Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2015, Article ID 238261, 7 pages
Research Article

On -Anisotropic Problems with Neumann Boundary Conditions

Department of Mathematics (ENSAH), Nonlinear Analysis Laboratory (FSO), University of Mohammed First, 60000 Oujda, Morocco

Received 6 July 2015; Revised 8 November 2015; Accepted 23 November 2015

Academic Editor: Emmanuel Hebey

Copyright © 2015 Anass Ourraoui. 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.


This work is devoted to the study of a general class of anisotropic problems involving -Laplace operator. Based on the variational method, we establish the existence of a nontrivial solution without Ambrosetti-Rabinowitz type conditions.

1. Introduction

The elliptic problems in anisotropic form concerning the Sobolev space with variable exponents have recently attracted the attention of many mathematicians; see [113] and the references therein. Such equations arise in connection with the equations describing electromagnetic fields and the plasma physics; see [14, 15] and various applications like those in thermorheological fluids [16], elastic mechanics [17], and image restoration [18]. They also appear in biology; see, for instance, Bendahmane et al. in [19], as a model for the propagation of epidemic diseases in heterogeneous domains.

In the present paper, we study the anisotropic nonlinear elliptic problem of the formwhere () is a bounded open set with smooth boundary and are the components of the outer normal unit vector and for , for all , , where the exponents are continuous functions such that .

We assume that the functions and are Carathéodory and satisfying the following conditions for all .There exists a positive constant such that fulfills for all and all , where (with ) is a nonnegative function and is the mapping which verifies There exists such that for all and all .The monotonicity condition takes place for all and all with .

Example 1. We take and then the operatorbecomes in particular -Laplace operatorThis is why operators (7) are often known as generalized -Laplace type operators.

On the other hand, the anisotropic equations with the variable exponent growth conditions enable the study of equations with more complicated nonlinearities since the differential operator allows a distinct behavior for partial derivatives in various directions.

This paper is organized as follows. In Section 2, we give the necessary notations; we also include some useful results involving the variable exponent Sobolev spaces in order to facilitate the reading of the paper. Finally, in Section 3, we prove the existence of nontrivial solution.

2. Preliminaries and Main Result

We introduce the setting of our problem with some auxiliary results. For convenience, we only recall some basic facts which will be used later; we refer to [20, 21].

For , we introduce the Lebesgue space with variable exponent defined by where This space, endowed with the Luxemburg norm,is a separable and reflexive Banach space. We also have an embedding result.

Proposition 2. Assume that is bounded and , such that in . Then, the embedding is continuous.

Furthermore, the Hölder-type inequalityholds for all and , where is the conjugate space of , with .

Moreover, we denote and, for , we have the following properties:To recall the definition of the isotropic Sobolev space with variable exponent, , we setendowed with the normThe space is a separable and reflexive Banach space.

Now, we consider to be the vectorial function with for all and we put The anisotropic space with variable exponent isand it is endowed with the norm The space is a reflexive Banach space. Furthermore, an embedding theorem takes place for all the exponents that are strictly less than a variable critical exponent, which is introduced with the help of the notations

Proposition 3. Let be a bounded open set for all and for all . If , for all , then one has the compact and continuous embedding .

Remark 4. We make the following notations: Then, by (14), (15), and (16), Thus,

Definition 5. One defines the weak solution for problem (1) as a function satisfying for all .

We suppose the following hypotheses.There exist and with for all , such that verifies for all and all . uniformly for . uniformly for .There exist two positive constants and such that where with . for all and all .

The function , where is an example of functions verifying the assumptions (F1)–(F4). In fact, we have and then we get which means that (F4) is satisfied since we have which is nondecreasing in and then when , so we take . Taking into account that , it follows that Obviously the other assumptions are held.

We report our main result.

Theorem 6. Under conditions (A1)–(A4) and (F1)–(F4), problem (1) has at least a nontrivial weak solution.

The purpose of this work is to improve the results of the above-mentioned papers and many others, without assuming the Ambrosetti-Rabinowitz type conditions (A-R) used, for instance, in [13, 10], where in (A-R) there exist , such that for any and we have

In fact, it is known that (F4) is much weaker than the (A-R) condition in the constant exponent case (see, for instance, [22]). We will use the mountain pass theorem with Cerami condition which is weaker than condition used, for example, in [4, 6].

The energy functional corresponding to (1) is defined as ,By a standard argument, we can see that the functional is well defined and of class , with its Gâteaux derivative being described byfor all .


Proposition 7. (i) By A3, the functional is of type; that is, if and , then in .
(ii) From F1, the functional is weakly strongly continuous; that is, .

The proof of the first assertion (i) is similar to that in [2]. The second assertion is well known.

3. Proof of the Main Result

We will use the mountain pass theorem (see [2325]), so we start by the condition of geometry in the form of the following lemma.

Lemma 8. (a) There exists with such that as .
(b) There exist , such that for .

Proof. (a) From (F2), we may choose a constant such thatLet large enough and with , and from (A1) and (39) we getwhere is a constant, taking sufficiently large to ensure that which implies that(b) By (A2), for , we haveOn the other side, from (F1), By the continuous embedding from into and ), there exist , such thatfor all . Hence,for all and all .
Therefore, since . Then for sufficiently small, we take such that

Definition 9. A sequence is called a Cerami sequence if is bounded and .

Lemma 10. If , then any sequence of Cerami of is bounded.

Proof. Let be a sequence of . We claim that is bounded; otherwise, up to a subsequence, we may assume that Putting , up to a subsequence, we have in in and in , . Almost everywhere .
Here, two cases appear, when , since we know that , that means Dividing (50) by , by using (A1), a straightforward computation leads to Meanwhile, in view of condition (F2) and Fatou’s lemma, which is contradictory.
In the case when , we choose a sequence satisfying If , since in and , by the continuity of the Nemitskii operator, we see that in as ; therefore,Given , since, for large enough, we have , using (54) with , from assumption (A2), and, considering Remark 4, we getThereby, . On the other hand, we know that , , so we can deduce that It yields Therefore,so we get Moreover,From (A2) and (F4), there exist such thatHence, , which is impossible.

Proof of Theorem 6. According to Lemma 8 and Lemma 10, we are to apply the mountain pass theorem, so seeing that the sequence (in Lemma 10) is strongly convergent to remains and it will be done.
Now, because the Banach space is reflexive (cf. [2, 3]), and regarding the boundedness of in , there exists such that . Since is the sum of type maps and which is weakly strongly continuous (cf. Proposition 7), is also of type. Thus, in .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.


The author would like to thank the referee for the suggestions and helpful comments which improved the presentation of the original paper.


  1. C. O. Alves and A. El Hamidi, “Existence of solution for an anisotropic equation with critical exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, pp. 611–624, 2005. View at Google Scholar
  2. M.-M. Boureanu, P. Pucci, and V. D. Radulescu, “Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent,” Complex Variables and Elliptic Equations, vol. 56, no. 7–9, pp. 755–767, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M.-M. Boureanu, “Critical point methods in degenerate anisotropic problems with variable exponent,” Studia Universitatis Babes-Bolyai, Mathematica, vol. 55, no. 4, pp. 27–39, 2010. View at Google Scholar
  4. N. T. Chung and H. Q. Toan, “On a class of anisotropic elliptic equations without Ambrosetti-Rabinowitz type conditions,” Nonlinear Analysis. Real World Applications, vol. 16, pp. 132–145, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. F. C. Cîrstea and J. Vétois, “Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates,” Communications in Partial Differential Equations, vol. 40, no. 4, pp. 727–765, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. S. 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
  7. A. El Hamidi and J. Vétois, “Sharp Sobolev asymptotics for critical anisotropic equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 1–36, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. X. Fan, “Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations,” Complex Variables and Elliptic Equations, vol. 56, no. 7-9, pp. 623–642, 2011. View at Publisher · View at Google Scholar
  9. I. Fragalà, F. Gazzola, and B. Kawohl, “Existence and nonexistence results for anisotropic quasilinear elliptic equations,” Annales de l'Institut Henri Poincaré C: Non Linear Analysis, vol. 21, no. 5, pp. 715–734, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. B. Kone, S. Ouaro, and S. Traore, “Weak solutions for anisotropic nonlinear elliptic equations with variable exponents,” Electronic Journal of Differential Equations, vol. 2009, no. 144, pp. 1–11, 2009. View at Google Scholar · View at MathSciNet
  11. A. Ourraoui, “On a nonlocal p(x)-Laplacian equations via genus theory,” Rivista di Matematica della Università di Parma, In press.
  12. J. Vétois, “Strong maximum principles for anisotropic elliptic and parabolic equations,” Advanced Nonlinear Studies, vol. 12, no. 1, pp. 101–114, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Vétoiss, “Existence and regularity for critical Anisotropic equations with critical directions,” Advances in Differential Equations, vol. 16, no. 1-2, pp. 61–83, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Batt and Y. Li, “The positive solutions of the Matukuma equation and the problem of finite radius and finite mass,” Archive for Rational Mechanics and Analysis, vol. 198, no. 2, pp. 613–675, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. A. J. Simmonds, “Electro-rheological valves in a hydraulic circuit,” IEE Proceedings D: Control Theory and Applications, vol. 138, no. 4, pp. 400–404, 1991. View at Publisher · View at Google Scholar · View at Scopus
  16. S. N. Antontsev and J. F. Rodrigues, “On stationary thermorheological viscous flows,” Annali dell'Università di Ferrara Sezione VII. Scienze Matematiche, vol. 52, pp. 19–36, 2006. View at Google Scholar
  17. V. V. Zhikov, “Averaging of functionals in the calculus of variations and elasticity,” Mathematics of the USSR-Izvestiya, vol. 29, no. 1, pp. 33–66, 1987. View at Publisher · View at Google Scholar
  18. Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. M. Bendahmane, M. Langlais, and M. Saad, “On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 4, pp. 617–636, 2003. View at Publisher · View at Google Scholar
  20. D. E. Edmunds and J. R'akosnk, “Sobolev embedding with variable exponent,” Studia Mathematica, vol. 143, no. 3, pp. 267–293, 2000. View at Google Scholar · View at MathSciNet · View at Scopus
  21. O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41, pp. 592–618, 1991. View at Google Scholar
  22. D. Geng, “Infinitely many solutions of p-Laplacian equations with limit subcritical growth,” Applied Mathematics and Mechanics, vol. 28, no. 10, pp. 1373–1382, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. P. Pucci, “Geometric description of the mountain pass critical points,” in Contemporary Mathematicians, vol. 2, pp. 469–471, Birkhuser, Basel, Switzerland, 2014. View at Google Scholar
  24. P. Pucci and V. Radulescu, “The impact of the mountain pass theory in nonlinear analysis: a mathematical survey,” Bollettino dell'Unione Matematica Italiana Series IX, vol. 3, pp. 543–584, 2010. View at Google Scholar
  25. M. Willem, Minimax Theorems, Birkhuser, 1996.