## Abstract

We study the boundary value problem in , *u*=0 on , where is a smooth bounded domain in and is a -Laplace type operator, with . 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.

## 1. Introduction and preliminary results

Most materials can be modelled with sufficient accuracy using classical Lebesgue and Sobolev spaces, and , 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, and , where *p* is a real-valued function.

This paper is motivated by phenomena which are described by nonlinear boundary value problems of the type(1.1)where is a bounded domain with smooth boundary, and . The interest in studying such problems consists of the presence of the -Laplace type operator . We remember that the -Laplace operator is defined by . The study of differential equations and variational problems involving -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 is(1.2)where is the electromagnetic field, is the velocity of the field, is the symmetric part of the gradient, 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 isfor all symmetric matrices *z* and where . The structure of the system allows the determination of so that it depends on *x* and thus, .

The extra stress tensor is chosen such that it is a monotone vector field satisfying the ellipticity conditionwhere , for any 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 case

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 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 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 and , where is a bounded domain in .

Throughout this paper, we assume that , with .

SetFor any , we defineFor any , we define the variable exponent Lebesgue spaceWe define a norm, the so-called *Luxemburg norm*, on this space by the formulaVariable 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 (Kováčik & Rákosník 1991; corollary 2.7) and continuous functions are dense, if (Kováčik & Rákosník 1991; theorem 2.11). The inclusion between Lebesgue spaces also generalizes naturally (Kováčik & Rákosník 1991; theorem 2.8): if and , are variable exponents so that almost everywhere in then there exists the continuous embedding , whose norm does not exceed .

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

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

Next, we define as the closure of under the normThe space is a separable and reflexive Banach space. We note that if and for all , then the embedding is compact and continuous, where if or if . 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 is the continuous derivative with respect to *ξ* of the mapping , , i.e. . Suppose that *a* and *A* satisfy the following hypotheses:

The following equality holdsfor all .

There exists a positive constant such thatfor all and .

The following inequality holdsfor all and , with equality if and only if

*ξ*=*ψ*.There exists

*k*>0 such thatfor all and .The following inequalities hold truefor all and .

Examples:

Set , , where . Then we get the -Laplace operator

Set , , where . Then we obtain the generalized mean curvature operator

In this paper, we study problem (1.1) in the particular casewith and . More precisely, we consider the degenerate boundary value problem(2.1)

We say that is a *weak solution* of problem (2.1), if a.e. in andfor all .

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* *such that for all* *problem* *(2.1)* *has at least two distinct non-negative,* *non-trivial weak solutions,* *provided that* .

By theorem 4.3 in Fan & Zhang (2003), problem (2.1) has at least a weak solution in the particular case . 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 .

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 -Laplace operator is not trivial, since the -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 -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 .

Define the energy functional bywhere .

We first establish some basic properties of *I*.

*The functional I is well*-*defined on E and* *with the derivative given by**for all u*, .

To prove proposition 3.1, we define the functional by

*The functional**is well*-*defined on E*.*The functional**is of class**and*

*for all* .

(i) For any and , we haveUsing hypothesis (A2), we get(3.1)The above inequality and (A5) implyUsing inequality (1.3) and relations (1.4) and (1.5), we deduce that is well defined on *E*.

(ii) *Existence of the Gâteaux derivative*. Let *u*, . Fix and . Then, by the mean value theorem, there exists such thatUsing condition (A2), we obtainNext, by inequality (1.3), we haveandThe above inequalities implyIt follows from the Lebesgue theorem that

Assume in *E*. Let us define . Using hypothesis (A2) and proposition 2.2 in Fan & Zhang (2003), we deduce that in , where . By inequality (1.3), we obtainand soThe proof of lemma 3.2 is complete. ▪

*If* *then* , *and**where* *for all* .

