A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

Mihai Mihăilescu, Vicenţiu Rădulescu

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Abstract

We study the boundary value problem Embedded Image in Embedded Image, u=0 on Embedded Image, where Embedded Image is a smooth bounded domain in Embedded Image and Embedded Image is a Embedded Image-Laplace type operator, with Embedded Image. We prove that if λ is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

Keywords:

1. Introduction and preliminary results

Most materials can be modelled with sufficient accuracy using classical Lebesgue and Sobolev spaces, Embedded Image and Embedded Image, where p is a fixed constant. For some materials with inhomogeneities, for instance electrorheological fluids (sometimes referred to as ‘smart fluids’), this is not adequate, but rather the exponent p should be able to vary. This leads us to the study of variable exponent Lebesgue and Sobolev spaces, Embedded Image and Embedded Image, where p is a real-valued function.

This paper is motivated by phenomena which are described by nonlinear boundary value problems of the typeEmbedded Image(1.1)where Embedded Image Embedded Image is a bounded domain with smooth boundary, Embedded Image and Embedded Image. The interest in studying such problems consists of the presence of the Embedded Image-Laplace type operator Embedded Image. We remember that the Embedded Image-Laplace operator is defined by Embedded Image. The study of differential equations and variational problems involving Embedded Image-growth conditions is a consequence of their applications. Materials requiring such more advanced theory have been studied experimentally since the middle of the last century. The first major discovery in electrorheological fluids was due to Willis Winslow in 1949. These fluids have the interesting property that their viscosity depends on the electric field in the fluid. Winslow noticed that in such fluids (for instance lithium polymetachrylate) viscosity in an electrical field is inversely proportional to the strength of the field. The field induces string-like formations in the fluid, which are parallel to the field. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect. For a general account of the underlying physics consult Halsey (1992) and for some technical applications Pfeiffer et al. (1999). Electrorheological fluids have been used in robotics and space technology. The experimental research has been done mainly in the USA, for instance in NASA laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids we refer to Halsey (1992), Acerbi & Mingione (2001), Diening (2002), Ruzicka (2002), Fan et al. (2005) and Chabrowski & Fu (2005).

We point out a recent mathematical model developed by Rajagopal & Ruzicka (2001). The model takes into account the delicate interaction between the electromagnetic fields and the moving fluids. Particularly, in the context of continuum mechanics, these fluids are seen as non-Newtonian fluids. The system modelling the phenomenon arising from this study isEmbedded Image(1.2)where Embedded Image is the electromagnetic field, Embedded Image is the velocity of the field, Embedded Image is the symmetric part of the gradient, Embedded Image is the extra stress tensor and π is the pressure (according to notations in Rajagopal & Ruzicka (2001)).

The constitutive relation for the extra stress tensor Embedded Image isEmbedded Imagefor all symmetric Embedded Image matrices z and where Embedded Image. The structure of the system allows the determination of Embedded Image so that it depends on x and thus, Embedded Image.

The extra stress tensor Embedded Image is chosen such that it is a monotone vector field satisfying the ellipticity conditionEmbedded Imagewhere Embedded Image, for any Embedded Image symmetric matrices z, λ with null trace.

For the system described above, Rajagopal & Ruzicka established an existence theory which is particularly satisfying in the steady caseEmbedded Image

Our paper can be regarded as a starting point for investigations of models like those described above, since we treat the existence and multiplicity of solutions for problems with Embedded Image growth as in equation (1.1). We point out that even if our results will be formulated in a variational context, our methods and techniques can be applied to systems as well (see e.g. the work of El Hamidi (2004) for a nice generalization of such results to the study of elliptic systems of gradient type with Embedded Image growth).

A complete description regarding the development of variable exponent spaces, based on a rich bibliography, can be found in the paper of Diening et al. (2004). We resume in what follows some basic facts from the above quoted study. According to that paper, variable exponent Lebesgue spaces had already appeared in the literature for the first time in a article by Orlicz (1931). In the 1950s, this study was carried on by Nakano who made the first systematic study of spaces with variable exponent (called modular spaces). Nakano explicitly mentioned variable exponent Lebesgue spaces as an example of more general spaces he considered, see Nakano (1950; p. 284). Later, the Polish mathematicians investigated the modular function spaces (e.g. Musielak 1983). Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. In that context, we refer to the work of Tsenov (1961), Sharapudinov (1978) and Zhikov (1987).

