Inﬁnitely Many Solutions for Fractional p -Laplacian Schrödinger–Kirchhoff Type Equations with Symmetric Variable-Order

: In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations (cid:18) , x ∈ ∂ Ω , where Ω is a bounded Lipschitz domain in R N , 1 < p < + ∞ , a , b > 0 are constants, s ( · ) : R N × R N → ( 0,1 ) is a continuous and symmetric function with N > s ( x , y ) p for all ( x , y ) ∈ Ω × Ω , λ > 0 is a parameter, ( − ∆ ) s ( · ) p is a fractional p -Laplace operator with variable-order, V ( x ) : Ω → R + is a potential function, and f ( x , ξ ) : Ω × R N → R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.


Abstract:
In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger-Kirchhoff type equations where Ω is a bounded Lipschitz domain in R N , 1 < p < +∞, a, b > 0 are constants, s(·) : R N × R N → (0, 1) is a continuous and symmetric function with N > s(x, y)p for all (x, y) ∈ Ω × Ω, λ > 0 is a parameter, (−∆)
For the variable-order fractional and some important results, we refer to [7][8][9][10][11][12]. The fractional derivatives of variable-order were introduced by Lorenzo et al. in [7]. Subsequently, Samko  differentiation to the fractional operator with variable-order, see [9,10] for more details with respect to this topic.
When p = 2, the fractional Laplace operator with variable-order was studied by Xiang et al. in [11], they investigated the following Laplacian equations where (−∆) s(·) is the fractional Laplacian operator. First of all, they proved the embedding theorem of variable-order fractional Sobolev space, and then they obtained a multiplicity result for a Schrödinger equation via variational methods.
Note that we also mention the work by Wang et al. in [12]; they also studied the fractional Laplace operator with variable-order, as follows where M is a model of Kirchhoff coefficient, and infinitely many solutions were obtained by using four different critical point theorems. The main feature for this kind of Kirchhoff-type problem is that M could be zero at zero. In recent decades, many scholars have extensively studied the existence of results for classical Schrödinger equations and fractional Schrödinger equations under reasonable assumptions of V and f . We refer the reader to [13][14][15][16][17]. Nyamoradi et al. in [15] studied the Schrödinger-Kirchhoff type equations by variational methods. Note that Teng in [16] established the existence of high or small energy solutions by applying a variant fountain theorem. Especially, in [17] César E. Torres Ledesma studied the existence of multiple solutions for Schrödinger-Kirchhoff type involving the non-homogeneous fractional p-Laplacian, which V and f are under some weaker assumptions.
On the other hand, the Kirchhoff equation was introduced by Kirchhoff in [18]. Kirchhoff proposed the following model where ρ, p 0 , h, E, L are constants with respect to some physical meanings, respectively. We call (P v ) a problem of Kirchhoff type because there is the Kirchhoff term which not only makes the study of (P v ) interesting but also becomes more delicate and causes some mathematical difficulties. The literature on Kirchhoff type problems about the existence and multiplicity of solutions is quite large; here we just list a few, for example, [4,6,17,[19][20][21][22][23][24] for further details. It is worth pointing out that if a > 0, b ≥ 0, M(t) = a + bt m is non-degenerate Kirchhoff type equations, for example, see [23,24]; if a = 0, b > 0,M(t) = a + bt m is degenerate Kirchhoff type equations, such as, see [6,20]. The fractional Kirchhoff type equation regarding non-local integro-differential operator was first introduced in [ where Ω is a bounded Lipschitz domain of R N . Moreover, in [20], Molica et al. investigated a kind of nonlocal fractional Kirchhoff type equations, and three solutions were obtained by applying the critical points theorem. Indeed, the works of literature on Kirchhoff equations, Schrödinger equations, and their applications are quite large. Kirchhoff equations model several physical and biological systems, for example, population density, see [25][26][27] for some related works. On the other hand, many scholars are interested in Schrödinger equations, which describe the dynamic behavior of particles in quantum mechanics, see [28], and the standing wave solutions, see [29]. Moreover, fractional Schrödinger-Kirchhoff type equations involving an external magnetic potential were studied in [30].
As is known, the (AR) condition plays a crucial role to guarantee that the Palais-Smale sequences are bounded. In the famous paper [31], Ambrosetti and Rabinowitz introduced the well-known (AR) condition, that is, there exist constants p 2 0 < µ 0 and 0 < M 0 such that In [32], Servadei et al. obtained the existence and multiplicity of nontrivial solutions and showed that the verification of the Palais-Smale compactness condition depends on the (AR) condition. However, there are a lot of functions where the (AR) condition is not satisfied, an example of such function is For this kind of problem, many people have attached much importance to finding new, reasonable conditions instead of the (AR) condition, see, for instance, [5,6,33,34].
Motivated by the above cited works, we find that there are some papers on Kirchhoff equations or Schrödinger equations involving the fractional p-Laplace operator; however, there are no results for Schrödinger-Kirchhoff type equations driven by the fractional p-Laplace operator with variable-order. Thus, we are devoted to investigating the existence of infinitely many solutions for Schrödinger-Kirchhoff type equations involving a variableorder fractional p-Laplace operator by applying the fountain theorem and symmetric mountain pass theorem, respectively.
Our work is different from the previous articles. To the best of our knowledge, this article is the first to discuss the existence of infinitely many solutions for the fractional p-Laplacian Schrödinger-Kirchhoff type equations without the (AR) condition. Under the reasonable hypothesis, we first establish an embedding theorem for the variable-order fractional Sobolev spaces. Second, compared with [15], we deal with the problem (P v ) generalized from fractional framework to variable-order fractional framework, and our results generalize Theorem 1 and Theorem 2 of [15] in some directions. Finally, relative to paper [11], for the case p = 2, we can deal with a general Kirchhoff-Schrödinger type equations and the nonlinearity with variable coefficients.
Throughout this paper, for simplicity, C i and K i , i = 1, 2, ..., N are used in various places to denote distinct constants, and we will specify them whenever it is necessary. Define the function space C + (Ω) where H − := min H(x) and H + := max H(x) for all x ∈ Ω. s(·) : R N × R N → (0, 1) is a continuous function, satisfying: (S1): s(·) is symmetric function, that is, s(x, y) = s(y, x) for all (x, y) ∈ R n × R n .
Regarding the potential function V(x) : Ω → R, we assume the following hypothesis: denotes the open ball of Ω centered at x and of radius R 0 > 0.
Furthermore, the nonlinearity f (x, t) : Ω × R → R is a continuous Carathéodory function, satisfying: (F1): There exist positive constants C 1 and C 2 such that