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

By equations (3.2) and (3.6), we conclude thatSince , theorem 2.6 and remark 2.9 in Fan & Zhao (2001) show that . Thus, , 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 haveThus, we deduce that . 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 such that . Next, by means of the Mountain Pass theorem, a second critical point with is obtained.

*The functional* *is weakly lower semi-continuous*.

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

*There exists* *such that*

We know that *E* is continuously embedded in . It follows that there exists *C*>0 such thatOn the other hand, by equation (1.4), we haveCombining the above inequalities, we obtainThe proof of lemma 3.5 is complete. ▪

*The functional I is bounded from below and coercive*.*The functional I is weakly lower semi-continuous*.

(i) Since , we haveThen for any *λ*>0, there exists such thatwhere is defined in lemma 3.5.

Condition (A5) and the above inequality show that for any with , we haveThis shows that *I* is bounded from below and coercive.

(ii) Using lemma 3.4, we deduce that is weakly lower semi-continuous. We show that *I* is weakly lower semi-continuous. Let be a sequence which converges weakly to *u* in *E*. Since is weakly lower semi-continuous, we have(3.7)On the other hand, since *E* is compactly embedded in and , it follows that converges strongly to both in and in . This fact together with relation (3.7) implyTherefore, *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 a global minimizer of *I*. The following result implies that , provided that *λ* is sufficiently large.

*There exists* *such that* .

Let be a compact subset, large enough and be such that in and in , where is chosen such thatWe haveand thus for *λ*>0 large enough. The proof of proposition 3.7 is complete. ▪

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

Fix . SetandDefine the functional byThe same arguments as those used for functional *I* imply that andfor all *u*, .

On the other hand, we point out that if is a critical point of *J* then . 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* .

We haveBy condition (A3), we deduce that the above equality holds if and only if . It follows that for all . Henceand thus,By relation (1.5), we obtainSince by lemma 3.3, we have that . Thus, we obtain that in , i.e. in . The proof of lemma 3.8 is complete. ▪

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

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

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

Let be fixed, such that . It is clear that there exists *δ*>1 such thatFor *δ* given above, we defineIf with , we haveIf with , then and we haveThus, we deduce thatProvided that by condition (A5) and relation (1.5), we get(3.8)Since , it follows that for all . Then there exists such that *E* is continuously embedded in . Thus, there exists a positive constant *C*>0 such thatUsing the definition of *G*, Hölder's inequality and the above estimate, we obtain(3.9)By equations (3.8) and (3.9), we infer that it is enough to show that as in order to prove lemma 3.9.

Let *ϵ*>0. We choose a compact subset, such that . We denote by . Then it is clear thatThe above inequality implies that as .

Since , we haveand *ϵ*>0 is arbitrary. We find that as . This concludes the proof of lemma 3.9. ▪

*The functional J is coercive*.

For each with by condition (A5), relation (1.4) and inequality (1.3), we havewhere , and are positive constants. Since the above inequality implies that as , 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* *converges weakly to u in E and**Then* *converges strongly to u in E*.

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

Taking into account that converges weakly to *u* in *E* and using lemma 3.4, we have(3.11)We assume by contradiction that does not converge to *u* in *E*. Then by (1.6), it follows that there exist *ϵ*>0 and a subsequence of such that(3.12)By condition (A4), we have(3.13)Relations (3.12) and (3.13) yieldLetting in the above inequality, we obtainand that is a contradiction with (3.11). It follows that 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 such that(3.14)whereandBy relation (3.14) and lemma 3.10, we obtain that is bounded and thus, passing eventually to a subsequence, still denoted by , we may assume that there exists such that converges weakly to . Since *E* is compactly embedded in for any , it follows that converges strongly to in for all . Hence,as . By lemma 3.11, we deduce that converges strongly to in *E* and using relation (3.14), we findTherefore, and . By lemma 3.8, we deduce that in . Therefore,and thus,We conclude that is a critical point of *I* and thus a solution of (2.1). Furthermore, and . Thus, is not trivial and . 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.

- © 2006 The Royal Society