In this paper, we deal with p-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 Ω a bounded smooth domain in ℝ^N(N ≥ 3), and consider the following p-Laplace equations with singular nonlinearities

(1)

where() 1 < p < N, 0 ∈ Ω, λ > 0, and q + 1 = p^∗ with p^∗≔pN/(N − p) is the critical Sobolev exponent, 0 ≤ α < N(q + β)/q, 0 < β < 1 and Δ_pu≔div(|∇u|^p−2∇u) is the p-Laplace operator which is degenerate if p > 2 and singular if p < 2.

In recent years, researchers have been interested in studying problems of the type:

(2)

where Ω is a bounded smooth domain in ℝ^N(N ≥ 3), 0 ∈ Ω, λ > 0, and q + 1 = p^∗ with p^∗≔pN/(N − p) is the critical Sobolev exponent and f is a suitable function containing singularities on x (see [1–4] and references therein). For p = 2 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 N ≥ 3 and f(x, u) satisfying (AR) condition in the case λ = 1. Very recently, M. E. O. El Mokhtar et al. [21] considered Problem (1) with p = 2.

The term represented by the function f(x, u)≔|x|^−αu|u|^−β with 0 < β < 1 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

(3)

We consider the following approximation equation:

(4)

for any θ > 0. The energy functional of (4) E_θ is defined by

(5)

for all , where u^± = max{u^±, 0}.

We know that E_θ is a C¹−function on

A point

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

(6)

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

Let

(7)

From [3], S is achieved.

Let λ_∗ and λ_∗∗ be positive numbers such that

(8)

and

(9)

where,

(10)

with

(11)

The main results are concluded as the following theorems.

Theorem 1. Assume that N ≥ 3, 0 ≤ α < (N/(q + 1))(q + β), 0 < β < 1, and λ verifying 0 < λ < λ_∗; then, the system (1) has at least one positive solution.

Theorem 2. In addition to the assumptions of the Theorem 1, there exists Λ_∗ ∈ (0, λ_∗∗) such that if λ satisfying 0 < λ < Λ_∗, then (1) has at least two positive solutions.

Theorem 3. Under the assumptions of Theorem 2 then, there exists a positive real λ^∗∗ such that if λ satisfies 0 < λ < λ^∗∗, 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 c ∈ ℝM be a Banach space and E ∈ C¹(M, ℝ).

(i)
is a Palais-Smale sequence at level c (in short (PS)_c) in M for E

(12)

where o_n(1) tends to 0 as n goes at infinity

(ii)
We say that E satisfies the (PS)_c condition if any (PS)_c sequence in M for E has a convergent subsequence

2.1. Nehari Manifold [32, 33]

It is well known that E is of class C¹ in

and the solutions of (1) are the critical points of E which is not bounded below on

. Consider the following Nehari manifold:

(13)

Note that

contains every nontrivial solution of the problem (1). Thus,

if and only if

and

(14)

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. E_θ is coercive and bounded from below on .

Proof. Let R₀ > 0 such that Ω ⊂ B(0, R₀) = {x ∈ ℝ^N : |x| < R₀}. If , then by (14) and the Hölder inequality, we obtain

(15)

(16)

where,

(17)

Therefore, we obtain that

(18)

for 0 ≤ α < (N/(q + 1))(q + β).

Thus, E_θ is coercive and bounded from below on .

Define

(19)

Then, for

(20)

Splitting

in three parts, we set

(21)

We have the following results.

Lemma 6. Suppose that u₀ is a local minimizer for E_θ on .If , then u₀ is a critical point of E_θ.

Proof. If u₀ is a local minimizer for E_θ on , then u₀ is a solution of the optimization problem:

(22)

Hence, there exists a Lagrange multipliers μ ∈ ℝ such that

(23)

Thus,

(24)

But ϕ^′(u₀), u₀ ≠ 0, since . Hence, μ = 0. This completes the proof.

Lemma 7. There exists a positive number λ_∗ such that for all λ verifying

(25)

we have

Proof. Let us reason by contradiction.

Suppose that for all λ such that 0 < λ < λ_∗. Then, by (20) and for , we have

(26)

Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain

(27)

(28)

From (26) and (27), we obtain λ ≥ λ_∗, which contradicts our hypothesis.

then

. Define

(29)

For the sequel, we need the following Lemma.

Lemma 8.

(i) For all λ such that 0 < λ < λ_∗, one has c ≤ c⁺ < 0

(ii) For all λ such that 0 < λ < λ_∗∗ there exists C₀ > 0 such hat c⁻ > −λ^{p/(p − 1 + β)}C₀

Proof.

