#### Abstract

We study the existence and multiplicity of solutions for the following fractional boundary value problem: , where are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.

#### 1. Introduction and Main Results

Consider the fractional boundary value problem (BVP for short) of the following form: where and are the left and right Riemann-Liouville fractional integrals of order , respectively, satisfies the following assumptions.(A) is measurable in for every and continuously differentiable in for a.e. , and there exist , , such that for all and a.e. . In particular, if , BVP (1.1) reduces to the standard second-order boundary value problem of the following form:

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not mere mathematical curiosities but rather has interesting applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Such differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Samko et al. [4], and the papers [5–20] and the references therein.

Recently, there are many papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or singular and nonsingular boundary) value problems of fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems [12–14], Leray-Schauder theory [15, 16], lower and upper solution method, monotone iterative method [17–19], Adomian decomposition method [20], etc.), see [12–20] and the references therein.

Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis. The existence and multiplicity of solutions for Hamilton systems, Schrödinger equations, and Dirac equations have been studied extensively via critical point theory, see [21–34].

In [32], Jiao and Zhou obtained the existence of solutions for BVP (1.1) by Mountain Pass theorem under the Ambrosetti-Rabinowitz condition (denoted by A.R. condition). Under the usual A.R. condition, it is easy to show that the energy functional associated with the system has the Mountain Pass geometry and satisfies the (PS) condition. However, the A.R. condition is so strong that many potential functions cannot satisfy it, then the problem becomes more delicate and complicated.

In this paper, in order to establish the existence and multiplicity of solutions for BVP (1.1) under distinct hypotheses on potential function by critical point theory, we introduce some functional space , where , and divide the problem into the following three cases.

##### 1.1. The Superquadratic Case

For the superquadratic case, we make the following assumptions. (A1), uniformly for some and a.e. .(A2) uniformly for some and a.e. .(A3) uniformly for some and a.e. ,

where and . We state our first existence result as follows.

Theorem 1.1. *Assume that (A1)–(A3) hold and that satisfies the condition (A). Then BVP (1.1) has at least one solution on .*

##### 1.2. The Asymptotically Quadratic Case

For the asymptotically quadratic case, we assume the following.(A2′) uniformly for some and a.e. .(A4)There exists such that for all and a.e. .(A5) for a.e. .

Our second and third main results read as follows.

Theorem 1.2. *Assume that satisfies (A), (A1), (A2'), (A4), and (A5). Then BVP (1.1) has at least one solution on .*

Theorem 1.3. *Assume that satisfies (A), (A1), (A2'), and the following conditions: *(A4′)* there exists such that for all and a.e. ;*(A5′)* for a.e. .**Then BVP (1.1) has at least one solution on .*

##### 1.3. The Subquadratic Case

For the subquadratic case, we give the following multiplicity result.

Theorem 1.4. *Assume that satisfies the following assumption: *(A6)*, where and is a constant.*

Then BVP (1.1) has infinitely many solutions on .

#### 2. Preliminaries

In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space are also listed for the convenience of readers.

*Definition 2.1 (see [1]). * Let be a function defined on and . The left and right Riemann-Liouville fractional integrals of order for function denoted by and , respectively, are defined by
provided the right-hand sides are pointwise defined on , where is the gamma function.

*Definition 2.2 (see [1]). *Let be a function defined on and . The left and right Riemann-Liouville fractional derivatives of order for function denoted by and , respectively, are defined by
where , and .

The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by . is the space of functions such that and . In particular, .

*Definition 2.3 (see [1]). *Let and . If and , then the left and right Caputo fractional derivative of order for function denoted by and , respectively, exist almost everywhere on . and are represented by
respectively, where .

*Property 2.4 (see [1]). *The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, that is,

*Definition 2.5 (see [32]). *Define and . The fractional derivative space is defined by the closure of with respect to the norm
where denotes the set of all functions with . It is obvious that the fractional derivative space is the space of functions having an -order Caputo fractional derivative and .

Proposition 2.6 (see [32]). *Let and . The fractional derivative space is a reflexive and separable space.*

Proposition 2.7 (see [32]). *Let and . For all , one has
**
Moreover, if and , then
*

According to (2.8), we can consider with respect to the norm

Proposition 2.8 (see [32]). *Define and . Assume that and the sequence converges weakly to in , that is, . Then in , that is, , as .*

Making use of Property 2.4 and Definition 2.3, for any , BVP (1.1) is equivalent to the following problem: where .

