Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2013, Article ID 476781, 14 pages
http://dx.doi.org/10.1155/2013/476781
Research Article
Entropy Solutions for Nonlinear Elliptic Anisotropic
Homogeneous Neumann Problem
B. K. Bonzi, S. Ouaro, and F. D. Y. Zongo
Laboratoire d’Analyse Mathématique des Equations (LAME), UFR, Sciences Exactes et Appliquées, Université de Ouagadougou,
03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
Correspondence should be addressed to S. Ouaro; ouaro@yahoo.fr
Received 22 June 2012; Accepted 1 February 2013
Academic Editor: Elena Braverman
Copyright © 2013 B. K. Bonzi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous
boundary value condition for 𝐿1 -data. We prove first, by using minimization techniques, the existence and uniqueness of weak
solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy
solution.
1. Introduction
Let Ω be an open bounded domain of R𝑁 (𝑁 ≥ 3) with
smooth boundary. Our aim is to prove the existence and
uniqueness of entropy solution for the anisotropic nonlinear
elliptic problem of the form
𝑁
𝜕
𝜕
𝑎𝑖 (𝑥,
𝑢) + |𝑢|𝑝𝑀 (𝑥)−2 𝑢 = 𝑓
𝜕𝑥𝑖
𝑖=1 𝜕𝑥𝑖
−∑
𝑁
𝜕
𝑢) 𝜈𝑖 = 0
∑𝑎𝑖 (𝑥,
𝜕𝑥
𝑖
𝑖=1
in Ω,
(1)
𝐴 𝑖 (𝑥, 0) = 0,
on 𝜕Ω,
where the right-hand side 𝑓 ∈ 𝐿1 (Ω) and 𝜈𝑖 , 𝑖 ∈ {1, . . . , 𝑁}
are the components of the outer normal unit vector.
For the rest of the functions involved in (1), we are going
to enumerate their properties after we make some notations.
For any Ω ⊂ R𝑁, we set
𝐶+ (Ω) = {ℎ ∈ 𝐶 (Ω) : inf ℎ (𝑥) > 1} ,
𝑥∈Ω
(2)
and we denote
ℎ+ = sup ℎ (𝑥) ,
𝑥∈Ω
ℎ− = inf ℎ (𝑥) .
𝑥∈Ω
Ω, we put 𝑝𝑀(𝑥) = max{𝑝1 (𝑥), . . . , 𝑝𝑁(𝑥)} and 𝑝𝑚 (𝑥) =
min{𝑝1 (𝑥), . . . , 𝑝𝑁(𝑥)}. Now, we can give the properties of the
rest of the functions involved in (1).
We assume that for 𝑖 = 1, . . . , 𝑁, the function 𝑎𝑖 :
Ω × R → R is Carathéodory and satisfies the following
conditions: 𝑎𝑖 (𝑥, 𝜉) is the continuous derivative with respect
to 𝜉 of the mapping 𝐴 𝑖 : Ω × R → R, 𝐴 𝑖 = 𝐴 𝑖 (𝑥, 𝜉), that is,
𝑎𝑖 (𝑥, 𝜉) = (𝜕/𝜕𝜉)𝐴 𝑖 (𝑥, 𝜉) such that the following equality and
inequalities holds
(3)
⃗ : Ω → R𝑁, 𝑝(⋅)
⃗ = (𝑝1 (⋅), . . . , 𝑝𝑁(⋅))
For the exponents, 𝑝(⋅)
with 𝑝𝑖 ∈ 𝐶+ (Ω) for every 𝑖 ∈ {1, . . . , 𝑁} and for all 𝑥 ∈
(4)
for almost every 𝑥 ∈ Ω.
There exists a positive constant 𝐶1 such that
󵄨󵄨
󵄨
󵄨 󵄨𝑝 (𝑥)−1
),
󵄨󵄨𝑎𝑖 (𝑥, 𝜉)󵄨󵄨󵄨 ≤ 𝐶1 (𝑗𝑖 (𝑥) + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝑖
(5)
for almost every 𝑥 ∈ Ω and for every 𝜉 ∈ R, where 𝑗𝑖 is a
󸀠
nonnegative function in 𝐿𝑝𝑖 (⋅) (Ω), with 1/𝑝𝑖 (𝑥) + 1/𝑝𝑖󸀠 (𝑥) = 1.
