Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces
UDC 517.5 We deal with the existence result for nonlinear elliptic equations related to the form$$Au + g(x, u,\nabla u) = f,$$where the term $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ is a Leray–Lions operator from a subset of $W^{1}_{0}L_M(\Omega)$ into its dual.  The growth and coercivity...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/373 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507016217231360 |
|---|---|
| author | Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. |
| author_facet | Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. |
| author_sort | Moussa, H. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:33Z |
| description | UDC 517.5
We deal with the existence result for nonlinear elliptic equations related to the form$$Au + g(x, u,\nabla u) = f,$$where the term $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ is a Leray–Lions operator from a subset of $W^{1}_{0}L_M(\Omega)$ into its dual.  The growth and coercivity conditions on the monotone vector field $a$ are prescribed by an $N$-function $M$ which does not have to satisfy a $\Delta_2$-condition. Therefore we use Orlicz–Sobolev spaces which are not necessarily reflexive and assume that the nonlinearity $g(x,u,\nabla u)$ is a Carathéodory function satisfying only a growth condition with no sign condition. The right-hand side~$f$ belongs to $W^{-1}E_{\overline{M}}(\Omega).$ |
| doi_str_mv | 10.37863/umzh.v72i4.373 |
| first_indexed | 2026-03-24T02:02:36Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i4.373
UDC 517.5
H. Moussa (Univ. Sultan Moulay Slimane, Morocco),
M. Rhoudaf (Univ. Moulay Ismail, Morocco),
H. Sabiki (Univ. Ibn Tofail, Morocco)
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM
WITHOUT SIGN CONDITION IN ORLICZ SPACES
РЕЗУЛЬТАТИ IСНУВАННЯ ДЛЯ ЗБУРЕНОЇ ЗАДАЧI ДIРIХЛЕ
ЗА ВIДСУТНОСТI УМОВ ЩОДО ЗНАКА В ПРОСТОРАХ ОРЛIЧА
We deal with the existence result for nonlinear elliptic equations related to the form
Au+ g(x, u,\nabla u) = f,
where the term - \mathrm{d}\mathrm{i}\mathrm{v}
\Bigl(
a(x, u,\nabla u)
\Bigr)
is a Leray – Lions operator from a subset of W 1
0LM (\Omega ) into its dual. The growth
and coercivity conditions on the monotone vector field a are prescribed by an N -function M which does not have to
satisfy a \Delta 2 -condition. Therefore we use Orlicz – Sobolev spaces which are not necessarily reflexive and assume that
the nonlinearity g(x, u,\nabla u) is a Carathéodory function satisfying only a growth condition with no sign condition. The
right-hand side f belongs to W - 1EM (\Omega ).
Розглядається задача iснування для нелiнiйних елiптичних рiвнянь у формi
Au+ g(x, u,\nabla u) = f,
де - \mathrm{d}\mathrm{i}\mathrm{v}
\Bigl(
a(x, u,\nabla u)
\Bigr)
— оператор Лере – Лiонса з пiдмножини W 1
0LM (\Omega ) у її дуальну множину. Умови зростання та
коерцитивностi в монотонному векторному полi a визначаються N -функцiєю M, яка не повинна задовольняти \Delta 2 -
умови. Тому ми використовуємо простори Орлiча – Соболєва, якi не обов’язково є рефлексивними, i припускаємо,
що нелiнiйнiсть g(x, u,\nabla u) є функцiєю Каратеодорi, що задовольняє лише умову зростання без умови знака. Права
частина f належить W - 1EM (\Omega ).
1. Introduction. In the last decade, there has been an increasing interest in the study of various
mathematical problems in modular spaces. These problems have many consideration in applications
[13, 33, 34] and have resulted in a renewal interest in Modular spaces, the origins of which can be
traced back to the work of Orlicz in the 1930s. In the 1950s, this study was carried on by Nakano
[29]. Later on, Polish and Czechoslovak mathematicians investigated the modular function spaces
(see, for instance, [25, 28]).
One of our motivations to study nonlinear problems in modular spaces comes from applications
to electro-rheological fluids as an important class of non-Newtonian fluids (sometimes referred to as
smart fluids). The electro-rheological fluids are characterized by their ability to highly change in their
mechanical properties under the influence of an external electromagnetic field. A mathematical model
of electro-rheological fluids was proposed by Rajagopal and Ru̇žička [32, 33], we refer for instance
to [4, 9, 21 – 24, 27] for different non-standard growth conditions. Another important application is
related to image processing [30] where this kind of diffusion operator is used to underline the borders
of the distorted image and to eliminate the noise.
In this paper, we are interesting to prove an existence result for a nonlinear elliptic problem with
nonlinearity. The studies will be undertaken for the case of rather general growth conditions for the
c\bigcirc H. MOUSSA, M. RHOUDAF, H. SABIKI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4 509
510 H. MOUSSA, M. RHOUDAF, H. SABIKI
highest order term. This formulation requires a general framework for the function space setting.
The problems will be considered in Orlicz spaces. The level of generality of our considerations will
have a crucial significance on the applied methods. This is a natural generalization of the numerous
recent studies appearing on Lebesgue and Sobolev spaces, which may be considered as a particular
case of our approach.
Let \Omega be a bounded open set of \BbbR N , N \geq 2, we consider the following nonlinear elliptic
problem:
Au+ g(x, u,\nabla u) = f in \Omega ,
u = 0 on \partial \Omega ,
(1.1)
where Au = - \mathrm{d}\mathrm{i}\mathrm{v}(a(x, u,\nabla u)) is a Leray – Lions operator defined on W 1,p(\Omega ), 1 < p < \infty .
Bensoussan, Boccardo, and Murat [8] proved the existence of solutions for the Dirichlet problem
associated to the problem (1.1), where g is a nonlinearity satisfying the following (natural) growth
condition:
| g(x, s, \xi )| \leq b(| s| )(c(x) + | \xi | p),
and the sign condition g(x, s, \xi )s \geq 0. In the case, where the right-hand side f is assumed to belong
to W - 1,p\prime (\Omega ) and g depends only on x and u, see the result of Brézis and Browder in [12]. In [31]
Porretta has proved the existence result for the problem (1.1) but the result is restricted to b(.) \equiv 1
in Sobolev spaces, and in the case with b(| s| ) \leq \beta | s| r - 1where 0 \leq r < p the same problem has
been studied by Benkirane et al. in [7]. A different approach (without sign condition) was used in
[10], under the assumption g(x, s, \xi ) = \lambda s - | \xi | 2, with \lambda > 0.
We recall also that the authors used in [10] the methods of lower and uper solutions. In the
literature of the same problems, the sign condition play a crucial role in the proof of the main result.
In [19], Gossez and Mustonen solved (1.1) in the case where g satisfies the classical sign condi-
tion g(x, s).s \geq 0 and data f in W - 1EM (\Omega ).
We find also some existence results in the same context for strongly nonlinear problem associated
to (1.1) proved in [3, 5, 6, 16] when data f belongs either to W - 1EM (\Omega ) or L1(\Omega ) with M satisfies
\Delta 2-condition. In the case where the \Delta 2-condition is not fulfilled the above problem was studied in
[2, 14, 15].
In the present paper, we deal with the existence result for the following problem:
(P )
\left\{
u \in W 1
0LM (\Omega ), g(x, u,\nabla u) \in L1(\Omega ),\int
\Omega
(a(x, u,\nabla u))\nabla Tk(u - v) dx+
\int
\Omega
g(x, u,\nabla u)Tk(u - v) dx \leq
\leq
\int
\Omega
fTk(u - v) dx
for every v \in W 1
0LM (\Omega ) \cap L\infty (\Omega ) and for all k > 0,
where f \in W - 1EM (\Omega ).
Note that the sign condition on a nonlinearity plays a crucial role to obtain a priori estimates and
existence of solution, to overcome the difficulty of the elimination of the sign condition we use the
following growth condition:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 511
| g(x, s, \xi )| \leq b(| s| ) + h(| s| )M(| \xi | ) (1.2)
with h \in L1(\BbbR +), b(| s| ) \leq P
- 1
P (| s| ) where M and P are two N -functions such that P \ll M.
In [2], the authors assume the same growth condition (1.2) on nonlinearities but the function b
depends only on x not on u and belongs to L1(\Omega ).
The main novelty of the paper is that the nonlinearity g does not have to satisfy any sign
condition, beside this we have the function b depends on solution u of our problem.
Our principal goal in this paper is to prove the existence result for the problem (P ) but without
assuming any sign condition on nonlinearities and any restriction on the N -function M of Orlicz
spaces.
This paper is organized as follows. Section 2 contains some preliminaries. In Section 3, we state
and prove our general results.
2. Preliminaries. Let M : \BbbR \rightarrow \BbbR be an N -function, i.e., M is even, continuous and convex
function, with M(t) > 0 for t > 0,
M(t)
t
\rightarrow 0 as t \rightarrow 0 and
M(t)
t
\rightarrow \infty as t \rightarrow \infty . Equivalently,
M admits the representation M(t) =
\int | t|
0
m(s) ds, where m : \BbbR + \rightarrow \BbbR + is non-decreasing, right
continuous, with m(0) = 0, m(t) > 0 for t > 0 and m(t) \rightarrow \infty as t \rightarrow \infty . The N -function
M conjugate to M is defined by M(t) =
\int | t|
0
m(s) ds, where m : \BbbR + \rightarrow \BbbR + is given by m(t) =
= \mathrm{s}\mathrm{u}\mathrm{p}\{ s : m(s) \leq t\} .
The N-function M is said to satisfy the \Delta 2-condition if, for some k > 0,
M(2t) \leq kM(t) for all t \geq 0.
When this inequality holds only for t \geq t0 > 0, M is said to satisfy the \Delta 2-condition near infinity.
Let P and M be two N -functions. P \ll M means that P grows essentially less rapidly than
M, i.e., for each \varepsilon > 0,
P (t)
M(\varepsilon t)
\rightarrow 0 as t \rightarrow \infty .
This is the case if and only if
M - 1(t)
P - 1(t)
\rightarrow 0 as t \rightarrow \infty .
We will extend these N -functions into even functions on all \BbbR . Let \Omega be an open subset of \BbbR N . The
Orlicz class \scrL M (\Omega ) (resp., the Orlicz space LM (\Omega )) is defined as the set of (equivalence classes of)
real-valued measurable functions u on \Omega such that\int
\Omega
M(u(x))dx < +\infty
\left( resp.,
\int
\Omega
M
\biggl(
u(x)
\lambda
\biggr)
dx < +\infty for some \lambda > 0
\right) .
Note that LM (\Omega ) is a Banach space under the norm
\| u\| M,\Omega = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \lambda > 0 :
\int
\Omega
M
\biggl(
u(x)
\lambda
\biggr)
dx \leq 1
\right\}
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
512 H. MOUSSA, M. RHOUDAF, H. SABIKI
and \scrL M (\Omega ) is a convex subset of LM (\Omega ). The closure in LM (\Omega ) of the set of bounded measurable
functions with compact support in \Omega is denoted by EM (\Omega ). The equality EM (\Omega ) = LM (\Omega ) holds if
and only if M satisfies the \Delta 2-condition, for all t or for t large, according to whether \Omega has infinite
measure or not.
The dual of EM (\Omega ) can be identified with LM (\Omega ) by means of the pairing
\int
\Omega
u(x)v(x)dx, and
the dual norm on LM (\Omega ) is equivalent to \| .\| M,\Omega . The space LM (\Omega ) is reflexive if and only if M
and M satisfy the \Delta 2-condition, for all t or for t large, according to whether \Omega has infinite measure
or not.
We define the Orlicz norm | | u| | (M) by
| | u| | (M) = \mathrm{s}\mathrm{u}\mathrm{p}
\int
QT
u(x)v(x) dx,
where the supremum is taken over all v \in EM(\Omega ) such that | | v| | M \leq 1, for which
| | u| | M \leq | | u| | (M) \leq 2| | u| | M
holds for all u \in LM(\Omega ).
We now turn to the Orlicz – Sobolev space. W 1LM (\Omega ) (resp., W 1EM (\Omega )) is the space of all
functions u such that u and its distributional derivatives up to order 1 lie in LM (\Omega ) (resp., EM (\Omega )).
This is a Banach space under the norm
\| u\| 1,M,\Omega =
\sum
| \alpha | \leq 1
\| \nabla \alpha u\| M,\Omega .
Thus W 1LM (\Omega ) and W 1EM (\Omega ) can be identified with subspaces of the product of N + 1 copies
of LM (\Omega ). Denoting this product by \Pi LM , we will use the weak topologies \sigma (\Pi LM ,\Pi EM ) and
\sigma (\Pi LM ,\Pi LM ). The space W 1
0EM (\Omega ) is defined as the (norm) closure of the Schwartz space \scrD (\Omega )
in W 1EM (\Omega ) and the space W 1
0LM (\Omega ) as the \sigma (\Pi LM ,\Pi EM ) closure of \scrD (\Omega ) in W 1LM (\Omega ).
We say that un converges to u for the modular convergence in W 1LM (\Omega ) if, for some \lambda > 0,\int
\Omega
M((D\alpha un - D\alpha u)/\lambda )dx \rightarrow 0 for all | \alpha | \leq 1 with | \alpha | = \alpha 1 + \alpha 2 + . . .+ \alpha N .
If M satisfies the \Delta 2 condition on \BbbR +(near infinity only when \Omega has finite measure), then
modular convergence coincides with norm convergence.
Let W - 1LM (\Omega ) (resp., W - 1EM (\Omega )) denote the space of distributions on \Omega which can be
written as sums of derivatives of order \leq 1 of functions in LM (\Omega ) (resp., EM (\Omega )). It is a Banach
space under the usual quotient norm.
If the open set \Omega has the segment property, then the space \scrD (\Omega ) is dense in W 1
0LM (\Omega ) for the
modular convergence and for the topology \sigma (\Pi LM ,\Pi LM ) (cf. [18]). Consequently, the action of a
distribution in W - 1LM (\Omega ) on an element of W 1
0LM (\Omega ) is well defined.
For k > 0, we define the truncation at height k, TK : \BbbR \rightarrow \BbbR by
Tk(s) = \mathrm{m}\mathrm{i}\mathrm{n}(K,\mathrm{m}\mathrm{a}\mathrm{x}(s, - k)) =
\left\{
s, if | s| \leq k,
ks
| s|
, if | s| > k.
The following abstract lemmas will be applied to the truncation operators.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 513
Lemma 2.1 (cf. [18]). Let F : \BbbR \rightarrow \BbbR be uniformly Lipschitzian with F (0) = 0. Let M be an
N -function and let u \in W 1LM (\Omega ) (resp., W 1EM (\Omega )).
Then F (u) \in W 1LM (\Omega ) (resp., W 1EM (\Omega )). Moreover, if the set of discontinuity points D of
F \prime is finite, then
\partial
\partial xi
F (u) =
\left\{ F \prime (u)
\partial u
\partial xi
, a.e. in \{ x \in \Omega : u(x) /\in D\} ,
0, a.e. in \{ x \in \Omega : u(x) \in D\} .
Lemma 2.2 (cf. [18]). Let F : \BbbR \rightarrow \BbbR be uniformly Lipschitzian with F (0) = 0. We sup-
pose that the set of discontinuity points of F \prime is finite. Let M be an N-function, then the map-
ping F : W 1LM (\Omega ) \rightarrow W 1LM (\Omega ) is sequentially continuous with respect to the weak* topology
\sigma (\Pi LM ,\Pi EM ).
Lemma 2.3 (cf. [20]). Let uk, u \in LM (\Omega ). If uk \rightarrow u with respect to the modular convergence,
then uk \rightarrow u for \sigma (LM , LM ).
Below, we will use the following technical lemmas.
Lemma 2.4 (cf. [5]). Let (fn), f \in L1(\Omega ), such that
(i) fn \geq 0 a.e. in \Omega ,
(ii) fn \rightarrow f a.e. in \Omega ,
(iii)
\int
\Omega
fn(x) dx \rightarrow
\int
\Omega
f(x) dx.
Then fn \rightarrow f strongly in L1(\Omega ).
Lemma 2.5 (Young’s inequality). Let M be an N -function and \=M its conjugate. Then we have
st \leq M(s) + \=M for all s, t \geq 0.
We give now the following lemma which concerns operators of the Nemytskii type in Orlicz
spaces (see [5]).
Lemma 2.6. Let \Omega be an open subset of \BbbR N with finite measure. Let M, P, Q be N -functions
such that Q \ll P and let f : \Omega \times \BbbR \rightarrow \BbbR be a Carathéodory function such that, for a.e. x \in \Omega and
all s \in \BbbR ,
| f(x, s)| \leq c(x) + k1P
- 1M(k2| s| ),
where k1, k2 are real constants and c(x) \in EQ(\Omega ). Then the Nemytskii operator Nf defined by
Nf (u)(x) = f(x, u(x)) is strongly continuous from
\scrP
\biggl(
EM (\Omega ),
1
k2
\biggr)
=
\biggl\{
u \in LM (\Omega ) : d(u,EM (\Omega )) <
1
k2
\biggr\}
into EQ(\Omega ).
Lemma 2.7 [17]. Let \Omega be a bounded open subset of \BbbR N and M is an N -function, so there
exist two positive constants \delta and \lambda such that\int
\Omega
M(\delta | v| )dx \leq
\int
\Omega
\lambda M(| \nabla v| )dx for all v \in W 1
0LM (\Omega ).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
514 H. MOUSSA, M. RHOUDAF, H. SABIKI
3. Main results. Throughout this paper, we assume that the following assumptions hold true:
Let \Omega be an open bounded subset of \BbbR N , N \geq 2, with the segment property, M be an N -
function and P be an N -function such that P \ll M. We consider the Leray – Lions operator
Au = - \mathrm{d}\mathrm{i}\mathrm{v}(a(x, u,\nabla u)), (3.1)
defined on \scrD (A) \subset W 1
0LM (\Omega ) into W - 1LM (\Omega ) where a : \Omega \times \BbbR \times \BbbR N \rightarrow \BbbR N is a Carathéodory
function such that, for a.e. x \in \Omega , for all \zeta , \xi \in \BbbR N (\zeta \not = \xi ) and for all s \in \BbbR ,
| a(x, s, \zeta )| \leq c(x) + k1M
- 1
P (k2| s| ) + k3M
- 1
M(k4| \zeta | ), (3.2)
(a(x, s, \zeta ) - a(x, s, \xi ), \zeta - \xi ) > 0, (3.3)
a(x, s, \zeta )\zeta \geq \alpha M(| \zeta | ) (3.4)
with \alpha > 0 k1, k2, k3, k4 \geq 0, c \in EM (\Omega ).
Furthermore, let g(x, s, \xi ) be a Carathéodory function satisfying the following assumptions:
| g(x, s, \xi )| \leq b(| s| ) + h(s)M(| \xi | ) (3.5)
with h \in L1(\BbbR +) and b(| s| ) \leq P
- 1
P (| s| ), where b : \BbbR \rightarrow \BbbR is a continuous function and P \ll M.
Let us give and prove the following lemmas which will be needed later.
Lemma 3.1 (cf. [1]). Assume that the assumptions (3.2) – (3.4) hold and let (zn) be a sequence
in W 1
0LM (\Omega ) such that
zn \rightharpoonup z in W 1
0LM (\Omega ) for \sigma (\Pi LM (\Omega ),\Pi EM (\Omega )), (3.6)
(a(x, zn,\nabla zn))n is bounded in (LM (\Omega ))N , (3.7)\int
\Omega
\Bigl[
a(x, zn,\nabla zn) - a(x, zn,\nabla z\chi s)
\Bigr] \Bigl[
\nabla zn - \nabla z\chi s
\Bigr]
dx - \rightarrow 0 (3.8)
as n and s tend to +\infty , and where \chi s is the characteristic function of
\Omega s =
\Bigl\{
x \in \Omega ; | \nabla z| \leq s
\Bigr\}
.
Then
\nabla zn \rightarrow \nabla z a.e. in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
a(x, zn,\nabla zn)\nabla zn dx =
\int
\Omega
a(x, z,\nabla z)\nabla z dx ,
M(| \nabla zn| ) \rightarrow M(| \nabla z| ) in L1(\Omega ).
Remark 3.1. It should be interesting to note that the condition (3.7) is not necessary in the case
where the N -function M satisfies the \Delta 2-condition.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 515
Lemma 3.2. Let define the function \varphi as follows:
\varphi (t) = c\prime t \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right)
with \alpha , c\prime > 0, s, t \in \BbbR and h \in L1(\BbbR +), then
\varphi \prime (t) - h(| t| )
\alpha
| \varphi (t)| \geq c\prime .
Proof. If t \geq 0, then
\varphi \prime (t) = c\prime \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) + c\prime t
h(| t| )
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) =
= c\prime \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) +
h(| t| )
\alpha
| \varphi (t)| ,
and if t \leq 0, then
\varphi \prime (t) = c\prime \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) + c\prime t
h(| t| )
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) =
= c\prime \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) +
h(| t| )
\alpha
| \varphi (t)|
which implies that
\varphi \prime (t) - h(| t| )
\alpha
| \varphi (t)| = c\prime \mathrm{e}\mathrm{x}\mathrm{p}
\left( | t| \int
0
h(| s| )
\alpha
ds
\right) \geq c\prime .
Lemma 3.2 is proved.
Remark 3.2. If we have u \in W 1LM (\Omega ) by using Lemma 2.2, we can conclude that \varphi (u)
belongs to W 1LM (\Omega ).
Theorem 3.1. Assume that the assumptions (3.2) – (3.5) hold and let f belongs to W - 1EM (\Omega ).
Then there exists a measurable function u solution of the following problem:
(P )
\left\{
u \in W 1
0LM (\Omega ), g(x, u,\nabla u) \in L1(\Omega ),\int
\Omega
(a(x, u,\nabla u))\nabla Tk(u - v) dx+
\int
\Omega
g(x, u,\nabla u)Tk(u - v) dx \leq
\leq
\int
\Omega
fTk(u - v) dx
for every v \in W 1
0LM (\Omega ) \cap L\infty (\Omega ) and for every k > 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
516 H. MOUSSA, M. RHOUDAF, H. SABIKI
Proof. The proof is divided into several steps, first we introduce a sequence of approximate
problems and derive a priori estimates for the approximate problem and we show two intermediate
results, namely, the the almost everywhere convergente of \nabla un and the strong convergence in L1(\Omega )
of the nonlinearity gn(x, un,\nabla un).
Let us consider the sequence of approximate problem
(Pn)
\left\{
un \in W 1
0LM (\Omega ),\int
\Omega
(a(x, un,\nabla un))\nabla Tk(un - v) dx+
\int
\Omega
gn(x, un,\nabla un)Tk(un - v) dx \leq
\leq
\int
\Omega
fTk(un - v) dx
for every v \in W 1
0LM (\Omega ),
where
gn(x, s, \xi ) =
g(x, s, \xi )
1 +
1
n
| g(x, s, \xi )|
.
Note that gn(x, s, \xi ) satisfies the following conditions:
| gn(x, s, \xi )| \leq | g(x, s, \xi )| , | gn(x, s, \xi )| \leq n.
By the classical result of [19], the approximate problem (Pn) has at least one solution.
Lemma 3.3. Let un be a solution of the problem (Pn), then we have\int
\Omega
M(| \nabla un| ) dx \leq C,
where C is a positive constant not depending on n.
Proof. Let v = un - \varphi (un), where \varphi (t) = t \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl( \int | t|
0
h(| s| )
\alpha
ds
\Biggr)
(the function h appears in
(3.5)). Since v \in W 1
0LM (\Omega ), v is admissible test function in (Pn), then\int
\Omega
a(x, un,\nabla un)\nabla (\varphi (un)) dx+
\int
\Omega
gn(x, un,\nabla un)\varphi (un) dx \leq
\int
\Omega
f\varphi (un) dx,
by (3.5) we get\int
\Omega
a(x, un,\nabla un)\nabla un\varphi
\prime (un) dx \leq
\int
\Omega
b(| un| )| \varphi (un)| dx+
\int
\Omega
h(| un| )| \varphi (un)| M(| \nabla un| ) dx+
+
\int
\Omega
| f | | \varphi (un)| dx.
Since \varphi \prime \geq 0 and by (3.4), we obtain
\alpha
\int
\Omega
M(| \nabla un| )
\Bigl(
\varphi \prime (un) -
h(| un| )
\alpha
| \varphi (un)|
\Bigr)
dx \leq c0
\int
\Omega
P
- 1
P (| un| )| un| dx+ c0
\int
\Omega
| f | | un| dx
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 517
with c0 = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \int +\infty
0
h(| s| )
\alpha
ds
\biggr)
.
By Lemma 3.2, we deduce
\alpha
\int
\Omega
M(| \nabla un| ) dx \leq c0
\int
\Omega
P
- 1
P (| un| )| un| dx+ c0
\int
\Omega
| f | | un| dx. (3.9)
Since P \ll M, for all \varepsilon > 0, there exists a constant that K\varepsilon depending on \varepsilon such that
P (t) \leq M(\varepsilon t) +K\varepsilon \forall t \geq 0 (3.10)
without loss of generality, we can assume that \varepsilon =
\alpha \delta
4c0\lambda + \alpha (\delta + 1)
< 1, where \delta and \lambda are two
positive constants in Lemma 2.7 and c0 = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \int +\infty
0
h(| s| )
\alpha
ds
\biggr)
, so, by convexity, we have
P (t) \leq \varepsilon M(t) +K\varepsilon \forall t \geq 0. (3.11)
By Young inequality and in view of (3.10), we deduce
c0
\int
\Omega
P
- 1
P (| un| )| un| dx \leq 2c0
\int
\Omega
M(\varepsilon | un| ) dx+ C1
\varepsilon ,
by Lemma 2.7 and the fact that
\varepsilon
\delta
< 1, we get
c0
\int
\Omega
P
- 1
P (| un| )| un| dx \leq 2c0\varepsilon \lambda
\delta
\int
\Omega
M(| \nabla un| ) dx+ C1
\varepsilon . (3.12)
On the other hand, f can be written as f = f0 - \mathrm{d}\mathrm{i}\mathrm{v}F, where f0 \in EM (\Omega ), F \in (EM (\Omega ))N , using
Lemma 5.7 in [18] and Young’s inequality we obtain\int
\Omega
f0un dx \leq C1 +
\alpha
4
\int
\Omega
M(| \nabla un| ) dx,
\int
\Omega
F\nabla un dx \leq C2 +
\alpha
4
\int
\Omega
M(| \nabla un| ) dx.
(3.13)
Using (3.17) and (3.13) in (3.9), we get
\alpha
\int
\Omega
M(| \nabla un| ) dx \leq 2c0\varepsilon \lambda
\delta
\int
\Omega
M(| \nabla un| ) dx+
\alpha
2
\int
\Omega
M(| \nabla un| ) dx+ C\varepsilon ,
which implies \biggl(
\alpha
2
- 2c0\varepsilon \lambda
\delta
\biggr) \int
\Omega
M(| \nabla un| ) dx \leq C\varepsilon ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
518 H. MOUSSA, M. RHOUDAF, H. SABIKI
we can easily verify that
\Bigl( \alpha
2
- 2c0\varepsilon \lambda
\delta
\Bigr)
> 0, where \varepsilon =
\alpha \delta
4c0\lambda + \alpha (\delta + 1)
.
Thus, \int
\Omega
M(| \nabla un| ) dx \leq C, (3.14)
where C is a positive constant independent on n.
It follows that the sequence \{ un\} is bounded in W 1
0LM (\Omega ). Consequently, there exist a subse-
quence on \{ un\} , still denoted by \{ un\} and a function u \in W 1
0LM (\Omega ) such that
un \rightharpoonup u in W 1
0LM (\Omega ) for \sigma (\Pi LM ,\Pi EM ), (3.15)
un \rightarrow u in EM (\Omega ) strongly and a.e. in \Omega . (3.16)
Lemma 3.4. Let un be a solution of the approximate problem (Pn). Then\Bigl(
a(x, un,\nabla un)
\Bigr)
n
is bounded in (LM (\Omega ))N .
Proof. Let \varphi \in (EM (\Omega ))N be arbitrary. In view of the monotonicity of a, one easily has\int
\Omega
a(x, un,\nabla un)\varphi dx \leq
\int
\Omega
a(x, un,\nabla un)\nabla un dx+
+
\int
\Omega
a(x, un, \varphi )(\nabla un - \varphi ) dx.
First, let take v = un - une
G(un) with G(r) =
\int | r|
0
h(| s| )
\alpha
ds as a test function in (Pn), then\int
\Omega
a(x, un,\nabla un)\nabla (une
G(un)) dx+
\int
\Omega
gn(x, un,\nabla un)une
G(un) dx \leq
\int
\Omega
fune
G(un) dx,
by (3.5), we get\int
\Omega
a(x, un,\nabla un)\nabla une
G(un) dx+
\int
\Omega
a(x, un,\nabla un)\nabla un
h(| un| )
\alpha
eG(un)un dx \leq
\leq
\int
\Omega
b(| un| )uneG(un) dx+
\int
\Omega
h(| un| )uneG(un)M(| \nabla un| ) dx+
\int
\Omega
| f | | uneG(un)| dx.
By (3.4) and the fact that 1 \leq eG(un) \leq c0, we obtain\int
\Omega
a(x, un,\nabla un)\nabla un dx+
\int
\Omega
M(| \nabla un| )h(| un| )eG(un)un dx \leq
\leq c0
\int
\Omega
b(| un| )| un| dx+
\int
\Omega
M(| \nabla un| )h(| un| )uneG(un) dx+
\int
\Omega
| f | | un| dx,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 519
with c0 = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \int +\infty
0
h(| s| )
\alpha
ds
\biggr)
, which gives
\int
\Omega
a(x, un,\nabla un)\nabla un dx \leq c0
\int
\Omega
b(| un| )| un| dx+ c0
\int
\Omega
| f | | un| dx,
as in (3.9) and by (3.14), we have\int
\Omega
a(x, un,\nabla un)\nabla un dx \leq Ca.
Otherwise, for \lambda large enough, we get, by using (3.2) and convexity of M,
\int
\Omega
M
\biggl(
a(x, un, \varphi )
\lambda
\biggr)
dx \leq
\int
\Omega
M
\left( \beta
\Bigl[
c(x) + k1M
- 1
P (k2| un| ) +M
- 1
M(k3| \varphi | )
\Bigr]
\lambda
\right) dx \leq
\leq \beta
\lambda
\int
\Omega
M(c(x)) +
\beta k1
\lambda
\int
\Omega
MM
- 1
P (k2| un| ) dx+
\beta
\lambda
\int
\Omega
MM
- 1
M(k3| \varphi | ) dx \leq
\leq \beta
\lambda
\int
\Omega
M(c(x)) +
\beta k1
\lambda
\int
\Omega
P (k2| un| ) dx+
\beta
\lambda
\int
\Omega
M(k3| \varphi | ) dx.
We have \varphi \in (EM (\Omega ))N and c \in EM (\Omega ), then\int
\Omega
M
\biggl(
a(x, TK(un), w)
\lambda
\biggr)
dx \leq C1 + C2
\int
\Omega
P (k2| un| ) dx.
In view of (3.10), we can take \varepsilon small enough, the way that
\varepsilon k2
\delta
\leq 1, thus by Lemma 2.7 and the
convexity of M, we get
C2
\int
\Omega
P (k2| un| ) dx \leq C2k2\varepsilon \lambda
\delta
\int
\Omega
M(| \nabla un| ) dx+ C\varepsilon (\Omega ). (3.17)
By (3.9) and (3.17), we obtain \int
\Omega
M
\biggl(
a(x, TK(un), w)
\lambda
\biggr)
dx \leq Cb.
Hence, since un is bounded in W 1
0LM (\Omega ), one easily deduce that a(x, un,\nabla un) is a bounded
sequence in (LM (\Omega ))N . Thus, up to a subsequence
a(x, un,\nabla un) \rightharpoonup \varphi k in (LM (\Omega ))N for \sigma (\Pi LM ,\Pi EM )
for some \varphi k \in (LM (\Omega ))N .
Lemma 3.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
520 H. MOUSSA, M. RHOUDAF, H. SABIKI
Lemma 3.5. Let un be a solution of the problem (Pn), then we have
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\{ j\leq | un| \leq j+1\}
a(x, un,\nabla un)\nabla un dx = 0.
Proof. Consider the function v = un - eG(un)T1(un - Tj(un))
+ for j > 1, where G(un) =
=
\int | un|
0
h(| s| )
\alpha
ds. Then we obtain
\int
\Omega
a(x, un,\nabla un)\nabla un
h(| un| )
\alpha
eG(un)T1(un - Tj(un))
++
+
\int
\Omega
a(x, un,\nabla un)\nabla T1(un - Tj(un))
+eG(un)+
+
\int
\Omega
gn(x, un,\nabla un)e
G(un)T1(un - Tj(un))
+ \leq
\int
\Omega
feG(un)T1(un - Tj(un))
+ dx.
From the growth condition (3.5), we have\int
\Omega
a(x, un,\nabla un)\nabla un
h(| un| )
\alpha
eG(un)T1(un - Tj(un))
++
+
\int
\Omega
a(x, un,\nabla un)\nabla T1(un - Tj(un))
+eG(un) \leq
\leq
\int
\Omega
P
- 1
P (| un| )eG(un)T1(un - Tj(un))
++
+
\int
\Omega
h(| un| )M(| \nabla un| )eG(un)T1(un - Tj(un))
+ +
\int
\Omega
feG(un)T1(un - Tj(un))
+ dx.
Thanks to (3.4), we get \int
\Omega
a(x, un,\nabla un)\nabla T1(un - Tj(un))
+eG(un) \leq
\leq
\int
\{ un>j\}
eG(un)P
- 1
P (| un| )T1(un - Tj(un))
+ +
\int
\Omega
feG(un)T1(un - Tj(un))
+ dx.
By Young’s inequality and the fact that
1 \leq eG(un) = \mathrm{e}\mathrm{x}\mathrm{p}
\left( | un| \int
0
h(s)
\alpha
ds
\right) \leq \mathrm{e}\mathrm{x}\mathrm{p}
\left( +\infty \int
0
h(s)
\alpha
ds
\right) = c0,
we obtain
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 521\int
\Omega
a(x, un,\nabla un)\nabla T1(un - Tj(un))
+ \leq
\leq c0
\int
\{ un>j\}
(P (| un| ) + P (T1(un - Tj(un))
+)) + c0
\int
\Omega
fT1(un - Tj(un))
+ dx.
By the inequality (3.11), we get\int
\Omega
a(x, un,\nabla un)\nabla T1(un - Tj(un))
+ \leq
\leq c0\varepsilon
\int
\{ un>j\}
\Bigl(
M(| un| ) +M(T1(un - Tj(un))
+) + 2K\varepsilon
\Bigr)
+ c0
\int
\Omega
fT1(un - Tj(un))
+ dx. (3.18)
In view of (3.15) and (3.16), we have
T1(un - Tj(un))
+ \rightharpoonup T1(u - Tj(u))
+ in W 1
0LM (\Omega ) for \sigma (\Pi LM ,\Pi EM ).
In addition, since (un)n\in \BbbN is bounded in LM (\Omega ), we deduce by Lebesgue’s theorem that the right-
hand side of the last inequality goes to zero as n and j tend to infinity. Then (3.18) becomes
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\{ j\leq un\leq j+1\}
a(x, un,\nabla un)\nabla un dx = 0. (3.19)
Furthermore, consider the test function v = un + e - G(un)T1(un - Tj(un))
- in (Pn), and reasoning
as in the proof of (3.19), we deduce that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\{ - j - 1\leq un\leq - j\}
a(x, un,\nabla un)\nabla un dx = 0. (3.20)
Finally, combining (3.19) and (3.20), we have
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\{ j\leq | un| \leq j+1\}
a(x, un,\nabla un)\nabla un dx = 0. (3.21)
Lemma 3.5 is proved.
Proposition 3.1. Let un be a solution of the approximate problem (Pn). Then we have ( for a
subsequence noted again un)
\nabla un \rightarrow \nabla u a.e. in \Omega
as n tends to +\infty .
Proof. We will use the following function of one real variable, which is defined as follows:
Sj(s) =
\left\{
1, if | s| \leq j,
0, if | s| \geq j + 1,
j + 1 - s, if j \leq s \leq j + 1,
s+ j + 1, if - j - 1 \leq s \leq - j,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
522 H. MOUSSA, M. RHOUDAF, H. SABIKI
with j a nonnegative real parameter.
Let \Omega s = \{ x \in \Omega , | \nabla Tk(u(x))| \leq s\} and denote by \chi s the characteristic function of \Omega s, clearly,
\Omega s \subset \Omega s+1 and meas(\Omega \setminus \Omega s) \rightarrow 0 as s \rightarrow 0. Let vi \in \scrD (\Omega ) which converges to Tk(u) for the
modular convergence in W 1
0LM (\Omega ). Using v = un - \eta e - G(un)(Tk(un) - Tk(vi))
+Sj(un) as a test
function in (Pn), we obtain, by using (3.4) and (3.5),\int
\{ Tk(un) - Tk(vi)\geq 0\}
eG(un)a(x, un,\nabla un)\nabla (Tk(un) - Tk(vi))Sj(un) dx -
-
\int
\{ j\leq un\leq j+1\}
eG(un)a(x, un,\nabla un)\nabla un(Tk(un) - Tk(vi))
+ dx \leq
\leq
\int
\Omega
P
- 1
P (| un| )(Tk(un) - Tk(vi))
+Sj(un)e
G(un) dx+
+
\int
\Omega
f(Tk(un) - Tk(vi))
+Sj(un)e
G(un) dx.
Thanks to (3.21) the second integral tend to zero as n and j tend to infinity, and by Lebesgue
theorem, we deduce that the right-hand side converge to zero as n and j goes to infinity.
Using the same argument as in [2], we get\int
\Omega
[a(x, Tk(un),\nabla Tk(un)) - a(x, Tk(un),\nabla Tk(u)\chi s)][\nabla Tk(un) - \nabla Tk(u)\chi s] dx \rightarrow 0.
By the Lemma 3.1, we obtain
M(| \nabla un| ) - \rightarrow M(| \nabla u| ) in L1(\Omega ). (3.22)
Thanks to (3.22), we obtain for a subsequence
\nabla un - \rightarrow \nabla u a.e. in \Omega .
Proposition 3.1 is proved.
Proof of Theorem 3.1. Step 1. Equi-integrability of the nonlinearities.
We show that
gn(x, un,\nabla un) - \rightarrow g(x, u,\nabla u) strongly in L1(\Omega ). (3.23)
Let v = un + e( - G(un))
\int 0
un
h(s)\chi \{ s< - l\} ds. Since v \in W 1
0LM (\Omega ), v is an admissible test function
in (Pn). Then
\int
\Omega
a(x, un,\nabla un)\nabla
\left( - e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds
\right) dx+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 523
+
\int
\Omega
gn(x, un,\nabla un)
\left( - e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds
\right) dx \leq
\leq
\int
\Omega
f
\left( - e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds
\right) dx,
which implies that, by using (3.5), we have
\int
\Omega
a(x, un,\nabla un)\nabla un
h(un)
\alpha
e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds dx+
+
\int
\Omega
a(x, un,\nabla un)\nabla une
( - G(un))h(un)\chi \{ un< - l\} dx \leq
\leq
\int
\Omega
P
- 1
P (| un| )e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds dx+
+
\int
\Omega
h(un)M(| \nabla un| )e( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds dx -
\int
\Omega
fe( - G(un))
0\int
un
h(s)\chi \{ s< - l\} ds dx.
By using (3.4) and since
\int 0
un
h(s)\chi \{ s< - l\} ds \leq
\int - l
- \infty
h(s) ds, we get
\int
\Omega
a(x, un,\nabla un)\nabla une
( - G(un))h(un)\chi \{ un< - l\} dx \leq
\leq e
\biggl( \| h\|
L1(\BbbR )
\alpha
\biggr) - l\int
- \infty
h(s) ds
\left( \int
\Omega
P
- 1
P (| un| ) +
\int
\Omega
f0 dx
\right) .
Applying Young inequality, (3.10) and using Lemma 2.7, one has\int
\Omega
a(x, un,\nabla un)\nabla une
( - G(un))h(un)\chi \{ un< - l\} dx \leq
\leq e(
\| h\|
L1(\BbbR )
\alpha
)
- l\int
- \infty
h(s) ds
\left( \varepsilon \lambda
\delta
\int
\Omega
M(| \nabla un| ) dx+K \prime
\varepsilon +
\int
\Omega
f0 dx
\right) .
By using again (3.4) and the fact that un is bounded in W 1
0LM (\Omega ), we obtain
\int
\{ un< - l\}
h(un)M(| \nabla un| ) dx \leq c5
- l\int
- \infty
h(s) ds,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
524 H. MOUSSA, M. RHOUDAF, H. SABIKI
and, since h \in L1(\BbbR ), we deduce
\mathrm{l}\mathrm{i}\mathrm{m}
l\rightarrow +\infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\int
\{ un< - l\}
h(un)M(| \nabla un| ) dx = 0. (3.24)
Otherwise, considering v = un - e( - G(un))
\int un
0
h(s)\chi \{ s>l\} ds as a test function in (Pn). Thus,
similarly to (3.24), we deduce
\mathrm{l}\mathrm{i}\mathrm{m}
l\rightarrow +\infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\int
\{ un>l\}
h(un)M(| \nabla un| ) dx = 0. (3.25)
Moreover, on the set \{ un > l\} we have 1 <
un
l
, then\int
\{ un>l\}
P
- 1
P (| un| ) dx \leq 1
l
\int
\Omega
P
- 1
P (| un| )un dx.
Applying Young inequality, one has\int
\{ un>l\}
P
- 1
P (| un| ) dx \leq 1
l
\int
\Omega
(P (| un| ) + P (| un| )) dx.
In view of (3.10), we get \int
\{ un>l\}
P
- 1
P (| un| ) dx \leq 2
l
\int
\Omega
(M(\varepsilon | un| ) +K\varepsilon ) dx.
By using Lemma 2.7, we have\int
\{ un>l\}
P
- 1
P (| un| ) dx \leq 2\varepsilon \delta
l\lambda
\int
\Omega
\Bigl(
M(| \nabla un| ) +K\varepsilon
\Bigr)
dx.
By (3.14), we obtain \int
\{ un>l\}
P
- 1
P (| un| ) dx \leq 2\varepsilon \delta (C + C(\Omega ))
l\lambda
,
then we conclude that
\mathrm{l}\mathrm{i}\mathrm{m}
l\rightarrow +\infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\int
\{ un>l\}
P
- 1
P (| un| ) dx = 0. (3.26)
Combining (3.5), (3.22), (3.24), (3.25), (3.26) and Vitali’s theorem, we get (3.23).
Step 2. Passing to the limit.
Let v \in W 1
0LM (\Omega ) \cap L\infty (\Omega ), we take un - Tk(un - v) as test function in (Pn), we can write
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM . . . 525\int
\Omega
a(x, un,\nabla un)\nabla Tk(un - v) dx+
\int
\Omega
gn(x, un,\nabla un)Tk(un - v) dx \leq
\leq
\int
\Omega
fTk(un - v) dx
which implies that\int
\{ | un - v| \leq k\}
a(x, un,\nabla un)\nabla un dx+
\int
\{ | un - v| \leq k\}
a(x, Tk+\| v\| \infty un,\nabla Tk+\| v\| \infty (un))\nabla v dx+
+
\int
\Omega
gn(x, un,\nabla un)Tk(un - v) dx \leq
\int
\Omega
fTk(un - v) dx.
By Fatou’s lemma and the fact that
a(x, Tk+\| v\| \infty (un),\nabla Tk+\| v\| \infty (un)) \rightharpoonup a(x, Tk+\| v\| \infty (u),\nabla Tk+\| v\| \infty (u))
weakly in (LM (\Omega ))N for \sigma (\Pi LM ,\Pi EM ), one easily see that\int
\{ | u - v| \leq k\}
a(x, u,\nabla u)\nabla u dx -
\int
\{ | u - v| \leq k\}
a(x, Tk+\| v\| \infty (u),\nabla Tk+\| v\| \infty (u))\nabla v dx+
+
\int
\Omega
g(x, u,\nabla u)Tk(u - v) dx \leq
\int
\Omega
fTk(u - v) dx.
Hence, \int
\Omega
a(x, u,\nabla u)\nabla Tk(u - v) dx+
\int
\Omega
g(x, u,\nabla u)Tk(u - v) dx \leq
\leq
\int
\Omega
fTk(u - v) dx \forall v \in W 1
0LM (\Omega ) \cap L\infty (\Omega ) \forall k > 0.
Theorem 3.1 is proved.
References
1. L. Aharouch, E. Azroul, M. Rhoudaf, Existence of solutions for unilateral problems in L1 involving lower order
terms in divergence form in Orlicz spaces, J. Appl. Anal., 13, 151 – 181 (2007).
2. L. Aharoucha, A. Benkirane, M. Rhoudaf, Existence results for some unilateral problems without sign condition with
obstacle free in Orlicz spaces, Nonlinear Anal., 68, 2362 – 2380 (2008).
3. L. Aharouch, M. Rhoudaf, Existence of solution for unilateral problems with L1 data in Orlicz spaces, Proyecciones,
23, № 3, 293 – 317 (2004).
4. D. Apushkinskaya, M. Bildhauer, M. Fuchs, Steady states of anisotropic generalized Newtonian fluids, J. Math. Fluid
Mech., 7, 261 – 297 (2005).
5. A. Benkirane, A. Elmahi, Almost every where convergence of the gradients of solutions to elliptic equations in Orlicz
spaces and application, Nonlinear Anal., 11, № 28, 1769 – 1784 (1997).
6. A. Benkirane, A. Elmahi, A strongly nonlinear elliptic equation having natural grouth terms and L1 data, Nonlinear
Anal., 39, 403 – 411 (2000).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
526 H. MOUSSA, M. RHOUDAF, H. SABIKI
7. A. Benkirane, J. Benouna, M. Rhoudaf, Some remarks on a sign condition for perturbations of nonlinear problems,
Recent Developments in Nonlinear Analysis, World Sci. Publ., Hackensack, NJ (2010), p. 30 – 42.
8. A. Bensoussan, L. Boccardo, F. Murat, On a non linear partial differential equation having natural growth terms
and unbounded solution, Ann. Inst. Henri Poincaré, 5, № 4, 149 – 169 (1988).
9. M. Bildhauer, M. Fuchs, X. Zhong, On strong solutions of the differential equations modeling the steady flow of
certain incompressible generalized Newtonian fluids, St.Petersburg Math. J., 18, 183 – 199 (2007).
10. L. Boccardo, F. Murat, J. P. Puel, L\infty estimate for some nonlinear elliptic partial differential equations and application
to an existence result, SIAM J. Math. Anal., 2, 326 – 333 (1992).
11. L. Boccardo, F. Murat, J.P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann.
Mat. Pura ed Appl., 183 – 196 (1988).
12. H. Brézis, F. Browder, A property of Sobolev spaces, Commun. Part. Different. Equat., 4, 1077 – 1083 (1979).
13. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math.,
66, 1383 – 1406 (2006).
14. A. Elmahi, D. Meskine, Existence of solutions for elliptic equations having natural growth terms in Orlicz spaces,
Abstr. and Appl. Anal., 12, 1031 – 1045 (2004).
15. A. Elmahi, D. Meskine, Non-linear elliptic problems having natural growth and L1 data in Orlicz spaces, Ann. Mat.,
107 – 114 (2004).
16. A. Fqayeh, A. Benkirane, M. El Moumni et al., Existence of renormalized solutions for some strongly nonlinear
elliptic equations in Orlicz spaces, Georgian Math. J., 22, № 3, 305 – 321 (2015).
17. J.-P. Gossez, Nonlinear elliptic boundary value problems for equation with rapidly or slowly increasing coefficients,
Trans. Amer. Math. Soc., 190, 217 – 237 (1974).
18. J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients,
Trans. Amer. Math. Soc., 190, 163 – 205 (1974).
19. J.-P. Gossez, V. Mustonen, Variational inequalities in Orlicz spaces, Nonlinear Anal., 11, 379 – 492 (1987).
20. J.-P. Gossez, Some approximation properties in Orlicz – Sobolev, Stud. Math., 74, 17 – 24 (1982).
21. P. Gwiazda, P. Wittbold, A. Wróblewska-Kamińska, A. Zimmermann, Renormalized solutions to nonlinear parabolic
problems in generalized Musielak – Orlicz spaces, Nonlinear Anal., 129, 1 – 36 (2015).
22. P. Gwiazda, I. Skrzypczak, A. Zatorska-Goldstein, Existence of renormalized solutions to elliptic equation in Musi-
elak – Orlicz space, J. Different. Equat., 264, № 1, 341 – 377 (2018).
23. P. Gwiazda, M. Bulíc̃ek, J. Málek, A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressi-
ble fluids, SIAM J. Math. Anal., 44, № 4, 2756 – 2801 (2012).
24. S. Hadj Nassar, H. Moussa, M. Rhoudaf, Renormalized Solution for a nonlinear parabolic problems with noncoercivity
in divergence form in Orlicz spaces, Appl. Math. and Comput., 249, 253 – 264 (2014).
25. O. Kováčik, J. Rákosník, On spaces Lp(x) and W k,p(x) , J. Czechoslovak Math., 41, 592 – 618 (1991).
26. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris (1969).
27. H. Moussa, F. O. Gallego, M. Rhoudaf, Capacity solution to a coupled system of parabolic-elliptic equations in
Orlicz – Sobolev spaces, Nonlinear Different. Equat. and Appl. (2018).
28. J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math., 1034 (1983).
29. H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Ltd., Tokyo (1950).
30. P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine
Intell., 12, 629 – 639 (1990).
31. A. Porretta, Nonlinear equations with natural growth terms and measure data, Electron. J. Different. Equat., Conf.
09, 181 – 202 (2002).
32. K. R. Rajagopal, M. Ru̇žička, Mathematical modeling of electrorheological materials, Contin. Mech. and Thermodyn.,
13, 59 – 78 (2001).
33. M. Ru̇žička, Electrorheological fluids: modeling and mathematical theory, Lect. Notes Math., Springer, Berlin (2000).
34. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR, Izvestiya, 29,
№ 1, 33 – 66 (1987).
Received 07.01.17,
after revision — 10.12.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
|
| id | umjimathkievua-article-373 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:36Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/09/a1677851c7f37b0eb038fe23d24d6009.pdf |
| spelling | umjimathkievua-article-3732022-03-26T11:01:33Z Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. UDC 517.5 We deal with the existence result for nonlinear elliptic equations related to the form$$Au + g(x, u,\nabla u) = f,$$where the term $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ is a Leray–Lions operator from a subset of $W^{1}_{0}L_M(\Omega)$ into its dual. &nbsp;The growth and coercivity conditions on the monotone vector field $a$ are prescribed by an $N$-function $M$ which does not have to satisfy a $\Delta_2$-condition. Therefore we use Orlicz–Sobolev spaces which are not necessarily reflexive and assume that the nonlinearity $g(x,u,\nabla u)$ is a Carathéodory function satisfying only a growth condition with no sign condition. The right-hand side~$f$ belongs to $W^{-1}E_{\overline{M}}(\Omega).$ УДК 517.5 Розглядається задача існування для нелінійних еліптичних рівнянь у формі$$Au + g(x, u,\nabla u) = f,$$ де $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ — оператор Лере–Ліонса з підмножини $W^{1}_{0}L_M(\Omega)$ у її дуальну множину. &nbsp;Умови зростання та коерцитивності в монотонному векторному полі $a$ визначаються $N$-функцією $M,$ яка не повинна задовольняти $\Delta_2$-умови. Тому ми використовуємо простори Орліча–Соболєва, які не обов'язково є рефлексивними, і припускаємо, що нелінійність $g(x,u,\nabla u)$ є функцією Каратеодорі, що задовольняє лише умову зростання без умови знака. &nbsp;Права частина $f$ належить $W^{-1}E_{\overline{M}}(\Omega).$ Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/373 10.37863/umzh.v72i4.373 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 4 (2020); 509-526 Український математичний журнал; Том 72 № 4 (2020); 509-526 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/373/8704 |
| spellingShingle | Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. Moussa, H. Rhoudaf, M. Sabiki , H. Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_alt | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_full | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_fullStr | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_full_unstemmed | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_short | Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces |
| title_sort | existence results for a perturbed dirichlet problem without sign condition in orlicz spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/373 |
| work_keys_str_mv | AT moussah existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT rhoudafm existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT sabikih existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT moussah existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT rhoudafm existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT sabikih existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT moussah existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT rhoudafm existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces AT sabikih existenceresultsforaperturbeddirichletproblemwithoutsignconditioninorliczspaces |