We recall in what follows some definitions and basic properties of the generalized Lebesgue–Sobolev spaces Embedded Image and Embedded Image, where Embedded Image is a bounded domain in Embedded Image.

Throughout this paper, we assume that Embedded Image, Embedded Image with Embedded Image.

SetEmbedded ImageFor any Embedded Image, we defineEmbedded ImageFor any Embedded Image, we define the variable exponent Lebesgue spaceEmbedded ImageWe define a norm, the so-called Luxemburg norm, on this space by the formulaEmbedded ImageVariable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces (Kováčik & Rákosník 1991; theorem 2.5), the Hölder inequality holds (Kováčik & Rákosník 1991; theorem 2.1), they are reflexive if and only if Embedded Image (Kováčik & Rákosník 1991; corollary 2.7) and continuous functions are dense, if Embedded Image (Kováčik & Rákosník 1991; theorem 2.11). The inclusion between Lebesgue spaces also generalizes naturally (Kováčik & Rákosník 1991; theorem 2.8): if Embedded Image and Embedded Image, Embedded Image are variable exponents so that Embedded Image almost everywhere in Embedded Image then there exists the continuous embedding Embedded Image, whose norm does not exceed Embedded Image.

We denote by Embedded Image the conjugate space of Embedded Image, where Embedded Image. For any Embedded Image and Embedded Image, the Hölder type inequalityEmbedded Image(1.3)holds true.

An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the Modular of the Embedded Image space, which is the mapping Embedded Image defined byEmbedded ImageIf Embedded Image, Embedded Image and Embedded Image then the following relations hold trueEmbedded Image(1.4)Embedded Image(1.5)Embedded Image(1.6)Spaces with Embedded Image have been studied by Edmunds et al. (1999).

Next, we define Embedded Image as the closure of Embedded Image under the normEmbedded ImageThe space Embedded Image is a separable and reflexive Banach space. We note that if Embedded Image and Embedded Image for all Embedded Image, then the embedding Embedded Image is compact and continuous, where Embedded Image if Embedded Image or Embedded Image if Embedded Image. We refer to Kováčik & Rákosník (1991), Edmunds & Rákosník (1992, 2000) and Fan & Zhao (2001) for further properties of variable exponent Lebesgue–Sobolev spaces.

The paper contains two sections. In §2, we describe the problem and we state the main result. Some remarks and connections regarding similar results are also included at the end of this section. In §3, we prove the main result of the paper. We also include some generalizations of standard results involving the generalized Lebesgue–Sobolev spaces in order to offer clarity and strictness to our paper. These auxiliary results aim to be a guide which facilitates the reading of the paper.

2. The main result

Assume that Embedded Image is the continuous derivative with respect to ξ of the mapping Embedded Image, Embedded Image, i.e. Embedded Image. Suppose that a and A satisfy the following hypotheses:

  1. The following equality holdsEmbedded Imagefor all Embedded Image.

  2. There exists a positive constant Embedded Image such thatEmbedded Imagefor all Embedded Image and Embedded Image.

  3. The following inequality holdsEmbedded Imagefor all Embedded Image and Embedded Image, with equality if and only if ξ=ψ.

  4. There exists k>0 such thatEmbedded Imagefor all Embedded Image and Embedded Image.

  5. The following inequalities hold trueEmbedded Imagefor all Embedded Image and Embedded Image.

Examples:

  1. Set Embedded Image, Embedded Image, where Embedded Image. Then we get the Embedded Image-Laplace operatorEmbedded Image

  2. Set Embedded Image, Embedded Image, where Embedded Image. Then we obtain the generalized mean curvature operatorEmbedded Image

In this paper, we study problem (1.1) in the particular caseEmbedded Imagewith Embedded Image and Embedded Image. More precisely, we consider the degenerate boundary value problemEmbedded Image(2.1)

We say that Embedded Image is a weak solution of problem (2.1), if Embedded Image a.e. in Embedded Image andEmbedded Imagefor all Embedded Image.

Our main result asserts that problem (2.1) has at least two non-trivial weak solutions provided that λ>0 is large enough and operators A and a satisfy conditions (A1)–(A5). More precisely, we prove the following.

Assume hypotheses (A1)–(A5) are fulfilled. Then there exists Embedded Image such that for all Embedded Image problem (2.1) has at least two distinct non-negative, non-trivial weak solutions, provided that Embedded Image.

