Research Article
An Approximation of Hedberg’s Type in Sobolev Spaces with
Variable Exponent and Application
Abdelmoujib Benkirane, Mostafa El Moumni, and Aziz Fri
Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah,
BP 1796, 30000 Atlas Fez, Morocco
Correspondence should be addressed to Aziz Fri; friazizon@gmail.com
Received 20 February 2014; Accepted 4 April 2014; Published 30 April 2014
Academic Editor: Juntao Sun
Copyright © 2014 Abdelmoujib Benkirane et al. Tis 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.
Te aim of this paper is to extend the usual framework of PDE with ()=−div (,,∇) to include a large class of cases with
() = ∑
||≤
(−1)
||
(,,∇,...,∇
), whose coefcient
satisfes conditions (including growth conditions) which
guarantee the solvability of the problem () = . Tis new framework is conceptually more involved than the classical one
includes many more fundamental examples. Tus our main result can be applied to various types of PDEs such as reaction-difusion
equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.
1. Introduction
Tis paper is motivated by the study of the unilateral problem
associated with the following equation:
()+(,)=. (1)
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:
()= ∑
||≤
(−1)
||
(,∇,...,∇
).
(2)
Te function satisfes a sign condition but has otherwise
completely unrestricted growth with respect to .
Equations of type (1) were frst considered by Browder
[1] as an application to the theory of not everywhere defned
mapping of monotone type. For =1, that is, of
second order, their solvability under fairly general and natural
assumptions was proved by Hess [2]. Te 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 intro-
duced in nonlinear potential theory by Hedberg [4] could
be used in place of truncation. Tis 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.
Tey 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].
Te 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 [8–10].
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.
Te paper is structured as follows. Afer some neces-
sary 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).
Hindawi Publishing Corporation
Chinese Journal of Mathematics
Volume 2014, Article ID 549051, 7 pages
http://dx.doi.org/10.1155/2014/549051