Existence of Solutions for p-Kirchhoff Problem of Brézis- Nirenberg Type with Singular Terms

and Lp∗ðΩ, jxj ∗Þ denotes the usual weighted Lp∗ðΩÞ space with the weight jxjβp∗ . Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the term KðuÞ which implies that the equation in (1) is no longer a pointwise identity. In the case p = 2 and α = β = γ = μ = 0, it is analogous to the stationary version of equations that arise in the study of string or membrane vibrations, namely,

Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the term KðuÞ which implies that the equation in (1) is no longer a pointwise identity. In the case p = 2 and α = β = γ = μ = 0, it is analogous to the stationary version of equations that arise in the study of string or membrane vibrations, namely, where u denotes the displacement and gðx, uÞ is the external force. Equations of this type were first proposed by Kirchhoff in 1883 [1] to describe the transversal oscillations of a stretched string. These problems serve also 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 problem (1) without nonlocal term ða = 0Þ and without singular terms ðα = β = γ = μ = 0Þ has been treated by Brézis and Nirenberg [10] for p = 2. Subsequently, an increasing number of researchers have paid attention to semilinear or quasilinear elliptic equations with critical exponent of Sobolev or Caffarelli-Kohn-Nirenberg, for example, see [11,12] and the references therein.
Thus, it is natural for us to consider the quasilinear Brézis-Nirenberg problem in [10] with nonlocal term and singular weights, ðp > 1, a ≠ 0 and ðα, β, γ, μÞ ≠ ð0, 0, 0, 0ÞÞ: The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of D α ðΩÞ into L p * ðΩ, jxj βp * Þ prevent us from using the variational methods in a standard way. So, motivated by all the works mentioned above, we prove existence results of our problem for large range of N and under some little, as possible, conditions on q. We show that the existence of solutions depends on the parameter λ and the position of q with respect to p * − p. Here, we need more delicate estimates.
To the best of our knowledge, many of the results are new for ,q > 1, and even in the case ðα, β, γ, μÞ = ð0, 0, 0, 0Þ.
Our technique is based on variational methods and concentration compactness argument [2].
The main result is concluded as the following theorem.
Then, the problem (1) has a positive solution in the following cases: Remark 2. In the case where λ = 0 and Ω is a star-shaped domain with respect to the origin, we can easily verify that the problem (1) has no nontrivial solution by using a Pohozaev-type identity. This paper is organized as follows. In Section 2, we study the variational framework and give some preliminary results. In Section 3, we show the existence result and we will prove Theorem 1.

Variational Framework and Preliminary Results
The starting point of the variational approach to problem (1) is the following Caffarelli-Kohn-Nirenberg inequality in [13] which is also called the Hardy-Sobolev inequality. Assume that 1 < p < N,0 ≤ α < ðN − pÞ/p and α ≤ β < α + 1, and then, for some positive constantC. In the case where β = α + 1, we have p * = p,C = 1/ μ and we have the following Hardy inequality: Definition 3. We say that u ∈ D α ðΩÞ \ f0g is a weak solution of equation ð1Þ if 2 Journal of Function Spaces for any v ∈ D α ðΩÞ: Next, we define the energy functional associated to problem (1), for all u ∈ D α ðΩÞ: Notice that the functional I λ is well defined in D α ðΩÞ and belongs to C 1 ðD α ðΩÞ, ℝÞ and a critical point of I λ is a weak solution of problem (1).
Let c ∈ ℝ: We say that I λ satisfies the Palais-Smale condition at level c, if any ðPSÞ c sequence contains a convergent subsequence in D α ðΩÞ: 1 , and q ≤ p * − p: Let c ∈ ℝ + and ðu n Þ ⊂ D α ðΩÞ be a ðPSÞ c sequence for I λ . Then, for some u ∈ D α ðΩÞ with I λ ′ ðuÞ = 0: Proof. We have That is, for any v ∈ D α ðΩÞ: Then, as n ⟶ ∞, it follows that As λ < bλ 1 and q ≤ p * − p, we obtain that ðu n Þ is bounded in D α ðΩÞ. Up to a subsequence if necessary, there exists a function u ∈ D α ðΩÞ such that u n ⇀ u in D α ðΩÞ, u n ⇀ u in L p * ðΩ, jxj p * β Þ,u n ⟶ u in L r ðΩ, jxj p * β Þ, for all r < p * and u n ⟶ u a.e on Ω: Then, and thus I λ ′ ðuÞ = 0. This completes the proof of Lemma 5.
The following lemma is very important for giving the local Palais-Smale condition. Then, (1) If σ = 1 and 0 < a < S −q+p/p , then the equation f ðyÞ = 0 has a unique positive solution and f ðyÞ ≥ 0 for all y ≥ y 1 : (2) If σ > 1, then the equation f ðyÞ = 0 has a unique positive solution y 2 >ỹ and f ðyÞ ≥ 0 for all y ≥ y 2: Proof.
(1) For σ = 1 and 0 < a < S −q+p/p , we have that is, the equation f ðyÞ = 0 has a unique positive solution and f ðyÞ ≥ 0 for all y ≥ y 1 :

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(2) For σ > 1, we have f ′ ðyÞ = σS −1 y σ−1 − aS q/p and Then, f ′ ðỹÞ = 0,f ′ ðyÞ < 0 for y <ỹ and f ′ ðyÞ > 0 for y >ỹ . Hence, f is concave function and For i = 1, 2, let y i be defined in Lemma 6 and define and C * ≔ Aðy * Þ: > 0, and 0 < λ < bλ 1 : Assume that q = p * − p and 0 < a < S −q+p/p or q < p * − p and a > 0: Then, the functional I λ satisfies ðPSÞ c condition for all c < C * : Proof. Let fu n g ⊂ D α ðΩÞ is a ðPSÞ c sequence for I λ with c < C * : By the proof of Lemma 5, we have fu n g is a bounded sequence in D α ðΩÞ: Hence, by the concentration compactness principle due to Lions [2], there exists a subsequence, still denoted by fu n g, such that where I is an at most countable index set andδ i is the Dirac mass at x i . Moreover, by the Sobolev-Hardy inequality we infer that We claim that I is finite and for any i ∈ I, for ε > 0, let ϕ ε,i ðxÞ be a smooth cut-off function centered at x i such that 0 ≤ ϕ ε,i ðx Þ ≤ 1, and Since fϕ ε,i u n g is bounded in W 1,p 0 ðΩÞ and I λ ′ ðu n Þ ⟶ 0 as n ⟶ ∞, it holds by Hölder's inequality Then, θ i ≥ bη i + aη q+p/p i . Therefore, by (29), we deduce that Assume by contradiction that there exists i 0 ∈ I such that θ i 0 ≠ 0: Set y = ðθ i 0 Þ q/p * and σ = p * − p/q, then by (32) we get It is clear that σ ≥ 1 thanks to q ≤ p * − p: So, from (33) and the definition of f in Lemma 6 we get We will discuss it in two cases: Case 1. q = p * − p,b > 0 and 0 < a < S −q+p/p : According to Lemma 6, we have f ðy 1 Þ = 0 and f ðyÞ ≥ 0 if y ≥ y 1 with which implies that Journal of Function Spaces Case 2. q < p * − p,b > 0, and a > 0. In this case, from Lemma 6, we get f ðy 2 Þ = 0 and f ðyÞ ≥ 0 if y ≥ y 2 with which implies that Hence, using (29), we deduce η i 0 ≥ Sθ ( By Young inequality we have we observe that ðq/ðq + pÞpÞðb − λ/λ 1 Þ > 0,p * − q − p ≥ 0 ; thus, for j ∈ f1, 2g we get since f ðy j Þ = 0 for j ∈ f1, 2g and C * defined in Lemma 7. Contradiction with c < C * : Then, I is empty, which implies that Now, set l = limku n k as n ⟶ +∞; then, we have for any v ∈ D α ðΩÞ: Let n ⟶ +∞, and then, from (42) and (43), we deduce that Taking the test function v = u in (45), we get Therefore, the equalities (44) and (45) imply that kuk = l: Consequently, fu n g converges strongly in D α ðΩÞ, which is the desired result: 5 Journal of Function Spaces