(i)
Let . By (20), we have

(30)

and so

(31)

since p + 1 > 2 and 0 < β < 1. Then, we conclude that c ≤ c⁺ < 0.

(ii)
Let . By(20), we get

(32)

By Sobolev embedding theorem, we obtain

(33)

This implies

(34)

By the proof of Lemma 5, we have

(35)

Thus, for all λ such that 0 < λ < λ_∗∗ we have E_θ(u) ≥ −λ^{p/(p − 1 + β)}C₀ with

(36)

As in [34] we have the following result.

Proposition 9.

(i) For all λ such that 0 < λ < λ_∗, there exists a sequence in

(ii) For all λ such that 0 < λ < λ_∗∗, there exists a sequence in and for each .

Define

(37)

Lemma 10. Suppose that 0 < λ < λ_∗. For each , there exists unique t⁺ and t⁻ such that 0 < t⁺ < t_M < t⁻, , and

(38)

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 E_θ on .

Proposition 11. For all λ such that 0 < λ < λ_∗, the functional E_θ has a minimizer , and it satisfies

(i)
(ii)
is a nontrivial solution of (1)

Proof. If 0 < λ < λ_∗, then by Proposition 9, (i) there exists a (u_n) sequence in ; , thus, it bounded by Lemma 5. Then, there exists and we can extract a subsequence which will denoted by (u_n) such that

(39)

By (15) and (39), we have

(40)

Thus, by (39), is a weak nontrivial solution of (1). Now, we show that (u_n) converges to strongly in . Suppose otherwise. By the lower semi-continuity of the norm, if we have and we obtain

(41)

We get a contradiction. Therefore, (u_n) 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

(42)

there exists

t⁻≤

such that

. Then, we get

(43)

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 E_θ on . For this, we require the following Lemmas.

Lemma 12. Let (u_n) be (PS)_c sequence for E_θ for some c ∈ ℝ with u_n⟶u in .

Then, and E_θ(u) ≥ −λ^{p/(1 + β)}C(q, β, Ψ, S), with C(q, β, Ψ, S) > 0.

Proof. Let (u_n) ⊂ S_u be a minimizing sequence for E_θ with S_u is the unit sphere. By Ekeland’ s variational principle [12], we may assume . So (u_n) is a (PS)_c sequence and therefore u_n⟶u after passing to a subsequence. Hence E_θ(u) = c and , which implies that , and

(44)

Therefore,

(45)

From (15) and considering ‖u‖ small enough, we get

(46)

which implies that

(47)

with

(48)

Set f(t) = Dt^p − λEt^1−β for all t > 0, with

(49)

Using (46), we obtain that

(50)

where

(51)

We have C(p, q, β, Ψ, S) > 0 since 0 < β < 1. Then, we conclude that

(52)

Lemma 13. Let λ ∈ (0, λ_∗∗); then, the functional E_θ satisfies the (PS)_c condition in with c < c^∗, where

(53)

Proof. If 0 < λ < λ_∗∗, then by Proposition 9, (ii) there exists a (u_n), (PS)_c sequence in ; thus, it bounded by Lemma 5. Then, there exists and we can extract a subsequence which will denoted by (u_n) such that

(54)

Then, u is a weak solution of (1). Let v_n = u_n − u; then, by Brézis-Lieb [36], we obtain

(55)

(56)

Since

(57)

and by (55) and (56), we deduce that

(58)

Hence, we may assume that

(59)

Moreover, by Sobolev inequality, we have

(60)

Combining (60) and (59), we obtain

(61)

Either,

(62)

Then from (58), (59), Lemma 13 and Lemma 12, we obtain

(63)

which is a contradiction. Therefore, l = 0, and we conclude that (u_n) converges to u strongly in

Thus,

(64)

Lemma 14. There exists and Λ_∗ > 0 such that for all λ ∈ (0, Λ_∗), one has

(65)

In particular, c < c^∗ for all λ ∈ (0, Λ_∗).

Proof. Let φ_ε(x) satisfies (4). Then, we have

(66)

We consider the two functions:

(67)

Then, for all for all λ ∈ (0, λ_∗∗),

(68)

By the continuity of f, there exists t₀ > 0 such that

(69)

On the other hand we have

(70)

Then, we obtain

(71)

Now, taking λ > 0 such that

(72)

we obtain

(73)

Set

(74)

We deduce that c⁻ < c^∗ for all λ ∈ (0, Λ_∗); then, there exists t_n > 0 such that with w_n satisfying (4) and for all λ ∈ (0, Λ_∗),

(75)

Lemma 15. For all λ such that 0 < λ < Λ_∗ = min{λ_∗∗, Λ₁}, the functional E_θ has a minimizer in and it satisfies

(76)

is a nontrivial solution of (1) in .

