Multiplicity of solutions for a fractional Kirchhoff type problem

In this paper, by using the (variant) Fountain Theorem, we obtain that there are infinitely many solutions for a Kirchhoff type equation that involves a nonlocal operator.


1.
Introduction. In this article, we investigate multiplicity of solutions for the following problem where Ω is an open bounded set in R N , N > 2s with s ∈ (0, 1), a, b > 0, f is a continuous function whose properties will be introduced later, · X0 is a functional norm which is defined in (9) and L K is a nonlocal operator defined as follows: Here K : R N \{0} → (0, +∞) is a measurable function which satisfies    γK(x) ∈ L 1 (R N ) with γ(x) = min{|x| 2 , 1}; there exists θ > 0 such that K(x) ≥ θ|x| −(N +2s) for any x ∈ R N \{0}; K(x) = K(−x) for any x ∈ R N \{0}. (3) A typical example is K(x) = |x| −(N +2s) . In this case L K u(x) = −(− ) s u(x) is the fractional Laplace operator which (up to normalization factors) can be defined as In the classical Laplace operator case, problem (1) is related to the stationary analogue of the equation proposed by Kirchhoff [14] 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. Some early studies of Kirchhoff equations were those of Bernstein [3] and Pohozaev [21]. Equation (5) received much attention only after Lions [16] proposed an abstract framework to the problem. More recently, by variational methods, Alves [1], Ma-Rivera [18] studied the existence of one positive solution, and He-Zou [12,13] obtained the existence of infinitely many positive solutions for problem (5) respectively. Perera-Zhang [20] studied the existence of nontrivial solutions of problem (5) via the Yang index theory; Zhang-Perera [35] and Mao-Zhang [19] established the existence of sign-changing solutions of problem (5) via invariant sets of descent flow. We refer to [5,6,7,8,15,17,31,33,34] for more existence results of the Kirchhoff type equations.
Recently, much attention has been focused on studying the existence and multiplicity of solutions to problem (1) when a = 1, b = 0. This type operator seems to have a prevalent role in physical situations such as combustion and dislocations inmechanical systems or in crystals. This problem has been studied by many authors, see [10,22,23,24,25,26,27,28,29] and the references therein.
The Kirchhoff type problem involving an integrodifferential operator was first studied by Fiscella and Valdinoci in a recent paper [11]. They considered the following Kirchhoff type problem: where Ω ⊂ R N is an open bounded set, 2 * = 2N N −2s with s ∈ (0, 1), N > 2s, L K is a nonlocal operator defined in (2), M and f are continuous functions. Based on the Mountain Pass Theorem, the authors in [11] obtained the existence of a nontrivial solution to (6).
Moreover, Sun and Teng [30] investigated problem (1) when K(x) = |x| −(N +2s) , which is a special case of problem (1). The authors proved the existence and multiplicity of solutions by using the Mountain Pass Theorem and the symmetric Mountain Pass Theorem together with truncation techniques.
Motivated by the above results, in the present paper, we establish that there are infinitely many solutions to problem (1) by the Fountain Theorem.
In order to state our main results, let us introduce the functional space that we will use in the following, which was introduced in [25]. For fixed s ∈ (0, 1) , N > 2s, Ω ⊂ R N is an open bounded set with Lipschitz boundary, and X is the linear space of Lebesgue measurable functions from R N to R such that the restriction to Ω of function g in X belongs to L 2 (Ω) and According to the conditions of K, by Lemma 11 in [24], we have that C 2 0 (Ω) ⊆ X 0 , and so X and X 0 are nonempty. The spaces X and X 0 is endowed, respectively, with the norms defined by and where Q = R 2N \ ((CΩ) × (CΩ)) ⊂ R 2N . Since g ∈ X 0 , then the integral in (9) can be extended to all R 2N . Moreover, the norm on X 0 given in (9) is equivalent to the usual one defined in (8), by Lemmas 6 and 7 in [25].
With the norm given in (9), X 0 is a Hilbert space with scalar product defined as see Lemma 7 in [25]. For the framework of functional Sobolev spaces and fractional Laplace, we refer the reader to the survey of Di Nezza, Palatucci and Valdinoci [9].
Definition 1.1. We say that u is a weak solution of problem (1), if u satisfies Let J : X 0 → R be the energy functional associated with problem (1) defined by where F is the primitive function of f with respect to the second variable, that is, We can see that J ∈ C 1 (X 0 , R) and for any φ ∈ X 0 , there holds Now, we suppose that the right hand side of (1) is a Caratheodory function f : Ω × R N → R N verifying the following conditions: (F 2 ) f ∈ C(Ω × R, R) and for some 2 < p < 2 * , with 2 * = +∞ for N ≤ 2s and In the seminal work of Ambrosetti and Rabinowitz [2], a common feature is that the nonlinearity f (x, u) satisfies the Ambrosetti-Rabinowitz's 2−superlinear condition: (AR) there exist µ > 2 and R > 0 such that The role of (AR) is to ensure the boundedness of the Palais-Smale sequences of the Euler-Lagrange functional. This condition is crucial in the applications of critical point theory. Indeed, (AR) condition implies that for some c, d > 0, However, the second term in the Euler-Lagrange functional J of problem (1) is the 4-power of norm · X0 . Therefore, in our first result, we assume that f (x, u) is the Ambrosetti-Rabinowitz's 4−superlinear condition: Then problem (1) has a sequence of solutions u k such that J(u k ) → ∞ as k → ∞.
Remark 1. From the assumption (F 3 ), after integrating, we obtain that there exists C > 0 such that Moreover, by simple calculations, we can see that ( Hence F (x, u) grows in a 4−superlinear rate as |u| → +∞.
It is well known that (F 3 ) guarantees the boundedness of the (P S) c sequence of the corresponding functional J. Then we can apply the Fountain Theorem in [32] to get the desired result. However, there are many functions which are 4−superlinear but do not satisfy (F 3 ) for any α > 4. For example f (x, u) = u 3 (4 + ln(1 + |u|)).
So without condition (F 3 ), it is difficult to derive the boundedness of the (P S) c sequence of the corresponding functional. In this case, inspired by the variant Fountain Theorem in [36], we have the following result. Theorem 1.3. In addition to conditions (F 1 ) and (F 2 ), suppose that the following conditions are satisfied: This paper is organized as follows. In Section 2, we give some preliminaries. We prove Theorems 1.2 and 1.3 in Section 3 and Section 4 respectively.

2.
Preliminaries. In this paper we use the following notations: X 0 denotes Hilbert space given by (7) with the norm · given in (9), X * 0 denotes the dual space for X 0 , L p (Ω) denotes Lebesgue space with the norm | · | p .
Taking into account Lemma 8 in [25], we have the following result.
Lemma 2.1. The embedding X 0 → L r (Ω) is continuous for any r ∈ [1, 2 * ], while it is compact whenever r ∈ [1, 2 * ). Moreover, there exists a positive constant c(θ) depending on θ (which is given in (3)), such that Furthermore, there is a constant c r > 0 such that for every u ∈ X 0 , Next, we introduce the definition of the (P S) c condition.
Definition 2.2. Let J ∈ C 1 (X 0 , R) and c ∈ R. The functional J satisfies the (P S) c condition if any sequence {u n } ⊂ X 0 such that has a convergent subsequence.
Since X 0 is a reflexive and separable Banach space, then there exist e j ∈ X 0 and e * j ∈ X * 0 (j = 1, 2, · · · ) such that e i , e * j = δ ij where δ ij = 1 for i = j and δ ij = 0 for i = j, and X 0 = span{e j | j = 1, 2, · · · }, Theorem 2.3. [32, Theorem 3.6: the Fountain Theorem] Let J ∈ C 1 (X 0 , R) be an even functional. Assume that for each k ∈ N, there exists ρ k > γ k > 0 such that (a 1 ) a k := max (a 3 ) J satisfies the (P S) c condition for every c > 0. Then J has an unbounded sequence of critical values.
So (a 2 ) holds. Next we prove J satisfies (P S) c condition. Indeed, let {u n } be the (P S) c sequence of J, that is, for every c > 0, Let η ∈ ( 1 α , 1 4 ), using (14) and (26), for n sufficiently large, we obtain This yields that {u n } is bounded in X 0 .
Going if necessary to a subsequence, still denote it u n , there exists u ∈ X 0 such that u n → u weakly in X 0 , that is Observe that It is clear that the left hand side of (28) tends to zero as n → ∞. From (27), we have b( u n 2 X0 − u 2 X0 ) u, u n − u X0 → 0 as n → ∞.
Hence we get (a 3 ). Applying Theorem 2.3, problem (1) has a sequence of solutions u k such that J(u k ) → ∞ as k → ∞.

4.
Proof of Theorem 1.3. In order to use Theorem 2.4 to prove Theorem 1.3, we define Next, we show that Φ λ satisfies the assumptions in Theorem 2.4.
Proof. We first prove (i). From (S 1 ), we get for any L > 0, there exists a constant C L such that F (x, u) ≥ L|u| 4 − C L for all u ∈ R. Since Y k is a finite dimensional space, then there exists c k,r > 0 such that |u| r ≥ c k,r u X0 for all u ∈ Y k . Therefore, for u ∈ Y k , where C 0 is a positive constant. Choosing L such that Lc 4 k,r > b 4 , then for u X0 = ρ k with ρ k > 0 large enough, we get that (i) holds.
Case (i), by Φ λn (u n ) = 0, we get Dividing both sides of the above equality by λ n u n 4 X0 , we obtain On the other hand, by Fatou's Lemma and condition (S 1 ), we have which is a contradiction. Case (ii), set Φ λn (t n u n ) = max For any given M > 0, define w * n = √ 4c * w n with c * > 0. By (21), we get as n → ∞. Then for n large enough, we have since c * > 0 can be large arbitrarily, we then get that lim n→∞ Φ λn (t n u n ) → +∞.

WENJING CHEN
Moreover, by the definition of t n , we have that Φ λn (t n u n ), t n u n X0 = 0. Thus, by (S 2 ), we have as n → ∞. This is a contradiction and this completes the proof of Lemma 4.2.
Lemma 4.3. The sequence {u n } has a convergent subsequence with the limit u k ∈ X 0 for k large.
Proof. By Lemma 4.2, {u n } is bounded in X 0 , then there is a subsequence of {u n }, denoted by itself, such that u n → u k weakly in X 0 as k → ∞. By the same argument given in Section 3, we get u n → u k strongly in X 0 as k → ∞.
Proof of Theorem 1.3. Recall that u n := u k (λ n ). From Lemmas 4.1-4.3 and Lemma 2.1, by the standard argument, we obtain that there is a convergent subsequence of {u n } when λ n → 1, such that u n u k for some u k ∈ X 0 . On the other hand, we have J(u n ) = Φ 1 (u n ) = Φ λn (u n ) + (λ n − 1) Ω F (x, u n )dx.
Using these facts and (31), we obtain and lim n→∞ J (u n ), v X0 = 0 for all v ∈ X 0 .
In view of J ∈ C 1 (X 0 , R), we have J (u n ) → J (u k ) in X * 0 . Therefore, for every v ∈ X 0 , J (u n ) − J (u k ), v X0 ≤ J (u n ) − J (u k ) X * 0 v X0 → 0 as n → ∞. This means that J (u k ), v X0 = 0 for all v ∈ X 0 , that is, J (u k ) = 0 in X * 0 . By (33) and β k (λ n ) → +∞ as k → ∞, we then get that {u k } ∞ k=1 is an unbounded sequence of critical points of J(u). This completes the proof of Theorem 1.3.