There exists a positive constant 𝐶2 such that
(𝑎𝑖 (𝑥, 𝜉) − 𝑎𝑖 (𝑥, 𝜂)) ⋅ (𝜉 − 𝜂)
󵄨𝑝 (𝑥)
󵄨
𝐶 󵄨󵄨𝜉 − 𝜂󵄨󵄨󵄨 𝑖
≥ { 2 󵄨󵄨
󵄨𝑝−
𝐶2 󵄨󵄨󵄨𝜉 − 𝜂󵄨󵄨󵄨 𝑖
󵄨
󵄨
if 󵄨󵄨󵄨𝜉 − 𝜂󵄨󵄨󵄨 ≥ 1,
󵄨
󵄨
if 󵄨󵄨󵄨𝜉 − 𝜂󵄨󵄨󵄨 < 1,
(6)
2
International Journal of Differential Equations
for almost every 𝑥 ∈ Ω and for every 𝜉, 𝜂 ∈ R, with 𝜉 ≠ 𝜂 and
󵄨󵄨 󵄨󵄨𝑝𝑖 (𝑥)
≤ 𝑎𝑖 (𝑥, 𝜉) ⋅ 𝜉 ≤ 𝑝𝑖 (𝑥) 𝐴 𝑖 (𝑥, 𝜉) ,
󵄨󵄨𝜉󵄨󵄨
(7)
for almost every 𝑥 ∈ Ω and for every 𝜉 ∈ R.
We also assume that the variable exponents 𝑝𝑖 (⋅) : Ω →
[2, 𝑁) are continuous functions for all 𝑖 = 1, . . . , 𝑁 such that
𝑝 (𝑁 − 1)
𝑝 (𝑁 − 1)
< 𝑝𝑖− <
,
𝑁−𝑝
𝑁 (𝑝 − 1)
𝑁
1
− > 1,
𝑝
𝑖=1 𝑖
∑
𝑝𝑖+ − 𝑝𝑖− − 1
𝑝−𝑁
<
,
𝑝𝑖−
𝑝 (𝑁 − 1)
(8)
−
where 1/𝑝 = (1/𝑁) ∑𝑁
𝑖=1 (1/𝑝𝑖 ).
We introduce the numbers
𝑞=
𝑁 (𝑝 − 1)
,
𝑁−1
𝑞∗ =
𝑁 (𝑝 − 1)
𝑁𝑞
=
.
𝑁−𝑞
𝑁−𝑝
(9)
A prototype example, that is, covered by our assumptions is
the following anisotropic equation:
Set 𝐴 𝑖 (𝑥, 𝜉) = (1/𝑝𝑖 (𝑥))|𝜉|𝑝𝑖 (𝑥) , 𝑎𝑖 (𝑥, 𝜉) = |𝜉|𝑝𝑖 (𝑥)−2 𝜉 where
𝑝𝑖 (𝑥) ≥ 2. Then, we get the following equation.
𝜕 󵄨󵄨󵄨󵄨 𝜕 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−2 𝜕
𝑢󵄨󵄨
𝑢) + |𝑢|𝑝𝑀 (𝑥)−2 = 𝑓.
(󵄨󵄨
󵄨
󵄨
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝑖 󵄨
𝑖 󵄨
𝑖
𝑖=1
𝑁
−∑
(10)
Actually, one of the topics from the field of PDEs that continuously gained interest is the one concerning the Sobolev space
1,𝑝(⋅)
with variable exponents, 𝑊1,𝑝(⋅) (Ω) or 𝑊0 (Ω) depending
on the boundary condition (see [1–23]). In that context,
problems involving the 𝑝(⋅)-Laplace operator
𝑝(𝑥)−2
Δ 𝑝(𝑥) 𝑢 = div (|∇𝑢|
∇𝑢)
(11)
or the more general operator
div 𝑎 (𝑥, ∇𝑢)
(12)
were intensively studied (see [13]). At the same time, some
authors was interested by PDEs involving anisotropic Sobolev
⃗
spaces with variable exponent 𝑊1,𝑝(⋅)
when the boundary
condition is the homogeneous Dirichlet boundary condition
(see [15, 16, 18, 20, 24–26]). In that context, the authors have
considered the anisotropic 𝑝(⋅)-Laplace operator
𝑁
󵄨󵄨
󵄨󵄨𝑝𝑖 (𝑥)−2
𝜕𝑥𝑖 𝑢)
Δ 𝑝(𝑥)
⃗ 𝑢 = ∑𝜕𝑥𝑖 (󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨
(13)
𝑖=1
or the more general variable exponent anisotropic operator
𝑁
∑𝜕𝑥𝑖 𝑎 (𝑥, 𝜕𝑥𝑖 𝑢) .
