Volume 2022, Issue 1 2187810

Research Article

Open Access

Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in ℝ⁴

Khadidja Sabri

University of Oran 2, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Atika Matallah,

Atika Matallah

Higher School of Management-Tlemcen, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Safia Benmansour,

Safia Benmansour

Higher School of Management-Tlemcen, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Mohammed El Mokhtar Ould El Mokhtar,

Corresponding Author

Mohammed El Mokhtar Ould El Mokhtar

[email protected]

orcid.org/0000-0002-0075-8837

Qassim University, College of Science, Department of Mathematics, BO 6644, Buraidah: 51452, Saudi Arabia qu.edu.sa

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Khadidja Sabri,

Khadidja Sabri

University of Oran 2, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Atika Matallah,

Atika Matallah

Higher School of Management-Tlemcen, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Safia Benmansour,

Safia Benmansour

Higher School of Management-Tlemcen, Laboratory of Analysis and Control of Partial Differential Equations of Sidi Bel Abbes-Algeria, Algeria

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Mohammed El Mokhtar Ould El Mokhtar,

Corresponding Author

Mohammed El Mokhtar Ould El Mokhtar

[email protected]

orcid.org/0000-0002-0075-8837

Qassim University, College of Science, Department of Mathematics, BO 6644, Buraidah: 51452, Saudi Arabia qu.edu.sa

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First published: 20 June 2022

https://doi.org/10.1155/2022/2187810

Citations: 1

Academic Editor: Pasquale Vetro

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Abstract

In this paper, we prove the existence of two positive solutions for a critical elliptic problem with nonlocal term and Sobolev exponent in dimension four.

1. Introduction

In this work, we are mainly concerned by the existence and the multiplicity of solutions for the following critical elliptic nonlocal problem:

(1)

where Ω is a smooth bounded domain of ℝ⁴, a and b are positive constants, and f belongs to H⁻¹

satisfying suitable condition specified afterward.

In our setting, the Laplacian operator is associated to Kirchhoff term a∫_Ω|∇u|²dx + b, which contains an integral over the entire domain Ω, this implies that the equation in is no longer a pointwise identity and so the problem turns to be nonlocal. This fact brings some mathematical difficulties in the search of the solution, and the solvability of this kind of problems has been under various authors’ attention; so, some classical investigations can be seen in the works [1, 2] and the references therein.

Such nonlinear Kirchhoff’s equations can be used for describing the dynamic for an axially moving string and was first formulated by Kirchhoff himself [3] in 1883, he take into account the changes in length of the strings produced by transverse vibrations, and his model can be seen as a generalization of the classical D’Alembert wave equation for free vibrations of elastic strings.

Problems which involve nonlocal operator have been widely studied due to their numerous and relevant applications in various fields of sciences. In particular, Kirchhoff type problems proved to be valuable tools for modeling several physical and biological phenomena, and many works have been made to ensure the existence of solutions for such problems; we quote in particular the article of Lions [4]. Since this famous paper, very fruitful development has given rise to many works on this advantageous axis, and in most of them, the used approach relies on topological methods. However, just few improvements were held concerning the multiplicity of solutions. In [5], Maia obtained a multiplicity of solutions for a class of p(x)-Choquard equations with a nonlocal and nondegenerate Kirchhoff term by using truncation arguments and Krasnoselskii’s genus. In [6], Vetro studied the existence of two different notions of solutions by using Galerkin approximation method, jointly with the theory of pseudomonotone operators.

With this regard, variational approach was solicited instead of topological methods to solve this kind of problems and also to prove the existence of multiple solutions; we refer interested readers to the works [7–10].

We begin by giving an overview about the previous research related to the problem

which can be written in the more general form,

(2)

where Ω is a smooth bounded domain of ℝ^N, N ≥ 3.

Without nonlocal term (a = 0), much interest has grown on problems involving critical exponents, and there are many publications dealing with the existence of solutions, starting from the celebrated paper by Brézis and Nirenberg [11] when and 2^∗ = 2N/N − 2 are the critical Sobolev exponent. For convenience of the reader, we give a brief summary of these results: they established existence results in dimension N = 3 when Ω is a ball namely, and they ensure the existence of a positive constant λ₀ such that the problem admits a positive solution for λ ∈ (λ₀/4, λ₁), where λ₁ is the first eigenvalue of the operator −Δ. In higher dimensions N ≥ 4, they proved the existence of a positive solution for λ sufficiently small, i.e., λ ≤ λ₀ and no positive solution for λ > λ₀ and Ω a star-shaped domain.