In the following, we will treat BVP (2.10) in the Hilbert space with the corresponding norm . The variational structure of BVP (2.10) on the space has been established.

Lemma 2.9 (see [32]). *Let be defined by
**
where satisfies the assumption (A).*

If , then the functional defined by is continuously differentiable on , and , we have

*Definition 2.10 (see [32]). *A function is called a solution of BVP (2.10) if (i) is derivative for almost every , (ii) satisfies (2.10), where .

Lemma 2.11 (see [32]). *Let and be defined by (2.12). If assumption (A) is satisfied and is a solution of corresponding Euler equation , then is a solution of BVP (2.10) which corresponding to the solution of BVP (1.1).*

By Lemma 2.11, it means that the solutions for BVP (1.1) correspond to the critical points of the functional . We need the following estimate and known results for the sequel.

Proposition 2.12 (see [32]). *If , then for any , one has
*

Lemma 2.13 (see [23]). *Let be a real Banach space, is differentiable. One says that satisfies the condition if any sequence in such that is bounded and as contains a convergent subsequence.*

Lemma 2.14 (Mountain Pass theorem [24]). *Let be a real Banach space and is differentiable and satisfies the (PS) condition. Suppose that *(i)*,*(ii)*there exist and such that for all with , *(iii)*there exists in with such that .**Then possesses a critical value . Moreover, can be characterized as
**
where .*

Lemma 2.15 (Clark theorem [24]). *Let be a real Banach space, with even, bounded below, and satisfying the (PS) condition. Suppose , there is a set such that is homeomorphic to , , by an odd map, and . Then possesses at least distinct pairs of critical points.*

#### 3. Proof of the Theorems

For , where is a reflexive Banach space with the norm defined by

It follows from Lemma 2.9 that the functional on given by is continuously differentiable on . Moreover, we have

Recall that a sequence is said to be a (C) sequence of if is bounded and as . The functional satisfies condition (C) if every (C) sequence of has a convergent subsequence. This condition is due to Cerami [21].

##### 3.1. Proof of Theorem 1.1

We will first establish the following lemma and then give the proof of Theorem 1.1.

Lemma 3.1. *Assume (A), (A2), and (A3) hold, then the functional satisfies condition (C).*

*Proof of Lemma 3.1. *Let be a (C) sequence of , that is, is bounded and as . Then there exists such that
for all .

By (A2), there exist positive constants and such that
for all and a.e. .

It follows from that
for all and a.e. . Therefore, we obtain
for all and a.e. .

Combining (2.14) and (3.8), we get

On the other hand, by (A3), there exist and such that
for a.e. and .

By (A), we have
for all and a.e. .

Therefore, we obtain
for all and a.e. .

It follows from (3.5) and (3.12) that
thus, is bounded.

If , then
which, combining (3.9), implies that is bounded.

If , then
where
by (2.8).

Since , it follows from (3.9) that is bounded too. Thus is bounded in .

By Proposition 2.8, the sequence has a subsequence, also denoted by , such that

Then we obtain in by use of the same argument of Theorem 5.2 in [32]. The proof of Lemma 3.1 is completed.

*Proof of Theorem 1.1. *By (A1), there exist and such that
for a.e. and with .

Let
Then it follows from (2.8) that
for all with .

Therefore, we have
for all with . This implies that (ii) in Lemma 2.14 is satisfied.

It is obvious from the definition of and (A1) that , and therefore, it suffices to show that satisfies (iii) in Lemma 2.14.

By (A1), there exist and such that
for all and a.e. .

It follows from (A) that
for all and a.e. .

Therefore, we obtain
for all and a.e. .

Choosing , then

For and noting that (3.24) and (3.25), we have
as , where is a positive constant. Then there exists a sufficiently large such that . Hence (iii) holds.

Finally, noting that while for critical point , . Hence is a nontrivial solution of BVP (1.1), and this completes the proof.

##### 3.2. Proof of Theorem 1.2

The following lemmata are needed in the proof of Theorem 1.2.

