A CRITICAL FRACTIONAL p –KIRCHHOFF TYPE PROBLEM INVOLVING DISCONTINUOUS NONLINEARITY

. The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional p -Kirchhoﬀ type problem involving the critical Sobolev exponent and discontinuous nonlinearity: where M ( t ) = a + bt θ − 1 for t ≥ 0, a ≥ 0 ,b > 0 ,θ > 1, ( − ∆) sp is the fractional p –Laplacian with 0 < s < 1 and 1 < p < N/s , p ∗ s = Np/ ( N − ps ) is the critical Sobolev exponent, λ > 0 is a parameter, and f : R N × R → R is a function. Under suitable assumptions on f , we show that there exists λ 0 > 0 such that the above equation admits at least one nontrivial nonnegative solution provided λ < λ 0 by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any k ∈ N , there exists Λ k > 0 such that the above equation has k pairs of nontrivial solutions if λ < Λ k . The main feature is that our paper covers the degenerate case, that is the coeﬃcient of ( − ∆) sp may be zero at zero. Moreover, the existence results are obtained when f is discontinuous. Thus, our results are new even in the semilinear case.

1. Introduction and main results. In this paper we are concerned with the existence and multiplicity of solutions for a critical elliptic equations involving the fractional p-Laplacian. More precisely, we consider where the Kirchhoff term M (t) = a + bt θ−1 for t ≥ 0, a ≥ 0, b > 0, θ > 1, N > sp with s ∈ (0, 1), and (−∆) s p is the fractional p-Laplacian which (up to normalization 414 MINGQI XIANG AND BINLIN ZHANG factors) may be defined for any x ∈ R N as |ϕ(x) − ϕ(y)| p−2 (ϕ(x) − ϕ(y)) |x − y| N +ps dy along any ϕ ∈ C ∞ 0 (R N ), where B ε (x) denotes the ball in R N centered at x with radius ε. In particular, (−∆) s p becomes the fractional Laplacian (−∆) s as p = 2, and (−∆) s p reduces to the standard p-Laplacian as s ↑ 1 in the limit sense of Bourgain-Brézis-Mironescu, as shown in [9]. For more details about the fractional Laplacian, we refer to [17,33,41,49] and the references therein.
In the last years, a great attention has been paid to the study of fractional and non-local problems involving critical nonlinearities. For example, a Brézis-Nirenberg type result for fractional Laplacian in bounded domain with homogeneous Dirichlet boundary datum is given in [44], see also [36] for related results for fractional p-Laplacian and the references cited there for further results. Nonexistence results for nonlocal equations involving critical and supercritical nonlinearities can be found in [43]. The multiplicity of solutions for critical fractional Laplacian equations is an interesting and difficult problem. For example, a multiplicity result for fractional Laplacian problems in R N is obtained in [5] by using the mountain pass theorem and the direct method in variational methods, where one of two superlinear nonlinearities could be critical or even supercritical. Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator are obtained in [22] by using a suitable truncation argument combined with Krasnoselskii's genus theory, see also [32,42] for more results in this direction. In [46], Wang and Xiang considered a superlinear fractional Choquard equation with fractional magnetic operators and critical exponent and obtained infinitely many solutions by critical point theory. It is worth mentioning that the interest in nonlocal fractional problems goes beyond the mathematical curiosity. Indeed, this type of operators arises in a quite natural way in many different applications, such as, continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, see for example [3,11,17,27] and the references therein. The literature on non-local operators and their applications is quite large, here we just quote a few, see [34,35,49] and the references therein. For the basic properties of fractional Sobolev spaces, we refer the readers to [17,33].
Recently, Fiscella and Valdinoci in [20] proposed a stationary Kirchhoff variational equation which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. More precisely, they established a model given by the following formulation: where M (t) = a + bt for all t ≥ 0, here a > 0, b ≥ 0. Note that M is this type, problem (2) is called non-degenerate if a > 0 and b ≥ 0, while it is named degenerate if a = 0 and b > 0, see [42] for some physical explanations about non-degenerate Kirchhoff problems. Hence problem (2) [48] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave and the Kirchhoff function is degenerate. Using the same methods as in [48], Pucci et al. in [41] obtained the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type in the whole space. Indeed, the fractional Kirchhoff problems have been extensively studied in recent years, for instance, we also refer to [12,40] about non-degenerate Kirchhoff type problems and to [4,21,42] about degenerate Kirchhoff type problems for the recent advances in this direction. When s = 1, a = 1 and b = 0, problem (1) reduces to the following equation where −∆ p u = −div(|∇u| p−2 ∇u) is the p-Laplacian operator and f is a discontinuous nonlinearity. In [45], Shang studied problem (3) by using variational methods and obtained the existence and multiplicity of solutions. In [1], Alves and Bertone got two nonnegative solutions for the following equation where H is the Heaviside function, i.e.
For more results on such kinds of discontinuous nonlinearity problems, we refer the readers to [2] and the references cited there. Motivated by above cited papers, we will study the existence and multiplicity of solutions for problem (1) involving the fractional p-Laplacian in R N and discontinuous nonlinearity. To the best of our knowledge, there is no result in the literature on problem (1). Certainly, such a setting raises extra difficulties due to the lack of compactness and the nonlocal nature of the p-fractional Laplacian, as well as the presence of discontinuous nonlinearity. For this purpose, we shall use the principle of concentration compactness of P.L. Lions in fractional Sobolev spaces.
Note that if a = 1, b = 0, λ = 1 and f ≡ 0, problem (1) reduces to the following fractional p-Laplaican problem where D s,p (R N ) denotes the completion of C ∞ 0 (R N ) with respect to the norm [5,41]. Let which is positive by the fractional Sobolev inequality. Here and throughout this paper, we shortly denote by | · | q the norm of L q (R N ) for any q ∈ (1, ∞). Recently, Brasco et al. in [8] obtained that there exists a radially symmetric nonnegative decreasing minimizer U = U (r) for S. The authors also showed that U is a weak 416 MINGQI XIANG AND BINLIN ZHANG solution of (4) and satisfies In this paper, we assume f : R N × R → R is a measurable function satisfying: for all x ∈ R N and for all t ∈ R, there exist the limits: A simple example of f which satisfies (f 1 )-(f 4 ) is given by where C(x) ∈ C ∞ 0 (Ω) with positive minimum and Ω ⊂ R N is bounded, sgn denotes the sign function, a 0 > 0, θp < q < p * s and t + = max{0, t}.
Since p * s > θp by θ < N/(N − sp) and q > θp, we deduce that there exists ρ > 0 such that max t≥0 g(t) = g(ρ) > 0. Based on this fact, we further assume that which will be used later. Note that by assumption (f 1 ), it is easy to see that We give the definition of weak solutions for problem (1).
Now we can state the first result as follows.
One of the main motivations is to consider a particular and relevant case associated to problem (1) given by H is the Heaviside function, λ and Θ are positive real parameters, θp < q < p * s and h : We are in a position to state the second result of our paper as follows: To study the multiplicity result, we need the following assumption: (f 1 ) f (x, 0) = 0 and there exist the limits: Then for each k > 0 there is λ k > 0 such that problem (1) has at least k pairs of nontrivial solutions for all λ ∈ (0, λ k ).
Finally, let us simply describe the main approach to obtain Theorem 1.2. The proof relies on some results of convex analysis since the functional I λ associated to problem (1) is locally Lipschitz. To get critical points for I λ , we use a version of the mountain pass theorem for locally Lipschitz functional. It is worth stressing that the major difficulties in this paper are as follows: first of all, because we are working with the fractional p-Laplacian, which is not linear and local, the growth of the nonlinear part is critical. Secondly, arguments involved the case that f (x, t) is continuous with respect to t (the classical case) could not be used easily in our context. In particular, we establish a new estimate which is a crucial ingredient to prove that the energy functional verifies the nonsmooth Palais-Smale condition at some levels.
This paper is organized as follows. In Section 2, we first recall some basic properties of generalized gradient of locally Lipschitz functionals and the concentration compactness lemma in fractional Sobolev spaces D s,p (R N ). In Section 3, using the mountain pass theorem combined with the principle of concentration compactness in D s,p (R N ), we first establish the existence of one nontrivial and nonnegative solutions for problem (1) with a suitable range of positive parameter λ. Then using the symmetric mountain pass theorem, we obtain the multiplicity of solutions for problem (1).

2.
Preliminaries. Throughout this paper, for any set A ⊂ R N , |A| is the Lebesgue measure of A. Set u ± = max{±u, 0}. Denote by o(1) a real vanishing sequence, and let C be various positive constants.
Let us now recall some definitions and properties of generalized gradient of locally Lipschitz functionals, which will be used later and can be found in [15]. Let X be a Banach space, X * be its topological dual and ·, · be the duality. A functional I : X → R is said to be locally Lipschitz if for all u ∈ X there exists a neighborhood U of u and a constant K > 0 depending on U such that For a locally Lipschitz functional I, the directional derivative at u ∈ X in the direction v ∈ X is defined by It is easy to know that I 0 (u; v) is subadditive and positively homogeneous. Moreover, there is a k > 0 such that |I 0 (u; v)| ≤ k v for all v ∈ X, that is, for each u ∈ X, the functional I 0 (u; ·) is a continuous on X.
The generalized gradient of I at u is defined as Then, for each v ∈ X, I 0 (u; v) = max{ ω, v : ω ∈ ∂I(u)}. A point u ∈ X is a critical point of I if 0 ∈ ∂I(u). If u ∈ X is a local minimum or maximum, then 0 ∈ ∂I(u).
The solutions of (1) are exactly the critical points of the functional given by We say that I λ satisfies the nonsmooth (P S) c condition on X, if any sequence {u n } n ⊂ X such that I λ (u n ) → c and m(u n ) = min{ ω (D s,p (R N )) * : ω ∈ ∂I λ (u n )} → 0 as n → ∞, possesses a convergent subsequence. To prove our main results, we need the generalizations of the mountain pass theorem [14] and of the symmetric mountain pass theorem [23].
Theorem 2.1. (see [14]) Let X be a reflexive Banach space, I : X → R is locally Lipschitz functional which satisfies the nonsmooth (P S) c condition, I(0) = 0 and there are ρ, α > 0 and e ∈ X with e > ρ, such that Then there exists u 0 ∈ X such that 0 ∈ ∂I(u 0 ) and I(u 0 ) = c, where
In [50], Xiang et al. established the principle of concentration compactness in D s,p (R N ), which can be regarded as a fractional counterpart of the principle of concentration compactness in classical Sobolev space W 1,p (R N ). Now, we recall the fractional concentration-compactness principle, which will be the keystone that enables us to verify that I λ satisfies the nonsmooth (P S) c condition.
Let C c (R N ) = {u ∈ C(R N ) : supp u is a compact subset of R N } and denote by C 0 (R N ) the closure of C c (R N ) with respect to the norm |η| ∞ = sup x∈R N |η(x)|. As well known, a finite measure on R N is a continuous linear functional on C 0 (R N ). Now we give a norm for measure µ µ = sup where (µ, η) = R N ηdµ. Definition 2.3. Let M(R N ) denote the finite nonnegative Borel measure space on R N . For any µ ∈ M(R N ), µ(R N ) = µ holds. We say that µ n µ weakly * in M(R N ), if (µ n , η) → (µ, η) holds for all η ∈ C 0 (R N ) as n → ∞.

Then
and S > 0 is the best constant of D s,p (R N ) → L p * s (R N ).
The following lemma is fundamental to prove Theorem 2.4.
Lemma 2.5. (see [50]) Assume {u n } n ⊂ D s,p (R N ) is the sequence given by Theorem 2.4 and let x 0 ∈ R N fixed and let ϕ ∈ Evidently, Theorem 2.4 does not provide any information about the possible loss of mass at infinity for a weakly convergent sequence. The following theorem expresses this fact in quantitative terms.
Theorem 2.6. (see [50]) Let {u n } n ⊂ D s,p (R N ) be a bounded sequence such that and define Then the quantities µ ∞ and ν ∞ are well defined and satisfy lim sup Moreover, the following inequality holds 3. Proofs of Theorems 1.2-1.4. Before giving the proof of main results, let us first show that the functional I λ is a locally Lipschitz functional. Set for all u ∈ D s,p (R N ). Then I λ (u) = Φ(u) − Ψ(u). Proof. Let u, v ∈ D s,p (R N ). It follows from (f 2 ), the mean value theorem and Hölder's inequality that Hence, Ψ(u) is locally Lipschitz in D s,p (R N ). By the definition of the directional derivative, there exist functions v ∈ C ∞ 0 (R N ), h n → 0 in D s,p (R N ) and δ n ↓ 0 such that and {v > 0} := {x ∈ R N : v(x) > 0}. Since h n → 0 in D s,p (R N ), up to a subsequence, we may assume that h n → 0 a.e. in R N . By h n → 0 in D s,p (R N ), we Observe that the mean value theorem yields Thus, by Fatou's lemma, one can deduce that Similarly, Therefore, we arrive at the following inequality Let Arguing by contradiction, we assume that there is a set A ⊂ R N with |A| > 0 such that Taking v = −χ A (x) in (7) and using the definition of the generalized gradient, we obtain which contradicts (8). Thus, f (x, u) ≤ ω(x) a.e. in R N . Similarly, we can obtain ω(x) ≤ f (x, u) a.e. in R N . Therefore, the proof is complete.
It is easy to see that Φ(u) ∈ C 1 (D s,p (R N ), R), this, together with Lemma 3.1, implies that I λ is a locally Lipschitz functional. Thus, we have ω ∈ ∂I λ (u) if and By the definition of D s,p (R N ), we know that (9) holds for any ϕ ∈ D s,p (R N ). .
Proof. Let {u n } n ⊂ D s,p (R N ) be such that I λ (u n ) → c and m(u n ) = min{ ω X * : ω ∈ ∂I λ (u n )} → 0 as n → ∞. Here X * = (D s,p (R N )) * . We first show that {u n } n is bounded in D s,p (R N ). Let ω n ∈ ∂I λ (u n ) such that ω n X * = m(u n ) = o(1). By (9), we have Hence, we deduce which together with σ > θp > p implies that {u n } n is bounded in D s,p (R N ). Thus, up to a subsequence, there exists a nonnegative function u ∈ D s,p (R N ) such that u n u in D s,p (R N ), u n → u in L σ loc for σ ∈ [1, p * s ), and u n → u a.e. in R N . By Theorem 2.4, up to a subsequence, there exists a (at most) countable set J, a non-atomic measure µ, points {x j } j∈J ⊂ R N and {µ j } j∈J , {ν j } j∈J ⊂ R + such that as n → ∞ and in the measure sense, where δ xj is the Dirac measure concentrated x j . Moreover, and S > 0 is the best constant of the embedding D s,p (R N ) → L p * s (R N ). Next we claim that J = ∅. Suppose by contradiction that and |∇ϕ ε,j | ≤ 2/ε. Obviously, ϕ ε,j u n ∈ D s.p (R N ). Replacing u n with ϕ ε,j u n in ω n , u n we have where u n , u n ϕ ε,j : |u n (x) − u n (y)| p−2 (u n (x) − u n (y))(ϕ ε,j (x)u n (x) − ϕ ε,j (y)u n (y)) |x − y| N +ps dxdy.

MINGQI XIANG AND BINLIN ZHANG
Using a similar discussion as in [50], we get Hence we deduce from (21), (24), (26) and (27) that thanks to the fact that θ > 1. By (f 2 ) and ω n ∈ [f (x, u n ), f (x, u n )], it is easy to see that Therefore, we conclude from (26)- (29) and (22) that bµ θ ∞ ≤ λν ∞ , which together with (23) yields ν ∞ = 0 or Assume that (30) holds. Then It follows from (30) that which is a contradiction. Hence ν ∞ = 0. In view of J = ∅ and (25), we have Now we show that u n → u in D s,p (R N ). To this aim, we first assume that d := inf n≥1 u n > 0.
Since ω n , u n − u → 0 as n → ∞, we have A similar discussion gives that lim n→∞ R N |u n | p * s −2 u n udx = R N |u| p * s dx.