Remark 1.
It is obvious that the condition (F3) is weaker than the well-known (AR) condition.
Before stating our main results, we need to present the corresponding variational framework and definition, which plays an important role to solve problem (P v ).
for any ϕ ∈ X 0 , where X 0 will be introduction in Section 2.
The functional I : X 0 → R, which is defined as Moreover, if (F1) and (V1) hold, then I : X 0 → R is of class C 1 (X 0 , R) and for any ξ, ϕ ∈ X 0 . Under our reasonable assumptions, the functional I is well defined. Hence, ξ ∈ X 0 is a (weak) solution of Schrödinger-Kirchhoff type equations (P v ) if and only if ξ ∈ X 0 is a critical point of the functional I.
hold, then the problem (P v ) has infinitely many nontrivial weak solutions in X 0 , whenever λ > 0 is sufficiently large.
The remainder of this paper is organized as follows. Some basic knowledge about the Lebesgue spaces with variable exponent and fractional Sobolev spaces with variable exponents and variable-order are given in Section 2. The functional I satisfying (PS) condition is proved in Section 3. In Section 4, by using the fountain theorem, we prove Theorem 1. Finally, in Section 5, we prove Theorem 1 by applying the symmetric mountain pass theorem.

Variable Exponent Lebesgue Spaces
In this subsection, we recall some preliminary knowledge of generalized Lebesgue spaces with variable exponent. The readers are invited to consult [35][36][37][38][39][40] for a detailed description.
Let Ω be a nonempty Lipschitz domain in R N , a measurable function ϑ(x) ∈ C + (Ω), and u be a measurable real-valued function. We introduce the Lebesgue spaces with variable exponent [38]), called generalized Lebesgue space.
The relation between modular and Luxemburg norm is clarified by the following properties.

Remark 2. Note that for any function
Especially, when ϑ(x) ≡ constant, the results of Lemmas 1 and 2 still hold.