By theorem 4.3 in Fan & Zhang (2003), problem (2.1) has at least a weak solution in the particular case Embedded Image. However, the proof in Fan & Zhang (2003) does not state the fact that the solution is non-negative and not even non-trivial in the case when Embedded Image.

We point out that our result is inspired by theorem 1.2 in Perera (2003), where a related property is proved in the case of the p-Laplace operators. We point out that the extension from p-Laplace operator to Embedded Image-Laplace operator is not trivial, since the Embedded Image-Laplacian has a more complicated structure than the p-Laplace operator, for example, it is inhomogeneous.

Finally, we mention that a similar study regarding the existence and multiplicity of solutions for a system of equations involving the Embedded Image-Laplace operator can be found in El Hamidi (2004). The arguments used by the author rely on the Mountain Pass theorem and Bartsch's Fountain theorem.

3. Proof of theorem 2.1

Let E denote the generalized Sobolev space Embedded Image.

Define the energy functional Embedded Image byEmbedded Imagewhere Embedded Image.

We first establish some basic properties of I.

The functional I is well-defined on E and Embedded Image with the derivative given byEmbedded Imagefor all u, Embedded Image.

To prove proposition 3.1, we define the functional Embedded Image byEmbedded Image

  1. The functional Embedded Image is well-defined on E.

  2. The functional Embedded Image is of class Embedded Image and

Embedded Imagefor all Embedded Image.

(i) For any Embedded Image and Embedded Image, we haveEmbedded ImageUsing hypothesis (A2), we getEmbedded Image(3.1)The above inequality and (A5) implyEmbedded ImageUsing inequality (1.3) and relations (1.4) and (1.5), we deduce that Embedded Image is well defined on E.

(ii) Existence of the Gâteaux derivative. Let u, Embedded Image. Fix Embedded Image and Embedded Image. Then, by the mean value theorem, there exists Embedded Image such thatEmbedded ImageUsing condition (A2), we obtainEmbedded ImageNext, by inequality (1.3), we haveEmbedded ImageandEmbedded ImageThe above inequalities implyEmbedded ImageIt follows from the Lebesgue theorem thatEmbedded Image

Assume Embedded Image in E. Let us define Embedded Image. Using hypothesis (A2) and proposition 2.2 in Fan & Zhang (2003), we deduce that Embedded Image in Embedded Image, where Embedded Image. By inequality (1.3), we obtainEmbedded Imageand soEmbedded ImageThe proof of lemma 3.2 is complete. ▪

If Embedded Image then Embedded Image, Embedded Image andEmbedded Imagewhere Embedded Image for all Embedded Image.

Let Embedded Image be fixed. Then there exists a sequence Embedded Image such thatEmbedded ImageSince Embedded Image for all Embedded Image, it follows that Embedded Image is continuously embedded in Embedded Image and thus,Embedded ImageHence Embedded Image. We obtainEmbedded Image(3.2)On the other hand, theorem 7.6 in Gilbarg & Trudinger (1998) impliesEmbedded ImageBy the above equalities, we deduce thatEmbedded Image(3.3)andEmbedded Image(3.4)Since Embedded Image, we haveEmbedded Image(3.5)By equations (3.3)–(3.5) and Lebesgue theorem, we obtain that Embedded Image, Embedded Image and Embedded Image, Embedded Image. It follows thatEmbedded Image(3.6)where Embedded Image (see Fan & Zhao (2001) for more details).

By equations (3.2) and (3.6), we conclude thatEmbedded ImageSince Embedded Image, theorem 2.6 and remark 2.9 in Fan & Zhao (2001) show that Embedded Image. Thus, Embedded Image, Embedded Image and the proof of lemma 3.3 is complete. ▪

By lemmas 3.2 and 3.3, it is clear that proposition 3.1 holds true.

If u is a critical point of I then using lemma 3.3 and condition (A5), we haveEmbedded ImageThus, we deduce that Embedded Image. It follows that the non-trivial critical points of I are non-negative solutions of (2.1).

The above remark shows that we can prove theorem 2.1 using the critical points theory. More exactly, we first show that for λ>0 large enough, the functional I has a global minimizer Embedded Image such that Embedded Image. Next, by means of the Mountain Pass theorem, a second critical point Embedded Image with Embedded Image is obtained.

The functional Embedded Image is weakly lower semi-continuous.

By corollary III.8 in Brezis (1992), it is enough to show that Embedded Image is lower semi-continuous. For this purpose, we fix Embedded Image and ϵ>0. Since Embedded Image is convex (by condition (A4)), we deduce that for any Embedded Image, the following inequality holdsEmbedded ImageUsing condition (A2) and inequality (1.3), we haveEmbedded Imagefor all Embedded Image with Embedded Image, where Embedded Image, Embedded Image, Embedded Image are positive constants and Embedded Image. We conclude that Embedded Image is weakly lower semi-continuous. The proof of lemma 3.4 is complete. ▪

There exists Embedded Image such thatEmbedded Image

We know that E is continuously embedded in Embedded Image. It follows that there exists C>0 such thatEmbedded ImageOn the other hand, by equation (1.4), we haveEmbedded ImageCombining the above inequalities, we obtainEmbedded ImageThe proof of lemma 3.5 is complete. ▪

  1. The functional I is bounded from below and coercive.

  2. The functional I is weakly lower semi-continuous.

(i) Since Embedded Image, we haveEmbedded ImageThen for any λ>0, there exists Embedded Image such thatEmbedded Imagewhere Embedded Image is defined in lemma 3.5.

Condition (A5) and the above inequality show that for any Embedded Image with Embedded Image, we haveEmbedded ImageThis shows that I is bounded from below and coercive.

(ii) Using lemma 3.4, we deduce that Embedded Image is weakly lower semi-continuous. We show that I is weakly lower semi-continuous. Let Embedded Image be a sequence which converges weakly to u in E. Since Embedded Image is weakly lower semi-continuous, we haveEmbedded Image(3.7)On the other hand, since E is compactly embedded in Embedded Image and Embedded Image, it follows that Embedded Image converges strongly to Embedded Image both in Embedded Image and in Embedded Image. This fact together with relation (3.7) implyEmbedded ImageTherefore, I is weakly lower semi-continuous. The proof of proposition 3.6 is complete. ▪

By proposition 2 and theorem 1.2 in Struwe (1996), we deduce that there exists Embedded Image a global minimizer of I. The following result implies that Embedded Image, provided that λ is sufficiently large.

There exists Embedded Image such that Embedded Image.

Let Embedded Image be a compact subset, large enough and Embedded Image be such that Embedded Image in Embedded Image and Embedded Image in Embedded Image, where Embedded Image is chosen such thatEmbedded ImageWe haveEmbedded Imageand thus Embedded Image for λ>0 large enough. The proof of proposition 3.7 is complete. ▪

Since proposition 3.7 holds true, it follows that Embedded Image is a non-trivial weak solution of problem (2.1).

Fix Embedded Image. SetEmbedded ImageandEmbedded ImageDefine the functional Embedded Image byEmbedded ImageThe same arguments as those used for functional I imply that Embedded Image andEmbedded Imagefor all u, Embedded Image.

On the other hand, we point out that if Embedded Image is a critical point of J then Embedded Image. The proof can be carried out as in the case of functional I.

Next, we prove the following.

If u is a critical point of J then Embedded Image.

We haveEmbedded ImageBy condition (A3), we deduce that the above equality holds if and only if Embedded Image. It follows that Embedded Image for all Embedded Image. HenceEmbedded Imageand thus,Embedded ImageBy relation (1.5), we obtainEmbedded ImageSince Embedded Image by lemma 3.3, we have that Embedded Image. Thus, we obtain that Embedded Image in Embedded Image, i.e. Embedded Image in Embedded Image. The proof of lemma 3.8 is complete. ▪

In the following, we determine a critical point Embedded Image of J such that Embedded Image via the Mountain Pass theorem. By the above lemma, we will deduce that Embedded Image in Embedded Image. Therefore,Embedded Imageand thus,Embedded ImageMore exactly we findEmbedded ImageThis shows that Embedded Image is a weak solution of problem (2.1) such that Embedded Image, Embedded Image and Embedded Image.

In order to find Embedded Image described above, we prove the following.

There exist Embedded Image and a>0 such that Embedded Image, for all Embedded Image with Embedded Image.

