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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 914210, 8 pages
http://dx.doi.org/10.1155/2013/914210
Research Article

Existence Results for a -Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

1Institute of Mathematics, Jilin University, Chang'chun 130012, China
2Department of Mathematics, Beihua University, Ji'lin 132013, China

Received 25 November 2012; Accepted 28 April 2013

Academic Editor: Jaime Munoz Rivera

Copyright © 2013 Libo Wang and Minghe Pei. 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

We consider the existence and multiplicity of solutions for the -Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.

1. Introduction

The Kirchhoff equation was introduced by Kirchhoff [1] in the study of oscillations of stretched strings and plates, where and are constants. The stationary analogue of the Kirchhoff equation, that is, (1), is as follows: After the excellent work of Lions [2], problem (2) has received more attention; see [310] and references therein.

The -Laplace operator arises from various phenomena, for instance, the image restoration [11], the electro-rheological fluids [12], and the thermoconvective flows of non-Newtonian fluids [13, 14]. The study of the -Laplace operator is based on the theory of the generalized Lebesgue space and the Sobolev space , which we called variable exponent Lebesgue and Sobolev space. We refer the reader to [1519] for an overview on the variable exponent Sobo-lev space, and to [2029] for the study of the -Laplacian-type equations.

Recently, there has been an increasing interest in studying the Kirchhoff equation involving the -Laplace operator. Autuori et al. [30, 31] have dealt with the nonstationary Kirchhoff-type equation involving the -Laplacian of the form

In [3235], applying variational method and Ambrosetti-Rabinowitz (AR) condition, Guowei Dai has studied the existence and multiplicity of solutions for the -Kirchhoff-type equations with Dirichlet or Neumann boundary condition. In [36], by using mapping theory and the Mountain Pass Lemma, Fan has discussed the nonlocal -Laplacian Dirichlet problem with the nonvariational form and the -Kirchhoff-type equation with the variational form under (AR) condition, where are two functionals defined on , and .

Motivated by the above works, the purpose of this paper is to study the -Kirchhoff-type equation without (AR) condition, where is a smooth bounded domain in , are two positive constants, , for some , and By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the -Kirchhoff-type equation (6) under no (AR) condition.

2. Preliminary

We recall in this section some definitions and properties of variable exponent Lebesgue-Sobolev space. The variable exponent Lebesgue space is defined by with the norm The variable exponent Sobolev space is defined by with the norm Denote by the closure of in . is an equivalent norm on . In this paper we use the notation for . Define We refer the reader to [3638] for the elementary properties of the space .

Proposition 1 (see [38]). Set . For any , then the following are given:(1)  if  ; (2);(3)  if  ; (4)  if    ; (5);(6).

3. Positive Energy Solution

In this section we discuss the existence of weak solutions of (6). For simplicity we write .

First, we state the assumptions on as follows. Let be a continuous function, and there exist positive constants such that where and for all . Let be a continuous function, and there exist positive constants such that where and for all ; for all . Let , uniformly for , where . There exists such that for and , where Let , uniformly on . There exists , such that for , . Let for and . Let , uniformly on , where satisfies for .

Remark 2. Condition was first introduced by Jeanjean [39] for the case . Let , then It is easy to see that the function does not satisfy (AR) condition, but it satisfies and .
Define , where Then .

Proposition 3 (see [38]). Assume that hold, then the functional is sequentially weakly lower semicontinuous, is sequentially weakly continuous, and is sequentially weakly lower semicontinuous.

Proposition 4 (see [37]). Assume that hold, and let be a local minimizer (resp., a strictly local minimizer) of in the topology. Then is a local minimizer (resp., a strictly local minimizer) of in the topology.

Definition 5. We say that is a weak solution of (6), if for any .

Definition 6. Let be a Banach space and . Given . we say that satisfies the Cerami condition (we denote by the condition), if (i)any bounded sequence such that and has a convergent subsequence; (ii)there exist constants such that

If satisfies condition for every , then we say that satisfies condition.

Remark 7. Although (PS) condition is stronger than condition, the Deformation Theorem is still valid under condition; we see that the Mountain Pass Lemma, the Fountain Theorem, and its dual are true under condition.

Lemma 8. Assume that conditions hold. Then satisfies condition.

Proof. From [36, Proposition 3.1], satisfies (i) of condition.
Now we check that satisfies (ii) of condition. Arguing by contradiction, we may assume that, for some , Then we have Let , then up to a subsequence we may assume that If , inspired by [13, 14], then we define For any , let . Since in and by the continuity of , in , thus, Then for large enough, and That is, . From and , we know that and Therefore, from , we have This contradicts (21).
If , from (20), when , Then from we have For , . By we have Note that the Lebesgue measure of is positive; using the Fatou Lemma, we have This contradicts (30).
The technique used in this lemma was first introduced by [39, 40].

Theorem 9. Assume that conditions and (or ) hold. Then (6) has a nontrivial solution with positive energy.

Proof. From Lemma 8, satisfies condition. Let us show that the functional has a Mountain-Pass-type geometry.
Note that . By , there exists ,  and  for  any with , This shows that 0 is a strictly local minimizer of in the topology, and hence 0 is a strictly local minimizer of in the topology. By [37, Theorem 1.1], 0 is a strictly local minimizer of in the topology. Thus there exists such that for every with .
We claim that . To prove this claim, arguing by contradiction, assume that there exists a sequence with such that as . We may assume that in . Since is sequentially weakly lower semicontinuous, we have that , and hence . Since is sequentially weakly continuous, then we have that , and hence . It follows from this that in which contradicts with .
Let with in and . By and , for we have We set . Then for large, we obtain Hence by the famous Mountain Pass Lemma, problem (6) has a nontrivial weak solution with positive energy.

4. Infinitely Many Solutions

Since is a reflexive and separable Banach space, then there exists and such that For convenience, we write , , .

Lemma 10 (see [21]). If ,   for any , denote Then .

Proposition 11 (Fountain Theorem). Assume that is an even functional. If, for any , there exists such that,   as  ,  satisfies    condition for every ,then    has an unbounded sequence of critical values.

Proposition 12 (Dual Fountain Theorem). Assume that is an even functional. If, for any , there exists such that, ,  as  ,  satisfies    condition for every  , then    has a sequence of negative critical values converging to .

Theorem 13. Assume that the conditions , hold. Then (6) has infinitely many solutions such that as .

Proof. By conditions , , and , for any , there exists such that For , when , Then for some large enough, On the other hand, by and , there exists such that Let . From Lemma 10,   as . For , when and small enough, If we choose as , then, for with , which implies that as .

Theorem 14. Assume that conditions , , , , and hold. Then (6) has infinitely many solutions such that as .

Proof. By conditions , , and , for any , there exists such that For , when is large enough, Then for some large enough, On the other hand, by , there exists such that Let , then as . For , when and small enough, If we choose as , then, for with , which implies that as .
Furthermore, if with , then which implies that as .

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (11126339, 11201008).

References

  1. G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
  2. J. L. Lions, “On some equations in boundary value problems of mathematical physics,” in Proceedings of the International Symposium on Contemporary Developments in Continuum Mechanics and Partial Differential Equations, vol. 30 of North-Holland Mathematics Studies, pp. 284–346, Universidade Federal Rural do Rio de Janeiro, North-Holland, Rio de Janeiro, Brazil, 1977-1978.
  3. C. O. Alves, F. J. S. A. Corrêa, and T. F. Ma, “Positive solutions for a quasilinear elliptic equation of Kirchhoff type,” Computers & Mathematics with Applications, vol. 49, no. 1, pp. 85–93, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B. Cheng, X. Wu, and J. Liu, “Multiplicity of nontrivial solutions for Kirchhoff type problems,” Boundary Value Problems, vol. 2010, Article ID 268946, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. He and W. Zou, “Infinitely many positive solutions for Kirchhoff-type problems,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 70, no. 3, pp. 1407–1414, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. B. Liu and S. J. Li, “Infinitely many solutions for a superlinear elliptic equation,” Acta Mathematica Sinica, vol. 46, no. 4, pp. 625–630, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. T. F. Ma and J. E. Muñoz Rivera, “Positive solutions for a nonlinear nonlocal elliptic transmission problem,” Applied Mathematics Letters, vol. 16, no. 2, pp. 243–248, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Mao and Z. Zhang, “Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 70, no. 3, pp. 1275–1287, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. Perera and Z. Zhang, “Nontrivial solutions of Kirchhoff-type problems via the Yang index,” Journal of Differential Equations, vol. 221, no. 1, pp. 246–255, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Z. Zhang and K. Perera, “Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 456–463, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Chen, S. Levine, and R. Rao, “Functionals with px-growth in image processing,” Technical Report 2004-01, Duquesne University, Department of Mathematics and Computer Science, Pittsburgh, Pa, USA, http://www.mathcs.duq.edu/sel/CLR05SIAPfinal.pdf.
  12. M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 60, no. 3, pp. 515–545, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Annali dell'Universitá di Ferrara, vol. 52, no. 1, pp. 19–36, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  16. X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces Wk,px(Ω),” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. Harjulehto and P. Hästö, “An overview of variable exponent Lebesgue and Sobolev spaces,” in Proceedings of the RNC Workshop on Future Trends in Geometric Function Theory, D. Herron, Ed., pp. 85–93, Jyväskylä, Finland, 2003. View at Zentralblatt MATH · View at MathSciNet
  18. S. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,” Integral Transforms and Special Functions, vol. 16, no. 5-6, pp. 461–482, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, Germany, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  20. J. Chabrowski and Y. Fu, “Existence of solutions for px-Laplacian problems on a bounded domain,” Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 604–618, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  21. X.-L. Fan and Q.-H. Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 52, no. 8, pp. 1843–1852, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  22. X. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of px-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  23. X. Fan and X. Han, “Existence and multiplicity of solutions for px-Laplacian equations in N,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 59, no. 1-2, pp. 173–188, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  24. X. Fan, “Solutions for px-Laplacian Dirichlet problems with singular coefficients,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 464–477, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  25. X. Fan, “Remarks on eigenvalue problems involving the px-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 85–98, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  26. D. Stancu-Dumitru, “Multiplicity of solutions for anisotropic quasilinear elliptic equations with variable exponents,” Bulletin of the Belgian Mathematical Society, vol. 17, no. 5, pp. 875–889, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. D. Stancu-Dumitru, “Two nontrivial solutions for a class of anisotropic variable exponent problems,” Taiwanese Journal of Mathematics, vol. 16, no. 4, pp. 1205–1219, 2012. View at Google Scholar · View at MathSciNet
  28. D. Stancu-Dumitru, “Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 1, pp. 117–130, 2013. View at Google Scholar · View at MathSciNet
  29. Z. Tan and F. Fang, “On superlinear px-Laplacian problems without Ambrosetti and Rabinowitz condition,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 75, no. 9, pp. 3902–3915, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. G. Autuori, P. Pucci, and M. C. Salvatori, “Asymptotic stability for anisotropic Kirchhoff systems,” Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 149–165, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. G. Autuori, P. Pucci, and M. C. Salvatori, “Global nonexistence for nonlinear Kirchhoff systems,” Archive for Rational Mechanics and Analysis, vol. 196, no. 2, pp. 489–516, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. G. Dai and D. Liu, “Infinitely many positive solutions for a px-Kirchhoff-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 704–710, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  33. G. Dai and J. Wei, “Infinitely many non-negative solutions for a px-Kirchhoff-type problem with Dirichlet boundary condition,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 73, no. 10, pp. 3420–3430, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  34. G. Dai and R. Hao, “Existence of solutions for a px-Kirchhoff-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 275–284, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  35. G. Dai and R. Ma, “Solutions for a px-Kirchhoff type equation with Neumann boundary data,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 12, no. 5, pp. 2666–2680, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  36. X. Fan, “On nonlocal px-Laplacian Dirichlet problems,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 72, no. 7-8, pp. 3314–3323, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  37. X. Fan, “A Brezis-Nirenberg type theorem on local minimizers for px-Kirchhoff Dirichlet problems and applications,” Differential Equations & Applications, vol. 2, no. 4, pp. 537–551, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  38. X. Fan and D. Zhao, “On the spaces Lpx and Wm,px,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on N,” Proceedings of the Royal Society of Edinburgh A, vol. 129, no. 4, pp. 787–809, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet