Research on Sturm–Liouville boundary value problems of fractional p-Laplacian equation

In this work we investigate the following fractional p -Laplacian differential equation with Sturm–Liouville boundary value conditions: { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e.  t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ where D t α 0 C ${}_{0}^{C}D_{t}^{\alpha }$ , D T α t ${}_{t}D_{T}^{\alpha }$ are the left Caputo and right Riemann–Liouville fractional derivatives of order α ∈ ( 1 2 , 1 ] $\alpha \in ( {\frac{1}{2},1} ]$ , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.


Introduction
Fractional differential equations have been extensively applied in mathematical modeling. Many scholars have developed a strong interest in this kind of problem and achieved some excellent results [1][2][3][4][5][6][7][8]. Especially, in the last several years, the investigations on the equations including both left and right fractional differential operators have got increasing attention. Left and right fractional differential operators are widely used in the physical phenomena of anomalous diffusion, such as fractional convection diffusion equation [9,10]. In [11], Ervin and Roop first proposed a class of steady-state fractional convectiondiffusion equations with variational structure Jiao and Zhou [12] considered the Dirichlet problems T (u (t))) + ∇F(t, u(t)) = 0, a.e. t ∈ [0, T], 0 ≤ β < 1, u(0) = u(T) = 0.
The authors gave the variational structure of the problem. Under the Ambrosetti-Rabinowitz condition, the existence results were obtained by employing the mountain pass theorem and the minimization principle. The following year, the authors [13] further studied the following problems: Under the Ambrosetti-Rabinowitz condition, the existence of weak solution was obtained by using the mountain pass theorem. In addition, the authors also discussed the regularity of weak solution.
Based on the above work, this article further studies problem (1.1) with the concaveconvex nonlinearity. In order to compare the results of this paper with Theorem 1.0, the main assumptions and conclusions of this paper are given below. In this paper, we study the case that the nonlinearity f ∈ C([0, T] × R, R) involves a combination of p-suplinear and p-sublinear terms. That is, where f 1 (t, u) is p-suplinear as |u| → ∞ and f 2 (t, u) is p-sublinear growth at infinity. Here we give some reasonable assumptions on f 1 and f 2 as follows: where F 1 (t, x) = Theorem 1.1 Assume that (H 1 )-(H 5 ) hold. Then problem (1.1) with λ = 1 has at least two nontrivial weak solutions. Remark 1.1 Clearly, conditions (H 2 ) and (H 3 ) are weaker than condition (F 1 ) of Theorem 1.0. In addition, the nonlinear function f studied in this paper is more general, it contains both p-suplinear and p-sublinear terms. Consequently, our conclusion generalizes Theorem 1.0 in [17].
Moreover, we also consider that the nonlinear function f satisfies p-sublinear growth. The specific assumptions are as follows: (H 6 ) There exist L > 0, 0 < β ≤ p such that

Preliminaries
For the convenience of readers, this section firstly introduces some basic definitions and lemmas of fractional calculus theory.
Let AC([a, b]) be the space of absolutely continuous functions within [a, b] (see [16]).
Let us recall that, for any fixed t ∈ [0, T] and 1 ≤ r < ∞, By combining Lemma 2.5, we can see that, for ∀u ∈ E α,p , if 1/p < α ≤ 1, then By multiplying the equation in problem (1.1) by any v ∈ E α,p and integrating on [0, T], one has (2.5) From Definitions 2.1, 2.2 and Lemma 2.9, we can get Getting the similar result for the second part of equation (1.1), we can give the definition of weak solution for problem (1.1).

Definition 2.4
The function u ∈ E α,p is a weak solution of problem (1.1) if the identity holds for any v ∈ E α,p .
Define the functional I : E α,p → R as follows: (2.7) According to the continuity of f , it is easy to prove I ∈ C 1 (E α,p , R). For ∀v ∈ E α,p , one has (2.8) Then Therefore, the critical point of functional I corresponds to the weak solution of problem (1.1). To prove our main results, we introduce the following tools. Let A ∈ , if there is an odd mapping G ∈ C(A, R n \{0}) and n is the smallest integer with this property, then we say that the deficit of A is n, and γ (A) = n.

Lemma 2.11 ([19])
Let I ∈ C 1 (X, R) be an even functional on X and I satisfy the (PS) condition. For any n ∈ N, let (1) If n = ∅ and c n ∈ R, then c n is the critical value of I.
Remark 2.2 According to Remark 7.3 in [19], if K c ∈ and γ (K c ) > 1, then K c contains an infinite number of different points. That is, I has an infinite number of different critical points on X.
Lemma 2.12 ([20]) Let X be a reflexive real Banach space, : X → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X * , : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist r > 0, x ∈ X with r < (x) such that the functionalλ is coercive. Then, for each λ ∈ r , the functionalλ has at least three distinct critical points in X.

Main results
In order to prove the theorems, the following lemma plays an essential role.