When 1 < q < 2^∗, Ambrosetti et al. [12] established a multiplicity result in a bounded domain of ℝ^N, N ≥ 3 indeed, they ensured the existence of a positive constant λ₀ such that the problem admits two positive solutions for λ ∈ (0, λ_∗), a positive solution for λ = λ_∗ and no positive solution for λ > λ_∗.

For the nonhomogeneous case, namely, when

Tarantello [13] proved the existence of at least two solutions when f satisfies

(3)

for all

We emphases that the extension of the previous results to the nonlocal case, namely, for elliptic problems driven by Kirchhoff type operator are not obvious in high dimensions N ≥ 4. Therefore, no improvement was hold concerning the multiplicity of solutions in this case.

For the case a > 0 and , Naimen in [14] treated the problem for N = 3 and obtained homologous results than the ones obtained by Brézis and Niremberg [11] in the nonlocal case under a suitable condition on a.

In dimension four, Naimen in [15] used variational methods to explore problem and showed that admits a positive solution when a > 0, b ≥ 0.

In the same order of ideas and still in the nonlocal case (a > 0), Lei et al. in [16] and Liao et al. in [17] extended the findings of [12] to a more general setting, namely, with the Kirchhoff operator; they established a multiplicity result in dimensions three and four, respectively.

Benmansour and Bouchekif [8] generalized the results obtained by Tarantello [13] to the nonlocal case in dimension three. Indeed, they have shown the existence of two solutions under a sufficient condition on f by introducing the Nehari manifold.

A natural question is to know whether the multiplicity result persists in the case of dimension four.

In the current paper, our main purpose inspired by [8] is to see that the result obtained in [8] can be extended to dimension four. We emphases that our results are new and complement the above works.

In order to study

we shall work with the space

endowed with the norm

we use also the following notation:

for 1 ≤ p < ∞, C and C_i denote generic positive constants whose exact values are not important, B_H(0, r)≔{u ∈ H : ‖u‖ < r} is the ball of center 0 and radius r, o_n(1) denotes any quantity which tends to zero as n tends to infinity, and S is the best Sobolev constant defined by

(4)

To state the main results, we define

(5)

where b > 0, a < S⁻² a small enough positive number and f belongs to H⁻¹\{0}.

The main results are concluded as the following theorems, which are news for the case when a ≠ 0.

Theorem 1. Assume that γ_a,b,f > 0. Then, the problem admits a local minimal solution u₀ with I(u₀) < 0. Furthermore u₀ ≥ 0 for f ≥ 0.

Theorem 2. Assume that γ_a,b,f > 0. Then, the problem admits another solution u₁ with I(u₁) > 0. Furthermore, u₁ ≥ 0 for f ≥ 0.

Notice that γ_a,b,f ≥ 0 if

(6)

Moreover, the assumption γ_a,b,f > 0 certainly holds if f satisfies certain conditions, for example,

(7)

for all u ∈ H with ‖u‖₄ = 1. Indeed, we have γ_0,b,f is achieved and strictly positive if f satisfies (f)₂ (see Lemma 2.2 in [13]). In order as

(8)

then, γ_a,b,f ≥ γ_0,b,f > 0.

This paper is structured as follows: in Section 2, we give some basic results useful for what follows. Section 3 is devoted to the proofs of our main results.

2. Some Preliminary Results

We consider the energy functional associated to problem

defined for u ∈ H and given by

(9)

Observe that I∈C¹(H, ℝ), whose derivative at the point u ∈ H is given by

(10)

Obviously, if u ∈ H is a critical point of the functional I; then, u is a weak solution of problem

In general, I is not bounded from below on H, to overcome this and achieve a multiplicity result, the key argument is to use an appropriate manifold called in mathematical literature the Nehari manifold, it is a suitable manifold who has a pertinent property to prove the distinction of two solutions. Indeed, a minimizer in this set may give rise to solution of the corresponding equation. This so called Nehari manifold is defined by

(11)

Lemma 3. Assume that b > 0, a ≥ 0, and f ∈ H⁻¹\{0}. Then, the functional I is coercive and bounded from below on

Proof. For , we have

(12)

Therefore

(13)

Thus, I is coercive and bounded from below on .

