An Approximation of Hedberg’s Type in Sobolev Spaces with Variable Exponent and Application

A (u) = ∑ |β|≤α (−1) |β| D β Aβ (x, u, ∇u, . . . , ∇ α u), whose coefficient Aβ satisfies conditions (including growth conditions) which guarantee the solvability of the problem A (u) = f. This new framework is conceptually more involved than the classical one includesmanymore fundamental examples.Thus ourmain result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.


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 2 on a domain Ω in R  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  = 1, that is,  of second order, their solvability under fairly general and natural assumptions was proved by Hess [2].The treatment of the case  > 1 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  > 1. 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  , (R  ), 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     (R  ), see [8][9][10].
It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent  ,(⋅) (R  ) of the harder higher order case  > 1.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).

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 Ω be an open subset of R  , by the symbol P(Ω), we denote the family of all measurable functions (⋅) : Ω → [1, ∞].
The Hardy-Little-wood maximal operator  is defined on locally integrable functions  on Ω by the following formula: where For more details, see [13][14][15][16], where various sufficient conditions for (⋅) ∈ M(Ω) can be found.

Main Results
First, we give the following results which will be used in our main result.
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.Proof.We assume that () < ∞; otherwise, there is nothing to prove.We then observe that there is a constant  1 such that   * ||() ≤  1 ().In fact, by ( 17), (18) Proof.For  integer.Assume that  ∈  ∞ 0 (R  ) and  ≥ 0. Set  =   *  and notice that () > 0 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 multiindices { 1 , . . .,   } such that But  is continuous, so it follows that lim On the other hand, the weak convergence of {  } ∞ 1 implies that the pointwise limit of {  *   } ∞ 1 (which is now known to exist a.e.) is   * .In fact, setting   −  = ℎ  , for an arbitrary  > 0, By weak convergence, the last term tends to zero, since   is in    (⋅) (R  ) away from the origin.Consider the following: , which implies that for all , there exists  ≥ , such that Then, Since  ≥ ‖lim inf  → ∞   * ℎ  ‖ (⋅) , then we have which is an arbitrary small number, and thus for a.e., This completes the proof of Lemma 5.

Proposition 7.
Let  ∈  ,(⋅) (R  ), there exist a sequence   , such that Proof.The proof of Proposition 7 is done in two steps as follows.
Remark 8.The sequence   constructed above satisfies with a constant  depending only on  and .

Existence
Result.This subsection is devoted to establish the following existence theorem.
(, ) denote the open ball in Ω with center  and radius , and |(, )| denotes the volume of (, ).