Let Embedded Image be fixed, such that Embedded Image. It is clear that there exists δ>1 such thatEmbedded ImageFor δ given above, we defineEmbedded ImageIf Embedded Image with Embedded Image, we haveEmbedded ImageIf Embedded Image with Embedded Image, then Embedded Image and we haveEmbedded ImageThus, we deduce thatEmbedded ImageProvided that Embedded Image by condition (A5) and relation (1.5), we getEmbedded Image(3.8)Since Embedded Image, it follows that Embedded Image for all Embedded Image. Then there exists Embedded Image such that E is continuously embedded in Embedded Image. Thus, there exists a positive constant C>0 such thatEmbedded ImageUsing the definition of G, Hölder's inequality and the above estimate, we obtainEmbedded Image(3.9)By equations (3.8) and (3.9), we infer that it is enough to show that Embedded Image as Embedded Image in order to prove lemma 3.9.

Let ϵ>0. We choose Embedded Image a compact subset, such that Embedded Image. We denote by Embedded Image. Then it is clear thatEmbedded ImageThe above inequality implies that Embedded Image as Embedded Image.

Since Embedded Image, we haveEmbedded Imageand ϵ>0 is arbitrary. We find that Embedded Image as Embedded Image. This concludes the proof of lemma 3.9. ▪

The functional J is coercive.

For each Embedded Image with Embedded Image by condition (A5), relation (1.4) and inequality (1.3), we haveEmbedded Imagewhere Embedded Image, Embedded Image and Embedded Image are positive constants. Since Embedded Image the above inequality implies that Embedded Image as Embedded Image, i.e. J is coercive. The proof of lemma 3.10 is complete. ▪

The following result yields a sufficient condition which ensures that a weakly convergent sequence in E converges strongly, too.

Assume that the sequence Embedded Image converges weakly to u in E andEmbedded ImageThen Embedded Image converges strongly to u in E.

Using relation (3.1), we have that there exists a positive constant Embedded Image such thatEmbedded ImageThe above inequality impliesEmbedded Image(3.10)The fact that Embedded Image converges weakly to u in E implies that there exists R>0 such that Embedded Image for all n. By relation (3.10) and inequalities (1.3)–(1.5), we deduce that Embedded Image is bounded. Then, up to a subsequence, we deduce that Embedded Image. By lemma 3.4, we obtainEmbedded ImageOn the other hand, since Embedded Image is convex, we haveEmbedded ImageNext, by the hypothesis Embedded Image, we conclude that Embedded Image.

Taking into account that Embedded Image converges weakly to u in E and using lemma 3.4, we haveEmbedded Image(3.11)We assume by contradiction that Embedded Image does not converge to u in E. Then by (1.6), it follows that there exist ϵ>0 and a subsequence Embedded Image of Embedded Image such thatEmbedded Image(3.12)By condition (A4), we haveEmbedded Image(3.13)Relations (3.12) and (3.13) yieldEmbedded ImageLetting Embedded Image in the above inequality, we obtainEmbedded Imageand that is a contradiction with (3.11). It follows that Embedded Image converges strongly to u in E and lemma 3.11 is proved. ▪

Using lemma 3.9 and the Mountain Pass theorem (see Ambrosetti & Rabinowitz (1973) with the variant given by theorem 1.15 in Willem (1996)) we deduce that there exists a sequence Embedded Image such thatEmbedded Image(3.14)whereEmbedded ImageandEmbedded ImageBy relation (3.14) and lemma 3.10, we obtain that Embedded Image is bounded and thus, passing eventually to a subsequence, still denoted by Embedded Image, we may assume that there exists Embedded Image such that Embedded Image converges weakly to Embedded Image. Since E is compactly embedded in Embedded Image for any Embedded Image, it follows that Embedded Image converges strongly to Embedded Image in Embedded Image for all Embedded Image. Hence,Embedded Imageas Embedded Image. By lemma 3.11, we deduce that Embedded Image converges strongly to Embedded Image in E and using relation (3.14), we findEmbedded ImageTherefore, Embedded Image and Embedded Image. By lemma 3.8, we deduce that Embedded Image in Embedded Image. Therefore,Embedded Imageand thus,Embedded ImageWe conclude that Embedded Image is a critical point of I and thus a solution of (2.1). Furthermore, Embedded Image and Embedded Image. Thus, Embedded Image is not trivial and Embedded Image. The proof of theorem 2.1 is now complete. ▪

Acknowledgments

V.R. has been partially supported by grant CEEX05-DE11-36/05.10.2005, grant CNCSIS ‘Nonlinearities and Singularities in Mathamatical Physics’ and grant EAR ‘Singular Problems of Lane–Emden–Fowler Type with convection’ of the Romanian Academy.

Footnotes

    • Received September 13, 2005.
    • Accepted November 29, 2005.

References

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