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

Abstract

In this article, we consider the new results for the Kirchhoff-type -Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.

1. Introduction

In this paper, we are concerned with the existence and multiplicity of nontrivial weak solutions for the Kirchhoff-type fractional Dirichlet problem with -Laplacian of the form where , , and are constants, and are the left and right Riemann-Liouville fractional derivatives of order , respectively, and is the -Laplacian [1] defined by and . 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 is nonlocal and reduces to the local first-order differential operator when . Moreover, the -Laplacian is nonlinear and reduces to the linear identity operator when . If , BVP (1) reduces to the following fractional -Laplacian BVP [2]:

In contrast to BVP (3), if , another nonlocal term, makes BVP (1) rough when one deals with it by the variational methods.

Recently, many important results on the fractional differential equations [3–18] and the Kirchhoff equations [19–24] have been obtained. Motivated by the above works, we study the solvability of BVP (1). More precisely, we prove that BVP (1) possesses at least one nontrivial weak solution when is -superlinear or -sublinear in at infinity and possesses infinitely many nontrivial weak solutions when is -sublinear in at infinity. The main ingredients used here are the mountain pass theorem and the genus properties in the critical point theory.

Note that, since the Kirchhoff-type -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 .

(H₁₁). There exist two constants such that where .

(H₁₂). as uniformly for .

(H₂₁). There exists a constant and a function such that

(H₂₂). There exists an open interval and three constants such that

(H₂₃). .

We are now to state our main results.

Theorem 1. Let (H₁₁) and (H₁₂) be satisfied. Then, BVP (1) possesses at least one nontrivial weak solution.

Theorem 2. Let (H₂₁) and (H₂₂) be satisfied. Then, BVP (1) possesses at least one nontrivial weak solution.

Theorem 3. Let (H₂₁)–(H₂₃) be satisfied. Then, BVP (1) possesses infinitely many nontrivial weak solutions.

2. 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 , the left and right Riemann-Liouville fractional integrals of order of a function are given by respectively, provided that the right-hand-side integrals are pointwise defined on , where is the gamma function.

Definition 5 (see [25]). For (), the left and right Riemann-Liouville fractional derivatives of order of a function are given by

Remark 6. When , one can obtain from Definitions 4 and 5 that where is the usual first-order derivative of .

Definition 7 (see [14]). For and , the fractional derivative space is defined by the closure of with respect to the following norm: where is the norm of .

Remark 8. It is obvious that, for , one has

Lemma 9 (see [14]). Let and . The fractional derivative space is a reflexive and separable Banach space.

Lemma 10 (see [14]). Let and . For , one has where is a constant. Moreover, if , then where is the norm of ,

Remark 11. By (13), we can consider the space with norm in what follows.

Lemma 12 (see [14]). Let and . The imbedding of in is compact.

2.2. Critical Point Theory

Now, we present some necessary definitions and theorems of the critical point theory [27, 28].

Let be a real Banach space, and which means that is a continuously Fréchet differentiable functional. Moreover, let be an open ball in and denote its boundary.

Definition 13 (see [27]). Let . If any sequence for which is bounded and as possesses a convergent subsequence in , then we say that satisfies the (PS)-condition.

Lemma 14 (see [28]). Let be a real Banach space, and satisfying the (PS)-condition. Suppose that and , there are constants such that ; there is an such that .
Then, possesses a critical value . Moreover, can be characterized as where

Lemma 15 (see [27]). Let be a real Banach space, and satisfies the (PS)-condition. If is bounded from below, then is a critical value of .

In order to find the infinitely many critical points of , we introduce the following genus properties. Let

Definition 16 (see [28]). For , we say that the genus of is denoted by if there is an odd map and is the smallest integer with this property.

Lemma 17 (see [28]). Let be an even functional on and satisfy the (PS)-condition. For any , set (i)If and , then is a critical value of (ii)If there exists such that , and , then

Remark 18. From Remark 7.3 in [28], we know that if and , then contains infinitely many distinct points; that is, has infinitely many distinct critical points in .

3. Proof of Theorem 1

In this section, we discuss the existence of nontrivial weak solutions of BVP (1) when the nonlinearity is -superlinear in at infinity.

