see Adams [14]). In , the Orlicz norm ‖u‖(Λ) is defined by
(13)
for all such that . One major inequality in is
(14)
We have the following inequality:
(15)
Moreover, the Hölder inequality holds
(16)
For Orlicz spaces, Young inequality reads as follows:
(17)
Also, in [14], for all s, t ≥ 0 and p ∈ [1, ∞), we have that
(18)
Let Λ1, Λ2 ∈ N. The notation Λ1 ≪ Λ2 means that for each ε > 0,
(19)
We now focus on the Orlicz–Sobolev space defined by
(20)
equipped with the norm
(21)
Thus, is a Banach space. Denote
(22)

Definition 1. Let and . in if for certain μ > 0, we have that ∫♌Λ((uk − u)/μ)dy⟶0.

The reason for this is that Λ ∈ Δ2 implies that

(23)

Theorem 1 (Theorem 3.2 in [17]). Let ♌⊂ℝN be an open, bounded set and satisfies a cone condition. Then,

  • (1)

    If , with continuous injection

  • (2)

    If , with continuous injection

Theorem 2. Let Λ ∈ N. Then, there is μ > 0 such that

(24)

In this work, we note by and by .

After that, we list few inequalities to be used in our proofs (see [14,18]).

Lemma 1. Let χ0(t) = min{tl, tn} and χ1(t) = max{tl, tn} for t ≥ 0 and Λ ∈ N, then these are equivalent:

  • (1)

    (25)

  • (2)

    χ0(t)Λ(s) ≤ Λ(st) ≤ χ1(t)Λ(s) for all t, s ≥ 0.

  • (3)

    Λ ∈ Δ2.

Lemma 2. If (25) holds, then

(26)

Lemma 3. Let be the complement of Λ and , for t ≥ 0 where and . If Λ is an N-function and (25) holds with l > 1, then satisfies the following:

  • (1)

  • (2)

  • (3)

Remark 1. We can see that and .

We assume that
(27)
Under the assumptions (A0) and (A∞), we may introduce Λ∗ ∈ N, say the conjugate given its inverse in ℝ+:
(28)

Lemma 4. Let , for t ≥ 0 where l∗ = (lN/(N − l)), and n∗ = (nN/(N − n)). If Λ ∈ N function γ satisfies the conditions (A0), (A∞). If (25) in Lemma 1 holds with l, n ∈ (1, N), then Λ∗ satisfies the following:

  • (1)

  • (2)

    χ4(t)Λ∗(s) ≤ Λ∗(st) ≤ χ5(t)Λ∗(s) ∀ t, s ≥ 0

  • (3)

Lemma 5. Λ ≪ Λ∗(t) near infinity, i.e., limt⟶∞(Λ(kt)/Λ∗(t)) = 0 ∀ k > 0.

2.1. Working Space

This subsection is dedicated to the definition of the subsite of Orlicz–Sobolev spaces .

Due to nature of ω(⋅)-Laplacian operator defined in (6), we need to consider the Orlicz–Sobolev framework and we will examine some specific techniques to Orlicz and the Orlicz–Sobolev spaces. For that we define
(29)
equipped with the following norm , where
(30)
with
(31)

Remark 2. If is a convex function, then we can see that is uniformly convex Banach space.

Remark 3. It can be noticed by means of Fatou’s lemma that

(32)

For any , we may assume u ≠ 0; by the definition of , there exists such that
(33)
By Fatou’s lemma,
(34)
Now, we check that is a norm in ; for that, it is sufficient to verify that is a norm in . Let us prove that is a norm in .
  • (i)

    If , then ∫♌ϱi(y)Λi(u(y)/Λ)dy = 0. Since Λi : ℝ+⟶ℝ+ and ϱi : ℝℕ⟶ℝ+ are tow continuous and positive functions, then ϱi(y)Λi(u(y)/Λ) = 0, and by the condition (A1), we infer that ϱi(y) > 0, and then Λi(u(y)/Λ) = 0. Or Λi(0) = 0 then (u(y)/Λ) = 0, which implies that u(y) = 0.

  • (ii)

    For α ∈ ℝ and , we have

    (35)
    • (i)

      Set K1 = K/α with α ≠ 0, then K = αK1, which gives that

      (36)

  • (iii)

    For the triangle inequality, let . Using Remark 3, then

(37)

Thus, .

Now, we recall some basic results which are needed to our study.

