Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 3713909, 10 pages
https://doi.org/10.1155/2019/3713909

Research Article

The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity

Jiangyan Yao and Wei Han

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Correspondence should be addressed to Wei Han; sh_hanweiwei1@126.com

Received 11 April 2019; Accepted 17 June 2019; Published 7 July 2019

Academic Editor: Marek Galewski

Copyright © 2019 Jiangyan Yao and Wei Han. 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.

In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a -linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that . The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter .

1. Introduction and the Main Results

In this paper, we are devoted to investigating the existence of the least energy sign-changing solutions for the following p-Kirchhoff-type problem with a (2p-1)-linear growth nonlinearity:

where is a bounded domain in , , , is the first eigenvalue of the following problem:

In fact, the related problems have been studied extensively, especially on the existence of the positive solutions, multiple solutions, ground state solutions, and least energy sign-changing solutions. In [1], Li and Sun studied the existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities; the mountain pass theorem was used in the paper. Guo, Ma, and Zhang [2] studied a class of autonomous Kirchhoff-type equation. By a simple transformation, they found that the solutions of autonomous Kirchhoff-type equation or system could be obtained by using the known solutions of the corresponding local equation or system, which is very interesting. In [3], Ying Li and Lin Li considered the existence and multiplicity of solutions to a class of p(x)-Laplacian-like equations. They introduced a revised Ambrosetti-Rabinowitz condition and obtained that the problem had a nontrivial solution and infinitely many solutions, respectively. Meanwhile, in [4], Luca Vilasi proved an eigenvalue theorem for a stationary p(x)-Kirchhoff problem by using variational techniques, and the author also provided an estimate for the range of such eigenvalues. For more details, we refer the reader to [5–30].

In [31, 32], the authors studied the following Kirchhoff-type problems in bounded domains:

under different assumptions on , the authors mainly use the quantitative deformation lemma and the degree theory to get the existence of the least energy sign-changing solution and its corresponding convergence property as the parameter From the assumptions on , we can easily find that both in [31, 32] satisfies 3-superlinear growth at infinity and superlinear growth at zero.

Later, some scholars made some expanding work; we can find some details in [33]. In [33], we know that the nonlinearity satisfies -superlinear growth condition at infinity.

Motivated by the above works, a natural question is that if there exists a ground state sign-changing solution for problem (1). However, up to now, no paper has appeared in the literature which discusses the existence and convergence property of the solution for the p-Kirchhoff-type problem with a -linear growth nonlinearity. This paper attempts to fill this gap in the literature.

Throughout this paper, we will make full use of the following notations. Let be the usual Sobolev space equipped with the following norm:

denotes the usual Lebesgue space norm. is the best Sobolev constant for the embedding of in ; that is,

From the above definition, we give the energy functional corresponding to problem (1) by

Clearly, is well defined on and is of class. For each , by a simple calculation, we have

Obviously, the critical points of are corresponding to the weak solutions of problem (1). If is a sign-changing solution of problem (1), then

(i) is a solution of problem (1), that is, is a critical point of ;

(ii) , where .

For , from (6) and (7), we have

When , problem (1) reduces to the following problem:

The corresponding energy functional is defined by

Also, we can compute that

For , problem (1) is called a nonlocal problem since the appearance of the nonlocal term . The differences posed by the nonlocal term make the method in solving problem (11) cannot be applied to solve problem (1), which makes the study of our paper very interesting and meaningful.

In our paper, we restrict in the following sets to find the ground state sign-changing solutions of (1) and (11),

and we define and .

To get the ground state solutions, we define the following sets:

and consider the following minimization problem:

Since , we can immediately get . The main results of the paper are described as follows.

Theorem 1. For and , problem (1) has at least one ground state sign-changing solution, which precisely has two nodal domains. Moreover, .

Theorem 2. For each , for any sequence small enough with as , there exists a subsequence still denoted by , such that convergent to strongly in , where is a ground state sign-changing solution of problem (11), which changes sign only once.

Our paper is organized as follows. In Section 2, some preliminary lemmas are given to prove the main results. In Sections 3 and 4, we are devoted to proving the main results of the paper.