Let h_u(t) = I(tu) for t ∈ ℝ^∗ and u ∈ H. These maps are known as fibering maps and were first introduced by Drábek and Pohozaev [18]. The set is closely linked to the behavior of h_u(t), for more details, see for example [19] or [20].

It is natural to split into three subsets:

(14)

(15)

where These subsets correspond to local minima, points of inflexion, and local maxima of I, respectively.

Definition 4. A sequence {u_n} ⊂ H is said to be a Palais Smale sequence at level c ((PS)_c sequence in short) for I if

(16)

I verifies Palais Smale condition at level c ((PS)_c condition in short) if any (PS)_c sequence has a convergent subsequence in H.

Next, for u ≠ 0, b > 0, and a, a small enough positive number set

(17)

then

Easy computations show that Φ_u is concave and achieves its maximum at the point

where

(18)

That is,

(19)

(20)

Now, for u ∈ H\{0} set

, that is

(21)

Fix t₁ > 0, then, for t ≥ t₁, we have Ψ(tu) = tΨ(u),

(22)

(23)

This is crucial for the following.

Lemma 5. Assume that γ_a,b,f > 0, then,

Proof. Arguing by contradiction we assume that there exists i.e., u ≠ 0 verifies

(24)

(25)

From (24) and (25), we derive that

(26)

As 0 ≤ a < S⁻², we get from (24), (26), and the definition of S

(27)

that is

(28)

Thus, as u ≠ 0 and a < S⁻², we derive that

(29)

therefore, from (25), we obtain ‖u‖₄ ≥ t₁, with

(30)

Then, from (23), (25), and (26), we get

(31)

which yields to a contradiction.

Lemma 6. Assume that γ_a,b,f > 0, then, for all u ∈ H\{0}, there exists unique positive value such that

(32)

Moreover, if ∫_Ωfudx > 0, then, there exists unique positive value such that

(33)

Proof. We have h_u(t) = I(tu), and Φ_u is concave and achieves its maximum at the point If γ_a,b,f > 0; then, there exists a unique such that and , which implies that and for all Moreover, if ∫_Ωfudx > 0, then, there exists a unique such that and , which implies that and for all

Set

(34)

(35)

In the following lemma, we prove that is closed and disconnects H in exactly two connected components E₁ and E₂.

Lemma 7. Assume that γ_a,b,f > 0. Then

(i)
is closed
(ii)
(iii)

Proof. Let and then, . Assume by contradiction that then

(36)

(37)

So, this implies that

From (36) and the definition of S, we get so which yields to a contradiction.

Let and v = u/‖u‖, then, t⁺(u) = 1, and there exists unique t⁺(v) such that As

(38)

then

(39)

Thus if u ∈ H\{0} and,

(40)

then, u∉

and

(41)

Let then

(42)

Since t⁺(u) > t⁻(u), it follows that

(43)

So, ‖u‖ < t⁺(u/‖u‖), and we conclude that

By Lemma 6, we know that

and

are not empty, so we can define θ₀ ≤ θ₁ with

(44)

Lemma 8. Assume that γ_a,b,f > 0, then, there exists t_∗ > 0 such that

(45)

Proof. Let then

(46)

Thus .

Set u_∗ ∈ H the unique solution of the equation −Δu = f, it follows

(47)

By Lemma 6, there exists a unique positive value such that . So

(48)

consequently

(49)

The following lemma is needed for prove the existence of Palais Smale sequences.

Lemma 9. Assume that γ_a,b,f > 0. Then, for any there exist ε > 0 and a differentiable function ζ : B_H(0, ε)⟶ℝ⁺\{0} such that

(50)

(51)

Proof. Let and define:ℝ × H⟶ℝ as follows

(52)

Clearly, (1, 0) = 0. Moreover, from Lemma 5, we derive that

(53)

Thus, we get our result by a straightforward application of the implicit function theorem to the function at the point (1, 0).

Lemma 10. Let θ ∈ {θ₀, θ₁}. There exist a Palais Smale sequences such that

(54)

Proof. Assume θ = θ₀, by Lemma 3, I is bounded from below in then by applying the Ekeland Variational Principle, we can obtain a minimizing sequence satisfying

(55)

(56)

for all

Thus, I(u_n)⟶θ₀.

