About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 281949, 6 pages
http://dx.doi.org/10.1155/2013/281949
Research Article

The Existence of a Nontrivial Solution for a -Kirchhoff Type Elliptic Equation in

1Science and Information College, Qingdao Agricultural University, Qingdao 266109, China
2College of Science, Hohai University, Nanjing 210098, China

Received 11 March 2013; Revised 18 July 2013; Accepted 18 July 2013

Academic Editor: Dumitru Motreanu

Copyright © 2013 Zonghu Xiu. 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

Using Mountain Pass lemma, under some appropriate assumptions, we establish the existence of one nontrivial solution for a class of p-Kirchhoff-type elliptic equations in .

1. Introduction and Main Result

In this paper, we consider the existence of solution for the following elliptic problem: where , , and are positive functions, is continuous function, and further assumptions will be listed later. Problems like (1) originally came from the stationary problem of a model introduced by Kirchhoff [1]. Due to the existence of the integration over the whole space, problems like (1) are also called nonlocal problems.

In recent years, the Kirchhoff-type equations with -Laplacian operator has been considered by many authors; see [26]. When , the authors in [6] considered the following similar Kirchhoff-type elliptic problem on the bounded domain

The function in (2) is required to meet the condition of for some constants . The authors proved that problem (2) has at least one positive solution under some other additional conditions. Problem (2) was also considered in [3], where the function is odd about . For , we refer to [79]. The authors in [8] studied the following Kirchhoff type problem:

By the Fountain theorem, the author proved the existence of infinitely many solutions. Note that one of the assumptions made on the function in (3) is that for any . In the present paper, however, the function is not required to be odd about as that of [3, 8]. When and , problem (1) becomes the -Laplacian elliptic equations without nonlocal term, and this kind of problem is also studied by many authors. For these works, we refer to [1014] and the references therein. In [15], Liu discussed the following elliptic problem:

The author proved that problem (4) has at least one ground state. The weight function is required to be bounded; more precisely, there exist constants such that

We point out that the weight function in problem (1) is permitted to tends to infinity.

In this paper, inspired by [4, 8, 15], we consider the existence of solution of problem (1). By the variational method, we will prove that problem (1) has at least one nontrivial weak solution. Since problem (1) is considered in the whole space , the loss of compactness of the Sobolev embedding renders the variational technique more delicate.

In this paper, we make the following hypothesis:  , and as , there exists such that , where , there exist constants and such that .

Remark 1. Throughout this paper, we denote by the constant which may vary from line to line but remains independent of the relevant quantities. Note that there exist many functions , and such that satisfy the assumptions of Theorem 3, for example, , , .
Let be the usual Sobolev space with the norm of
Denote
Then, is a Sobolev space with the norm of
We give another space endowed with the norm
It is not difficult to check that is a Banach space. The Euler functional of problem (1) is where . Then, the assumptions imply that , and for any , there holds
Particularly,

Definition 2. A function is said to be a weak solution of (1) if and only if (12) holds for any .

Our main result is listed later.

Theorem 3. Assume . Then, problem (1) has at least one nontrivial weak solution in .

This paper is organized as follows. In Section 2, we introduce some definitions and prove several lemmas which will be used later. In Section 3, we give the proof of Theorem 3 by making use of the Mountain Pass lemma.

2. Preliminary Results

In this section, we give some important lemmas, which will be needed in the proof of our main result. Particularly, one result of compact embedding on unbounded domain will be proved.

Lemma 4. Assume and . Then, the embedding is compact.

Proof. We split the proof into two cases.
(). Let and ; then shows that as . Furthermore, one gets that which implies that
It follows from (16) that
Let be a bounded sequence of such that . Then, there exists such that . For any , it follows from (17) that there exists large enough such that
Note that the embedding is compact. Therefore, for these and , there exists large enough such that for all . Thus, one can get from (18) and (19) that which implies that strongly in .
  (). By the Hölder and Young inequalities, one can get from that
Since and as , then
Therefore, similar to the proof of  , the embedding is compact.

Now, we give the definitions of Palais-Smale (simply ) sequence and condition.

Definition 5. Let , , and be a Banach space. The sequence is said to be a sequence if there holds
A functional is said to satisfy the condition if any sequence in contains a convergent subsequence.

Next, we will prove that the functional satisfies the condition.

Lemma 6. Assume   . Then, satisfies the condition on for any .

Proof. Let be an any sequence in . We divide the proof into two steps.
( is bounded in ). Note that ; then, it follows from (23) and that which implies that is bounded in .
( converges strongly in ). Since is bounded in the separable , there exist and a subsequence of , still denoted by , such that in . Now, we want to prove that
In fact, it follows from that
Thus, in order to prove (25), we need only to prove that
Here, we divide the proof (27) into two cases:
(i)
By the Hölder inequality, we obtain that
It is easy to check that . Then, the compact embedding in Lemma 4 shows that which gives (27).
(ii)
Let . Since and , there exist such that
The Hölder inequality shows that
Therefore, the embedding in Lemma 4 implies that which also gives (27).
Note that and
Therefore, the previous cases (i) and (ii) imply that (25) holds for all .
Denote
Then, it follows from (23) and (25) that and
Since in , one obtains from the Hölder inequality that
Thus, it follows from (37) and (38) that
Since , , and , are positive functions, we get from (25) and (39) that
Therefore, we can deduce from (40) and the following standard inequalities in [16]: that
Then, we complete the proof.

3. Existence of Solution

In this section, the proof of Theorem 3 is mainly based on the following Mountain Pass lemma [17] (also see [18]).

Lemma 7 (Mountain Pass lemma). Let be a Banach space and satisfies condition. Suppose that and there are constant such that , there is an such that .Then, possesses a critical value . Moreover, can be characterized as where

In view of Lemma 6, the functional satisfies the condition. It is obvious that ; then, in order to apply Lemma 7, we need only to prove that satisfies the geometric conditions of the Mountain Pass lemma.

Lemma 8. Assume . Then, there exist such that with , there is an such that and .

Proof. It follows from , , and the embedding in Lemma 4 that Since , there exists a sufficient small and such that with . On the other hand, for fixed and , it follows from that Then, there is a large such that and . One may choose ; then, and . Thus, the proof of Lemma 8 is complete.

Proof of Theorem 3. It follows from Lemmas 7 and 8 that the functional has a critical point such that ; that is, problem (1) has at least one weak solution. On the other hand, ; then, problem (1) has at least one nontrivial weak solution.

Acknowledgment

The author is very grateful to the anonymous reviewer for valuable comments and suggestions.

References

  1. G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
  2. H. Bueno, G. Ercole, W. Ferreira, and A. Zumpano, “Existence and multiplicity of positive solutions for the p-laplacian with nonlocal coefficient,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 151–158, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  3. F. J. S. A. Corrêa and G. M. Figueiredo, “On a p-kirchhoff equation via Krasnoselskii's genus,” Applied Mathematics Letters, vol. 22, no. 6, pp. 819–822, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. F. J. S. A. Corrêa and G. M. Figueiredo, “On an elliptic equation of p-kirchhoff type via variational methods,” Bulletin of the Australian Mathematical Society, vol. 74, no. 2, pp. 263–277, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  5. F. J. S. A. Corrêa and R. G. Nascimento, “On a nonlocal elliptic system of p-kirchhoff-type under Neumann boundary condition,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 598–604, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  6. D. Liu and P. Zhao, “Multiple nontrivial solutions to a p-kirchhoff equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 13, pp. 5032–5038, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Heidarkhani, “Infinitely many solutions for systems of n two-point kirchhoff-type boundary value problems,” Annales Polonici Mathematici, vol. 107, no. 2, pp. 133–152, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Jin and X. Wu, “Infinitely many radial solutions for Kirchhoff-type problems in N,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 564–574, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. Ricceri, “On an elliptic kirchhoff-type problem depending on two parameters,” Journal of Global Optimization, vol. 46, no. 4, pp. 543–549, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. Cammaroto, A. Chinnì, and B. Di Bella, “Some multiplicity results for quasilinear Neumann problems,” Archiv der Mathematik, vol. 86, no. 2, pp. 154–162, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Cingolani and G. Vannella, “On the multiplicity of positive solutions for p-laplace equations via Morse theory,” Journal of Differential Equations, vol. 247, no. 11, pp. 3011–3027, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Franca, “Radial ground states and singular ground states for a spatial-dependent p-laplace equation,” Journal of Differential Equations, vol. 248, no. 11, pp. 2629–2656, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Kristály and W. Marzantowicz, “Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in N,” Nonlinear Differential Equations and Applications, vol. 15, no. 1-2, pp. 209–226, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. Su and Z.-Q. Wang, “Sobolev type embedding and quasilinear elliptic equations with radial potentials,” Journal of Differential Equations, vol. 250, no. 1, pp. 223–242, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. B. Liu, “On ground states of superlinear p-Laplacian equations in N,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 48–58, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. 106 of Research Notes in Mathematics, Pitman Advanced Publishing, Boston, Mass, USA, 1985, Elliptic equations. View at MathSciNet
  17. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973. View at Zentralblatt MATH · View at MathSciNet
  18. M. Struwe, Variational Methods, Springer, Berlin, Germany, 3rd edition, 2000, Applications to nonlinear partial differential equations and Hamiltonian systems. View at MathSciNet