2. Some Critical Preliminaries

The following several lemmas are crucial to prove our main results.

Lemma 3. If , , satisfies and

then there exists a unique pair of positive numbers such that
(i) ;
(ii) .

Proof. (i) If , then from (7), (9), and (10), we have

and

Let and , the above equations correspond to the following system:

Obviously, if we can prove that system (20) has the unique solution , then is the unique solution for (18) and (19).
Let

For , we have . Since , then

Similarly, we have

From (21)-(23), we have , and is the unique solution for system (20). Accordingly, is the unique positive solution for (18) and (19). Thus, (i) is proved.
(ii) Next, we give the proof of (ii).
From (6), we have

By a simple computation, we have

and

From , we have

and

We consider the Hessian matrix of ; then from (17), we have

The above deduction implies that is a maximal point of for . Since we cannot get the maximal point of on the boundary of , is the unique maximal point; that is,

Lemma 4. Assume that and , then (17) holds.

Proof. For , we have

Since , we have . Thus, we have

Lemma 5. Assume that , with and , there exists a unique pair such that .

Proof. If with and , we have

Since , then

From Lemma 3, there is a unique pair of positive numbers such that , which implies that is the solution of system (20). Then, we have

Therefore, we have . Similarly, we have . Thus, there exists a unique pair such that .

Lemma 6. If for any with , there exists a unique such that . Moreover, for all and .

Proof. If and satisfies , implies that

Thus, there exists a unique satisfying (35). From (6), we have

By a simple deduction, we have

Thus, attains its maximal point at . In other words, we have for all and .

Lemma 7. Assume ; we have that
(i) if , is attained by some and is a constant sign critical point of , where is given by (5);
(ii) if , is attained by some and is a sign-changing critical point of .

Proof. (i) Firstly, we will show that for all , there exists such that , which implies . From (5), we know that there exists such that For , we have

Thus, we have .
For each , it follows from and (5) that

Then, . Thus, we have

that is, and is coercive and bounded below on for and .
Let be a minimizing sequence for . From and , we assume that in for all . Since is coercive and bounded below on , the sequence is bounded in , so that, up to subsequences, in and . Next, we will prove that strongly in . We suppose by contradiction that . Therefore, we have

If , the above inequality makes a contradiction. Thus, we have in . From the fact that , we have . By Lemma 6, there exists a unique such that and for all Thus, we have

which leads to a contradiction. Therefore, we have , strongly in and . Then, by a standard argument, which is similar to the discussion in [34], we can deduce that is a constant sign critical point of .
(ii) From a similar deduction as (i), we know that for , there exists such that

Obviously, if such that satisfies (43), then also satisfies (43). Therefore, we assume that a.e. in . We let and define for all , where and . Then, from (43), we have

Let ; we can obtain that and and

that is,

Similarly, we also have

By Lemma 3, we know that for .
Assume that is a minimizing sequence for , such that . Since is coercive on , the sequence is bounded in ; going if necessary to a subsequence, still denoted by , we can assume that there exists a such that for sufficiently large,

From , we have ; that is,

Therefore,

In the same way, we have and Passing to the limit, we have

which implies that and

From , Lemmas 3 and 5, there exists a unique pair such that

From the definition of , we have

Thus, , , and , is the required minimizer.
Next, we will prove that is indeed a sign-changing solution; that is, . We mainly use the quantitative deformation lemma [35] to prove the results.
If , there exists and , such that

Let , , and . It follows from Lemma 3 that

Let and ; there exists a deformation such that
(i) if ;
(ii) ;
(iii) .
From (56), Lemma 3 and (ii), we can easily get

We prove that , which contradicts the definition of . We define and

Lemma 3 and the degree theory yield . From (56), we know that on . Consequently, we have . Therefore, for some ; that is, , which is a contradiction. From this point, is a sign-changing critical point of and .

3. The Existence of the Sign-Changing Solutions

In this part, we are devoted to proving Theorem 1.