3. Hypotheses

We are going to show that the representation given by
(38)
where ωi=1;2(⋅) verified (8) exists and it is an N-function (see [16]). We suppose through our paper that Λi ∈ Δ2. Then, by Lemma 1, we have for all t > 0 that
(39)

Related to the function , our hypotheses are the following.

satisfies the following:

(F1): F : ♌×ℝ × ℝ⟶ℝ is a C1 function such that for all y ∈ ♌ and there exist Ψi=1,2 ∈ N such that Ψi ≪ Λi=1,2 and satisfy that
(40)
where
(41)
Moreover,
(42)
for every (y, u, v) ∈ ♌×ℝ × ℝ, where c1 > 0 and denote the complements of Ψi=1,2, respectively.
(F2):
(43)
(44)
(F3): γ : [0, ∞)⟶ℝ is a continuous function, such that
(45)
where
(46)
is in N and functions and satisfies
(47)
where c2, r > 0 and
(48)
Now, the functional is defined by
(49)

Lemma 6. These properties are true:

  • (1)

  • (2)

Proof.

  • (1)

    See Remark 3.

  • (2)

    Let , by choosing in Lemma 6, we obtain

(50)
and then
(51)

We deduce that from the definition of the norm (30),

(52)

Furthermore, let β > 0 and in Lemma 6, we get

(53)
and then
(54)

For ϵ⟶0 in the above inequality, we obtain that

(55)

This completes the proof.

Theorem 3 (see [19].)Let Λi, Φi ∈ N. If (A1) and (A2) and (25) hold then is continuous, and is compact for Φi ≪ Λi.

4. Main Results

In the following, we show the existence of a weak solution to problem (5).

Theorem 4. Let us suppose that (A1), (A2), (F1), (F2), and (F3) hold. Then, system (5) has a nontrivial weak solution.

Remark 4. Under assumption (39) and Lemma 1, satisfies Δ2.

Remark 5. Under assumptions (F1) and (F3), by Theorem 3, the following embeddings , , and are compact where and .

Remark 6. By (2) in Lemma 1, (44) and assumptions (F3) show

(56)

Remark 7. By Young’s inequality (17),

(57)

By (42) and (2) in Lemma 1, it is shown that there is c3 > 0 such that

(58)
which together with (25) and (40) show that there is c4 > 0 such that
(59)

We now have the tools to investigate problem (5). For this purpose, we define our working space under the following norm:
(60)
Notice that, according to Remark 2, is a separable and reflexive Banach space. We observe that the energy function I on corresponding to system (5) is
(61)
. Label by Ii=1,2: , the functionals
(62)

Then, I(u, v) = I1(u, v) − I2(u, v).

The ω(⋅)-Laplacian operator defined in (6) is well-defined from to its dual space . By (39) and Theorem 3, by means of similar arguments in [12], we can show that I1 is well-defined and and
(63)
for all . Then, the critical points of I on are weak solutions of system (5).

Lemma 7. The functional ℱi is weak lower semicontinuous.

Proof. By Corollary III.8 in [20], it is suffice to show that ℱi is lower semicontinuous.

Let such that uk⇀u in . By Theorem 3, if Φi(t) = |t|μ, then is compactly embedded in so it follows that uk⟶u in . As far as a subsequence,

(64)

With Fatou’s lemma, we obtain ℱi(u) ≤ liminfk⟶∞ℱi(uk). This ends the proof.

Definition 2. Let . A sequence uk in is a Cerami sequence (abbreviated to (C)c) at the level c ∈ ℝ for the functional J when

(65)

Remark 8. By means of arguments in [21], it appears that(PS)c due to Palais–Smale may be replaced by (C)c due to Cerami sequence in the mountain pass theorem.

Lemma 8 (Theorem 2.2 in [13]). Consider A as a real Banach space and I ∈ C1(A, R) fulfilling (PS)c. Assume that I(0) = 0 and

  • (i)

    (I1) We have for certain β, α > 0

  • (ii)

    (I2) We have I(e) ≤ 0 for some e ∈ A\Bβ

Then, I has a critical value c ≥ α.

Lemma 9. By (39), (F1), and (43), there is β, α > 0 such that .

Proof. By (43), we have that for each ϵ > 0, there is η > 0 such that

(66)

Hence,

(67)

As Ψi ∈ N, it also means that

(68)

Therefore, K > 0 exists so that

(69)

When K > η, it results from the continuity of t ↦ (Ψi(t)/|t|) in η ≤ |t| ≤ K that a > 0 exists such that (Ψi(t)/|t|) ≥ a. Therefore, using (59), we infer that

(70)

Under these conditions, it follows from (67) and (70) and continuity of that for ϵ = ϱ0 > 0 there is C > 0 in such a way that

(71)

When ‖u, v‖ ≤ 1, for a given ε ∈ (0,1) by monotony Λi, (24), and (2) in Lemma 6, we infer that

(72)
where C′ > 0 is obtained by Poincaré’s Inequality and compact embedding. From , we may choose β, α > 0 sufficiently small that I(u, v) > α for all with ‖(u, v)‖ = β.

Lemma 10. There is a point (u, v) ∈ W∖Bβ such that I(u, v) < 0.