Lemma 3.2. *Assume (A5), then for any , there exists a subset with meas such that
**
uniformly for .*

*Proof of Lemma 3.2. * The proof is similar to that of Lemma 2 in [29] and is omitted.

Lemma 3.3. *Assume (A), (A2’), (A4), and (A5), then the functional satisfies condition (C).*

*Proof of Lemma 3.3. * Suppose that is a (C) sequence of , that is, is bounded and as . Then we have
which implies that

We only need to show that is bounded in . If is unbounded, we may assume, without loss of generality, that as . Put , we then have . Going to a sequence if necessary, we assume that weakly in , strongly in and .

By (A2), it follows that there exist constants and such that
for all and a.e. .

By assumption (A), it follows that
for all and a.e. . Therefore, we obtain
for all and a.e. . Therefore, we have
from which, it follows that

Passing to the limit in the last inequality, we get
which yields . Therefore, there exists a subset with meas such that on .

By virtue of Lemma 3.2, for ) meas, we can choose a subset with meas such that
uniformly for .

We assert that meas. If not, meas.

Since , it follows that
which leads to a contradiction and establishes the assertion.

By (A4), we obtain thye following:

By (3.36), (3.38), and Fatou's lemma, it follows that
which contradicts (3.29). This contradiction shows that is bounded in , and this completes the proof.

By virtue of Lemmas 3.2 and 3.3, the rest of the proof is similar to Theorem 1.1. Theorem 1.3 can be proved similarly.

##### 3.3. Proof of Theorem 1.4

The proof of Theorem 1.4 is divided into a sequence of lemma.

Lemma 3.4. * The functional is bounded below on .*

*Proof of Lemma 3.4. *By (2.8) and (2.14), for every , we have
where . The proof of Lemma 3.4 is complete.

Lemma 3.5. * The functional satisfies the (PS) condition.*

*Proof of Lemma 3.5. *Let be a Palais-Smale sequence in , that is,

Suppose that is unbounded in , that is, as . Since

However, from (3.42), we have
thus is a bounded sequence in . Since is a reflexive space, going, if necessary, to a subsequence, we can assume that in , thus we have
as . Moreover, according to (2.8) and Proposition 2.8, we have that is bounded in and as .

Noting that
Combining (3.44) and (3.45), it is easy to verify that as , and hence that in . Thus, admits a convergent subsequence. The proof of Lemma 3.5 is complete.

Lemma 3.6. *For any , there exists a set which is homeomorphic to by an odd map, and . *

*Proof of Lemma 3.6. * For every , define
where is a positive number to be chosen later.

For any , there exist , , such that
where and is a real quadratic form.

Since
So, is a real positive definite quadratic form. Then there exist an invertible matrix and , , such that

It is easy to prove that the odd mapping defined by
is a homeomorphism between and .

Since is a finite dimensional space, there exists such that

Otherwise, for any positive integer , there exists such that

Set , then for all and

Since , it follows from the compactness of the unit sphere of that there exists a subsequence, denoted also by , such that converges to some in . It is obvious that .

By the equivalence of the norms on the finite dimensional space, we have in , that is,

By (3.54) and Hölder inequality, we have

Thus, there exist such that

In fact, if not, we have
for all positive integer .

It implies that
as . Hence which contradicts that . Therefore, (3.56) holds.

Now let
and .

By (3.53) and (3.56), we have
for all positive integer . Let be large enough such that
then we have
for all large , which is a contradiction to (3.55). Therefore, (3.51) holds.

For any , we have
by (3.51), where .

Choosing , we conclude which completes the proof.

Now from the assertion of Lemma 2.15, we know that has at least distinct pairs of critical points for every , therefore, BVP (1.1) possesses infinitely many solutions on . The proof of Theorem 1.4 is completed.

#### 4. Examples

In this section, we give some examples to illustrate our results.

*Example 4.1. *In BVP (1.1), let

These show that all conditions of Theorem 1.1 are satisfied, where

By Theorem 1.1, BVP (1.1) has at least one solution .

*Example 4.2. *In BVP (1.1), let and , where and will be specified below.

Let . Noting that , we see that (A) and (A2′) hold. It is also easy to see that (A1) holds for