Define the functional by

It is easy to verify from (15), (17), and that the functional is well defined on and is a continuously Fréchet differentiable functional; that is, . Furthermore, we have which yields

In the following, for simplicity, let

Lemma 19. Assume that (H₁₁) holds. Then, satisfies the (PS)-condition in .

Proof. Let be a sequence such that where is a constant. We first prove that is bounded in . From the continuity of and (H₁₁), we obtain that there exists a constant such that

Thus, by (22) and (24), we have which together with as yields

Then, it follows from that is bounded in .

Since is a reflexive Banach space (see Lemma 9), going if necessary to a subsequence, we can assume in . Hence, from as and the definition of weak convergence, we have

In addition, we obtain from (15), (17), and Lemma 12 that is bounded in and as . Thus, there exists a constant such that which yields

Moreover, by the boundedness of in , one has where is the Fréchet derivative of defined by

From (23), we have which together with (30)–(33) yields as .

Following (2.10) in [29], there exist two constants such that

When , based on the Hölder inequality, we get where is a constant, which together with (37) implies

When , by (37), we have

Then, it follows from (36), (39), and (40) that

Hence, satisfies the (PS)-condition.

Proof of Theorem 1. From (H₁₂), there exist two constants such that where is a constant defined in (13). Let and , where is a constant defined in (15). Then, by (15) and (17), we have which together with (13), (17), (22), and (42) implies which means that the condition in Lemma 14 is satisfied.

From (H₁₁), a simple argument using the very definition of the derivative shows that there exist two constants such that

Then, for any , we can obtain from (22) and that

Thus, taking large enough and letting , we have . Hence, the condition in Lemma 14 is also satisfied.

Finally, by , Lemmas 14 and 19, we get a critical point of satisfying , and so is a nontrivial solution of BVP (1).

4. Proofs of Theorems 2 and 3

In this section, we discuss the existence and multiplicity of nontrivial weak solutions of BVP (1) when the nonlinearity is -sublinear in at infinity.

Lemma 20. Suppose that (H₂₁) is satisfied. Then, is bounded from below in .

Proof. From (H₂₁), one has which together with (15)–(22) yields

Since , (48) yields as . Hence, is bounded from below.

Lemma 21. Assume that (H₂₁) holds. Then, satisfies the (PS)-condition in .

Proof. Let be a sequence such that where is a constant. Then, (48) implies that is bounded in . The remainder of proof is similar to the proof of Lemma 19, so we omit the details.

Proof of Theorem 2. From Lemmas 15, 20, and 21, we obtain which is a critical value of ; that is, there exists a critical point such that .

Now, we show . Let and , from (22) and (H₂₂), we get

Since , (50) implies I(su₀) < 0 for s > 0 small enough. Then, ; hence, is a nontrivial critical point of , and so is a nontrivial solution of BVP (1).

Proof of Theorem 3. From Lemmas 20 and 21, we obtain that is bounded from below and satisfies the (PS)-condition. In addition, (22) and (H₂₃) show that is even and .

Fixing , we take disjoint open intervals such that .

Let and , and

For , there exists , such that

Thus, we get

From (15)–(22), (52), and (H₂₂), for , one has where is a constant. Since , it follows from (54) that there exist constants such that

Let

Then, we obtain from (55) that which, together with the fact that is even and , yields

From (53), it is seen that the mapping from to is odd and homeomorphic. Hence, by some properties of the genus (Propositions 7.5 and 7.7 in [28]), we deduce that

Thus, and so . Let

It follows from is bounded from below that . That is, for any , is a real negative number. Hence, by Lemma 17, admits infinitely many nontrivial critical points, and so BVP (1) possesses infinitely many nontrivial negative energy solutions.

Obviously the following assumption implies (H₂₁)–(H₂₃).

(H₂₄). , where is a constant, , and there exists an open interval such that .

As a direct result, we have the following result.

Corollary 22. Let (H₂₄) be satisfied. Then, BVP (1) possesses infinitely many nontrivial weak solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities (2019XKQYMS90).