Nonlinear elliptic equations with measure data in Orlicz spaces
UDC 517.5 In this article, we study the existence result of the unilateral problem\begin{gather*}Au-\mbox{div} (\Phi(x,u))+H(x,u,\nabla u)=\mu,\end{gather*}where $Au = -\mbox{div}(a(x,u,\nabla u))$ is a Leray–Lions operator defined on Sobolev–Orlicz space $D(A)\subset W_{0}^{1}L_{M}(\Omega),$ $\mu \...
Gespeichert in:
| Datum: | 2021 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1290 |
| 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_ | 1860507161921060864 |
|---|---|
| author | Aberqi, A. Bennouna , J. Elmassoudi, M. Aberqi, A. Bennouna , J. Elmassoudi, M. |
| author_facet | Aberqi, A. Bennouna , J. Elmassoudi, M. Aberqi, A. Bennouna , J. Elmassoudi, M. |
| author_sort | Aberqi, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:46:08Z |
| description | UDC 517.5
In this article, we study the existence result of the unilateral problem\begin{gather*}Au-\mbox{div} (\Phi(x,u))+H(x,u,\nabla u)=\mu,\end{gather*}where $Au = -\mbox{div}(a(x,u,\nabla u))$ is a Leray–Lions operator defined on Sobolev–Orlicz space $D(A)\subset W_{0}^{1}L_{M}(\Omega),$ $\mu \in L^{1}(\Omega)+W^{-1}E_{\overline{M}}(\Omega),$ where $M$ and $\overline{M}$ are two complementary $N$-functions, the first and the second lower terms $\Phi$ and $H$ satisfies only the growth condition and any sign condition is assumed and $u\geq \zeta,$ where $\zeta$ is a measurable function. |
| doi_str_mv | 10.37863/umzh.v73i12.1290 |
| first_indexed | 2026-03-24T02:04:55Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i12.1290
UDC 517.5
A. Aberqi, J. Bennouna, M. Elmassoudi (Sidi Mohammed Ben Abdellah Univ., Laboratory LAMA, Morocco)
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA
IN ORLICZ SPACES
НЕЛIНIЙНI ЕЛIПТИЧНI РIВНЯННЯ З ДАНИМИ МIРИ
У ПРОСТОРАХ ОРЛIЧА
In this article, we study the existence result of the unilateral problem
Au - div(\Phi (x, u)) +H(x, u,\nabla u) = \mu ,
where Au = - div(a(x, u,\nabla u)) is a Leray – Lions operator defined on Sobolev – Orlicz space D(A) \subset W 1
0LM (\Omega ),
\mu \in L1(\Omega )+W - 1EM (\Omega ), where M and M are two complementary N -functions, the first and the second lower terms \Phi
and H satisfies only the growth condition and any sign condition is assumed and u \geq \zeta , where \zeta is a measurable function.
Вивчено питання iснування для односторонньої задачi
Au - div(\Phi (x, u)) +H(x, u,\nabla u) = \mu ,
де Au = - div(a(x, u,\nabla u)) — оператор Лере – Лiонса, який визначено у просторi Соболєва – Орлiча D(A) \subset
\subset W 1
0LM (\Omega ), \mu \in L1(\Omega ) + W - 1EM (\Omega ), M i M — двi додатковi N -функцiї, перший i другий члени \Phi i H
задовольняють лише умову зростання та будь-яку умову знака, u \geq \zeta , \zeta — вимiрна функцiя.
1. Introduction. Let \Omega be a bounded open domain in IRN , N \geq 2, and consider the following
strongly nonlinear Dirichlet problem with an obstacle:
u \geq \zeta a.e. in \Omega ,
- div(a(x, u,\nabla u)) - div(\Phi (x, u)) +H(x, u,\nabla u) = \mu on \Omega , (1)
u = 0 on \partial \Omega ,
where \mu \in L1(\Omega ) +W - 1EM (\Omega ).
Under our assumptions, the problem (1) does not admit, in general, a weak solution since the
term \Phi may not belong to (L1(\Omega ))N and a lack of coercivity for the two terms \Phi and H. Thus to
overcome this difficulty, we use in this paper the framework of entropy solution, which need less
regularity than the usual weak solution. Knowing that the notion of entropy solutions have been
developed by P. Bénilan et al. [9] for the study of nonlinear elliptic problems.
In fact, in the classical Sobolev space W 1,p
0 (\Omega ), the paper [10] where \Phi = 0, H has polynomial
growth and \mu is a measure in \scrM b(\Omega ), L. Boccardo et al. proved the existence results and in [11]
have demonstrated the decomposition theorem of the Radon measure and studied the existence and
uniqueness of entropy solution. For more results we refer to [3, 4].
c\bigcirc A. ABERQI, J. BENNOUNA, M. ELMASSOUDI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12 1587
1588 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
To our knowledge, it is not yet possible to decompose the Radon measure in setting Orlicz space,
therefore the authors studied the problem (1) with the second member as the sum of an element
from W - 1EM (\Omega ) and of a function from L1(\Omega ). In the first J. P. Gossez and V. Mustonen in [13]
proved the existence of entropy solutions for nonlinear problem (1) where \mu \in W - 1EM (\Omega ) and H
satisfies sign condition. A. Benkirane and J. Bennouna in [7] proved the existence and uniqueness
of the solution of unilateral problem where \Phi = H = 0 and \mu \in L1(\Omega ), L. Ahrouch et al. in [5]
have proved the existence results where H = 0, \Phi \in \scrC 0(IRN , IRN ) and \mu \in L1(\Omega ) +W - 1EM (\Omega ).
Recently A. Aberqi et al. in [2] proved the existence and uniqueness results for problem (1) in the
parabolic case and L1-sources.
In this article, we are interested in proving the existence of entropy solution for unilateral problem
associated to (1) where \Phi depends on x, u and satisfies only the growth condition and H is a
nonlinear lower-order term having natural growth with respect to | \nabla u| . The second member of (1)
as \mu = f - div(F ) with f \in L1(\Omega ) and F \in (EM (\Omega ))N .
The main difficulties of this problem are in the first the lack of coercivity lower order term \Phi
that makes the operator that governs the equation, non coercive. The second lower order term H
is controlled by a non-polynomial growth (see (8)) and no sign condition is assumed. Finally, the
anisotropic function M defining Orlicz space W 1LM (\Omega ) does not satisfy the \Delta 2-condition.
We are not concerned here with the uniqueness of the solution. In fact, the uniqueness problem
being a rather delicate one, due to a counter-example of J. Serrin [15]. Note that our result generalizes
that of [5, 7, 10], to the case of Orlicz – Sobolev spaces.
This paper is organized as follows. Section 2 contains some preliminaries of Orlicz spaces and
a technical lemmas. Section 3 is devoted to the specification of the assumptions on a,\Phi ,K and \mu .
Main results are stated in Section 4, where we give and prove the principal theorem.
2. Orlicz spaces and technical lemmas. Let M : IR+ \rightarrow IR+ be an N -function, that is, M
is continuous, convex 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
a(s)ds, where a : IR+ \rightarrow IR+ is
nondecreasing, right continuous with a(0) = 0, a(t) > 0 for t > 0, and a(t) \rightarrow +\infty as t\rightarrow +\infty .
The N -function M conjugate to M is defined by M(t) =
\int t
0
a(s)ds, where a : IR+ \rightarrow IR+ is
given by a(t) = \mathrm{s}\mathrm{u}\mathrm{p}\{ s : a(s) \leq t\} . (See [1] for more details.) We will extend these N -functions
into even functions on all IR.
Example 1. For M(t) =
| t| p
p
, M(t) =
| t| q
q
, where
1
p
+
1
q
= 1 and p, q \in (1;+\infty ). For
M(t) = \mathrm{e}\mathrm{x}\mathrm{p}(t) - 1 - | t| , M(t) = (1 + | t| ) \mathrm{l}\mathrm{n} (1 + | t| ) - | t| .
Let P and Q be two N -functions. P \ll Q means that P grows essentially less rapidly than Q,
that is, for each \varepsilon > 0, \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow +\infty
P (t)
Q(\varepsilon t)
= 0.
Proposition 1. P \ll M if and only if, for all \varepsilon > 0, there exists a constant c\varepsilon such that
P (t) \leq M(\varepsilon t) + c\varepsilon for all t \geq 0. (2)
Proof. Let \varepsilon > 0, then, by the definition of P \ll M, there exists t\varepsilon > 0 such that, for all t > t\varepsilon ,
P (t) \leq M(\varepsilon t). On the other hand, for t \in [0, t\varepsilon ], by the continuity of P, there exists a constant C\varepsilon
such that P (t) \leq C\varepsilon , where C\varepsilon = \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,t\varepsilon ] P (t). So, from the above we have (2).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1589
The Orlicz class KM (\Omega ) (resp., the Orlicz space LM (\Omega )) is defined as the set of (equivalence
classes) 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) .
The set LM (\Omega ) is 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\}
and KM (\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 dual EM (\Omega ) can be identified with
LM (\Omega ) by means of the pairing
\int
\Omega
uvdx, and the dual norm of LM (\Omega ) is equivalent to \| u\| M,\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 )
\bigl(
resp., EM (\Omega )
\bigr)
.
It is a Banach space under the norm
\| u\| 1,M = \| u\| M,\Omega +
\sum
1\leq i\leq N
\bigm\| \bigm\| \bigm\| \bigm\| \partial u\partial xi
\bigm\| \bigm\| \bigm\| \bigm\|
M,\Omega
.
Thus, W 1LM (\Omega ) and W 1EM (\Omega ) can be identified with subspaces of 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 ).
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 (see [1]). We recall the following lemmas.
Lemma 1 [14]. For all u \in W 1
0LM (\Omega ) with \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s} (\Omega ) < +\infty one has\int
\Omega
M
\biggl(
| u|
\delta
\biggr)
dx \leq
\int
\Omega
M(| \nabla u| )dx, (3)
where \delta = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\Omega ) is the diameter of \Omega .
Lemma 2 [8]. Let \Omega be an open subset of IRN with finite measure. Let M,P and Q be N -
functions such that Q\ll P, and let f : \Omega \times IR\rightarrow IR be a Carathéodory function such that
| f(x, s)| \leq c(x) + k1P
- 1M(k2| s| ) a.e. x \in \Omega , for all s \in IR,
where k1, k2 are real constants and c(x) \in EQ(\Omega ).
Then the Nemytskii operator defined by Nf (u)(x) = f(x, u(x)) is strongly continuous from
P
\biggl(
EM (\Omega ),
1
k2
\biggr)
=
\biggl\{
u \in LM (\Omega ) : d(u,EM (\Omega )) <
1
k2
\biggr\}
into EQ(\Omega ).
Lemma 3 [14]. Let un and u belong to LM (\Omega ). If un \rightarrow u with respect to the modular
convergence, then un \rightarrow u for \sigma (\Pi LM ,\Pi LM ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1590 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
Lemma 4 [14]. Let \Omega has the segment property. Then, for each v \in W 1
0LM (\Omega ), there exists a
sequence vn \in \scrD (\Omega ) such that vn converges to v for the modular convergence in v \in W 1
0LM (\Omega ).
Furthermore, if v \in W 1
0LM (\Omega ) \cap L\infty (\Omega ), then
\| vn\| L\infty (\Omega ) \leq (N + 1)\| v\| L\infty (\Omega ).
3. Formulation of the problem. Let \Omega be an open bounded subset of IRN , N \geq 2, and let M
and P be two N -functions such that P \ll M.
a : \Omega \times IR \times IRN \rightarrow IRN is Carathéodory function such that, for a.e. x \in \Omega and for all s \in IR,
\xi , \xi \ast \in IRN , \xi \not = \xi \ast ,
| a(x, s, \xi )| \leq \beta (a0(x) +M
- 1
P (k1| s| )) +M
- 1
M(k2| \xi | )) (4)
with \beta , k1, k2 > 0 and a0(\cdot ) \in EM (\Omega ),
(a(x, s, \xi ) - a(x, s, \xi \ast ))(\xi - \xi \ast ) > 0, (5)
a(x, s, \xi ).\xi \geq \alpha M(| \xi | ) +M(| s| ). (6)
\Phi : \Omega \times IR\rightarrow IRN is a Carathéodory function such that
| \Phi (x, s)| \leq c(x)M
- 1
M(\alpha 0| s| ), (7)
where c(\cdot ) \in L\infty (\Omega ) such that \| c(\cdot )\| L\infty (\Omega ) <
\alpha
2
and 0 < \alpha 0 < \mathrm{m}\mathrm{i}\mathrm{n}
\biggl(
1,
1
\alpha
\biggr)
.
H : \Omega \times IR\times IRN \rightarrow IR is a Carathéodory function such that
| H(x, s, \xi )| \leq h(x) + \rho (s)M(| \xi | ), (8)
\rho : IR \rightarrow IR+ is a continuous positive function which belongs L1(IR) \cap L\infty (IR) and h belongs
to L1(\Omega ).
Let \mu \in L1(\Omega ) +W - 1EM (\Omega ) such that
\mu = f - div(F ) with f \in L1(\Omega ) and F \in (EM (\Omega ))N . (9)
Given a negative measurable obstacle function \zeta : \Omega \rightarrow IR,
K\zeta = \{ u \in W 1
0LM (\Omega ) : u \geq \zeta a.e. in \Omega \} , (10)
and we suppose that K\zeta \cap L\infty (\Omega ) \not = \varnothing .
Throughout the paper, Tk denotes the truncation at height k \geq 0:
Tk(r) = max ( - k;min (k, r)).
Definition 1. A measurable function u, defined on \Omega , is said an entropy solution of problem
(1), if it satisfies the following conditions:
u \in D(A) \cap W 1
0LM (\Omega ), u \geq \zeta ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1591\int
\Omega
a(x, u,\nabla u)\nabla Tk(u - v)dx+
\int
\Omega
\Phi (x, u)\nabla Tk(u - v)dx +
+
\int
\Omega
H(x, u,\nabla u)\nabla Tk(u - v)dx \leq
\int
\Omega
fTk(u - v)dx+
\int
\Omega
F\nabla Tk(u - v)dx
\forall v \in K\zeta \cap L\infty (\Omega ) \forall k > 0. (11)
4. Main results.
Theorem 1. Assume that (4) – (10) hold true. Then there exists at least one solution of the
unilateral problem (1) in the sense of the Definition 1.
Remark 1. 1. The condition (6) can be replaced by the weaker one
a(x, s, \xi )\xi \geq \alpha M(| \xi | ) +M(| s| ) - b(x),
where b(x) is in L1-function.
2. The results obtained in Theorem 1, remain true if we replace (7) by the general growth
condition
| \Phi (x, s)| \leq c(x)P
- 1
P (| s| ), (12)
where c(\cdot ) \in EP (\Omega ) and P \ll M.
3. For any s \in IR and \alpha \prime > 0, we have
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
-
\| \rho \| L1(IR)
\alpha \prime
\biggr)
\leq \mathrm{e}\mathrm{x}\mathrm{p} (\pm G(s)) \leq \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr)
, (13)
where G(s) =
\int s
0
\rho (r)
\alpha \prime dr.
Remark 2. 1. For the sake of the simplification, the explicit dependence on x of the functions
a,\Phi and H will be omited so that a(x, u,\nabla u) = a(u,\nabla u), \Phi (x, u) = \Phi (u) and H(x, u,\nabla u) =
= H(u,\nabla u).
2. We will denote by Ci with i = 1, 2, . . . any constant which depends on the various quantities
of the problem but not on n.
Proof of Theorem 1. Step 1: Approximate problem. For each n > 0, we define the approxima-
tions
an(x, s, \xi ) = a(Tn(s), \xi ), \Phi n(x, s) = \Phi (Tn(s)), Hn(x, s, \xi ) =
H(s, \xi )
1 +
1
n
| H(s, \xi )|
a.e. x \in \Omega ,
for all s \in IR and \xi \in IRN .
Let (fn)n be a sequence of a smooth functions such that fn \rightarrow f strongly in L1(\Omega ). Let us now
consider the approximate problem
un \in K\zeta \cap D(A),
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1592 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI\int
\Omega
an(un,\nabla un)\nabla (un - v)dx+
\int
\Omega
\Phi n(un)\nabla (un - v)dx +
+
\int
\Omega
Hn(un,\nabla un)(un - v)dx \leq
\int
\Omega
fn(un - v)dx+
\int
\Omega
F\nabla (un - v)dx \forall v \in K\zeta . (14)
Since Hn is bounded for any fixed n > 0, there exists at least one solution un \in W 1
0LM (\Omega ) of (14)
(see [13]).
Now, let show that un converges to a function u, where u is the solution of the unilateral
problem (1).
Step 2: A priori estimates.
Lemma 5. Let choose \{ un\} n be a solution of the approximate problem (14). Then, for all
k > 0, there exist two positive constants C1 and C2 such that\int
\Omega
M(| \nabla Tk(un)| )dx \leq kC1 + C2. (15)
Proof. Let v0 \in K\zeta \cap L\infty (\Omega )\cap W 1
0EM (\Omega ) and fix k > 0. Let un - \mathrm{e}\mathrm{x}\mathrm{p} (G(un))Tk(un - v0)+ as
a test function in problem (14), where G(s) =
\int s
0
\rho (r)
\alpha \prime dr, and \alpha \prime > 0 is a parameter to be specified
later. We get \int
\Omega
an(un,\nabla un)\nabla
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+
\bigr)
dx +
+
\int
\Omega
\Phi n(un)\nabla
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+
\bigr)
dx +
+
\int
\Omega
Hn(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+dx \leq
\leq k \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr)
\| fn\| L1(\Omega ) +
\int
\Omega
F\nabla
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+
\bigr)
dx. (16)
In the second term of the left-hand side of (16) we use (7), Lemma 1 and Young inequality to get\int
\Omega
\Phi n(un)\nabla (\mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+)dx \leq
\leq
\| c(\cdot )\| L\infty (\Omega )
\alpha \prime
\left[ \alpha 0
\int
\Omega
M(un)\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+dx +
+
\int
\Omega
M(\nabla un)\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+dx
\right] +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1593
+2\alpha 0\| c(\cdot )\| L\infty (\Omega )
\int
\{ 0\leq un - v0\leq k\}
M(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))dx +
+\| c(\cdot )\| L\infty (\Omega )
\int
\{ 0\leq un - v0\leq k\}
M(| \nabla Tk(un - v0)
+| ) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))dx+ c1.
For the third term of the left-hand side of (16), we have\int
\Omega
Hn(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+dx \leq k \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr) \int
\Omega
| h(x)| dx +
+
\int
\Omega
\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )Tk(un - v0)
+dx.
For the second term of the right-hand side of (16), we obtain\int
\Omega
F\nabla
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+
\bigr)
dx \leq k
\alpha \prime \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr)
\| \rho \| L\infty
\int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
dx +
+
\varepsilon 1
\alpha \prime
\int
\Omega
\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )Tk(un)+dx +
+2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr) \int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
dx+ \varepsilon 1
\int
\{ 0\leq un - v0\leq k\}
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )dx+ c2.
Finally, by using the above and (6) in (16), we get
1 - \alpha 0\| c(\cdot )\| L\infty (\Omega )
\alpha \prime
\int
\Omega
M(| un| )\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Tk(un - v0)
+dx +
+
\biggl[
\alpha
\alpha \prime -
\| c(\cdot )\| L\infty (\Omega )
\alpha \prime - 1 - \varepsilon 1
\alpha \prime
\biggr] \int
\Omega
\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )Tk(un - v0)
+dx +
+
\int
\Omega
a(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Tk(un - v0)
+)dx \leq
\leq 2\alpha 0\| c(\cdot )\| L\infty (\Omega )
\int
\{ 0\leq un - v0\leq k\}
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )dx +
+(\| c(\cdot )\| L\infty (\Omega ) + \varepsilon 1)
\int
\{ 0\leq un - v0\leq k\}
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )dx+ c3k + c4,
where c4 = c1 + c2.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1594 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
If we choose \alpha \prime and \varepsilon 1 such that \alpha \prime =
\alpha
2
, \varepsilon 1 <
\alpha
2
- \| c(\cdot )\| L\infty (\Omega ), by using that Tk(un - v0)+ =
= un - v0 for x \in \{ x \in \Omega : 0 \leq un - v0 \leq k\} , we obtain\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla undx \leq
\leq
\alpha 0(\varepsilon 1 + \| c(\cdot )\| L\infty (\Omega ))
\alpha
\left[ \int
\{ 0\leq un - v0\leq k\}
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| un| )dx
\right] +
+
\left[ (\varepsilon 1 + \| c(\cdot )\| L\infty (\Omega ))
\int
\{ 0\leq un - v0\leq k\}
\mathrm{e}\mathrm{x}\mathrm{p}(G(un))M(| \nabla un| )dx
\right] +
+
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla v0dx+ c3k + c4.
Since \alpha 0\alpha < 1, by using (6), we get\biggl[
1 -
\varepsilon 1 + \| c(\cdot )\| L\infty (\Omega )
\alpha
\biggr] \int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla undx \leq
\leq
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla v0dx+ c3k + c4.
By using (5), we get, for any \alpha 1 > 0 (\alpha 1 is a parameter to be specified later),
\alpha 1
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))
\nabla v0
\alpha 1
dx \leq
\leq \alpha 1
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla undx -
-
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla
\biggl(
un - \nabla v0
\alpha 1
\biggr)
dx.
Then
c\prime
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla undx \leq
\leq \alpha 1
\int
\{ 0\leq un - v0\leq k\}
\bigm| \bigm| \bigm| \bigm| an\biggl( un, \nabla v0\alpha 1
\biggr) \bigm| \bigm| \bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}(G(un))| \nabla un| dx +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1595
+\alpha 1
\int
\{ 0\leq un - v0\leq k\}
\bigm| \bigm| \bigm| \bigm| an\biggl( un, \nabla v0\alpha 1
\biggr) \bigm| \bigm| \bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}(G(un)) \bigm| \bigm| \bigm| \bigm| \nabla v0\alpha 1
\bigm| \bigm| \bigm| \bigm| dx. (17)
Taking c\prime =
\biggl[
1 -
\varepsilon 1 + \| c(\cdot )\| L\infty (\Omega )
\alpha
- \alpha 1
\biggr]
such that \alpha 1 < 1 -
\varepsilon 1 + \| c(\cdot )\| L\infty (\Omega )
\alpha
, using (2), (4) and
Young inequality, we have \bigm| \bigm| \bigm| \bigm| an\biggl( un, \nabla v0\alpha 1
\biggr) \bigm| \bigm| \bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}(G(un))| \nabla un| \leq
\leq \beta \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr) \biggl[
M(a(x)) +M
\biggl(
| un|
\delta
\biggr)
+ C \prime +M
\biggl(
k2
| \nabla v0|
\alpha 1
\biggr)
+ 3M(| \nabla un| )
\biggr]
(18)
and \bigm| \bigm| \bigm| \bigm| an\biggl( un, \nabla v0\alpha 1
\biggr) \bigm| \bigm| \bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}(G(un)) \bigm| \bigm| \bigm| \bigm| \nabla v0\alpha 1
\bigm| \bigm| \bigm| \bigm| \leq
\leq \beta \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr) \biggl[
M(a(x)) +M
\biggl(
| un|
\delta
\biggr)
+ C \prime +M
\biggl(
k2
| \nabla v0|
\alpha 1
\biggr)
+ 3M
\biggl(
| \nabla v0|
\alpha 1
\biggr) \biggr]
. (19)
Since
\nabla v0
\alpha 1
\in (EM (\Omega ))N , one pass to the integral in (18), (19), by using Lemma 1, (13) and (17),
we get
c\prime
\int
\{ 0\leq un - v0\leq k\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla undx \leq
\leq 4\alpha 1\beta \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(\Omega )
\alpha \prime
\biggr) \int
\{ 0\leq un - v0\leq k\}
M(| \nabla un| )dx+ c5k + c6.
Taking also \alpha 1 such that 4\alpha 1\beta \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(\Omega )
\alpha \prime
\biggr)
<
c\prime
2
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
-
\| \rho \| L1(\Omega )
\alpha \prime
\biggr)
and using (6), we obtain
\int
\{ 0\leq un - v0\leq k\}
M(| \nabla un| )dx \leq c7k + c8. (20)
Similarly, taking un + \mathrm{e}\mathrm{x}\mathrm{p} ( - G(un))Tk(un - v0)
- as a test function in problem (14), we get
c\prime
\int
\{ - k\leq un - v0\leq 0\}
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\nabla undx \leq
\leq c\prime
2
\int
\{ - k\leq un - v0\leq 0\}
\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))M(| \nabla un| )dx+ c9k + c10 (21)
and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1596 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI\int
\{ - k\leq un - v0\leq 0\}
\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))M(| \nabla un| )dx \leq c11k + c12.
Combining now (20) and (21), we deduce (15).
Since \{ x \in \Omega ; | un| \leq k\} \subset \{ x \in \Omega ; | un - v0| \leq k + | v0| \infty \} , we obtain\int
\Omega
M(| \nabla Tk(un)| )dx =
\int
\{ | un| \leq k\}
M(| \nabla Tk(un)| )dx \leq
\leq
\int
\{ | un - v0| \leq k+\| v0\| \infty \}
M(| \nabla Tk(un)| )dx \leq kC1 + C2.
We conclude that \{ Tk(un)\} n is bounded in W 1
0LM (\Omega ) independently of n and for any k > 0, so
there exists a subsequence still denoted by un such that
Tk(un)\rightharpoonup \xi k weakly in W 1
0LM (\Omega ). (22)
On the other hand, by using Lemma 1, we have
M
\biggl(
k
\delta
\biggr)
meas\{ | un| > k\} \leq
\int
\{ | un| >k\}
M
\biggl(
| Tk(un)|
\delta
\biggr)
dx \leq
\leq
\int
\Omega
M(| \nabla Tk(un)| )dx \leq kC1 + C2.
Then
meas \{ | un| > k\} \leq kC3 + C4
M
\biggl(
k
\delta
\biggr) for all n and k.
Thus, we get
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
meas \{ | un| > k\} = 0.
Step 3: Now we turn to prove the almost every convergence of \{ un\} n and the convergence of
an(Tk(un),\nabla Tk(un)).
Proposition 2. Let un be a solution of the approximate problem (14), then
un \rightarrow u a.e. in \Omega , (23)
an(Tk(un),\nabla Tk(un))\rightharpoonup \varpi k in (LM (\Omega ))N for \sigma (\Pi LM ,\Pi EM ), (24)
for some \varpi k \in (LM (\Omega ))N .
Proof. Proof of (23). Let \eta > 0, \varepsilon > 0 and k > 0, then
meas\{ | un - um| > \eta \} \leq meas\{ | un| > k\} + meas\{ | um| > k\} +
+meas\{ | Tk(un) - Tk(um)| > \eta \} ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1597
by (22), we can assume that (Tk(un))n is a Cauchy sequence in measure in \Omega . Thus, there exists
k(\varepsilon ) > 0 such that meas\{ | Tk(un) - Tk(um)| > \eta \} < \varepsilon for all n,m > n0. Hence, \{ un\} n is a Cauchy
sequence in measure in \Omega and then converges almost everywhere to some measurable function u.
Proof of (24). We shall prove that \{ a(Tk(un),\nabla Tk(un))\} n is bounded in
\bigl(
LM (\Omega )
\bigr) N
for all
k > 0.
Let w \in (EM (\Omega ))N be arbitrary. By (5) we have
(a(un,\nabla un) - a(un, w))(\nabla un - w) > 0.
Then \int
\{ | un| \leq k\}
a(un,\nabla un)wdx \leq
\int
\{ | un| \leq k\}
a(un,\nabla un)\nabla undx+
\int
\{ | un| \leq k\}
a(un, w)(w - \nabla un)dx.
By using (4), the convexity of M and the definition of Tk, we get
\int
\{ | un| \leq k\}
M
\left( a
\biggl(
un,
w
k2
\biggr)
3\beta
\right) dx \leq \beta
3\nu
\int
\Omega
\Bigl[
M(a0(x)) + P (k1| Tk(un)| ) +M(| w| )
\Bigr]
dx \leq
\leq \beta
3\nu
\int
\Omega
\Bigl[
M(a0(x)) + P (k1k)dx+M(| w| )
\Bigr]
dx for \nu > \beta .
Thus,
\biggl\{
a
\biggl(
Tk(un),
w
k2
\biggr) \biggr\}
n
is bounded in (LM (\Omega ))N . By (15), (21) and by Banach – Steinhaus the-
orem, the sequence \{ a(Tk(un),\nabla Tk(un))\} n remains bounded in (LM (\Omega ))N and we conclude (24).
Step 4: Almost everywhere convergence of the gradients. To have that the gradient converges
almost everywhere, we need to prove this proposition.
Proposition 3. Let \{ un\} n be a solution of the approximate problem (14), then
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\int
\{ m\leq | un| \leq m+1\}
a(un,\nabla un)\nabla undx = 0 (25)
and, for a subsequence as n\rightarrow \infty ,
\nabla un \rightarrow \nabla u a.e. in \Omega .
Proof. Choosing, in the equation (14), the test function Zm(un) = un+\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- ,
where \psi m(un) = T1(un - Tm(un)), we get\int
\Omega
an(un,\nabla un)\nabla (\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- )dx +
+
\int
\Omega
\Phi n(un)\nabla (\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- )dx +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1598 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
+
\int
\Omega
Hn(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- dx \geq
\geq
\int
\Omega
fn \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- dx+
\int
\Omega
F\nabla (\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- )dx. (26)
In the second term of the left-hand side of (26), we use (3), (7) and Young inequality to get\int
\Omega
\Phi n(un)\nabla (\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- )dx \leq
\leq
\| c(\cdot )\| L\infty (\Omega )
\alpha \prime \alpha 0
\int
\Omega
M(| un| )\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- dx +
+
\| c(\cdot )\| L\infty (\Omega )
\alpha \prime
\int
\Omega
M(| \nabla un| ) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- dx +
+\alpha 0\| c(\cdot )\| L\infty (\Omega )
\int
\{ - (m+1)\leq un\leq - m\}
M(| un| ) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))dx +
+\| c(\cdot )\| L\infty (\Omega )
\int
\Omega
M(| \nabla \psi m(un)
- | ) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))dx.
For the second term of the right-hand side of (26), we have\int
\Omega
F\nabla (\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))\psi m(un)
- )dx \leq \| \rho \| L\infty
\alpha \prime \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr) \int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
\psi m(un)
- dx +
+
\varepsilon 1
\alpha \prime
\int
\Omega
\rho (un) \mathrm{e}\mathrm{x}\mathrm{p}( - G(un))M(| un| )\psi m(un)
- dx +
+\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr) \int
\{ - (m+1)\leq un\leq - m\}
M
\biggl(
| F |
\varepsilon 1
\biggr)
dx +
+\varepsilon 1
\int
\Omega
\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))M(| \nabla \psi m(un)
- | )dx.
By using the same argument is step 2, we obtain\int
\Omega
an(un,\nabla un)\nabla (\psi m(un)
- )dx \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1599
\leq \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr)
C
\left( \int
\Omega
fnZm(un)dx+
\int
\Omega
h(x)Zm(un)dx
\right) +
+C
\| \rho \| L1
\alpha \prime \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr) \left[ \int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
Zm(un)dx+
\int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
dx
\right] ,
where
1
C
=
\biggl[
1 -
\| c(x)\| L\infty (\Omega ) + \varepsilon 1
\alpha
\biggr]
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- \| \rho \| L1
\alpha \prime
\biggr)
.
Passing to limit as n \rightarrow +\infty , since the pointwise convergence of un and strongly convergence
in L1(\Omega ) of fn, we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\Omega
an(un,\nabla un)\nabla (\psi m(un)
- )dx \leq
\leq \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\| \rho \| L1
\alpha \prime
\biggr)
C
\left[ \int
\Omega
fZm(u) dx+
\int
\Omega
h(x)Zm(u)dx +
+
\| \rho \| L\infty
\alpha \prime
\int
\Omega
M
\biggl(
| F |
\varepsilon 1
\biggr)
Zm(u)dx+
\int
\{ - (m+1)\leq u\leq - m\}
M
\biggl(
| F |
\varepsilon 1
\biggr)
dx
\right] .
By using Lebesgue’s theorem and passing to limit as m \rightarrow +\infty , in the all term of the right-hand
side, we get
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\{ - (m+1)\leq un\leq - m\}
an(un,\nabla un)\nabla undx = 0.
In the same way, we take Zm(un) = un - \mathrm{e}\mathrm{x}\mathrm{p} (G(un))\psi m(un)
+ and choosing in approximation
equation (14), the test function Zm(un), we also obtain
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\{ m\leq un\leq m+1\}
an(un,\nabla n)\nabla un = 0.
On the above we get (25).
For the almost everywhere convergence of the gradient (see Appendix).
Step 5: Compactness of the nonlinearities. We shall prove that Hn(un,\nabla un) \rightarrow H(u,\nabla u)
strongly in L1(\Omega ).
Let \chi be the characteristic function and consider g0(un) =
\int un
0
\rho (s)\chi \{ s>h\} ds. Choosing un +
+ \mathrm{e}\mathrm{x}\mathrm{p}(G(un))g0(un) in the equation (14), we get, after using the same technique in step 2,\int
\{ un>h\}
an(un,\nabla un)\rho (un)\chi \{ un>h\} \nabla undx \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1600 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
\leq C
\left( +\infty \int
h
\rho (s)dx
\right) \left[ \| f\| L1(\Omega ) + \| h(x)\| L1(\Omega ) +
\| \rho \| L\infty (IR)
\alpha \prime
\int
\Omega
M
\biggl(
F
\varepsilon 1
\biggr)
dx
\right] +
+\mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr)
\| \rho \| L1(IR)
\int
\{ un>h\}
M
\biggl(
F
\varepsilon 1
\biggr)
dx,
where
1
C
=
\biggl[
1 -
\| c(x)\| L\infty (\Omega ) + \varepsilon 1
\alpha
\biggr]
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- 2\| \rho \| L1
\alpha \prime
\biggr)
.
Since \rho \in L1(IR) and by (6), we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in IN
\int
\{ un>h\}
\rho (un)M(\nabla un)dx = 0.
Similarly, let g0(un) =
\int 0
un
\rho (s)\chi \{ s< - h\} dx and choosing in (14) the test function un +
+\mathrm{e}\mathrm{x}\mathrm{p}( - G(un))g0(un), we have also
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in IN
\int
\{ un< - h\}
\rho (un)M(\nabla un)dx = 0.
We conclude that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in IN
\int
\{ | un| >h\}
\rho (un)M(\nabla un)dx = 0.
Let D \subset \Omega , then \int
D
\rho (un)M(\nabla un)dx \leq \mathrm{m}\mathrm{a}\mathrm{x}
\{ | un| \leq h\}
(\rho (x))
\int
D\cap \{ | un| \leq h\}
M(\nabla un)dx +
+
\int
D\cap \{ | un| >h\}
\rho (un)M(\nabla un)dx.
Consequently, \rho (un)M(\nabla un) is equiintegrable. Then \rho (un)M(\nabla un) converges to \rho (u)M(\nabla u)
strongly in L1(IR). Hence, by (8), we get our result.
Step 6: We show that u satisfies (11). Let v \in K\zeta \cap L\infty (\Omega ), then by Lemma 4 there exists
vj \in \scrD (\Omega ) such that vj \rightarrow v in for the modular convergence in W 1
0LM (\Omega ) , with \| vj\| L\infty (\Omega ) \leq
\leq (N + 1)\| v\| L\infty (\Omega ) and we can take vj \in K\zeta .
By choosing, in the approximate equation (14), the test function Tk(un - vj), we get\int
\Omega
an(un,\nabla un)\nabla Tk(un - vj)dx+
\int
\Omega
\Phi n(un)\nabla Tk(un - vj)dx +
+
\int
\Omega
Hn(un,\nabla un)Tk(un - vj)dx =
\int
\Omega
fnTk(un - vj)dx -
\int
\Omega
F\nabla Tk(un - vj)dx. (27)
We pass to the limit in (27), as n\rightarrow +\infty and j \rightarrow +\infty :
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1601
We can follow same way as in [7] to prove that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\int
\Omega
a(un,\nabla un)\nabla Tk(un - vj)dx \geq
\int
\Omega
a(u,\nabla u)\nabla Tk(u - v)dx.
For n \geq K, where K = k + (N + 1)\| v\| L\infty (\Omega ), we have
\Phi n(un)\nabla Tk(un - vj) = \Phi (TK(un))\nabla Tk(un - vj).
The pointwise convergence of un to u as n \rightarrow +\infty and (12), gives \Phi (TK(un))\nabla Tk(un - vj) \rightharpoonup
\rightharpoonup \Phi (TK(u))\nabla Tk(u - vj) weakly for \sigma (\Pi LM ,\Pi LM ). In a similar way, we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\int
\Omega
\Phi (TK(u))\nabla Tk(u - vj)dx =
\int
\Omega
\Phi (TK(u))\nabla Tk(u - v)dx =
=
\int
\Omega
\Phi (u)\nabla Tk(u - v)dx.
Limit of Hn(un,\nabla un)Tk(un - vj): Since Hn(un,\nabla un) converges strongly to H(u,\nabla u) in
L1(\Omega ) and the pointwise convergence of un to u as n \rightarrow +\infty , it is possible to prove that
Hn(un,\nabla un)Tk(un - vj) converges to H(u,\nabla u)Tk(u - vj) in L1(\Omega ) and
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\int
\Omega
H(u,\nabla u)Tk(u - vj)dx =
\int
\Omega
H(u,\nabla u)Tk(u - v)dx.
Since fn converges strongly to f in L1(\Omega ) and Tk(un - vj) \rightarrow Tk(u - vj) *-weakly in L\infty (\Omega ),
we have
\int
\Omega
fnTk(un - vj)dx \rightarrow
\int
\Omega
fTk(u - vj)dx as n \rightarrow \infty and also
\int
\Omega
fTk(u - vj)dx \rightarrow
\rightarrow
\int
\Omega
fTk(u - v)dx as j \rightarrow \infty , then it easy to get
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
F\nabla Tk(un - vj)dx =
\int
\Omega
F\nabla Tk(u - v)dx.
Theorem 1 is proved.
Example 2. As an examples of equations to which the present result entropy solutions can be
applied, we give
1) for M(t) =
1
p
| u| p, a(x, u,\nabla u) = | \nabla u| p - 2\nabla u, \Phi (x, u) = c(x)| \alpha 0u|
p
q , c(\cdot ) \in L\infty (\Omega ) and
F \in (EM (\Omega ))N ,
- div(| \nabla u| p - 2\nabla u) - div(\Phi (x, u)) = f - div(F ) in \Omega ,
u = 0 on \partial \Omega ;
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1602 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
2) for M(t) = t \mathrm{l}\mathrm{o}\mathrm{g}(1 + t) and a(x, u,\nabla u) = (1 + | u| 2)\nabla u \mathrm{l}\mathrm{o}\mathrm{g}(1 + | \nabla u| )
| \nabla u|
, c(\cdot ) \in L\infty (\Omega ) and
F = 0,
- div(a(x, u,\nabla u)) - div
\biggl(
c(x) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\eta
\| x\| + 1
\biggr)
M
- 1
M(\alpha 0| u| )
\biggr)
= f in \Omega ,
u = 0 on \partial \Omega .
Appendix. First call the following lemma.
Lemma 6 [6]. Under assumptions (4) – (9), and let (zn) be a sequence in W 1
0LM (\Omega ) such that
zn \rightharpoonup z for \sigma (\Pi LM ,\Pi EM ),
\{ a(zn,\nabla zn)\} n is bounded in (LM (\Omega ))N ,\int
\Omega
[a(zn,\nabla zn) - a(zn,\nabla z\chi s)][\nabla zn - \nabla z\chi s]dx\rightarrow 0
as n and s tend to +\infty , and where \chi s is the characteristic function of \Omega s = \{ x \in \Omega ; | \nabla z| \leq s\} .
Then
\nabla zn \rightarrow \nabla z a.e. in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\Omega
a(zn,\nabla zn)\nabla zndx =
\int
\Omega
a(z,\nabla z)\nabla zdx,
M(| \nabla zn| ) \rightarrow M(| \nabla z| ) in L1(\Omega ).
Now, we show that \nabla un \rightarrow \nabla u a.e. in \Omega , where un is the solution of the approximate prob-
lem (14).
Indeed, we introduce a sequence of increasing \bfC 1(IR)-functions Sm such that, for any m \geq 1,
Sm(r) = 1 for | r| \leq m,
Sm(r) = m+ 1 - | r| for m \leq | r| \leq m+ 1,
Sm(r) = 0 for | r| \geq m+ 1,
and we denote by \varepsilon (n, \eta , j,m) the quantities (possibly different) such that
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
\eta \rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
\mu \rightarrow +\infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\varepsilon (n, \eta , j,m) = 0.
Let \upsilon j \in \scrD (\Omega ) be a sequence such that \upsilon j \rightarrow u in W 1
0LM (\Omega ) for the modular convergence. For
fixed k \geq 0, let Wn,j
\eta = T\eta (Tk(un) - Tk(\upsilon j))
+ and W j
\eta = T\eta (Tk(u) - Tk(\upsilon j))
+.
Choosing in the approximating equation the test function \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un) and using (6)
and (8), we obtain
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1603\int
\Omega
an(un,\nabla un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta )Sm(un)dx +
+
\int
\Omega
an(un,\nabla un)\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un) dx -
-
\int
\Omega
\Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta )Sm(un) dx -
-
\int
\Omega
\Phi n(un)\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un)dx \leq
\leq
\int
\Omega
fn \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un) dx+
\int
\Omega
h(x) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un)dx +
+
\int
\Omega
F \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta )Sm(un) dx+
\int
\Omega
F\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un)dx. (28)
Now we pass to the limit in (28) for k real number fixed.
In order to perform this task we prove below the following results for any fixed k \geq 0:
\int
\Omega
\Phi n(un)Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta ) dx = \varepsilon (n, j) for any m \geq 1, (29)
\int
\Omega
\Phi n(un)\nabla unS\prime
m(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta dx = \varepsilon (n, j) for any m \geq 1, (30)
\int
\Omega
an(un,\nabla un)\nabla unS\prime
m(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta dx \leq \varepsilon (n,m), (31)
\int
\Omega
an(un,\nabla un)Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta ) dx \leq C\eta + \varepsilon (n, j,m), (32)
\int
\Omega
fnSm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta dx+
\int
\Omega
h(x) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un) dx \leq C\eta + \varepsilon (n, \eta ), (33)
\int
\Omega
F \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta )Sm(un) dx+
\int
\Omega
F\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un)dx \leq
\leq \varepsilon (n,m, j, \eta ), (34)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1604 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI\int
\Omega
[a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u))] [\nabla Tk(un) - \nabla Tk(u)] dx\rightarrow 0. (35)
Proof of (29). If we take n > m+ 1, we get
\Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) = \Phi (Tm+1(un)) \mathrm{e}\mathrm{x}\mathrm{p}(G(Tm+1(un)))Sm(Tm+1(un)).
Then \Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) is bounded in LM (Q), thus, by using the pointwise convergence
of un and Lebesgue’s theorem, we obtain
\Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) \rightarrow \Phi (u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))Sm(u) as n\rightarrow +\infty
with the modular convergence.
Then \Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) \rightarrow \Phi (u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))Sm(u) for \sigma (\Pi LM ,\Pi LM ).
In the other hand, \nabla Wn,j
\eta = \nabla Tk(un) - \nabla (Tk(\upsilon j)), for | Tk(un) - (Tk(\upsilon j)| \leq \eta , converges to
\nabla Tk(u) - \nabla (Tk(\upsilon j)) weakly in (LM (\Omega ))N , then\int
\Omega
\Phi n(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un)\nabla Wn,j
\eta dx\rightarrow
\int
\Omega
\Phi (u)Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))\nabla W j
\eta dx
as n\rightarrow +\infty .
By using the modular convergence of W j
\eta as j \rightarrow +\infty and letting \mu tends to infinity, we get (29).
Proof of (30). For n > m + 1 > k , we have \nabla unS\prime
m(un) = \nabla Tm+1(un) a.e. in \Omega . By the
almost every where convergence of un, we have \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta \rightarrow \mathrm{e}\mathrm{x}\mathrm{p}(G(u))W j
\eta in L\infty (\Omega )
weak-* and since the sequence (\Phi n(Tm+1(un)))n converges strongly in EM (\Omega ), then
\Phi n(Tm+1(un)) \mathrm{e}\mathrm{x}\mathrm{p}(G(un)) W
n,j
\eta \rightarrow \Phi (Tm+1(u)) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))W j
\eta
converges strongly in EM (\Omega ) as n \rightarrow +\infty . By virtue of \nabla Tm+1(un) \rightarrow \nabla Tm+1(u) weakly in
(LM (\Omega ))N , we have \int
\{ m\leq | un| \leq m+1\}
\Phi n(Tm+1(un))\nabla unS\prime
m(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta dx\rightarrow
\rightarrow
\int
\{ m\leq | u| \leq m+1\}
\Phi (u)\nabla u \mathrm{e}\mathrm{x}\mathrm{p}(G(u))W j
\eta dx as n\rightarrow +\infty .
With the modular convergence of W j
\eta as j \rightarrow +\infty and letting \mu \rightarrow +\infty , we get (30).
Proof of (31). For (31), we have\int
\Omega
an(un,\nabla un)\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un) dx =
=
\int
\{ m\leq | un| \leq m+1\}
an(un,\nabla un)\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un) dx \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1605
\leq \eta C
\int
\{ m\leq | un| \leq m+1\}
an(un,\nabla un)\nabla un dx.
By using (25), we get\int
\Omega
an(un,\nabla un)\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un) dx \leq \varepsilon (n,m).
Proof of (33). Since Sm(r) \leq 1 and Wn,j
\eta \leq \eta , we get\int
\Omega
fnSm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta dx \leq \varepsilon (n, \eta )
and \int
\Omega
h(x) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un)dx \leq C\eta .
Proof of (34). We obtain\int
\Omega
F \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla (Wn,j
\eta )Sm(un) dx+
\int
\Omega
F\nabla un \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Wn,j
\eta S\prime
m(un)dx = KF,1 +KF,2.
For the first integral, we have
KF,1 \leq \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr) \int
\Omega
F \nabla Wn,j
\eta dx \leq \varepsilon (\eta ).
Since Tk(un) and Tk(vj) converges weakly in W 1
0LM (\Omega ), we deduce
KF,1 \leq \varepsilon (n, j, \eta ).
For the second integral, we know that \nabla unS\prime
m(un) = \nabla Tm+1(un) and using (6), we get
KF,2 \leq \mathrm{e}\mathrm{x}\mathrm{p}
\biggl( \| \rho \| L1(IR)
\alpha \prime
\biggr) \left[ \varepsilon 1 \int
\Omega
M
\biggl(
F
\varepsilon 1
\biggr)
Wn,j
\eta dx+ \varepsilon 1\eta
\int
m\leq | un| \leq m+1
an(un,\nabla un)\nabla undx
\right] \leq
\leq \varepsilon (n,m, j, \eta ).
Proof of (32). We obtain\int
\Omega
an(un,\nabla un)Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla Wn,j
\eta dx =
=
\int
\{ | un| \leq k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
an(Tk(un),\nabla Tk(un))Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\times
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1606 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
\times (\nabla Tk(un) - \nabla Tk(\upsilon j)) dx -
-
\int
\{ | un| >k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
an(un,\nabla un)\nabla Tk(\upsilon j) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) dx. (36)
Since an(Tk+\eta (un),\nabla Tk+\eta (un)) is bounded in (LM (\Omega ))N , there exist some \varpi k+\eta \in (LM (\Omega ))N
such that an(Tk+\eta (un),\nabla Tk+\eta (un)) \rightarrow \varpi k+\eta weakly in (LM (\Omega ))N . Consequently,\int
\{ | un| >k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
an(un,\nabla un)\nabla Tk(\upsilon j) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))Sm(un) dx =
=
\int
\{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
\varpi k+\eta .\nabla Tk(\upsilon j)Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u)) dx+ \varepsilon (n), (37)
where we have used the fact that
Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))\nabla Tk(\upsilon j)\chi \{ | un| >k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} \rightarrow
\rightarrow Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))\nabla Tk(\upsilon j)\chi \{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
strongly in (EM (\Omega ))N .
Letting j \rightarrow +\infty , we obtain\int
\{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))\varpi k+\eta .\nabla Tk(\upsilon j) dx =
=
\int
\{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(u)\leq \eta \}
Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))\varpi k+\eta .\nabla Tk(u) dx+ \varepsilon (n, j).
One easily has, \int
\{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(u)\leq \eta \}
Sm(u) \mathrm{e}\mathrm{x}\mathrm{p}(G(u))\varpi k+\eta .\nabla Tk(u) dx = \varepsilon (n, j, \mu ).
By (28) – (33), (36) and (37), we have\int
\{ | un| \leq k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
Sm(un) \mathrm{e}\mathrm{x}\mathrm{p}(G(un))an(Tk(un),\nabla Tk(un))\times
\times (\nabla Tk(un) - \nabla Tk(\upsilon j)) dx \leq
\leq C\eta + \varepsilon (n, j, \mu ,m).
We know that \mathrm{e}\mathrm{x}\mathrm{p}(G(un)) \geq 1 and Sm(un) = 1 for | un| \leq k, then
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1607\int
\{ | un| \leq k\} \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
an(x, Tk(un),\nabla Tk(un))(\nabla Tk(un) - \nabla Tk(\upsilon j)) dx \leq
\leq C\eta + \varepsilon (n, j, \mu ,m). (38)
Proof of (35). Setting for s > 0, \Omega s = \{ x \in \Omega : | \nabla Tk(u)| \leq s\} and \Omega s
j = \{ x \in \Omega :
| \nabla Tk(\upsilon j)| \leq s\} and denoting by \chi s and \chi s
j the characteristic functions of \Omega s and \Omega s
j , respectively,
we deduce that letting 0 < \delta < 1, define
\Theta n,k = (a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))(\nabla Tk(un) - \nabla Tk(u)).
For s > 0, we have
0 \leq
\int
\Omega s
\Theta \delta
n,k dx =
\int
\Omega s
\Theta \delta
n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx+
\int
\Omega s
\Theta \delta
n,k\chi \{ Tk(un) - Tk(\upsilon j)>\eta \} dx.
The first term of the right-hand side with the Hölder inequality
\int
\Omega s
\Theta \delta
n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx \leq
\left( \int
\Omega s
\Theta n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx
\right) \delta \left( \int
\Omega s
dx
\right) 1 - \delta
\leq
\leq C1
\left( \int
\Omega s
\Theta n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx
\right) \delta
.
Also, by using the Hölder inequality and second term of the right-hand side, we have
\int
\Omega s
\Theta \delta
n,k\chi \{ Tk(un) - Tk(\upsilon j)>\eta \} dx \leq
\left( \int
\Omega s
\Theta n,k dx
\right) \delta
\left( \int
\{ Tk(un) - Tk(\upsilon j)>\eta \}
dx
\right)
1 - \delta
.
Since \{ a(Tk(un),\nabla Tk(un))\} n is bounded in (LM (\Omega ))N , while \{ \nabla Tk(un)\} n is bounded in
(LM (\Omega ))N , then\int
\Omega s
\Theta \delta
n,k\chi \{ Tk(un) - Tk(\upsilon j)>\eta \} dx \leq C2 (meas\{ x \in \Omega : Tk(un) - Tk(\upsilon j) > \eta \} )1 - \delta .
We obtain
\int
\Omega s
\Theta \delta
n,k dx \leq C1
\left( \int
\Omega s
\Theta n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx
\right) \delta
+
+ C2 (meas\{ x \in \Omega : Tk(un) - Tk(\upsilon j) > \eta \} )1 - \delta .
On the other hand,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1608 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI\int
\Omega s
\Theta n,k\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} dx \leq
\leq
\int
\{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)\chi s))\times
\times (\nabla Tk(un) - \nabla Tk(u)\chi s) dx.
For each s > r, r > 0, one has
0 \leq
\int
\Omega r\cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))\times
\times (\nabla Tk(un) - \nabla Tk(u)) dx \leq
\leq
\int
\Omega s\cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))\times
\times (\nabla Tk(un) - \nabla Tk(u)) dx =
=
\int
\Omega s\cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)\chi s))\times
\times (\nabla Tk(un) - \nabla Tk(u)\chi s) dx \leq
\leq
\int
\Omega \cap \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)\chi s))\times
\times (\nabla Tk(un) - \nabla Tk(u)\chi s) dx =
=
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(\upsilon j)\chi s
j))\times
\times (\nabla Tk(un) - \nabla Tk(\upsilon j)\chi s
j) dx +
+
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(un))(\nabla Tk(\upsilon j)\chi s
j - \nabla Tk(u)\chi s) dx +
+
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
(a(Tk(un),\nabla Tk(\upsilon j)\chi s
j) - a(Tk(un),\nabla Tk(u)\chi s))\nabla Tk(un) dx -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1609
-
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(\upsilon j)\chi s
j)\nabla Tk(\upsilon j)\chi s
j dx +
+
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(u)\chi s)\nabla Tk(u)\chi s dx =
= I1(n, j, s) + I2(n, j) + I3(n, j) + I4(n, j) + I5(n).
We will go to the limit as n, j, \mu and s\rightarrow +\infty :
I1 =
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(un))(\nabla Tk(un) - \nabla Tk(\upsilon j)) dx -
-
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(un))(\nabla Tk(\upsilon j)\chi s
j - \nabla Tk(\upsilon j)) dx -
-
\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(\upsilon j)\chi s
j)(\nabla Tk(un) - \nabla Tk(\upsilon j)\chi s
j) dx.
Using (38) and the first term of the right-hand side, we get\int
\{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \}
a(Tk(un),\nabla Tk(un))(\nabla Tk(un) - \nabla Tk(\upsilon j)) dx \leq
\leq C\eta + \varepsilon (n,m, j, s) -
\int
\{ | u| >k\} \cap \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
a(Tk(u), 0)\nabla Tk(\upsilon j) dx \leq
\leq C\eta + \varepsilon (n,m, j, \mu ).
The second term of the right-hand side tends to\int
\{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
\varpi k(\nabla Tk(\upsilon j)\chi s
j - \nabla Tk(\upsilon j)) dx.
Since \{ a(Tk(un),\nabla Tk(un))\} n is bounded in (LM (\Omega ))N , there exists \varpi k \in (LM (\Omega ))N such that
(for a subsequence still denoted by un)
a(Tk(un),\nabla Tk(un)) \rightarrow \varpi k in (LM (\Omega ))N for \sigma (\Pi LM ,\Pi EM ).
In view of the fact that
(\nabla Tk(\upsilon j)\chi s
j - \nabla Tk(\upsilon j))\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} \rightarrow
\rightarrow (\nabla Tk(\upsilon j)\chi s
j - \nabla Tk(\upsilon j))\chi \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
1610 A. ABERQI, J. BENNOUNA, M. ELMASSOUDI
strongly in (EM (\Omega ))N as n\rightarrow +\infty . The third term of the right-hand side tends to\int
\{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
a(Tk(u),\nabla Tk(\upsilon j)\chi s
j)(\nabla Tk(u) - \nabla Tk(\upsilon j)\chi s
j) dx.
Since
a(Tk(un),\nabla Tk(\upsilon j)\chi s
j)\chi \{ 0\leq Tk(un) - Tk(\upsilon j)\leq \eta \} \rightarrow a(Tk(u),\nabla Tk(\upsilon j)\chi s
j)\chi \{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
in (EM (\Omega ))N while (\nabla Tk(un) - \nabla Tk(\upsilon j)\chi s
j) \rightharpoonup (\nabla Tk(u) - \nabla Tk(\upsilon j)\chi s
j) in (LM (\Omega ))N for
\sigma (\Pi LM ,\Pi EM ). Passing to limit as j \rightarrow +\infty and \mu \rightarrow +\infty and using Lebesgue’s theorem, we
have
I2 = \varepsilon (n, j).
Similar ways as above give
I3 = \varepsilon (n, j),
I4 =
\int
\{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
a(Tk(u),\nabla Tk(u))\nabla Tk(u) dx+ \varepsilon (n, j, \mu , s,m),
I5 =
\int
\{ 0\leq Tk(u) - Tk(\upsilon j)\leq \eta \}
a(Tk(u),\nabla Tk(u))\nabla Tk(u) dx+ \varepsilon (n, j, \mu , s,m).
Finally, we obtain \int
\Omega s
\Theta n,k dx dt \leq C1 (C\eta + \varepsilon (n, \mu , \eta ,m))\delta + C2(\varepsilon (n))
1 - \delta ,
which yields, by passing to the limit supremum over n, j, \mu , s and \eta ,\int
\{ T\eta (Tk(un) - Tk(\upsilon j))\geq 0\} \cap \Omega r
\Bigl[
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))\times
\times (\nabla Tk(un) - \nabla Tk(u))
\Bigr] \delta
dx = \varepsilon (n). (39)
On the other hand, taking the functions Wn,j
\eta = T\eta (Tk(un) - Tk(\upsilon j))
- and W j
\eta = T\eta (Tk(u) -
- Tk(\upsilon j))
- . Choosing in the approximate equation the test function \mathrm{e}\mathrm{x}\mathrm{p}(G(un))W
n,j
\eta Sm(un), we
obtain \int
\{ T\eta (Tk(un) - Tk(\upsilon j))\leq 0\} \cap \Omega r
\Bigl[
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))\times
\times (\nabla Tk(un) - \nabla Tk(u))
\Bigr] \delta
dx = \varepsilon (n). (40)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES 1611
By (39) and (40), we get\int
\Omega r
\Bigl[
(a(Tk(un),\nabla Tk(un)) - a(Tk(un),\nabla Tk(u)))(\nabla Tk(un) - \nabla Tk(u))
\Bigr] \delta
dx = \varepsilon (n).
Thus, passing to a subsequence if necessary, \nabla un \rightarrow \nabla u a.e. in \Omega r, and since r is arbitrary,
\nabla un \rightarrow \nabla u a.e. in \Omega .
References
1. R. A. Adams, Sobolev spaces, Acad. Press, New York (1975).
2. A. Aberqi, J. Bennouna, M. Elmassoudi, M. Hammoumi, Existence and uniqueness of a renormalized solution of
parabolic problems in Orlicz spaces, Monatsh. Math. (2019); DOI: 10.1007/s00605-018-01260-8.
3. A. Aberqi, J. Bennouna, M. Mekkour, H. Redwane, Nonlinear parabolic inequality with lower order terms, Appl.
Anal. (2016); DOI: 10.1080/0036811.2016.1205186.
4. A. Aberqi, J. Bennouna, H. Redwane, Nonlinear parabolic problems with lower order terms and measure data, Thai
J. Math., 14, № 1, 115 – 130 (2016).
5. L. Aharouch, E. Azroul, M. Rhoudaf, Nonlinear unilateral problems in Orlicz spaces, Appl. Math., 33, № 2, 217 – 241
(2006).
6. E. Azroul, H. Redwane, M. Rhoudaf, Existence of a renormalized solution for a class of nonlinear parabolic equations
in Orlicz spaces, Port. Math., 66, № 1, 29 – 63 (2009).
7. A. Benkirane, J. Bennouna, Existence and uniqueness of solution of unilateral problems with L1 -data in Orlicz
spaces, Abstr. and Appl. Anal., 7, № 2, 85 – 102 (2002).
8. A. Benkirane, A. Elmahi, Almost everywhere convergence of the gradients of solutions to elliptic equations problems
in Orlicz spaces and application, Nonlinear Anal., 28, 1769 – 1784 (1997).
9. P. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J. L. Vazquez, An L1 -theory of existence and uniqueness
of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Super. Pisa, 22, 241 – 273 (1995).
10. L. Boccardo, T. Gallouet, L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J.
Funct. Anal., 87, 149 – 169 (1989).
11. L. Boccardo, T. Gallouet, L. Orsina, Existence and nonexistence of entropy solutions for nonlinear elliptic equations
with measure data, Ann. Inst. Henri Poincaré C, 13, № 5, 539 – 551 (1996).
12. M. Elmassoudi, A. Aberqi, J. Bennouna, Nonlinear parabolic problem with lower order terms in Musielack – Orlicz
spaces, ASTES J., 2, № 5, 109 – 123 (2017).
13. J. P. Gossez, V. Mustonen, Variational inequalities in Orlicz – Sobolev spaces, Nonlinear Anal., 11, 379 – 492 (1987).
14. J. P. Gossez, Some approximation properties in Orlicz – Sobolev spaces, Stud. Math., 74, № 1, 17 – 24 (1982).
15. J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Super. Pisa, 18, 189 – 258
(1964).
Received 04.12.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 12
|
| id | umjimathkievua-article-1290 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:55Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7b/780688104476199bb6feb4e92003987b.pdf |
| spelling | umjimathkievua-article-12902025-03-31T08:46:08Z Nonlinear elliptic equations with measure data in Orlicz spaces Nonlinear elliptic equations with measure data in Orlicz spaces Aberqi, A. Bennouna , J. Elmassoudi, M. Aberqi, A. Bennouna , J. Elmassoudi, M. Nonlinear elliptic problem Unilateral problem Weak solution Orlicz spaces Measure data UDC 517.5 In this article, we study the existence result of the unilateral problem\begin{gather*}Au-\mbox{div} (\Phi(x,u))+H(x,u,\nabla u)=\mu,\end{gather*}where $Au = -\mbox{div}(a(x,u,\nabla u))$ is a Leray–Lions operator defined on Sobolev–Orlicz space $D(A)\subset W_{0}^{1}L_{M}(\Omega),$ $\mu \in L^{1}(\Omega)+W^{-1}E_{\overline{M}}(\Omega),$ where $M$ and $\overline{M}$ are two complementary $N$-functions, the first and the second lower terms $\Phi$ and $H$ satisfies only the growth condition and any sign condition is assumed and $u\geq \zeta,$ where $\zeta$ is a measurable function. UDC 517.5 Нелінійні еліптичні рівняння з даними міри у просторах Орліча Вивчено питання існування для односторонньої задачі \begin{gather*}Au-\mbox{div} (\Phi(x,u))+H(x,u,\nabla u)=\mu,\end{gather*}де $Au = -\mbox{div}(a(x,u,\nabla u))$ – оператор Лере–Ліонса, який визначено у просторі Соболєва–Орліча $D(A)\subset W_{0}^{1}L_{M}(\Omega),$ $\mu \in L^{1}(\Omega)+W^{-1}E_{\overline{M}}(\Omega),$ $M$ і $\overline{M}$ – дві додаткові $N$-функції, перший і другий члени $\Phi$ і $H$ задовольняють лише умову зростання та будь-яку умову знака, $u\geq \zeta,$ $\zeta$ – вимірна функція. Institute of Mathematics, NAS of Ukraine 2021-12-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1290 10.37863/umzh.v73i12.1290 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 12 (2021); 1587 - 1611 Український математичний журнал; Том 73 № 12 (2021); 1587 - 1611 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1290/9157 Copyright (c) 2021 Mhamed Elmassoudi |
| spellingShingle | Aberqi, A. Bennouna , J. Elmassoudi, M. Aberqi, A. Bennouna , J. Elmassoudi, M. Nonlinear elliptic equations with measure data in Orlicz spaces |
| title | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_alt | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_full | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_fullStr | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_full_unstemmed | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_short | Nonlinear elliptic equations with measure data in Orlicz spaces |
| title_sort | nonlinear elliptic equations with measure data in orlicz spaces |
| topic_facet | Nonlinear elliptic problem Unilateral problem Weak solution Orlicz spaces Measure data |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1290 |
| work_keys_str_mv | AT aberqia nonlinearellipticequationswithmeasuredatainorliczspaces AT bennounaj nonlinearellipticequationswithmeasuredatainorliczspaces AT elmassoudim nonlinearellipticequationswithmeasuredatainorliczspaces AT aberqia nonlinearellipticequationswithmeasuredatainorliczspaces AT bennounaj nonlinearellipticequationswithmeasuredatainorliczspaces AT elmassoudim nonlinearellipticequationswithmeasuredatainorliczspaces |