The Existence Result for a p -Kirchhoff-Type Problem Involving Critical Sobolev Exponent

In this paper


Introduction and Main Result
In this article, we study the existence of a positive solution for the following p-Kirchhoff-type problem: where Ω ⊂ ℝ N is a bounded domain, 1 < p < N, M a,b u = a u p r−1 + b, a > 0, b ≥ 0, λ > 0, p < rp < q < p * , is the usual norm in W 1,p 0 Ω given by u p = Ω ∇u p dx, and p * = pN/ N − p is the critical Sobolev exponent corresponding to the noncompact embedding of W 1,p 0 Ω into L p * Ω .This problem contains an integral over Ω, and it is no longer a pointwise identity; therefore, it is often called nonlocal problem.It is also called nondegenerate if b > 0 and a ≥ 0, while it is named degenerate if b = 0 and a > 0.
In the past several decades, much attention has been paid to the Kirchhoff-type problem which is closely related to the stationary analog of the following equation: proposed by Kirchhoff in [1] as an extension of the classical d'Alembert's wave equation by considering the effects of the changes in the length of the strings during the vibrations, where ρ, ρ 0 , h, E, and L are constants.Kirchhoff's model takes into account the changes in length of the strings produced by transverse vibrations.These problems also serve to model other physical phenomena as biological systems where u describes a process which depends on the average of itself (for example, population density).The presence of the nonlocal term makes the theoretical study of these problems so difficult; then, they have attracted the attention of many researchers in particular after the work of Lions [2], where a functional analysis approach was proposed to attack them.
In the last few years, great attention has been paid to the study of Kirchhoff problems involving critical nonlinearities.These problems create many difficulties in applying variational methods because of the lack of the compactness of the Sobolev embedding.It is worth mentioning that the first work on the Kirchhoff-type problem with critical Sobolev exponent is Alves et al. in [3].After that, many researchers dedicated to the study of several kinds of elliptic Kirchhoff equations with critical exponent of Sobolev in bounded domain or in the whole space ℝ N ; some interesting studies can be found in [4][5][6][7][8][9] and the references therein.More precisely, Naimen in [8] generalized the results of [10] to the semilinear Kirchhoff problem: where , and λ ∈ ℝ.Under several conditions on f and λ, he proved the existence and nonexistence of solutions.For larger dimensional case, Figueiredo in [5] considers the case N ≥ 3 if λ > 0 is sufficiently large.Matallah et al. in [7] studied the existence and nonexistence of solutions for the following p-Kirchhoff problem: where M R + ⟶ R + is a continuous function satisfying some extra assumptions and f Ω × R ⟶ R is a continuous function satisfying some conditions.Benaissa and Matallah in [4] discussed the problem where f satisfies some conditions.Very recently, Benchira et al. in [11] have generalized the results of [12] to the nonlocal problem (1) with q = p, λ ∈ 0, bλ 1 , b > 0, N ≥ p 2 , a > 0 if r < N/ N − p , and 0 < a < S −r if r = N/ N − p (S is the best Sobolev constant for the imbedding W 1,p 0 Ω ↪L p * Ω .Inspired by the above works, especially by [8,11], we are devoted to studying the existence of positive solutions for problem (1) for all λ positive.In our problem, a typical difficulty occurs in proving the existence of solutions because of the lack of the compactness of the Sobolev embedding W 1,p 0 Ω ⟶ L p * Ω .Furthermore, in view of the corresponding energy, the interaction between the Kirchhofftype perturbation u p r−1 and the critical nonlinearity Ω u p * dx is crucial.
The main result of this paper is the following.
Remark 2. If N ≥ p 2 , then max rp, N p − 1 / N − p = rp In the case where q ≤ rp, it is difficult to show that a Palais-Smale sequence of the corresponding energy is bounded; in this case, the authors in [7,9] used the truncation method to show the existence of solution under the condition " sufficiently large."Our objective in this paper is the existence of solution for all λ > 0.
Let us simply give a sketch of the Proof of Theorem 1.The main tool is variational methods; more precisely, by using the mountain pass theorem [13], we obtain a critical point of the corresponding energy.The main difficulties appear in the fact that problem (1) contains the critical Sobolev exponent; then, the functional energy does not satisfy the Palais-Smale condition in all range; to overcome the lack of compactness, we need to determine a good level of the Palais-Smale condition, and we must verify that the critical value is contained in the range of this level.This is the key point to obtain the existence of a mountain pass solution.
This paper is composed of two sections in addition to the introduction.In Section 2, we give some preliminary results which we will use later.Section 3 is devoted to the proof of main result.