Proof of Theorem 1. In view of Lemma 7, we know that for and , there exists a such that and ; that is, is a ground state sign-changing solution for problem (1). Then by Lemma 4, we have that

which implies

Then from Lemma 6, there exist such that . Therefore, we have

Therefore, the energy doubling property is proved.
Next, we prove that changes sign only once; that is, has exactly two nodal domains. We assume by contradiction that with and for .
Since , we can get

Then, by Lemma 5, there exists a pair such that and .
From , , and , we have

which leads to a contradiction; thus, has exactly two nodal domains.

4. The Convergence Property of as

In this part, we regard as a small parameter in (1) and discuss the convergence property of the least energy sign-changing solution , where and changes sign only once.

Proof of Theorem 2. We choose a nonzero function and such that and

Thus, for any , we have . It follows from Lemma 5 that there is a unique pair such that . Thus, we have

For any sequence with as , there exists such that is a ground state sign-changing critical point of and

The above inequality shows that is bounded in ; then there exists a subsequence of , still denoted by , such that weakly in . By the compactness of the embedding for , using a standard argument, we have that in and . Moreover, we have that for all ,

which implies that

Secondly, in the Proof of Theorem 1, is allowed; then, there exists a such that

and is a sign-changing solution for (11) which changes sign only once. Similarly, we can pick up which is independent of such that

According to Lemma 3, there exists a unique pair of positive numbers such that .
Let ; we know that

In the same way, we can obtain that

It follows from Lemma 5 that for all , there exists a unique pair such that . Then, for any sequence with as , we have as ,

According to , we have

From , we have

Combining (74) with (75), we have that as , , . Lastly, we only need to show ; then by (68), is a ground state sign-changing solution for (11), which changes sign only once. In fact,

Then, the proof of Theorem 2 is complete.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the Outstanding Youth Foundation of North University of China (No. JQ201604), the Program for the Innovative Talents of Higher Education Institutions of Shanxi, the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (201802085), and the Natural Science Foundation of Shanxi Province, China (201801D121158).

References

L. Li and J. Sun, “Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities,” Nonlinear Analysis: Real World Applications, vol. 26, pp. 391–399, 2015.

J. Guo, S. Ma, and G. Zhang, “Solutions of the autonomous Kirchhoff type equations in RN,” Applied Mathematics Letters, vol. 82, pp. 14–17, 2018.

Y. Li and L. Li, “Existence and multiplicity of solutions for p(x)-Laplacian equations in RN,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 40, no. 4, pp. 1455–1463, 2017.

L. Vilasi, “Eigenvalue estimates for stationary p(x)-Kirchhoff problems,” Electronic Journal of Differential Equations, vol. 186, pp. 1–9, 2016.

F. Cammaroto and L. Vilasi, “Three solutions for some Dirichlet and Neumann nonlocal problems,” Applicable Analysis: An International Journal, vol. 92, no. 8, pp. 1717–1730, 2013.

J. Zhang, Z. Lou, Y. Ji, and W. Shao, “Ground state of Kirchhoff type fractional Schrödinger equations with critical growth,” Journal of Mathematical Analysis and Applications, vol. 462, no. 1, pp. 57–83, 2018.

A. Mao, L. Yang, A. Qian, and S. Luan, “Existence and concentration of solutions of Schrödinger-Possion system,” Applied Mathematics Letters, vol. 68, pp. 8–12, 2017.

A. Mao and X. Zhu, “Existence and multiplicity results for Kirchhoff problems,” Mediterranean Journal of Mathematics, vol. 58, pp. 1–14, 2017.

Y. Sun, L. Liu, and Y. Wu, “The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrödinger equations on infinite domains,” Journal of Computational and Applied Mathematics, vol. 321, pp. 478–486, 2017.

Y. Wang and J. Jiang, “Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian,” Advances in Difference Equations, vol. 337, pp. 1–19, 2017.

A. Mao and H. Chang, “Schrödinger-Poisson systems with radial potentials and discontinuous quasilinear nonlinearity,” Topological Methods in Nonlinear Analysis, vol. 51, pp. 79–89, 2017.

X. S. Du and A. M. Mao, “Existence and multiplicity of nontrivial solutions for a class of semilinear fractional schrödinger equations,” Journal of Function Spaces, vol. 2017, no. 7, Article ID 3793872, 2017.

