Multiple solutions for a fractional p-Kirchho ﬀ equation with critical growth and low order perturbations

: In this article, we deal with the following fractional p -Kirchho ﬀ type equation


< β <
N(p * α −1)+α p * α , 1 < p < pk < p * α = p(N−α) N−ps is the fractional critical Hardy-Sobolev exponent.Problem (1.1) reduces to the following stationary analogue of the Kirchhoff equation which was proposed by Kirchhoff in [12] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.Here, L is the length of the string, f (x, u) is the area of the cross section, E is the Young modulus of the material, ρ is the mass density and P 0 is the initial tension.The appearance of nonlocal term Ω |∇u| 2 dx in the equations make its importance in many physical applications.It was pointed out that such nonlocal problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself (see [1]).Recently a great attention has been focused on studying the problems involving fractional Sobolev spaces and corresponding nonlocal equations.Indeed, nonlocal fractional problems arise in a quite natural way in many different contexts, such as, optimization, finance, phase transitions, stratified materials, anomalous diffusion, semipermeable membranes and flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, water waves and so on, we refer to [15] for more details.
In particular, Chen et al. in [6] considered the following fractional p-Laplacian equation with subcritical and critical growths They obtained the existence of positive solutions, ground state solutions and sign-changing solutions of the fractional p-Laplacian Eq (1.4) by using the variational method.
In [22], Xiang et al. studied the following fractional p-Laplacian Kirchhoff type equation with critical growth where By the variational method, the authors proved that Eq (1.5) admits at least two nonnegative solutions.
Recently, the following fractional p-Laplacian Kirchhoff type equation with critical growth has been well studied by various authors where u, Fiscella and Pucci in [10] deal with the existence and the asymptotic behavior of nontrivial solutions for Eq (1.6) with pk < q < p * α .In [5], Chen and Gui obtained the existence of multiple solutions to Eq (1.6) with w(x) = 1 and 1 < q < p < pk.When f (x, u) = (I µ * F(x, u)) f (u), I µ = |x| −µ is the Riesz potential of order µ ∈ (0, min{N, 2ps}), Chen [4] established the existence of positive solutions to Eq (1.6).Chen, Rǎdulescu and Zhang in [7] obtained the existence of a positive weak solution of Eq (1.6) with f (x, u) = |u| q−2 u and 1 < q < N p N−ps .Many papers studied the existence of infinitely many weak solutions and nontrivial solutions of Eq (1.6), we refer the readers to [3, 9, 11, 16-21, 23, 24] and the reference therein.
In this paper, we are interested in the existence and multiplicity of positive solutions of Problem (1.1) with critical growth.Our technique based on the Ekland variational principle and the Mountain pass lemma.Since the Problem (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding W s,p (Ω) → L p * α (Ω), we overcome this difficulty by using the concentration compactness principle.Now we state our main result.