By using Lemma 8, we get for n large enough

(57)

this implies that

(58)

then

(59)

and by Holder inequality, we get

(60)

Consequently, u_n ≠ 0 and

(61)

Now, we show that ‖I^′(u_n)‖ tend to 0 as n goes to +∞. Arguing by contradiction and fix n with ‖I^′(u_n)‖ ≠ 0.

Then, by Lemma 9, there exist ε > 0 and a function ζ_n : B_H(0, ε)⟶ℝ such that with v_n = δI^′(u_n)/‖I^′(u_n)‖ and 0 < δ < ε. By (56) and the Taylor expansion of I, we have

(62)

Then

(63)

We have

(64)

(65)

This, together with (61) implies

(66)

for a suitable constant C₃ > 0. Now, we must show that

is uniformly bounded in n: indeed, since {u_n} is a bounded sequence, we have from Lemma 8

(67)

for a suitable constant C₄ > 0. Assume by contradiction that for a subsequence still called {u_n}, we have

(68)

Then, as a is a small enough positive number, we get

(69)

So from (61), we derive that

(70)

Also, as , we get from (68)

(71)

then

(72)

which is absurd. At this point, we conclude that I^′(u_n)⟶0 in H⁻¹.

For θ = θ₁, adopting exactly the same way as in the case where θ = θ₀.

In the following, we will prove our results.

3. Proofs of the Main Results

The proof of our main results is divided in two parts.

3.1. Existence of a Solution in

In this subsection, we prove that I has a solution in

Proposition 11. Assume that γ_a,b,f > 0. Then, the minimization problem

(73)

attaints its infimum at a point

. Moreover, u₀ is a local minimizer for I in H.

Proof. By using Lemma 10, there exists a bounded minimizing sequence such that I(u_n)⟶θ₀ and I^′(u_n)⟶0 in H⁻¹. So, we deduce that {u_n} is bounded in H.

Passing to a subsequence if necessary, we have u_n⇀u₀ weakly in H, then, 〈I^′(u₀), w〉 = 0, for all w ∈ H. In addition, from (60), we get ∫_Ωfu₀dx > 0. So, u₀ is a weak solution for and

Thus

(74)

then

It follows that {u_n} converges strongly to u₀ in H, then,

and necessarily

(75)

To conclude that u₀ is a local minimum of I, let us recall that we have from Lemma 6

(76)

Choose ε > 0 sufficiently small to have

(77)

and t(w) satisfying

for every ‖w‖ < ε. Since t(w)⟶1 as ‖w‖⟶0, we can always assume that

(78)

and for

, we have

(79)

from (75), we can take s = 1 and conclude that I(u₀ − w) ≥ I(u₀), for all w ∈ H such that ‖w‖ < ε. Thus, u₀ is a local minimum of I.

If f ≥ 0, we have ∫_Ω fu₀dx ≤ ∫_Ω f|u₀|dx and clearly I(|u₀|) ≤ I(u₀), and from (75) necessarily t⁻(|u₀|) ≥ 1. Therefore, as , we get

(80)

so, we can always take u₀ ≥ 0.

3.2. Existence of a Solution in

The following part is devoted to prove the existence of a second solution u₁ such that First, we determine the good level for covering the Palais Smale condition.

We have the following important result.

Lemma 12. Assume that γ_a,b,f > 0. Then, I satisfies the (PS)_c condition for with

(81)

Proof. Let {u_n} be a (PS)_c sequence with then

(82)

Hence, {u_n} is a bounded sequence in H. Thus for a subsequence still denoted {u_n} and we can find u₁ ∈ H such that u_n⇀u₁ in H, ∫_Ωfu_ndx⟶∫_Ωfu₁dx and u_n⟶u₁ a.e in Ω. Therefore, u₁ ≠ 0, , and I(u₁) ≥ θ₁.

Let w_n = u_n − u₁. From Brézis-Lieb Lemma [21], one has

(83)

(84)

this implies that

(85)

Assume that ‖w_n‖⟶l with l > 0, then, by (85) and the Sobolev inequality, we obtain

(86)

this implies that

(87)

Hence

(88)

On the other hand, we have

(89)

consequently, we obtain

(90)

which is a contradiction. Therefore, l = 0 and u_n⟶u₁ strongly in H.

