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Abstract and Applied Analysis
Volume 2010, Article ID 385048, 12 pages
http://dx.doi.org/10.1155/2010/385048
Research Article

Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

Received 5 May 2010; Accepted 19 December 2010

Academic Editor: Stephen Clark

Copyright © 2010 Bilal Cekic and Rabil A. Mashiyev. 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, by means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of -Kirchhoff type with nonstandard growth condition.

1. Introduction

Let be smooth bounded domain of , and let be a subdomain with smooth boundary satisfying . Writing and we have and (Figure 1).

385048.fig.001
Figure 1

We are concerned with the existence of positive solutions to the following system of nonlinear elliptic equations: with the transmission conditions where   and are positive continuous functions, is outward normal to and is inward to , and . By means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of -Kirchhoff type with nonstandard growth condition. We investigate the problem (P) when , , where and , such that for all , where if or if .

The operator − is called the -Laplacian which is a natural generalization of the -Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than the -Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1, 2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [3].

Transmission problems arise in several applications in physics and biology. For instance, one of the important problems of the electrodynamics of solid media is the electromagnetic process research in ferromagnetic media with different dielectric constants. These problems appear as well as in solid mechanics if a body consists of composite materials. The existence and regularity results for linear transmission problems are well known, and a complete study can be found in [4]. We refer the reader to [5] for nonlinear elliptic transmission problems, to [6] for a nonlinear nonlocal elliptic transmission problem. Furthermore, uniqueness and regularity of the solutions to the thermoelastic transmission problem were investigated in [7].

We note that problem (P) with the transmission condition is a generalization of the stationary problem of two wave equations of Kirchhoff type, which models the transverse vibrations of the membrane composed by two different materials in and . Controllability and stabilization of transmission problems for the wave equations can be found in [8, 9]. We refer the reader to [10] for the stationary problems of Kirchhoff type, to [11] for elliptic equation -Kirchhoff type, and to [12, 13] for -Kirchhoff type equation.

2. Auxiliary Results

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces , , and . In that context we refer to [14, 15] for the fundamental properties of these spaces.

By we always denote a nonempty open subset of . Set For any we define We define the variable exponent Lebesgue space to consist of all measurable functions for which the modular is finite. We define the Luxemburg norm on this space by the formula Equipped with this norm, is a separable and reflexive Banach space. Define the variable exponent Sobolev space by and the norm makes a separable and reflexive Banach space. The space is denoted by the closure of in . is a separable and reflexive Banach space.

Proposition 2.1 (see [14, 15]). Let . Then conjugate space of is , where . For any and , one has

The next proposition illuminates the close relation between the and the convex modular .

Proposition 2.2 (see [14, 15]). If , then one has(i) , (ii) , (iii) ,(iv) .

Proposition 2.3 (see [14, 15]). If , , then the following statements are equivalent to each other:(i) ; (ii) ; (iii) in measure in and .

Proposition 2.4 (see [14]). In the Poincaré inequality holds, that is, there exists a positive constant such that
Consequently,   are equivalent norms on . In what follows, , with , will be considered as endowed with the norm . We will use for in the following discussions.

Proposition 2.5 (see [14, 16]). Assume that is bounded, the boundary of possesses the cone property and . If and for all then the embedding is the compact and continuous.

Lemma 2.6 (see [17]). Assume that is bounded and has a Lipschitz boundary with the cone property and . Then there is a continuous boundary trace embedding .

Lemma 2.7 (see [17]). For any , let Then, is a norm on , equivalent to the standard norm of .

Our analysis is based on the Sobolev space where Then, we have following lemma that will permit the variational setting of the problem (P).

Lemma 2.8. is a closed subspace of , and defines a norm in , equivalent to the standard norm of .

Proof. It is clear that (2.12) defines a seminorm. Then suppose that . Since defines a norm on and Poincaré inequality holds in , thus . From transmission condition, on , and therefore . This shows that , and hence (2.12) defines a norm in . Now, applying the trace theorem in , there exists such that for all But it is known that defines an equivalent norm in from Lemma 2.7. Then combining these two remarks the result follows.

3. Main Results

Let us precisely describe our assumptions in order to establish the main result. The energy functional corresponding to problem (P) is defined as , where , and . It is not difficult to show that , and as a matter of fact, is of class and weakly lower semicontinuous. In particular we have, for all , ,

We assume the following hypotheses for and .