Preliminary results
Define W s,p (Ω), the usual fractional Sobolev space endowed with the norm The space X is endowed with the norm , where the norm in L p (Ω) is denoted by • p .The space X 0 is defined as X 0 = {u ∈ X : u = 0 on CΩ} or equivalently the closure of C ∞ 0 in X, for all p > 1, it is a uniformly convex Banach space endowed with the norm The dual space of X 0 will be denoted by X * 0 .Since u = 0 in R N \ Ω, the integral in (2.1) can be extended to R N × R N .
The energy functional I λ : X 0 → R associated with Eq (1.1) is We say that u is a weak solution of Eq (1.1), if u satisfies |x − y| N+ps dxdy.
Then, we can obtain the following useful Lemma.
Proof.(i) Let R be a constant such that Ω ⊂ B(0, R) = {x ∈ R N : |x| < R}, by the Hölder inequality and the Sobolev inequality, for all β < α −1 and ω denotes the N-dimensional measure of the unit sphere.Combining with (2.2) and the Sobolev inequality, one has , there exists a constant r > 0, for all λ ∈ (0, Λ 0 ), we obtain that I λ (u) ≥ r > 0 with u = ρ.
For all u ∈ X 0 \{0}, we get ( Hence, we obtain that I λ (tu) < 0 with t > 0 small enough, when u ≤ ρ, one has (ii) For every u ∈ X 0 \{0}, we have as t → +∞.Consequently, we can find e ∈ X 0 such that I λ (e) < 0 provided with e > ρ.The proof is complete.
Next, we assume that a, b and k satisfy one of the following cases: Then, we have the following compactness result.Proof.Let {u n }⊂ X 0 be a (PS ) c λ sequence for (2.5) It follows form (2.2) and (2.5) that This implies that {u n } is bounded in X 0 .Up to a subsequence, still denote by {u n }, there exists u ∈ X 0 such that (2.6) By using the concentration compactness principle ( [6], Lemma 4.5), there exist u ∈ X 0 , two Borel regular measures σ and ν, J denumerable, at most countable set {x j } j∈J ⊂ Ω, and non-negative numbers {σ j } j∈J , {ν j } j∈J ⊂ [0, ∞), for all j ∈ J, such that as )) be a smooth cut-off function centered at x j such that 0 ≤ φ ε, j ≤ 1, ∇φ ε, j ∞ ≤ C ε , and (2.8) From the first term in (2.8), we have |x − y| N+ps dxdy. (2.9) Note that {u n } is bounded in X 0 , the third term in (2.9), by using the Hölder inequality and Lemma 2.3 in [22], we get where C > 0 is a positive constant.Letting ε → 0, by (2.2) and (2.7), we get (2.10) Combining the above facts with (2.8), we have (2.11) Hence, taking the limit for n → ∞ and ε → 0 in (2.8), it follows from (2.10) and (2.11) that ν j ≥ aσ j + bσ k j .This together with (2.7) implies that either ν j = 0 or (2.12) From (2.7) and (2.12), we obtain that σ j = 0 or

3). (2.13)
To proceed further we show that (2.12) and (2.13) are impossible.Indeed, by contradiction, we assume that there exists j 0 ∈ J such that (2.12) and (2.13) hold.Applying (2.2), (2.7) and the Sobolev inequality, we get By using the Young inequality, when a > 0, we have Consequently, we deduce that c λ ≥ c * = Λ − Dλ p p−1 .This is a contradiction.Hence σ j = ν j = 0 for all j ∈ J, which implies that as n → ∞.Now, we prove that u n → u in X 0 , let ϕ ∈ X 0 be fixed and B ϕ be the linear functional on X 0 defined by for every v ∈ X 0 .By using the Hölder inequality, one has According to I λ (u n ) → 0 in X * 0 and u n u in X 0 , we have (2.15) Since {u n } is bounded in X 0 , by (2.6), one has ) By a, b > 0, we get lim Moreover, it follows from the Brezis-Lieb Lemma that This together with (2.14) and the Hölder inequality, one has Let us now recall the well-known Simon inequalities.That is, for every Indeed, since u n p and u p are bounded in X 0 , by the subadditivity inequality, for all ξ, η ≥ 0 and 1 < p < 2, one has (ξ + η) Thus, we obtain that u n → u in X 0 .The proof is complete.
From [13], let 0 ≤ α < ps < N, for all minimizer U α for S α , there exist x 0 ∈ R N and a nonincreasing u : R + → R such that U α = u(|x − x 0 |).Next, we fix a positive radially symmetric decreasing minimizer U α = U α (r) for S α , multiplying U α by a positive constant, we assume that Then, we have the following Lemma.
Applying the Mountain pass Lemma According to Lemma 2.2, we know that {u n } ⊂ X 0 has a convergent subsequence, still denoted by {u n }, we may assume that u n → u * in X 0 as n → ∞.
which implies that u * 0. Similarly, we can obtain that u * is a positive solution of Eq (1.1) with I λ (u * ) > 0. That is, the proof of Theorem 1.1 is complete.

Conclusions
In this paper, we consider a class of fractional p-Kirchhoff type equations with critical growth.Under some suitable assumptions, by using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of positive solutions.
Case 2.3.if p ≥ 2, it follows from (2.17) and (2.18) as n → ∞ that u n .18)where c p , C p > 0 depending only on p.According to (2.18), we distinguish two cases: