Research Article | Open Access

Libo Wang, Minghe Pei, "Existence Results for a -Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition", *Journal of Applied Mathematics*, vol. 2013, Article ID 914210, 8 pages, 2013. https://doi.org/10.1155/2013/914210

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

**Academic Editor:**Jaime Munoz Rivera

#### 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 [3–10] 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 [15–19] for an overview on the variable exponent Sobo-lev space, and to [20–29] 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 [32–35], 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 [36–38] 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

- G. Kirchhoff,
*Mechanik*, Teubner, Leipzig, Germany, 1883. - 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. View at: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Chen, S. Levine, and R. Rao, “Functionals with $p\left(x\right)$-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. View at: Google Scholar - M. Ružička,
*Electrorheological Fluids: Modeling and Mathematical Theory*, Springer, Berlin, Germany, 2000. View at: Publisher Site | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | MathSciNet - X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces ${W}^{k,p\left(x\right)}$(Ω),”
*Journal of Mathematical Analysis and Applications*, vol. 262, no. 2, pp. 749–760, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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: Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. V. Jikov, S. M. Kozlov, and O. A. Oleinik,
*Homogenization of Differential Operators and Integral Functionals*, Springer, Berlin, Germany, 1994. View at: Publisher Site | MathSciNet - J. Chabrowski and Y. Fu, “Existence of solutions for $p\left(x\right)$-Laplacian problems on a bounded domain,”
*Journal of Mathematical Analysis and Applications*, vol. 306, no. 2, pp. 604–618, 2005. View at: Publisher Site | Google Scholar | MathSciNet - X.-L. Fan and Q.-H. Zhang, “Existence of solutions for $p\left(x\right)$-Laplacian Dirichlet problem,”
*Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods*, vol. 52, no. 8, pp. 1843–1852, 2003. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of $p\left(x\right)$-Laplacian Dirichlet problem,”
*Journal of Mathematical Analysis and Applications*, vol. 302, no. 2, pp. 306–317, 2005. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan and X. Han, “Existence and multiplicity of solutions for $p\left(x\right)$-Laplacian equations in ${\mathbb{R}}^{N}$,”
*Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods*, vol. 59, no. 1-2, pp. 173–188, 2004. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan, “Solutions for $p\left(x\right)$-Laplacian Dirichlet problems with singular coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 312, no. 2, pp. 464–477, 2005. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan, “Remarks on eigenvalue problems involving the $p\left(x\right)$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 352, no. 1, pp. 85–98, 2009. View at: Publisher Site | Google Scholar | MathSciNet - 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 | Zentralblatt MATH | MathSciNet - 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 | MathSciNet - 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 | MathSciNet - Z. Tan and F. Fang, “On superlinear $p\left(x\right)$-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 Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Dai and D. Liu, “Infinitely many positive solutions for a $p\left(x\right)$-Kirchhoff-type equation,”
*Journal of Mathematical Analysis and Applications*, vol. 359, no. 2, pp. 704–710, 2009. View at: Publisher Site | Google Scholar | MathSciNet - G. Dai and J. Wei, “Infinitely many non-negative solutions for a $p\left(x\right)$-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 Site | Google Scholar | MathSciNet - G. Dai and R. Hao, “Existence of solutions for a $p\left(x\right)$-Kirchhoff-type equation,”
*Journal of Mathematical Analysis and Applications*, vol. 359, no. 1, pp. 275–284, 2009. View at: Publisher Site | Google Scholar | MathSciNet - G. Dai and R. Ma, “Solutions for a $p\left(x\right)$-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 Site | Google Scholar | MathSciNet - X. Fan, “On nonlocal $p\left(x\right)$-Laplacian Dirichlet problems,”
*Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods*, vol. 72, no. 7-8, pp. 3314–3323, 2010. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan, “A Brezis-Nirenberg type theorem on local minimizers for $p\left(x\right)$-Kirchhoff Dirichlet problems and applications,”
*Differential Equations & Applications*, vol. 2, no. 4, pp. 537–551, 2010. View at: Publisher Site | Google Scholar | MathSciNet - X. Fan and D. Zhao, “On the spaces ${L}^{p\left(x\right)}$ and ${W}^{m,p\left(x\right)}$,”
*Journal of Mathematical Analysis and Applications*, vol. 263, no. 2, pp. 424–446, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbb{R}}^{N}$,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 129, no. 4, pp. 787–809, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Zou, “Variant fountain theorems and their applications,”
*Manuscripta Mathematica*, vol. 104, no. 3, pp. 343–358, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.