Abstract

The aim of this paper is to extend the usual framework of PDE with to include a large class of cases with , whose coefficient satisfies conditions (including growth conditions) which guarantee the solvability of the problem . This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.

1. Introduction

This paper is motivated by the study of the unilateral problem associated with the following equation:

We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order on a domain in in generalized divergence form as follows:

The function satisfies a sign condition but has otherwise completely unrestricted growth with respect to .

Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type. For , that is, of second order, their solvability under fairly general and natural assumptions was proved by Hess [2]. The treatment of the case is more involved due to the lack of a simple truncation operator in higher order Sobolev spaces. Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation. This yielded the solvability of (1) for . Brezis and Browder [5] then used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution. They also showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). In a more general case, Boccardo et al. studied inequations associated with (1), see [7].

The functional setting in all the results mentioned above is that of the usual Sobolev spaces , and the functions in (2) are supposed to satisfy polynomial growth conditions with respect to and its derivatives. Benkirane and Gossez established this result in the Orlicz-Sobolev spaces , see [8–10].

It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent of the harder higher order case . We consider problem (1) as well as Hedberg’s approximation theorem and Brezis-Browder’s question on the action of some distributions.

The paper is structured as follows. After some necessary preliminaries, in Section 3, we give the proof of the approximation theorem. In addition, Section 4 forms a useful supplement to some applications of (1).

2. Preliminaries

In this section we list briefly some definitions and well known facts about Sobolev spaces with variable exponent and Bessel potential spaces with variable exponent. Standard references are [11, 12].

Let be an open subset of , by the symbol , we denote the family of all measurable functions .

For , put Furthermore, we introduce a class by Let , and consider the functional on all measurable function on . The Lebesgue space with variable exponent is defined as the set of all measurable functions on such that, for some , equipped with the norm The space is a separable Banach space. Moreover, if , then is uniformly convex, hence, reflexive, and its dual space is isomorphic to , where .

Finally, we have the Hölder type inequality as follows: for all and .

The Hardy-Little-wood maximal operator is defined on locally integrable functions on by the following formula: where denote the open ball in with center and radius , and denotes the volume of .

Definition 1. By , denote the class of all functions for which the operator is bounded on ; that is, with a positive constant independent of .

Remark 2. For example, if the following two conditions are satisfied: For more details, see [13–16], where various sufficient conditions for can be found.
Let and ; we define the Sobolev space with variable exponent by equipped with the norm where is a multi-index, , and .
Next, we define as the closure of in and the dual space of , where .
The Bessel kernel (see [17]) of order is defined by The Riesz kernel (see [18]) of order is defined by where is a certain constant, whose exact value is It follows easily that and an examination of (14) shows without much effort that for any , Writing , , this implies that, for , with and .

3. Main Results

First, we give the following results which will be used in our main result.

3.1. Useful Results

Let and . The Bessel potential space with variable exponent is defined, for , by and is equipped with the norm

Lemma 3 (see [12]). If and , then and the corresponding norms are equivalent.

Lemma 4 (see [12]). Suppose that and . Then there exists a positive constant such that

We will now verify that satisfies all the required properties in Proposition 7. The argument relies heavily on the following Lemmas 5 and 6. In the sequel, we need the following two technical lemmas.

Lemma 5. If and for any multi-index , there exists a constant such that for any ,

Proof. We assume that ; otherwise, there is nothing to prove. We then observe that there is a constant such that . In fact, by (17), (18), and ([11], Lemma 6.1.4), there exists a constant such that Then, by (19) and (18), for any , we have Now choosing ; then, and the result follows.

Lemma 6. Suppose that and . Let for some , if satisfies Then for every , and there exists a constant , depending only on and , such that

Proof. For integer. Assume that and . Set and notice that for all , so that is defined. If is a multi-indix with , we find by the chain rule that where the interior sum is over all ordered -tuples of multi-indices such that , and all . The are coefficients, whose exact value is of no consequence to us. Thus, by assumption of Lemma 3.4 we get For , we estimate these derivatives by means of Lemma 5. By the positivity of , we have Thus, since , Taking the term with into account, we obtain But we already know from (10) that and that for .
This finishes the proof for smooth .
Now we pass to the general case and let be an arbitrary function in . Then there are nonnegative functions , , such that By the first part of the proof, for all sufficiently large .
Thus, setting , we can assume that converges weakly in to an element , with .
We have to prove that .
The strong convergence of and the fact that imply, by Lemma 4, that converges strongly in to . After extraction of a subsequence, we can assume that But is continuous, so it follows that
On the other hand, the weak convergence of implies that the pointwise limit of (which is now known to exist a.e.) is . In fact, setting , for an arbitrary , By weak convergence, the last term tends to zero, since is in away from the origin. Consider the following: We deduce that, for all , , which implies that for all , there exists , such that Then, Since , then we have which is an arbitrary small number, and thus for a.e. , This completes the proof of Lemma 5.

Proposition 7. Let , there exist a sequence , such that (i)) is compact;(ii) and a.e. in ;(iii) in as .

Proof. The proof of Proposition 7 is done in two steps as follows.
Step  1. Let be a fixed function such that 1 and in a neighborhood of the origin.
Let = ; then, = satisfies all the required properties, using the fact that by Sobolev’s theorem [19].
Step  2 (. We assume that has compact support, if necessary by multiplying with a suitable . We represent as a Bessel potential, , so that . Set and let be a function such that , for and for . Then, set We First observe that, on the set , which includes , and so we have a.e., and thus . It remains to prove that and that converges to as tends to .
Let be any multi-index with . If = , , and all , we find by the same arguments as in the proof of Lemma 6 that By Lemma 5, we have, for any multi-index with , On the open set , using the fact that ; then, for , and for = .
By using Leibniz’s formula, we have for = , if , again using Lemma 5, that If , we have . It follows that a.e.
The functions on the right hand side belong to , so the theorem follows by applying the dominated convergence.