(14)
𝑖=1
When the homogeneous Dirichlet boundary condition is
replaced by the Neumann boundary condition, one has to
work with the anisotropic variable exponent Sobolev space
⃗
1,𝑝(⋅)
⃗
(Ω) instead of 𝑊0 (Ω). The main difficulty which
𝑊1,𝑝(⋅)
appears is that the famous Poincaré inequality does not apply
and then it is very difficult to get a priori estimates which
are necessary for the proof of the existence result of entropy
solutions. Sometimes one can use the Wirtinger inequality
which does not apply, in some problems like (1). The first
systematic study of anisotropic Neumann problem was done
by Fan (see [11]). In a second time, Boureanu and Rădulescu
studied an anisotropic nonhomogeneous Neumann problem
with obstacle (see [2]). In the two papers, the authors were
interested by the existence and multiplicity results of weak
solution even if in [2], Boureanu and Rădulescu have showed
some conditions under which we can get uniqueness of weak
solution. In this paper, we are interested to the existence and
uniqueness of entropy solution. For the proof of the existence
of entropy solution of (1), we follow [27] and derive a priori
estimates for the approximated solutions 𝑢𝑛 and the partial
derivatives 𝜕𝑢𝑛 /𝜕𝑥𝑖 in the Marcinkiewicz spaces M𝑝̃ and
M𝑝̃ 𝑖 , respectively (see Section 2 or [27, 28] for definition and
properties of Marcinkiewicz spaces).
The study of anisotropic problems are motivated, for
example, by their applications to the mathematical analysis
of a system of nonlinear partial differential equations arising
in a population dynamics model describing the spread of an
epidemic disease through a heterogeneous habitat.
The paper is organized as follows. In Section 2, we
introduce some notations/functional spaces. In Section 3, we
prove for the problem (1), the existence and uniqueness of
weak solution when the data is bounded, and the existence
and uniqueness of entropy solution when the data is in 𝐿1 (Ω).
2. Preliminaries
In this section, we define Lebesgue, Sobolev, and anisotropic
spaces with variable exponent and give some of their
properties (see [29] for more details about Lebesgue and
Sobolev spaces with variable exponent). Roughly speaking,
anistropic Lebesgue and Sobolev spaces are functional spaces
of Lebesgue’s and Sobolev’s type in which different space
directions have different roles.
Given a measurable function 𝑝(⋅) : Ω → [1, ∞), we
define the Lebesgue space with variable exponent 𝐿𝑝(⋅) (Ω) as
the set of all measurable functions 𝑢 : Ω → R for which the
convex modular
𝜌𝑝(⋅) (𝑢) := ∫ |𝑢|𝑝(𝑥) 𝑑𝑥
Ω
(15)
is finite. If the exponent is bounded, that is, if 𝑝+ < ∞, then
the expression
𝑢
|𝑢|𝑝(⋅) := inf {𝜆 > 0 : 𝜌𝑝(⋅) ( ) ≤ 1}
𝜆
(16)
defines a norm in 𝐿𝑝(⋅) (Ω), called the Luxembourg norm. The
space (𝐿𝑝(⋅) (Ω), | ⋅ |𝑝(⋅) ) is a separable Banach space. Moreover,
if 𝑝− > 1, then 𝐿𝑝(⋅) (Ω) is uniformly convex, hence reflexive,
󸀠
and its dual space is isomorphic to 𝐿𝑝 (⋅) (Ω), where 1/𝑝(𝑥) +
1/𝑝󸀠 (𝑥) = 1. Finally, we have the Hölder type inequality.
International Journal of Differential Equations
3
Proposition 1 (generalized Hölder inequality, see [10]). (i)
󸀠
For any 𝑢 ∈ 𝐿𝑝(⋅) (Ω) and V ∈ 𝐿𝑝 (⋅) (Ω), we have
󵄨󵄨
󵄨
󵄨󵄨∫ 𝑢 V 𝑑𝑥󵄨󵄨󵄨 ≤ ( 1 + 1 ) |𝑢| |V| 󸀠 .
󵄨󵄨
󵄨󵄨
𝑝(⋅) 𝑝 (⋅)
𝑝− 𝑝−󸀠
󵄨 Ω
󵄨
(17)
(ii) If 𝑝1 , 𝑝2 ∈ C+ (Ω), 𝑝1 (𝑥) ≤ 𝑝2 (𝑥) for any 𝑥 ∈ Ω, then
𝐿
(Ω) 󳨅→ 𝐿𝑝1 (𝑥) (Ω) and the imbedding is continuous.
Moreover, the application 𝜌𝑝(⋅) : 𝐿𝑝(⋅) (Ω) → R called the
𝑝(⋅)-modular of the 𝐿𝑝(⋅) (Ω) space is very useful in handling
these Lebesgue spaces with variable exponent. Indeed we
have the following properties (see [10]). If 𝑢 ∈ 𝐿𝑝(⋅) (Ω) and
𝑝 < ∞ then
𝑝+
𝑝−
𝑝−
𝑝+
|𝑢|𝑝(⋅) > 1 󳨐⇒ |𝑢|𝑝(⋅) ≤ 𝜌𝑝(⋅) (𝑢) ≤ |𝑢|𝑝(⋅) ,
(18)
(19)
|𝑢|𝑝(⋅) < 1 (= 1; > 1) 󳨐⇒ 𝜌𝑝(⋅) (𝑢) < 1 (= 1; > 1) ,
(20)
|𝑢|𝑝(⋅) 󳨀→ 0 (󳨀→ ∞) ⇐⇒ 𝜌𝑝(⋅) (𝑢) 󳨀→ 0 (󳨀→ ∞) .
(21)
If, in addition, (𝑢𝑛 )𝑛 ⊂ 𝐿𝑝(⋅) (Ω), then
lim𝑛 → ∞ |𝑢𝑛 − 𝑢|𝑝(⋅) = 0 ⇔ lim𝑛 → ∞ 𝜌𝑝(⋅) (𝑢𝑛 − 𝑢) =
0 ⇔ (𝑢𝑛 )𝑛 converges to 𝑢 in measure and lim𝑛 → ∞ 𝜌𝑝(⋅) (𝑢𝑛 ) =
𝜌𝑝(⋅) (𝑢).
Now, let us introduce the definition of the isotropic
Sobolev space with variable exponent, 𝑊1,𝑝(⋅) (Ω).
We set
𝑊1,𝑝(⋅) (Ω) := {𝑢 ∈ 𝐿𝑝(⋅) (Ω) : |∇𝑢| ∈ 𝐿𝑝(⋅) (Ω)} ,
(22)
which is a Banach space equipped with the norm
‖𝑢‖1,𝑝(⋅) := |𝑢|𝑝(⋅) + |∇𝑢|𝑝(⋅) .
(23)
Now, we present a natural generalization of the variable
exponent Sobolev space 𝑊1,𝑝(⋅) (Ω) that will enable us to study
the problem (1) with sufficient accuracy.
⃗
The anisotropic variable exponent Sobolev space 𝑊1,𝑝(⋅)
(Ω)
is defined as follows:
⃗
𝑊1,𝑝(⋅) (Ω)
= {𝑢 ∈ 𝐿𝑝𝑀 (⋅) (Ω) ;
𝜕𝑢
∈ 𝐿𝑝𝑖 (⋅) (Ω) , ∀𝑖 ∈ {1, . . . , 𝑁}} .
𝜕𝑥𝑖
(24)
⃗
1,𝑝(⋅)
(26)
we have the compact embedding
⃗
𝑊1,𝑝(⋅) (Ω) 󳨅→ 𝐿𝑟(⋅) (Ω) .
