Nontrivial Solutions of the Kirchhoff-Type Fractional p-Laplacian Dirichlet Problem

and f ∈ Cð1⁄20, T ×R,RÞ. It should be pointed out that the weak solutions of the boundary value problem (BVP for short) (1) mean the critical points of the associated energy functional. The fractional derivative 0D α t is nonlocal and reduces to the local first-order differential operator when α = 1. Moreover, the p-Laplacian φp is nonlinear and reduces to the linear identity operator when p = 2. If b = 0, BVP (1) reduces to the following fractional p-Laplacian BVP [2]:


Introduction
In this paper, we are concerned with the existence and multiplicity of nontrivial weak solutions for the Kirchhoff-type fractional Dirichlet problem with p-Laplacian of the form where a, b > 0, and p > 1 are constants, 0 D t α and t D T α are the left and right Riemann-Liouville fractional derivatives of order α ∈ ð1/p, 1, respectively, and ϕ p : ℝ ⟶ ℝ is the p-Laplacian [1] defined by and f ∈ Cð½0, T × ℝ, ℝÞ. It should be pointed out that the weak solutions of the boundary value problem (BVP for short) (1) mean the critical points of the associated energy functional. The fractional derivative 0 D α t is nonlocal and reduces to the local first-order differential operator when α = 1. Moreover, the p-Laplacian ϕ p is nonlinear and reduces to the linear identity operator when p = 2. If b = 0, BVP (1) reduces to the following fractional p-Laplacian BVP [2]: In contrast to BVP (3), if b ≠ 0, another nonlocal term, makes BVP (1) rough when one deals with it by the variational methods.
Note that, since the Kirchhoff-type p-Laplacian is a nonlinear operator, it is usually difficult to verify the Palais-Smale condition ((PS)-condition for short). Now, we make the following assumptions on the nonlinearity f .
(H 21 ). There exists a constant 1 < r 1 < p 2 and a function d ∈ There exists an open interval I ⊂ ½0, T and three constants η, δ > 0, 1 < r 2 < p 2 such that We are now to state our main results.

Preliminaries
2.1. Fractional Sobolev Space. In this subsection, we present some basic definitions and notations of the fractional calculus [25,26]. Moreover, we introduce a fractional Sobolev space and some properties of this space [14].
Definition 4 (see [25]). For β > 0, the left and right Riemann-Liouville fractional integrals of order β of a function u : ½a, b ⟶ ℝ are given by respectively, provided that the right-hand-side integrals are pointwise defined on ½a, b, where Γð⋅Þ is the gamma function.
Definition 5 (see [25]). For n − 1 ≤ β < n (n ∈ ℕ), the left and right Riemann-Liouville fractional derivatives of order β of a function u : ½a, b ⟶ ℝ are given by Remark 6. When β = 1, one can obtain from Definitions 4 and 5 that where u 0 is the usual first-order derivative of u.
Let X be a real Banach space, and I ∈ C 1 ðX, ℝÞ which means that I is a continuously Fréchet differentiable functional. Moreover, let B ρ ð0Þ be an open ball in X and ∂B ρ ð0Þ denote its boundary.
Definition 13 (see [27]). Let I ∈ C 1 ðX, ℝÞ. If any sequence fu k g ⊂ X for which fIðu k Þg is bounded and I ′ ðu k Þ ⟶ 0 as k ⟶ ∞ possesses a convergent subsequence in X, then we say that I satisfies the (PS)-condition.
Then, I possesses a critical value c ≥ σ. Moreover, c can be characterized as where Lemma 15 (see [27]). Let X be a real Banach space, and I ∈ C 1 ðX, ℝÞ satisfies the (PS)-condition. If I is bounded from below, then c = inf X I is a critical value of I.
In order to find the infinitely many critical points of I, we introduce the following genus properties. Let closed in X and symmetric with respect to 0 f g , Definition 16 (see [28]). For A ∈ Σ, we say that the genus of A is n denoted by γðAÞ = n if there is an odd map G ∈ CðA, ℝ n \ f0gÞ and n is the smallest integer with this property.
Lemma 17 (see [28]). Let I be an even C 1 functional on X and satisfy the (PS)-condition. For any n ∈ ℕ, set Remark 18. From Remark 7.3 in [28], we know that if K c ∈ Σ and γðK c Þ > 1, then K c contains infinitely many distinct points; that is, I has infinitely many distinct critical points in X.

Proof of Theorem 1
In this section, we discuss the existence of nontrivial weak solutions of BVP (1) when the nonlinearity f ðt, xÞ is ðp 2 − 1Þ-superlinear in x at infinity. Define the functional I : E α,p It is easy to verify from (15), (17), and f ∈ Cð½0, T × ℝ, ℝÞ that the functional I is well defined on E α,p 0 and is a continuously Fréchet differentiable functional; that is, which yields In the following, for simplicity, let where K > 0 is a constant. We first prove that fu k g is Thus, by (22) and (24), we have which together with I ′ ðu k Þ ⟶ 0 as k ⟶ ∞ yields Then, it follows from μ > p 2 that fu k g is bounded in E α,p 0 . Since E α,p 0 is a reflexive Banach space (see Lemma 9), going if necessary to a subsequence, we can assume u k ⟶ u in E α,p 0 . Hence, from I ′ ðu k Þ ⟶ 0 as k ⟶ ∞ and the definition of weak convergence, we have In addition, we obtain from (15), (17), and Lemma 12 that fu k g is bounded in Cð½0, T, ℝÞ and ku k − uk ∞ ⟶ 0 as k ⟶ ∞. Thus, there exists a constant c 1 > 0 such that which yields Moreover, by the boundedness of fu k g in E α,p 0 , one has where I 1 ′ is the Fréchet derivative of I 1 : E α,p 0 → ℝ defined by From (23), we have which together with (30)-(33) yields as k ⟶ ∞. Following (2.10) in [29], there exist two constants c 2 , c 3 > 0 such that Journal of Function Spaces When 1 < p < 2, based on the Hölder inequality, we get where c 4 = 2 ðp−1Þð2−pÞ/2 > 0 is a constant, which together with (37) implies When p ≥ 2, by (37), we have Then, it follows from (36), (39), and (40) that Hence, I satisfies the (PS)-condition.