Now, it is natural to show that As it is well know, (seeHL), the best Sobolev constant S defined above is attained in ℝ⁴ by

(91)

For x₀ ∈ Ω let such that ϕ(x) = 1 for for and 0 ≤ ϕ ≤ 1, |∇ϕ| ≤ C. Now, we shall give some useful estimates of the extremal functions U_ε. Let and The following estimates are obtained in [22] as ε tends to 0

(92)

Let Ω^′ ⊂ Ω a set of positive measure such that u₀ > 0 on Ω^′ (if not replace u₀ and f by −u₀ and −f, respectively).

Lemma 13. Let b > 0, 0 ≤ a < S⁻² a small enough positive number. Assume that f ∈ H⁻¹\{0} satisfies γ_a,b,f > 0; then, for every t > 0 and a.e. x₀ ∈ Ω^′ ⊂ Ω, there exists ε₀ such that for every 0<ε < ε₀.

Proof. For ε small enough, let us consider the functional g defined by

(93)

We have by (92)

(94)

On the other hand, since is a solution of problem , we have ‖u₀‖ ≤ C, I(u₀) = θ₀ and 〈I^′(u₀), tu_ε〉 = 0.

Then we obtain by (92)

(95)

where

(96)

Since a < S⁻², we have h(t)⟶−∞ as t goes to ∞ and h(t)⟶0 as t goes to 0. This implies that there exist 0 < T₁ < T₂ such that

(97)

Then, for a = ε^σ with σ > 1, we conclude

(98)

Therefore,

(99)

for a small enough positive number.

Proposition 14. Let b > 0, 0 ≤ a < S⁻² a small enough positive number. Assume that f ∈ H⁻¹\{0} satisfies γ_a,b,f > 0, then, I has a local minimizer u₁ on such that I(u₁) = θ₁. Moreover u₁ is a local minimizer for I on H.

Proof. By Lemma 6, for every u ∈ H such that ‖u‖ = 1, there exists unique t⁺(u) > 0 such that t⁺(u)u∈ and I(t⁺u) ≥ I(tu), for all Then, for suitable constant T₃ > 0, we deduce that

(100)

Therefore for t₀ > 0 carefully chosen, the estimate (97) holds for ε small enough.

Thus, we derive that

(101)

Then, from Lemma 7, we conclude that u₀ + t₀u_ε ∈ E₂.

Set

(102)

It is obvious that ξ : [0, 1]⟶H given by ξ(t) = u₀ + tt₀u_ε belongs to Γ. We conclude from Lemma 13 that

(103)

As the range of any ξ∈Γ intersects one has

(104)

From Lemma 10, we can obtain a minimizing sequence such that

(105)

We also deduce that

Consequently, we obtain a subsequence still denoted {u_n}, and we can find u₁ ∈ H such that

(106)

This implies that u₁ is a critical point for I, and I(u₁) = θ₁.

Finally, for f ≥ 0, let t⁺(|u₁|) > 0 satisfying From Lemma 6, we have

(107)

So we conclude that u₁ ≥ 0.

4. 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 is a solution of our problem. Under some sufficient conditions, we split in two disjoint subsets and . Thus, we consider the minimization problems on and , respectively. If γ_a,b,f > 0, then, the problem has a local minimal solution u₀ with I(u₀) < 0. Furthermore, u₀ ≥ 0 for f ≥ 0 and if γ_a,b,f > 0. The problem has another solution u₁ with I(u₁) > 0. Furthermore, u₁ ≥ 0 for f ≥ 0.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge (1) Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1124) during the academic year 1443AH/2022 AD and (2) Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number (1124) during the academic year 1443AH/2022 AD.

Open Research

Data Availability

The functions, functionals, and parameters used to support the findings of this study are included within the article.

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Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in ℝ⁴

Abstract

1. Introduction

2. Some Preliminary Results

3. Proofs of the Main Results

3.1. Existence of a Solution in

3.2. Existence of a Solution in

4. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in ℝ4

Abstract

1. Introduction

2. Some Preliminary Results

3. Proofs of the Main Results

3.1. Existence of a Solution in

3.2. Existence of a Solution in

4. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in ℝ⁴