Multiplicity of solutions for the Dirichlet boundary value problem to a fractional quasilinear differential model with impulses

This paper aims to consider the multiplicity of solutions for a kind of boundary value problem to a fractional quasilinear diﬀerential model with impulsive eﬀects. By establishing a new variational structure and overcoming the diﬃculties brought by the inﬂuence of impulsive eﬀects, some new results are acquired via the symmetry mountain-pass theorem, which extend and enrich some previous results. MSC: 26A33; 34G20; 34B15


Introduction
In this paper, we are concerned with the following fractional quasilinear differential model with impulsive effects.
which plays an important role in some research fields of physics (see [1,2] and the references therein). An interesting question as to whether the existence or multiplicity of solutions to this fractional quasilinear differential model with suitable boundary conditions generated by impulsive effects can be obtained naturally comes to mind. It is well known that the impulsive differential models describe the discontinuous process and originate from some important research fields. In recent years, critical-point theory has been successfully applied to deal with the existence and multiplicity of solutions of boundary value problems (BVPs for short) to differential equations with impulsive effects. Based on some critical-point theorems, Nieto and O'Regan [3] considered the impulsive Dirichlet BVP (u (t j )) = I j (u(t j )), j = 1, 2, . . . , m, and obtained some existence results. Subsequently, more and more scholars have paid attention to this problem, such as Sun and Chen [4], Zhou and Li [5], Zhang and Yuan [6], etc. Moreover, for the case of impulsive BVPs with p-Laplacian operator, one can refer to [7,8] and references therein. On the other hand, recently, Jiao and Zhou [9] proved that under the Dirichlet boundary condition u(0) = u(T) = 0, the operator c t D α T 0 D α t has a variational structure. Also, by the mountain-pass theorem, the existence of solutions to the following systems was obtained under the Ambrosetti-Rabinowitz condition: where α ∈ ( 1 2 , 1]. After that, Bonanno, Rodríguez-López and Tersian [10] discussed the existence of three solutions to the following problem with impulsive effects and parameters: where α ∈ ( 1 2 , 1]. Nyamoradi and Rodríguez-López [11] extended the scalar model of (1.5) to the case of Hamiltonian systems and obtained the multiplicity of solutions by the variant Fountain theorems. Moreover, by the gene property and the mountain-pass theorem, Ledesma and Nyamoradi [12] investigated the eigenvalue problem t D α T φ p ( 0 D α t u) = λφ p (u) with the Dirichlet boundary conditions u(0) = u(T) = 0 and obtained the existence of solutions to the Dirichlet boundary problem of a fractional p-Laplacian equation with impulsive effects. Liu, Wang and Shen [13] extended the results of [12] to the case of combined nonlinearity. Furthermore, for the Dirichlet BVPs and other BVPs of fractional differential equations with or without impulsive effects, one can refer to [14][15][16][17][18][19][20][21] and references therein.
Motivated by the works mentioned above, we are concerned with the multiplicity of solutions to the fractional quasilinear differential model with impulsive effects (1.1). Let us present our paper's contribution: To begin with, the variational structure of (1.1) is established, which makes the critical-point theory applicable to discuss the existence and multiplicity of solutions to this problem. Moreover, the impulsive effects produced by the , which make this problem challenging. Furthermore, there are few papers considering this problem.
In order to describe our main conclusion, the following assumptions are presented: (I1) For any t ∈ R, I 1j (t) and I 2j (t) are odd on t and (I2) There exist constants a 1j , a 2j , d 1j , d 2j > 0 such that (I3) For any t ∈ R, I 1j (t) and I 2j (t) satisfy Now, we state our main results. Theorem 1.1 Assuming that the conditions (I1)-(I3) and (G1)-(G5) are satisfied, there exists a constant ζ * > 0 such that the problem (1.1) has infinitely many nontrivial weak solutions, provided that ζ ∈ [0, ζ * ).
Remark 1.2 It should be pointed out that if ζ = 0, the condition (G3) can be removed. Moreover, the conditions (G1) and (G2) are weaker than the following classical Ambrosetti-Rabinowitz condition: Then, the conclusion of Theorem 1.1 is also true.
Remark 1.5 It should be pointed out that the impulsive nonlinearity I 1j could be superlinear growth when θ > 4.

Preliminaries
For the definitions of fractional integrals and derivatives relating to the well-known left and right Riemann-Liouville and Caputo, one can refer to references [22,23]. Next, some of the necessary results and properties will be presented. Define the Sobolev space Let By the method of [24], the space E α 0 can be decomposed as follows. In fact, based on the Riesz representation theorem, we can find a linear self-adjoint operator Q : which implies that Noting that the embedding E α 0 → C is compact (see [9]), it implies that Q is compact. In view of the well-known compact operator's spectral theory, for the operator I -Q, we can decompose the Sobolev space E α 0 into the orthogonal sum of invariant subspaces as where Eand E + are negative and positive spectral subspaces corresponding to the operator I -Q, E 0 = N(I -Q). Moreover, letting = {1, 2, . . . , ι} with ι ∈ N, Q possesses only finitely many eigenvalues {λ i } i∈ satisfying λ i > 1 because Q is compact on E α 0 , which implies that the dimension of subspace Eis finite. By the classical self-adjoint operator theory, for I -Q that can be viewed as a compact perturbation relating to the self-adjoint operator I, it is clear that 0 is excluded in the essential spectrum of I -Q. Thus, the dimension of subspace E 0 is also finite. Furthermore, there exists a positive constant κ such that   Based on (2.4), clearly, the norm of E α 0 is equivalent to 0 D α t u L 2 .
If u ∈ E α 0 is a solution of the problem (1.1), for v ∈ E α 0 , based on Lemma 2.1 and Proposition 2.3, it implies that Similarly, one has As a conclusion, the definition of a weak solution is shown as follows.
holds for any v ∈ E α 0 .
Define the functional : where G(t, u) = u 0 g(t, s) ds. Since g, I 1j and I 2j are continuous, by the standard arguments, one can obtain that (u) ∈ C 1 (E α 0 , R). Moreover, it is clear that the critical points of (u) are weak solutions of the problem (1.1).

Lemma 2.7 ([25]) Let E be a Banach space and
satisfies the (PS)condition and the following conditions.
(ii) For each finite-dimensional subspace X ⊂ E, there exists an l = l( X) such that ≤ 0 on X \ B l . Then, has an unbounded sequence of critical values.

Main results
In order to prove our main conclusions, we need the following lemmas. First, in E α 0 , let V = E -⊕ E 0 and X = E + , then the dimension of subspace V is finite and E α 0 = V ⊕ X.
Based on Fatou's lemma, it follows that lim inf which is in contradiction to (3.8). Thus, {u n } is bounded, which implies that {u n } possesses a convergent subsequence (named again {u n }) such that u n = u + n + un + u 0 n u = u + + u -+ u 0 and u + n u + in E α 0 . Moreover, u n → u and u + n → u + uniformly in C. It should be