There are positive constants , , , , and such that( ): ,( ):

for all and .

Now we state our main result.

Theorem 3.1. Let us assume that and hold. If , then the problem (P) with the transmission condition has at least one nonnegative solution.

Lemma 3.2. There exists such that for any there exist and such that

Proof. Since for all it follows that (Proposition 2.5) By Lemma 2.8 we have We fix such that . Then the above relation implies By Proposition 2.2(ii) and 2.5 and Lemma 2.8, we deduce that Using , , Proposition 2.2(iii), and (3.8), we obtain that for any with the following inequalities hold true: By the above inequality if we define then for any and there exists such that .
The proof of Lemma 3.2 is complete.

Lemma 3.3. There exists such that and and for small enough.

Proof. Let and . Moreover, let choose and in ,   in . Then, for any , by and it follows Let Then Therefore, we conclude for providing that The proof of Lemma 3.3 is complete.

Proof of Theorem 3.1. From Lemma 3.3, we infer that there exists a ball centered at the origin , such that Furthermore, by Lemma 3.3, we know that there exists such that for small enough. Therefore, considering also inequality (3.14), we obtain that Let us choose . Then, it follows that Now, if we apply the Ekeland's variational principle [18] to the functional , it follows that there exists such that By the fact that we can infer that .
Now, let us define by . It is not difficult to see that is a minimum point of , and thus for small enough and any . By the above expression, we have Letting , we have and this implies that . So, we infer that there exists a sequence such that
It is obvious that is bounded in . Therefore, there exists such that, up to a subsequence, converges weakly to in and converges strongly to in and in (Proposition 2.5). Thus and By Proposition 2.1, it follows that Since converges strongly to in , that is, as , we get and similarly Hence, From and , it follows that Eventually, by [19, Theorem  3.1], we get that converges strongly to in , so we conclude that is a nontrivial weak solution for problem (P). The proof of Theorem 3.1 is complete.

Acknowledgments

The authors would like to thank the referees for their many helpful suggestions and corrections, which improve the presentation of this paper. This research was supported by DUBAP grant no. 10-FF-15, Dicle University, Turkey.

References

  1. M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar
  2. V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Mathematics of the USSR-Izvestiya, vol. 9, no. 4, pp. 33–66, 1986. View at Google Scholar · View at Zentralblatt MATH
  3. 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, vol. 60, no. 3, pp. 515–545, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY, USA, 1968.
  5. K. Pflüger, “Nonlinear transmission problems in bounded domains of n,” Applicable Analysis, vol. 62, no. 3-4, pp. 391–403, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. 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
  7. A. Marzocchi, J. E. Muñoz Rivera, and M. G. Naso, “Transmission problem in thermoelasticity with symmetry,” IMA Journal of Applied Mathematics, vol. 68, no. 1, pp. 23–46, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. E. Muñoz Rivera and H. P. Oquendo, “The transmission problem of viscoelastic waves,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 1–21, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Y. Park, J. J. Bae, and I. H. Jung, “Uniform decay of solution for wave equation of Kirchhoff type with nonlinear boundary damping and memory term,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 7, pp. 871–884, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. C. O. Alves and F. J. S. A. Corrêa, “On existence of solutions for a class of problem involving a nonlinear operator,” Communications on Applied Nonlinear Analysis, vol. 8, no. 2, pp. 43–56, 2001. View at Google Scholar
  11. 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
  12. G. Dai and R. Hao, “Existence of solutions for a p(x)-Kirchhoff-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 275–284, 2009. View at Publisher · View at Google Scholar
  13. G. Dai and D. Liu, “Infinitely many positive solutions for a p(x)-Kirchhoff-type equation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 704–710, 2009. View at Publisher · View at Google Scholar
  14. X. Fan and D. Zhao, “On the spaces Lp(x)(Ω) and Wm,p(x)(Ω),” 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
  15. O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41(116), no. 4, pp. 592–618, 1991. View at Google Scholar · View at Zentralblatt MATH
  16. X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces Wk,p(·)(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
  17. S.-G. Deng, “Positive solutions for Robin problem involving the p(x)-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 360, no. 2, pp. 548–560, 2009. View at Publisher · View at Google Scholar
  18. I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. X.-L. Fan and Q.-H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 8, pp. 1843–1852, 2003. View at Publisher · View at Google Scholar