International Journal of Differential Equations
Volume 2016 (2016), Article ID 9503073, 9 pages
http://dx.doi.org/10.1155/2016/9503073

Research Article

Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on

Yu Duan1,2 and Chun-Lei Tang¹

¹School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
²College of Science, Guizhou University of Engineering Science, Bijie, Guizhou 551700, China

Received 7 July 2016; Accepted 19 September 2016

Academic Editor: Wenming Zou

Copyright © 2016 Yu Duan and Chun-Lei Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter on is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.

1. Introduction and Main Results

The purpose of this article is to investigate the multiplicity of positive solutions to the following nonlocal Kirchhoff type equations:

where , is a positive constant, is a parameter, and is a continuous function.

In recent years, the following Kirchhoff type equation

has been studied by many researchers under variant assumptions on and . Problem (2) is often referred to as nonlocal problem because of the appearance of the term which implies that (2) is no longer a pointwise identity. This causes some mathematical difficulties which make the study of (2) particularly interesting. Problem (2) arises in an interesting physical context. Indeed, replacing by a smooth bounded domain and setting , then problem (2) becomes the following Kirchhoff type Dirichlet problem:

which is related to the stationary analogue of the Kirchhoff equation

that was presented by Kirchhoff [1] as a generalization of the well-known d’Alembert’s equation

for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. The readers can learn some early classical research of Kirchhoff equations from [2, 3]. However, (4) received great attention only after Lions [4] proposed an abstract framework to the problem. Some interesting results for problem (4) can be found in [5–7] and the references therein. There have been many works about the existence and multiplicity of nontrivial solutions to problem (3) using variational methods (see [8–18] and the references therein). Nevertheless, the problems they studied were based on a bounded domain of . Very recently, some authors had studied the Kirchhoff type equation on the whole space . Many solvability conditions with near zero and infinity for problem (2) have been considered, such as the superlinear case (see [19–28]); the asymptotically linear case (see [29, 30]); the sublinear case (see [31–33]).

Particularly, the following Kirchhoff type problem has been studied widely by some authors under various conditions on and :

When , , Huang and Liu [34] considered (6) and studied existence and nonexistence of positive solution by variational methods; they also discussed the energy doubling property of nodal solutions by Nehari manifold; Wu et al. [35] gave a total description on the positive solutions to (6), and they make an observation on the sign-changing solutions. The results of [34], respectively, complement the corresponding results of [25, 36]. Li and Ye [25] showed that problem (6) has no nontrivial solution provided , when is sufficiently large. If , Liu et al. [37] studied the existence of a positive solution for problem (6) involving subcritical growth, which unifies and sharply improves the results of [36]. Fan and Liu [38] studied (6) with concave-convex nonlinearities and showed that problem (6) has at least two positive solutions for sufficiently small. When is asymptotically linear with respect to at infinity, Ye and Yin [39] studied (6) and proved the existence of positive solution for sufficiently small and the nonexistence result for sufficiently large. When , is asymptotically linear with respect to at infinity; Li and Sun [40] showed the existence, nonexistence, and multiplicity to (6) in radial space . When the nonlinearities is sublinear or local sublinear, [41, 42] considered the existence and multiplicity of nontrivial solutions to problem (6). Recently, some authors extend problem (6) to the -Kirchhoff elliptic equations (see, e.g., [43–49] and the references therein). In all works for (6) mentioned above except for [39, 40], we found that the nonlinearities are superlinear, sublinear, or local sublinear. To the best of our knowledge, there is little information on the multiplicity of solution for (6) with the nonlinearities satisfying the asymptotically linear condition at infinity. In this paper, we will try to study multiplicity of positive solutions for problem (1) when is asymptotically linear at infinity.

In order to reduce our statements, we make the following assumptions:

  satisfies , where is a constant. , , and for all .  There exists such that .  There exists such that . , where will be given below. .

Before stating our main results, we give several notations. Set

with the usual norm

Let

with the inner product and the norm

Since satisfies , it is easy to see that is equivalent to . Obviously, the embedding is continuous for any . We denote by the usual norm.

Define the functional by

where . Clearly, by the assumptions imposed on , , and , we know that and are well defined on , and with the derivative given by

It is standard to verify that the weak solutions of (1) correspond to the critical points of the functional . Our first result for (1) without is as follows.

Theorem 1. Assume that , is a positive constant, and is a parameter. If the conditions and hold, then there exists such that, for any , problem (1) has at least two positive solutions.

Remark 2. Compared with the works mentioned above except [39, 40], where the nonlinearities are superlinear, sublinear, or local sublinear, here we consider problem (1) with asymptotically linear nonlinearities. So, our problem is different and extend the abovementioned results to some extent.

Remark 3. In [39], the authors only studied the existence of positive solutions. In this paper, we give multiplicity results when the potential is different from the conditions of in [39] and our method is simpler than that used in [39]. When satisfied some assumptions, Li and Sun [40] showed the existence, nonexistence, and multiplicity of radial solutions. Here, we get multiplicity results in nonradial space.

Remark 4. Indeed, it is not difficult to find some functions , , and such that the conditions of Theorem 1 are satisfied. For example, for any fixed , let

Choosing , it is easy to know that and are satisfied for any Moreover, in and . To verify the condition , we have to choose a special . Indeed, for , take such that

and for all , where is a constant independent of . Because of , thus for , we have

where is a constant independent of . So, choosing sufficiently large such that , the condition holds for .

In our second result, we consider the case of the perturbed Kirchhoff equations; that is, , and we obtain the following result.

Theorem 5. Assume that , is a positive constant, and is a parameter. If the conditions and hold, then there exist two constants and such that, for any , problem (1) has at least three positive solutions when .

Remark 6. In the aforementioned papers, the nonlinearities satisfy . Indeed, this condition is not necessary. Here, the nonlinearity may not be at zero because of .

In order to obtain our results, we have to overcome various difficulties. On the one hand, it is well known that Sobolev embedding is continuous but not compact for , and then it is usually difficult to prove that a minimizing sequence or a Palais-Smale sequence is strongly convergent if we seek solutions of problem (1) by variational methods. To overcome this difficulty, we make full use of integrability of potential function and perturbation . On the other hand, as we all know, the (PS) sequence is bounded if the nonlinearity satisfies a variant of Ambrosetti-Rabinowitz type condition ((AR) in short) or 4-superlinearity. However, for the asymptotically linear case of problem (1), we can adopt a simple method to verify the boundedness of (PS) sequence. The conditions and are crucial to obtain the boundedness of (PS) sequence.

This paper is organized as follows. We give some previous results and prove Theorem 1 in Section 2. Section 3 is devoted to giving the proof of Theorem 5. Throughout this paper, and are used in various places to denote distinct constants.

2. Proof of Theorem 1

In the following, we give some lemmas which are important to prove our main result.

Lemma 7. Suppose that and hold; then, is coercive on .

Proof. By and , we see that is bounded in . So, setting , thus and for any

Then,

Because of , we have

Furthermore, by (17), (18), , and the Hölder and Sobolev inequalities, we deduce that for any

Thus, we obtain

which shows that is coercive on .

It follows from Lemma 7 that is bounded from below on and thus we may define .

Lemma 8. Assume that and are satisfied; then, satisfies the condition.

Proof. Suppose that is the (PS) sequence for the functional ; that is,

By Lemma 7, the sequence is bounded in . Going if necessary to a subsequence, we can assume that weakly in for some . Now, we begin to prove strongly in . As we all know, it is sufficient to show that as . By (21), we see that

So, we have

Thus, to show that is equivalent to proving that

By , for any , there exists such that

By (16), (18), (25), , and the Sobolev and Hölder inequalities, we have

This implies

So, . It is easy to see that . Hence, ; that is, strongly in .