M. Shao and A. Mao, “Signed and sign-changing solutions of Kirchhoff type problems,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, pp. 1–20, 2018.

J. M. Liu and A. X. Qian, “Ground state solution for a Schrödinger-Poisson equation with critical growth,” Nonlinear Analysis: Real World Applications, vol. 40, pp. 428–443, 2018.

J. Liu and Z. Q. Zhao, “Existence of positive solutions to a singular boundary-value problem using variational methods,” Electronic Journal of Differential Equations, vol. 135, pp. 1–9, 2014.

X. Hao, H. Wang, L. Liu, and Y. Cui, “Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator,” Boundary Value Problems, vol. 182, pp. 1–18, 2017.

A. Mao and S. Luan, “Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 383, no. 1, pp. 239–243, 2011.

A. M. Mao, R. A. Jing, S. X. Luan, J. L. Chu, and Y. Kong, “Some nonlocal elliptic problem involving positive parameter,” Topological Methods in Nonlinear Analysis, vol. 42, no. 1, pp. 207–220, 2013.

A. Mao and H. Chang, “Kirchhoff type problems in RN with radial potentials and locally Lipschitz functional,” Applied Mathematics Letters, vol. 62, pp. 49–54, 2016.

A. M. Mao, Y. Zhu, and S. X. Luan, “Existence of solutions of elliptic boundary value problems with mixed type nonlinearities,” Boundary Value Problems, vol. 97, pp. 1–11, 2012.

A. X. Qian, “Sing-changing solutions for nonlinear problems with strong resonance,” Electronic Journal of Differential Equations, vol. 17, pp. 1–8, 2012.

A. X. Qian, “Sign-changing solutions for some nonlinear problems with strong resonance,” Boundary Value Problems, vol. 2011, no. 1, pp. 1–9, 2011.

A. M. Mao and W. Q. Wang, “Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in R3,” Journal of Mathematical Analysis and Applications, vol. 1, article no. 459, pp. 556–563, 2018.

M. Shao and A. Mao, “Multiplicity of solutions to Schrödinger-Poisson system with concave-convex nonlinearities,” Applied Mathematics Letters, vol. 83, pp. 212–218, 2018.

Q. Gao, F. Li, and Y. Wang, “Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation,” Central European Journal of Mathematics, vol. 9, no. 3, pp. 686–698, 2011.

A. Qian, “Infinitely many sign-changing solutions for a Schrödinger equation,” Advances in Difference Equations, vol. 2011, article no. 39, pp. 1–6, 2011.

A. M. Mao and Y. S. Chen, “Existence and concentration of solutions for sublinear Schrödinger-Poisson equations,” Indian Journal of Pure and Applied Mathematics, vol. 49, no. 2, pp. 339–348, 2018.

F. Sun, L. Liu, and Y. Wu, “Infinitely many sign-changing solutions for a class of biharmonic equation with p-Laplacian and Neumann boundary condition,” Applied Mathematics Letters, vol. 73, pp. 128–135, 2017.

Y. Sang, “An exact estimate result for a semilinear equation with critical exponent and prescribed singularity,” Journal of Mathematical Analysis and Applications, vol. 447, no. 1, pp. 128–153, 2017.

J. Zhang, “Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth,” Applied Mathematics Letters, vol. 63, pp. 53–58, 2017.

W. Shuai, “Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains,” Journal of Differential Equations, vol. 259, no. 4, pp. 1256–1274, 2015.

X. H. Tang and B. T. Cheng, “Ground state sign-changing solutions for Kirchhoff type problems in bounded domains,” Journal of Differential Equations, vol. 261, no. 4, pp. 2384–2402, 2016.

S. H. Rasouli, H. Fani, and S. Khademloo, “Existence of sign-changing solutions for a nonlocal problem of p-Kirchhoff type,” Mediterranean Journal of Mathematics, vol. 185, no. 5, pp. 1–14, 2017.

K. J. Brown and Y. Zhang, “The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,” Journal of Differential Equations, vol. 193, no. 2, pp. 481–499, 2003.

M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996.