⃗
⃗
T1,𝑝(⋅) (Ω) = {𝑢 : Ω → R; 𝑇𝑘 (𝑢) ∈ 𝑊1,𝑝(⋅) (Ω) , ∀𝑘 > 0} .
(28)
Finally, in this paper, we will use the Marcinkiewicz spaces
M𝑞 (Ω) (1 < 𝑞 < ∞) with constant exponent. Note that
the Marcinkiewicz spaces M𝑞(⋅) (Ω) in the variable exponent
setting were introduced for the first time by Sanchon and
Urbano (see [23]).
Marcinkiewicz spaces M𝑞 (Ω) (1 < 𝑞 < ∞) contain the
measurable functions ℎ : Ω → R for which the distribution
function
󵄨
󵄨
𝜆 ℎ (𝛾) = 󵄨󵄨󵄨{𝑥 ∈ Ω : |ℎ (𝑥)| > 𝛾}󵄨󵄨󵄨 ,
(Ω), ‖ ⋅ ‖𝑝(⋅)
the space (𝑊
⃗ ) is a reflexive Banach space (see
[11, Theorems 2.1 and 2.2]).
We have the following result.
𝛾≥0
(29)
satisfies an estimate of the form
𝜆 ℎ (𝛾) ≤ 𝐶𝛾−𝑞 ,
for some finite constant 𝐶 > 0.
(30)
The space M𝑞 (Ω) is a Banach space under the norm
1 𝑡
‖ℎ‖∗M𝑞 (Ω) = sup 𝑡1/𝑞 ( ∫ ℎ∗ (𝑠) 𝑑𝑠) ,
𝑡 0
𝑡>0
(31)
where ℎ∗ denotes the nonincreasing rearrangement of ℎ:
ℎ∗ (𝑡) = inf {𝛾 > 0 : 𝜆 ℎ (𝛾) ≤ 𝑡} .
(32)
We will use the following pseudonorm
‖ℎ‖M𝑞 (Ω) = inf {𝐶 : 𝜆 ℎ (𝛾) ≤ 𝐶𝛾−𝑞 , ∀𝛾 > 0} ,
(33)
which is equivalent to the norm ‖ℎ‖∗M𝑞 (Ω) (see [27]).
We need the following Lemma (see [28, Lemma A.2]).
Lemma 3. Let 1 ≤ 𝑞 < 𝑝 < +∞. Then, for every measurable
function 𝑢 on Ω, we have
𝑝
(i) ((𝑝 − 1)𝑝 /𝑝𝑝+1 )‖𝑢‖M𝑝 (Ω) ≤ sup{𝜆𝑝 meas[𝑥 ∈ Ω :
|𝑢(𝑥)| > 𝜆]} ≤ ‖𝑢‖M𝑝 (Ω) .
(25)
(27)
Next, we define
𝑝
Endowed with the norm
𝑁 󵄨󵄨
󵄨
󵄨 𝜕 󵄨󵄨󵄨
:= |𝑢|𝑝𝑀 (⋅) + ∑󵄨󵄨󵄨
𝑢󵄨󵄨 ,
‖𝑢‖𝑝(⋅)
⃗
󵄨
󵄨
𝑖=1 󵄨 𝜕𝑥𝑖 󵄨𝑝𝑖 (⋅)
ess inf (𝑝𝑀 (𝑥) − 𝑟 (𝑥)) > 0,
𝑥∈Ω
𝑝2 (𝑥)
|𝑢|𝑝(⋅) < 1 󳨐⇒ |𝑢|𝑝(⋅) ≤ 𝜌𝑝(⋅) (𝑢) ≤ |𝑢|𝑝(⋅) ,
Theorem 2 (see [11, Corollary 2.1]). Let Ω ⊂ R𝑁 (𝑁 ≥ 3) be
a bounded open set and for all 𝑖 ∈ {1, . . . , 𝑁}, 𝑝𝑖 ∈ 𝐿∞ (Ω),
𝑝𝑖 (𝑥) ≥ 1 a.e. in Ω. Then, for any 𝑟 ∈ 𝐿∞ (Ω) with 𝑟(𝑥) ≥ 1
a.e. in Ω such that
𝜆>0
Moreover,
(ii) ∫𝐾 |𝑢|𝑞 𝑑𝑥
≤
𝑞
(𝑝/(𝑝 − 𝑞)) (𝑝/𝑞)𝑞/𝑝 ‖𝑢‖M𝑝 (Ω)
(meas(𝐾))𝑝−𝑞/𝑝 , for every measurable subset 𝐾 ⊂ Ω.
𝑞
In particular, M𝑝 (Ω) ⊂ Lloc (Ω) with continuous injection
and 𝑢 ∈ M𝑝 (Ω) implies |𝑢|𝑞 ∈ M𝑝/𝑞 (Ω).
4
International Journal of Differential Equations
The following result is due to Troisi (see [30]).
Not also that
𝑝𝑖−
󵄩󵄩 𝜕𝑇 (𝑔) 󵄩󵄩
󵄩󵄩
󵄩󵄩 𝛾
󵄩󵄩
󵄩󵄩
󵄩󵄩
−
+ 󵄩󵄩󵄩
)
(󵄩󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩𝐿𝑝𝑀
(Ω)
󵄩󵄩 𝜕𝑥𝑖 󵄩󵄩󵄩 𝑝𝑖−
󵄩𝐿 (Ω)
󵄩
Theorem 4. Let 𝑝1 , 𝑝2 , . . . , 𝑝𝑁 ∈ [1, +∞); 𝑔 ∈ 𝑊1,(𝑝1 ,𝑝2 ,...,𝑝𝑁 )
(Ω) and let
∗
𝑞=𝑝
∗
if 𝑝 < 𝑁,
𝑞 ∈ [1, +∞)
(34)
∗
if 𝑝 ≥ 𝑁.
(35)
󵄩󵄩 𝜕𝑇 (𝑔) 󵄩󵄩𝑝𝑖−
󵄩󵄩
󵄩󵄩 𝛾
󵄩󵄩
󵄩󵄩𝑝𝑖−
󵄩󵄩
−
(󵄩󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩 𝑝𝑀
+ 󵄩󵄩󵄩
)
𝐿 (Ω)
󵄩󵄩 𝜕𝑥𝑖 󵄩󵄩󵄩 𝑝𝑖−
󵄩𝐿 (Ω)
󵄩
(𝑝𝑖− −1)
󵄩󵄩 𝜕𝑇 (𝑔) 󵄩󵄩𝑝𝑖−
−
󵄩󵄩
󵄩󵄩
󵄩󵄩𝑝𝑀
󵄩󵄩 𝛾
󵄩󵄩
−
(󵄩󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩 𝑝𝑀
+ 󵄩󵄩󵄩
).
𝐿 (Ω)
󵄩󵄩 𝜕𝑥𝑖 󵄩󵄩󵄩 𝑝𝑖−
󵄩𝐿 (Ω)
󵄩
≤2
Then, there exists a constant 𝐶 > 0 depending on
𝑁, 𝑝1 , 𝑝2 , . . . , 𝑝𝑁 if 𝑝 < 𝑁 and also on 𝑞 and meas(Ω) if
𝑝 ≥ 𝑁 such that
1/𝑁
𝑁
󵄩󵄩 𝜕𝑔 󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩 󵄩󵄩
] ,
󵄩
󵄩󵄩𝑔󵄩󵄩𝐿𝑞 (Ω) ≤ 𝐶∏[󵄩󵄩𝑔󵄩󵄩𝐿𝑝𝑀 (Ω) + 󵄩󵄩
󵄩󵄩 𝜕𝑥𝑖 󵄩󵄩󵄩𝐿𝑝𝑖 (Ω)
𝑖=1
(𝑝𝑖− −1)
≤2
Therefore, by using (35), we obtain for 𝛾 > 1,
𝑁
−
−
−
󵄩󵄩
󵄩
󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩 𝑞 ≤ 𝐶∏ [2(𝑝𝑖 −1)/𝑁𝑝𝑖 𝛾1/𝑁𝑝𝑖 ]
󵄩
󵄩𝐿 (Ω)
where 𝑝𝑀 = max {𝑝1 , 𝑝2 , . . . , 𝑝𝑁} and 1/𝑝 = (1/𝑁)
∑𝑁
𝑖=1 (1/𝑝𝑖 ).