Lemma 3.1 Under the assumption given in Theorem 1.1, I satisfies the (PS) condition.
Proof Assuming that {u k } k∈N ⊂ E α,p is a sequence such that {I(u k )} k∈N is bounded and I (u k ) → 0 as k → ∞, then there exists D > 0 such that The first step, we prove that {u k } k∈N is bounded in E α,p . If not, we assume that u k a → +∞ as k → ∞. Let z k = u k u k a , then z k a = 1. Since E α,p is a reflexive Banach space, there exists a subsequence of {z k } (still denoted as {z k }) such that z k z 0 (k → ∞) in E α,p , then z k → z 0 in C([0, T], R). By (H 4 ) and (H 5 ), one has The following two cases are discussed. Case 1: z 0 = 0. Let = {t ∈ [0, T]||z 0 (t)| > 0}, then meas( ) > 0. Because u k a → +∞ (k → ∞) and |u k (t)| = |z k (t)| · u k a , so for t ∈ , one has |u k (t)| → +∞ (k → ∞). On the one hand, by (2.3), (2.7), (3.1), (3.2), we have Since θ > p > r > 1, so On the other hand, according to Fatou's lemma, the properties of and (H 3 ), one has This is a contradiction to (3.3). Case 2: z 0 ≡ 0. According to the continuity of f , there exists d 0 > 0 such that Combined with condition (H 2 ), we get It is a contradiction. Therefore, {u k } k∈N is bounded in E α,p . Suppose that the sequence {u k } k∈N has a subsequence, still denoted as {u k } k∈N , there exists u ∈ E α,p such that u k u in E α,p , then u k → u in C([0, T], R). So The proof process of Theorem 1.1 is given below, which is structured into four steps.
Step 2. We will prove that condition (i) in Lemma 2.10 holds. By (H 1 ), for ∀ε > 0, there exists a constant δ > 0 such that (3.5) Step 4. It is known from (3.6) that I is bounded below in B ρ . Choose ϕ ∈ E α,p such that ϕ(t) = 0 in [0, T]. For ∀l ∈ (0, +∞), by (2.7), (H 3 ), and (H 4 ), we have (3.9) Thus, from 1 < r < p, we know that, for small enough l 0 satisfying l 0 ϕ a ≤ ρ, one has I(l 0 ϕ) < 0. Let u = l 0 ϕ, one has where ρ is defined in Step 2. Then, according to the Ekeland variational principle, there exists a minimization sequence {v k } k∈N ⊂ B ρ such that That is, {v k } k∈N is a (PS) sequence. According to Lemma 3.1, I satisfies the (PS) condition. Therefore, c (2) < 0 is another critical value of I. So there exists 0 = u (2) ∈ E α,p such that The proof of Theorem 1.2 is given below.
Proof of Theorem 1.2 The functionals : E α,p → R and : E α,p → R are defined as follows: then I(u) = (u)λ (u). Through simple calculation, we get Furthermore, and are continuous Gâteaux differential and Thus is weakly semicontinuous. In addition, we will show that is coercive and has a continuous inverse on (E α,p ) * . For u ∈ E α,p \{0}, by (3.10), one has then is coercive. For ∀u, v ∈ E α,p , by (3.10), we obtain From [22], we can see that there exist constants c p , d p > 0 such that (3.12) If p ≥ 2, then Since 1 < r 1 < p, (3.15) indicates that I(u) → ∞ as u a → ∞, so I is bounded below.
In the second step, I satisfies the (PS) condition on E α,p . Assume that {u k } ⊂ E α,p is a sequence such that where D > 0 is a constant. Then (3.15) shows that {u k } k∈N is bounded on E α,p . Suppose that the sequence {u k } k∈N has a subsequence, still recorded as {u k } k∈N , there exists u ∈ E α,p such that u k u in E α,p , then u k → u in C([0, T], R). So so u ku a → 0 (k → ∞). This means that u k → u in E α,p . That is, I satisfies the (PS) condition on E α,p . In addition, (2.7) and (H 10 ) indicate that I is an even functional and I(0) = 0. Fix n ∈ N, then take n disjoint open intervals i such that ∪ n i=1 i ⊂ . Let u i ∈ (W 1,2 ( i , R) ∩ E α,p )\{0} satisfy u i a = 1, and remember E n = span{u 1 , u 2 , . . . , u n }, S n = u ∈ E n | u a = 1 .
Therefore, for u ∈ E n , there exists λ i ∈ R such that then |λ i | p , ∀u ∈ E n . (3.17)

Conclusions
This paper mainly explores the multiplicity of solutions for a fractional p-Laplacian differential equation with Sturm-Liouville boundary value conditions. By employing variational methods, the multiplicity results of weak solutions are obtained under the conditions of p-suplinear growth, p-sublinear growth, and the combination of p-suplinear growth and p-sublinear growth. Compared with the existing related work, the research results of this paper weaken the existing related conditions and improve and enrich the related results to a certain extent.