Proof. By (44) and since is continuous, for every given constant K > 0, there is a constant CK > 0 such that

(73)

Now, choose with 0 ≤ u0(y) ≤ 1. Then, , and by (73) and (2) in Lemma 1, when t > 0, we have

(74)

Since K > 0 is arbitrary and limt⟶∞Λ1(t) = +∞, we can choose

(75)
and large t such that I(tu0, 0) ≤ 0 and ‖(tu0, 0)‖ > ρ.

Lemma 11. Any (C)c-sequence in is bounded.

Proof. Let be a (C)c-sequence of I on , then for large enough k by (39) and (65), we obtain

(76)

As a contradiction, we demonstrate the boundedness of the sequence {(uk, vk)}. Consider that there exists a subsequence of {(uk, vk)}, which always referred as {(uk, vk)}, so that

(77)

Next, we discuss the problem in two cases.

Case 1. Suppose that and . Let and Then, is bounded in a separable and reflexive space . By switching to a subsequence by Remark 5, there is a point so that

  • (a)

    in , in , and in a.e in ♌

  • (b)

    in , in , and , in a.e in ♌

First of all, we suppose that or has a nonzero Lebesgue measure. It is clear that

(78)

Then, by Fatou’s lemma, (76), and Remark 6, we have

(79)
which is a contradiction. Secondly, we assume that the two and have zero Lebesgue measure, that is, in and in . By Lemma 2, we have
(80)

For a large k, the inequality (80) becomes as follows:

(81)

By (18), we infer that

(82)
then
(83)
which is equivalent to
(84)
where R is given in (F3) and
(85)

By (43), there is CR > 0 so that

(86)
then
(87)

Furthermore, it is clear from Hölder’s inequality that

(88)
where χ denotes the characteristic function which satisfies For k large enough, by (47) and (76) and whether it is true that is continuous, we have
(89)

So, for k being sufficiently large, by Lemma 2, it exists S > 0 such that

(90)

Moreover, it is clear that

(91)

By Lemmas 2 and 3, (F3) implies that . Then, by (23), as . It results from Lemma 3(a) and (b) that

(92)
which implies
(93)

By combining (67), (88), (90), and (93) with (84), we obtain a contradiction.

Case 2. Assume that or for certain C > 0 and all k ∈ ℕ. In order to be general, we suppose that and for some C > 0 and all k ∈ ℕ. Let and , then and by Remark 5, and there is a point so that

  • (c)

    in , in , and in a.e in ♌

  • (d)

    in , in and , in a.e in ♌

Likewise, we first assume that has a nonzero Lebesgue measure. We may see that

(94)

Then, by Remark 6, (76), and Fatou’s lemma, we get a contradiction by

(95)

Secondly, let us assume that has a null Lebesgue measure, i.e., in . By Lemma 3(c) and (d), we have

(96)

This shows that a constant L > 0 exists such that

(97)

If k is large enough, (83) turns into

(98)
where K > 0 with K > 4SL (see (90) and (97)). Then, by (86), (90), (93), (97), and Hölder’s inequality, the estimate above means
(99)
which is a contradiction.

Lemma 12. The energy function I satisfies (C)c-condition.

Proof. Consider {(uk, vk)} as any (C)c sequence of I in . Lemma 8 proves that {(uk, vk)} is bounded. By switching to a subsequence {(uk, vk)}, by Remark 5, there is a point such that

  • (e)

    uk⇀u in , uk⇀u in , uk⇀u a.e ♌

  • (f)

    vk⇀v in , vk⇀v in , vk⇀v a.e ♌

Now, define the operators by

(100)

Then, we have

(101)

Equation (65) shows that

(102)

From (F1) and Hölder’s inequality, we get

(103)

In fact inequality (44) the functions and , which with the convexity of N-function, Lemma 2, Remark 5, and the boundedness of {(uk, vk)} implies that

(104)
which with Lemma 2 again shows that
(105)
for some C > 0. Furthermore, (e) and (f) give evidence that
(106)

Then, combining (101)–(103), (105), and (106), we obtain

(107)

According to the following lemma, we obtain the result.

Lemma 13 (see [22].)Suppose that uk⇀u in and

(108)

Then, uk⟶u in . Similarly, we can obtain that, vk⟶v in . Therefore, {(uk, vk)}⟶(u, v) in .

Proof of Theorem 4. By Lemmas 9, 11, and 12 and the evident fact that I(0) = 0, all conditions of Lemma 8 hold. Then, the functional I possesses a critical point which is a weak solution of system (5) satisfying . Lemma 10 implies that (u, v) ≠ 0. Thus, system (5) possesses at least one nontrivial weak solution.

5. Conclusion

In this paper, we established a Schrödinger system involving ω(⋅)-Laplacian operator. Our results are extensions and generalizations of some existing results in the literature.

Conflicts of Interest

The authors of this paper declare that they have no conflicts of interest.

Data Availability

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