Nontrivial solutions of a class of fractional differential equations with p-Laplacian via variational methods

In this paper, a class of boundary value problems for fractional differential equations with a parameter is studied via the variational methods. Firstly, we present a result that the boundary value problems have at least one weak solution under the quadratic condition and the superquadratic condition, respectively. Additionally, we obtain the existence of at least one nontrivial solution by using the famous mountain pass lemma without the Ambrosetti–Rabinowitz condition. Finally, by a recent critical points theorem of Bonanno and Marano, the existence of at least three solutions is established.


Introduction
Fractional-order derivatives and integrals are more suitable to describe the properties of real materials than those of integer-order, so fractional-order differential equations are more and more widely used in simulating the mechanical and electrical characteristics of real materials, dynamic system control theory, rock rheological properties, and many other fields (see [1][2][3][4][5][6][7][8][9][10] and their references). The existence of solutions for boundary value problems of fractional differential equations is also studied in various ways, such as some fixed point theorems, the fixed point index theory in cones, methods of upper and lower solutions, coincidence degree theory, topological degree theory, etc. (see [11][12][13][14][15][16]). Later, with the introduction of the mountain pass theorem, variational methods have become a new and effective tool to study the existence of solutions for boundary value problems (see [17][18][19] and their references). In addition, nonlinear terms are often required to satisfy the Ambrosetti-Rabinowitz (A-R for short) condition when variational methods are applied. For example, in [20], Zhao and Tang studied a class of boundary value problems for fractional differential equations and obtained the existence of at least one weak solution by using the critical point theory under satisfying the A-R condition.
Chen and Liu [21] considered the following boundary value problems of fractional differential equations and obtained the existence of at least one weak solution under the A-R condition: where 0 D α t and t D α T are the left and right Riemann-Liouville fractional derivatives of order α ∈ (0, 1] respectively. f : [0, T] × R → R. The result is as follows. Then BVP (1.1) admits at least one nontrivial weak solution.
Motivated by the above mentioned works, we are interested in the following fractional differential boundary value problem: where 0 < α ≤ 1, 0 D α t and t D α T denote the left and right Riemann-Liouville fractional derivatives of order α, respectively. λ > 0 is a parameter.
The existence and multiplicity of solutions for boundary value problem (BVP for short) (1.2) will be derived by the critical point theory.
In the following proofs, we assume that f (t, x) and F(t, x) satisfy some of the following conditions: is measurable in t for every x ∈ R and continuously differentiable in x for a.e. t ∈ [0, T], and there exist k 1 ∈ C(R + , R + ) and for all x ∈ R and a.e. t ∈ [0, T]. (A 1 ) There exist a constant 0 ≤ θ < p and a function η 1 (t) ∈ C([0, T]) with ess inf η 1 > 0 such that uniformly for a.e. t ∈ [0, T].
Remark 1.1 By (A 1 ), there exists a constant L 1 > 0 such that for a.e. t ∈ [0, T] and x ∈ R.
Similarly, (A 2 ) implies that there is a constant L 2 > 0 such that for a.e. t ∈ [0, T] and x ∈ R.
The following arrangement of the article is as follows. In the second part, the basic definitions, some properties and lemmas of fractional calculus are given. Additionally, the related fractional derivative space and the variational structure of BVP (1.2) is established. In the third part, we give the existence and multiplicity theorems of nontrivial solutions of BVP (1.2) under some appropriate conditions.

Preliminaries
In order to obtain the existence of solutions to BVP (1.2), we need to recall some necessary definitions and related properties of fractional calculus, which will be used in the proofs later in this paper.
where u (n-1) (t) is the usual derivative of order n -1. Now, we present the rule for fractional integration by parts.
In order to deal with BVP (1.2), we introduce appropriate function spaces and the related variational structure.
and C([0, T], R) be the p-Lebesgue space and the continuous function space, respectively, with the norms To set up the variational structure of BVP (1.2), we need to construct the appropriate function space first.
is a reflexive and separable Banach space.
If α > 1 p and 1 p + 1 q = 1, then In the following discussion, we denote x = x α,p .
For the sake of convenience, the solution of BVP (1.2) mentioned later is the weak solution of BVP (1.2).
For ∀x ∈ E α,p 0 , we consider the functional I : E α,p 0 → R as follows: From [25], we can know that I ∈ C 1 (E α,p 0 , R) and

Existence and multiplicity of solutions of BVP (1.2)
Definition 3.1 Let E be a real Banach space and J ∈ C 1 (E, R). If any sequence {x n } ⊂ E for which {J(x n )} is bounded and J (x n ) → 0 as n → +∞ possesses a convergent subsequence, then we say that J satisfies the Palais-Smale condition (P.S. condition for short).  (2) There exist ρ > 0 and σ > 0 such that

Lemma 3.3 ([26]
) Let E be a real reflexive Banach space, Φ : E → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on E * , Ψ : E → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist r > 0 and ω ∈ E, with r < Φ(ω) such that , the functional Φ -λΨ has at least three distinct critical points in E. Suppose that {x n } is unbounded. Passing to a subsequence, we may assume that x n → ∞. Let n = x n x n , so that n = 1. By Lemma 2.2, also passing to a subsequence, we can suppose that as n → ∞. By (3.1) there exists a constant K 1 > 0 such that Notice that x n → ∞, we have In view of (A 0 ) and (A 5 ), there exists Ω 0 ⊂ [0, T] with meas(Ω 0 ) = 0 such that for all x ∈ R and t ∈ [0, T] \ Ω 0 and for t ∈ [0, T] \ Ω 0 . Otherwise, there exist t 0 ∈ [0, T] \ Ω 0 and a subsequence of {x n } such that lim sup n→∞ ϑF(t 0 , x n (t 0 ))f (t 0 , x n (t 0 ))x n (t 0 ) x n p > 0. (3.6) If {x n (t 0 )} is bounded, then there exists a positive constant K 2 such that |x n (t 0 )| ≤ K 2 for all n ∈ N. By (3.4) we find as n → ∞, which contradicts (3.6). So, there is a subsequence of {x n (t 0 )} such that |x n (t 0 )| → ∞ as n → ∞. Since ϑ > p, we get n p → 0 as n → ∞, (3.7) but n = 1. This is a contradiction. Hence, {x n } is a bounded sequence in E α,p 0 . Since E α,p 0 is a reflexive space, there exists a weakly convergent subsequence such that x n x in E α,p 0 . Hence, we have as n → ∞. Combining Lemma 2.2 with (2.3), we obtain that {x n } converges to x strongly in C([0, T], R), i.e., x nx → 0 as n → ∞, which implies that as n → ∞. Note that Thus, from (3.8) and (3.9) we have as n → ∞. For any s 1 , s 2 ∈ R, it is well known that (see Lemma 4.2 in [27]) there exists a 1 > 0 such that (3.11) Following (3.11), we obtain that there exist c 1 , c 2 > 0 such that ≥ c 1 x nx p , p ≥ 2. (3.14) It follows from (3.10), (3.13), and (3.14) that The proof is completed. (1) (A 1 ) holds and λ ∈ (0, ∞).
Hence, using λ < λ r , we have On the other hand, by λ > λ l , we get