Variable-Order Fractional Sobolev Spaces
Let 1 < p < +∞, s(·) is a continuous symmetric function, and let the Gagliardo seminorm with variable-order be denoted as where ξ : Ω → R is a measurable function. Now, we define the fractional Sobolev spaces with variable-order by W = W s(·),p (Ω) := ξ ∈ L p (Ω) : ξ is a measurable and [ξ] p s(·),p < ∞ , then (W, · W ) is a reparable, reflexive Banach space and assume that it is endowed with the norm The variable-order fractional critical exponent is defined by We denote by W 0 the closure of C ∞ 0 (Ω) in W and with the norm then, the (W 0 , · W 0 ) is also a reparable, reflexive Banach space. W * 0 denotes the dual space of W 0 . Now let us give a very crucial lemma, which the proof process is similar to the one of Lemma 2.1 of [11].
Lemma 3. Let 0 < s 0 < s(·) < s 1 < 1 < p < +∞ and W s j 0 (j = 0, 1) in W 0 with s(x, y) = s j , the embeddings W Proof. First, for any ξ ∈ W 0 , we obtain where we used the fact that the kernel 1/|z| N+ps(x,y) is integrable since N + ps(x, y) > N. Taking into account the above estimate, it follows hence, for any ξ ∈ W s 1 0 and s 0 < s(·), we have On the other hand Thus, combining with (3)-(5), we get which gives the desired estimate, up to relabeling the constant C(N, s 0 , p). From this inequality, we can obtain the following continuous embedding Similarly, we can also deduce the following continuous embedding Finally, by Theorem 6.7 and Theorem 6.9 of [41], we know that for any constant exponent p ∈ (1, N p/(N − ps 0 )), the following continuous embedding holds W s 0 0 → L p (Ω). Therefore, the embedding W 0 → L p (Ω) is continuous.

Theorem 2.
Let Ω ⊂ R N be a smooth bounded domain, 1 < p < +∞ and s(·) : R N × R N → (0, 1) is a continuous function satisfying (S1) with N > ps(x, y) for all (x, y) ∈ Ω × Ω. Assume that ϑ(x) ∈ C + (Ω) such that ϑ(x) < p * s(·) for all x ∈ Ω. Then, there exists C ϑ = C ϑ (N, p, s, ϑ, Ω) such that for any ξ ∈ W, it holds that That is, the space W is continuously embedded in L ϑ(x) (Ω). Furthermore, this embedding is compact. If ξ ∈ W 0 , then there exists a constant C ϑ = C ϑ (N, p, s, ϑ, Ω) such that Proof. Here the process of proof is similar to [39]. Since 1 < p < +∞, ϑ(x) ∈ C + (Ω), s(x, y) are continuous symmetric functions, there is a constant k 1 > 0 such that Hence, we find a positive constant and K numbers of disjoint hypercubes Ω i such that Ω = K i=1 Ω i and diam(Ω i ) < , that verify that for all (z, y) ∈ Ω i × Ω i and x ∈ Ω i , i = 1, 2, ..., K. Let s i = inf (z,y)∈Ω i ×Ω i s(z, y) and the fractional Sobolev critical exponent p * s i = N p N−s i p . Then for all x ∈ Ω i and N > s i p. According to the Sobolev embedding theorem (Theorem 6.7 and Theorem 6.9 of [41]), there exists a positive constant k 2 such that In addition, from Lemma 3, we obtain Since |ξ| = ∑ K i=1 |ξ|χ Ω i , from (6), we get ϑ(x) < p * s i . There exist a i (x) ∈ C + (Ω) such that 1/ϑ(x) = 1/p * s i + 1/a i (x). By utilizing the Hölder's inequality, we have Note that by (7)-(9), we have where the constants C ϑ = C ϑ (N, p, s, ϑ, Ω) > 0. Finally, we recall that the previous embedding is compact since in the constant s i case we have that for subcritical exponents the embedding is compact. This completes the proof.
Moreover, in order to investigate the Schrödinger-Kirchhoff type equations (P v ), we consider the variable-order fractional Sobolev linear subspace X 0 with potential function, which is defined as follows The mapping ρ : X 0 → R is defined: , then the embeddings Proof. From the definition of ξ X 0 and (V1), we can get the following inequality is an equivalent norm on W 0 and the embedding X 0 → W 0 is continuous. According to Corollary 7.2 of [41], the embedding is compact. Therefore, also the embedding X 0 → L ϑ(x) (Ω) is compact by the first part of the lemma.