Proof. By Proposition 9 (ii), there exists a sequence (u_n) for E, in for all λ ∈ (0, λ_∗∗). From Lemmas 13, 18, and 8(ii), for λ ∈ (0, Λ₁), E_θ satisfies condition and c⁻ > 0. Then, we get that (u_n) is bounded in . Therefore, there exist a subsequence of (u_n) still denoted by (u_n) and such that (u_n) converges to strongly in and for all λ ∈ (0, Λ_∗).

Finally, by using the same arguments as in the proof of Proposition 11 for all λ ∈ (0, λ_∗), 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

(77)

Thus,

if and only if

(78)

Firstly, we need the following Lemmas.

Lemma 16. Under the hypothesis of theorem 3, there exist Λ₂ > 0 such that is nonempty for any λ ∈ (0, Λ₂).

Proof. Fix and let

(79)

Clearly g(0) = 0 and g(t)⟶−∞ as t⟶+∞. Moreover, we have

(80)

for t ≥ 0, put φ(t) = t^β − (S)^{−(q + 1)/p}t^q+β; then, we obtain

since q > 2 with t₁ = (β/(q−+β))^{(q + 1)/(q − 1)}(S)^{(q + 1)/(p(q − 1))}. Thus, we obtain

(81)

if λ < (((S)^{(1 − β)/p})/Ψ)φ(t₁)≔Λ₂.

Then, there exists t₀ > 0 such that g(t₀) = 0. Thus, and is nonempty for any λ ∈ (0, Λ₂).

Lemma 17. There exist δ, λ^∗ positive real numbers such that ϕ^′(u), u?<−?<0, for and any λ verifying

(82)

Proof. Let then by (14), (20) and the Holder inequality, it allows us to write

(83)

Thus, if

(84)

and choosing λ^∗≔min(Λ₂, Λ₃) with Λ₂ defined in Lemma 16, then we obtain that

(85)

Lemma 18. Suppose q > 2, β ∈ (0, 1) and 0 < λ < min(Λ₂, Λ₃, Λ₄) when

(86)

Then, there exist ε and η positive constants such that

(i)
We have

(87)

(ii)
There exists when ‖v‖ > ε, with ε = ‖u‖, such that E_θ(v) ≤ 0

Proof. We can suppose that the minima of E_θ 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 get

(88)

By exploiting the function ϕ(t) = at^p − bt^1−β which achieve its maximum at the point t₁ = (1 − β/p)^{p/(β − 1 + p)}(a/b)^{(p − 1)/(q − 1)} such that if

(89)

and the fact that, q > 2, β ∈ (0, 1) then, we obtain that

(90)

(ii)
Let t > 0, then we have for all

(91)

Letting v = tφ for t large enough, we obtain E_θ(v) ≤ 0. For t large enough, we can ensure ‖v‖ > ε.

Let Γ and c defined by

(92)

Proof of Theorem 19. If

(93)

then, by the Lemma 5 and Proposition 9 (ii), E_θ verifying the Palais-Smale condition in

. Moreover, from the Lemmas 6, 17, and 18, there exists u_c such that

(94)

Thus, u_c is the third solution of our system such that and . Since (1) is odd with respect u, we obtain that −u_c is also a solution of (1).

Finally, for every θ ∈ (0, 1), problem (4) has solution such that E_θ(u_θ) = 0. Thus, there exist {θ_n} ⊂ (0, 1) with θ_n⟶0 as n⟶∞. 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 N ≥ 3, 0 ≤ α < ((N(q + β))/((q + 1))) and β ∈ (0, 1), 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 0 < λ < λ^∗∗≔min(λ^∗, Λ₄) if N ≥ 3 and β ∈ (0, 1).

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.

Open Research

Data Availability

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

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On p-Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

Abstract

1. Introduction

2. Preliminaries

2.1. Nehari Manifold [32, 33]

3. Proof of Theorem 1

4. Proof of Theorem 2

5. Proof of Theorem 3

6. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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On p-Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

Abstract

1. Introduction

2. Preliminaries

2.1. Nehari Manifold [32, 33]

3. Proof of Theorem 1

4. Proof of Theorem 2

5. Proof of Theorem 3

6. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

Information