Proof of Theorem 1. The proof of this theorem is divided into two steps.
Step 1. In this step, we will show that problem (1) has a mountain pass solution.
By , we see that , and thus . So, we may choose a constant such that . For any , it follows from and that there exists and such that

and, then,

Using (29), , and Sobolev and Hölder inequalities, we deduce that for any

Thus, we obtain

So, fixing and letting sufficiently small, it is easy to see that there exists a constant such that

By , there is such that , , and . Combining with Fatou’s lemma, we deduce that

which implies that there exist with such that . Since as , we see that there exists such that , and then

for all . From (32), (34), and Mountain Pass Theorem, there is a sequence such that

Using Lemma 8, we know satisfies (PS)-condition. So, by Theorem   in [50], possess a critical point with . Setting , since , then by we have

which implies a.e. in . By the strong maximum principle, is positive on and .
Step 2. Problem (1) has a global minimum; that is, there exists a positive function such that and .
From Lemmas 7 and 8, we know that is bounded from below and satisfies (PS)-condition, and then by Theorem   in [51], is a critical value of ; that is, there exists a function such that and In view of (34), we know , which implies that , and using the same arguments as in Step , it is easy to know that is positive.
Because of , we get two different critical points ; that is, problem (1) has two positive solutions, and then the proof of Theorem 1 is completed.

3. Proof of Theorem 5

First, we need the following lemmas which are important to prove Theorem 5.

Lemma 9. Suppose that and hold; then, is coercive on .

Proof. The proof is similar to the proof of Lemma 7, so we omit it here.

Lemma 10. Assume that and are satisfied; then, satisfies the condition.

Proof. By Lemma 8, we only need to show By , for the above-given , there exists such that

By (37) and the Sobolev and Hölder inequalities, we have

that is

Using (27) and (39), we obtain

So, . It is easy to see that . Hence, ; that is, strongly in .

Proof of Theorem 5. The proof of this theorem is divided into four steps.
Step 1. In this step, we will show that problem (1) has a positive mountain pass solution.
Set

where is given in Theorem 1. By Lemma 7, we known that is coercive on . So we can define

Using (34), we have . By (31), we know

So, choosing and setting

for , we see that there exists a constant sufficiently small such that and , where and is given by (32). Taking , it then follows that there exists a constant such that

for all satisfying .
Using the similar proof of (34), we can obtain that there exists a constant and a function with such that

for all .
From (45), (46), and Mountain Pass Theorem, there is a sequence such that

It follows from Lemma 10 that satisfies (PS)-condition. So, using Theorem   in [50], possess a critical point with when Let . Since , then by and we have

which implies a.e. in . So, by the strong maximum principle, is positive on .

Step 2. In this step, we prove the existence of local minimum solution for problem (1).

Since and , we can choose a function such that

Hence, we obtain

Thus, we have

where is given by (45) and . By Ekeland’s variational principle, there exists a sequence such that

for all . Then, by a standard procedure, we can show that is a bounded Palais-Smale sequence of . Therefore, Lemma 10 implies that there exists a function such that and . Similarly, .

Step 3. Problem (1) has a global minimum.

It follows from Lemmas 9 and 10 that is bounded from below and satisfies the (PS) condition, so we may define . Using Theorem   in [51], is a critical value of ; that is, there exists a critical point such that and . By (46), , which implies . Similarly, .

Step 4. , , and are different from each other; that is, problem (1) has three positive solutions.

Since and , thus and . Next, we claim that and then the proof of Theorem 5 is completed. Using the proof of Theorem 1, we know and . Using (31), for all . Then,

uniformly in when So, when and , we deduce that

Since , using (54), we obtain

when . Thus, we have

So, ; that is, . Set . From the discussion above, we can obtain that problem (1) has three positive solutions when

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors wish to express their gratitude to the referees for valuable comments and suggestions on the manuscript. This paper is supported by National Natural Science Foundation of China (nos. 11471267, 11361003) and the Science and Technology Foundation of Guizhou Province (nos. LH[]7595, LH[]7054, LKB[]24, LH[]7535).

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