Lemma 5.
Assume that Ω ⊂ R N is a bounded open domain, then (X 0 , · X 0 ) is a separable and reflexive Banach space.
Proof. First, we prove that (X 0 , ξ X 0 ) is a Banach space. Set (V1) holds, {ξ n } n∈N be a Cauchy sequence in X 0 , for all ς > 0, there is a natural number N such that if n, k ≥ N Since and then ξ ∈ X 0 . Combining Fatou's lemma and (10), we deduce for all n, n j ≥ N this implies that ξ n → ξ strongly in X 0 as n → ∞. Hence, (X 0 , · X 0 ) is a Banach space. Next, we show that (X 0 , · X 0 ) is a separable and reflexive space. We define the operator G : W s(·),p (Ω) → L p (Ω) × L p (Ω × Ω) by clearly G(ξ) is an isometry, the rest of the proof is similar to Theorem 8.1 of [42]. Thus, we get (X 0 , · X 0 ) is reflexive space (see [42], Proposition 3.20), and we get (X 0 , · X 0 ) is separable space (see [42], Proposition 3.25).
) and {ξ n } n∈N be a bounded sequence in X 0 . Then there exists u ∈ X 0 L ϑ(x) (Ω) such that up to a subsequence ξ n → ξ strongly in L ϑ(x) (Ω) as n → ∞.
Theorem 3 (Fountain Theorem, see [43]). Let X 0 be a real Banach space and an even functional I ∈ C 1 (X 0 , R) satisfies the Palais-Smale ((PS) for short) condition, for every c > 0, and that there is k 0 > 0, such that for every k ≥ k 0 there exists ρ k > r k > 0, so that the following properties hold: Then I has a sequence of critical points ξ k such that I(ξ k ) → +∞. Theorem 4 (Symmetric Mountain Pass Theorem, see [44]). Let X 0 be an infinite dimensional Banach space and let I ∈ C 1 (X 0 , R) be even, satisfy (PS) condition, and I(0) = 0. If X 0 = A k B k , where A k is finite dimensional and I satisfies: (i) there exist constants ρ, α > 0 such that I| ∂B ρ B k ≥ α; (ii) for any finite dimensional subspace X ∈ X 0 , there is R = R( X) > 0 such that I(ξ) ≤ 0 on X\B R , then I possesses an unbounded sequence of critical values.

Palais-Smale Condition
In what follows, we need the following definition and prove a lemma which will play a critical role. Definition 2. Let X 0 be a Banach space, I ∈ C 1 (X 0 , R). We say that I satisfies the (PS) condition, if any (PS) c sequence {ξ n } n∈N ⊂ X 0 with I(ξ n ) → c, I (ξ n ) → 0, as n → ∞, possesses a convergent subsequence in X 0 .
Suppose that {ξ n } n∈N is a bounded sequence in X 0 . Combining Theorem 2 with Lemma 6, there exists ξ ∈ X 0 such that Lemma 7. Let the conditions of Theorem 1 hold, then I satisfies the (PS) condition for large λ > 0. Proof.
Step 1. We show that {ξ n } n∈N is bounded in X 0 . Assume that {ξ n } n∈N ⊂ X 0 is a sequence, from Definition 2, there exists a positive constant C such that for every n ∈ N. We prove this by contrary arguments. Supposing {ξ n } n∈N is unbounded in X 0 , that is Let ω n := ξ n ξ n X 0 , then ω n ∈ X 0 with ω n X 0 = 1, ω n s ≤ C s ω n X 0 = C s for ∈ Ω × R and for all t > 1.
Since µ > p 2 , letting λ > 0 be so large that the term µ−p 2 p 2 − C 6 λd is positive, we get a contradiction. Therefore, the sequence {ξ n } n∈N is bounded in X 0 .
Now, we describe the Simon inequalities, that is for all η 1 , η 2 ∈ R N , where c p > 0 and c p > 0 are constants which depend on p. We also need the following elementary inequalities, that is to say , for all 1 2 ≥ 0 and 1 < p < 2.
To prove Theorem 1 by using the fountain theorem, we first prove the following two lemmas. Lemma 9. Let the conditions of Theorem 1 hold, then there exist constants ρ k > 0 such that max ξ∈A k , ξ =ρ k I(ξ) ≤ 0.