Preliminary Results
In this paper, we use the following notations: ⟶ resp ⇀ denotes strong (resp., weak) convergence, o n 1 denotes o n 1 ⟶ 0 as n ⟶ +∞, B R x 0 is the ball centred at x 0 and of radius R, u − = max −u, 0 , and C, C 1 , C 2 , ⋯, denote various positive constants.We define the best Sobolev constant for the imbedding W Recall that the infimum S is attained in ℝ N by the functions of the form Let R be a positive constant and set Then, we have the well-known estimates as ε ⟶ 0: Journal of Function Spaces (See [14,15]).
The energy function corresponding to problem (1) is given by Hence, a critical point of functional E is a weak solution of problem (1).
Let c ∈ ℝ.We say that E satisfies the Palais-Smale condition at level c, if any PS c sequence contains a convergent subsequence in W 1,p 0 Ω By [11], we have the following result.Lemma 4. Let a > 0, b ≥ 0, r, θ > 1, and y = a/θ S r 1/ θ−1 .For y ≥ 0, we consider the function f θ ℝ + ⟶ ℝ * , given by Then, (1) when b ≠ 0, the equation f θ y = 0 has a unique positive solution y 0 > y and f θ y ≥ 0 for all y ≥ y 0 (2) when b = 0, the equation f θ y = 0 has a unique positive solution y 1 = aS r 1/ θ−1 and f θ y ≥ 0 for all y ≥ y 1

Proof of Main Result
To prove our main result, we use the mountain pass theorem.First, we will verify that the functional E exhibits the mountain pass geometry.
Lemma 5. Suppose that a > 0, b ≥ 0, 1 < p < N, and rp < q < p * .Then, there exists e ∈ W Let ρ = u , from ( 14), one has As rp < q < p * and a > 0 there exists a sufficiently small positive numbers ρ 1 and δ 1 such that where e is taken from Lemma 5. Now, we prove the following lemma which is important to ensure the local compactness of PS sequences for E. with Then, u n contains a subsequence converging strongly in W 1,p 0 Ω .

Proof.
As n ⟶ ∞ and rp < q < p * , we have Then, u n is bounded in W 1,p 0 Ω .Hence, by the concentration compactness principle due to Lions (see [6,16]), there exists a subsequence, still denoted by u n , such that where I is an at most countable index set, η i , γ i are nonnegative numbers, and δ x i is the Dirac mass at x i .Moreover, by the Sobolev inequality, we infer that for all i ∈ I 23 We now claim that I = ∅.To this end, by contradiction, suppose that I ≠ ∅; then, there exists j ∈ I.For ε > 0, let ϕ ε,j be a smooth cut-off function centered at x j such that 0 ≤ ϕ ε,j ≤ 1,ϕ ε,j B ε x j = 1, ϕ ε,j Ω\B 2ε x j = 0, and ∇ϕ ε,j x ≤ 2/ε.

24
On the one hand, by Hölder's inequality and (6), we have Moreover, by using Hölder's inequality, we find By ( 22), ( 26), (28), and Hölder's inequality, we obtain that is, It is clear that θ > 1 thanks to p * > rp.So, from (32) and the definition of f θ in Lemma 4, we get According to Lemma 4, there exist y 0 > ap r − 1 / p * − p S r p r−1 / p * −rp and y 1 = aS r p r−1 / p * −rp such that f θ y * = 0 and f θ y ≥ 0 if y ≥ y * with Moreover, using (23), we conclude that

36
On the other hand, by the fact p < rp < q < p * , one can get This is a contradiction.Hence, I is empty and so On the other hand, we have

41
for any v ∈ W 1,p 0 Ω .Set l = lim u n as n ⟶ +∞; then, from (40) and (41), we deduce that Taking the test function v = u in (43), we get Therefore, the equalities (42) and (44) imply that u = l.Consequently, u n converges strongly in W 1,p 0 Ω , which is the desired result.

Journal of Function Spaces
The energy functional E satisfies the condition at level c for any c < C * .So, the existence of the solution follows immediately from the following lemma.Proof.We define the functions g and h such that Note that lim t⟶+∞ g t = −∞ and g t > 0 when t is close to 0, so sup t≥0 g t is attained for some T ε > 0. Furthermore, from g ′ T ε = 0, it follows that By multiplying the equation in (49) by T 1−p ε , we obtain By applying (9), we have for ε small enough On the other hand, we multiply the equation in (50) by T 1−rp ε and by recalling (53), we obtain 54 By applying (9), we have for ε small enough Now, we estimate g T ε .
It follows from h ′ t = 0 that and θ = p * − p / p r − 1 As θ > 1, then by (57), we get which implies from Lemma 4 that f θ y * = 0 with y * defined in (34).Therefore, h ′ t * = 0, where As f θ y is concave, then h ′ t is convex and so  9) and as q > N p − 1 / N − p , we have

64
Taking ε small enough, we obtain sup t≥0 E tz ε < C * .Thus, the proof of this lemma is completed.Now, we can proof the existence of a mountain pass-type solution.
Proof of Theorem 1. Applying Lemma 5, we get that E possesses a mountain pass geometry.Then, from the mountain pass theorem [13], there exists a PS c sequences u n ⊂ W 1,p 0 Ω of E. According to Lemmas 6 and 7, u n has a subsequence (still denoted by u n ) such that u n ⟶ u in W 1,p 0 Ω .Hence, u is a critical point of E and therefore a solution of (1).Now, we show that u > 0. To obtain a contradiction assume that u = u − .We have Then, u − = 0.By the strong maximum principle [17], one has u > 0. Theorem 1 can be concluded.