We will use through the paper, the truncation function 𝑇𝛾
at height (𝛾 > 0), that is
𝑠
𝑇𝛾 (𝑠) = {
𝛾 sign (𝑠)
𝑖=1
≤ 𝐷𝛾
∫
󵄨 󵄨
𝛾𝑞 meas ({󵄨󵄨󵄨𝑔󵄨󵄨󵄨 > 𝛾}) ≤ 𝐷𝛾𝑞/𝑝 .
(45)
󵄨 󵄨
meas ({󵄨󵄨󵄨𝑔󵄨󵄨󵄨 > 𝛾}) ≤ 𝐷𝛾−𝑞(𝑝−1)/𝑝 .
(46)
(Ω).
Lemma 5. Let 𝑔 be a nonnegative function in 𝑊
Assume 𝑝 < 𝑁 and there exists a constant 𝐶 > 0 such that
Therefore,
(37)
∗
Since, 𝑞 = 𝑝 = 𝑁𝑝/(𝑁 − 𝑝) we get
∀𝛾 > 0.
󵄨 󵄨
meas ({󵄨󵄨󵄨𝑔󵄨󵄨󵄨 > 𝛾}) ≤ 𝐷𝛾−𝑁(𝑝−1)/(𝑁−𝑝)
Then, there exists a constant 𝐷, depending on 𝐶, such that
(38)
̃ = 𝑁(𝑝 − 1)/(𝑁 − 𝑝).
where 𝑝
−
Step 1 (||𝑇𝛾 (𝑔)||𝐿𝑝𝑀
≤ 1). Then, obviously we have
(Ω)
‖𝑔‖M𝑝̃ (Ω) ≤ 𝐷, for some positive constant 𝐷. Indeed, since
−
̃ ≤ 𝑝 ≤ 𝑝𝑀
, according to Proposition 1 there exists a
1 < 𝑝
positive constant 𝐶 such that
(39)
It follows that there exists a positive constant 𝐷 such that
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑔󵄩󵄩M𝑝̃ (Ω) ≤ 𝐷.
(40)
(47)
which implies that ‖𝑔‖M𝑝̃ (Ω) ≤ 𝐷.
For 0 < 𝛾 ≤ 1 we have
̃
󵄨 󵄨
meas ({󵄨󵄨󵄨𝑔󵄨󵄨󵄨 > 𝛾}) ≤ meas (Ω) ≤ meas (Ω) 𝛾−𝑝 .
Proof. Consider the following
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩 𝑝̃ ≤ 𝐶󵄩󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩󵄩 𝑝𝑀
−
󵄩
󵄩𝐿 (Ω)
󵄩
󵄩𝐿 (Ω) ≤ 𝐶.
(44)
which is equivalent to
⃗
1,𝑝(⋅)
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑔󵄩󵄩M𝑝̃ (Ω) ≤ 𝐷,
(43)
= 𝐷𝛾1/𝑝 .
󵄨󵄨
󵄨𝑞
󵄨󵄨𝑇𝛾 (𝑔)󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐷𝛾𝑞/𝑝
󵄨
󵄨
{|𝑔|>𝛾}
(36)
We need the following lemma.
≤ 𝐶 (𝛾 + 1) ,
−
∑𝑁
𝑖=1 (1/𝑁𝑝𝑖 )
It follows that
if |𝑠| ≤ 𝛾,
if |𝑠| > 𝛾.
𝑁
󵄨󵄨 𝜕𝑔 󵄨󵄨𝑝𝑖−
−
󵄨󵄨
󵄨󵄨𝑝𝑀
󵄨󵄨
󵄨󵄨
∫ 󵄨󵄨󵄨𝑇𝛾 (𝑔)󵄨󵄨󵄨 𝑑𝑥 + ∑ ∫
󵄨󵄨
󵄨󵄨 𝑑𝑥
󵄨
󵄨󵄨
𝜕𝑥
Ω
{|𝑔|≤𝛾}
󵄨
𝑖
𝑖=1
(42)
(48)
So,
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑔󵄩󵄩M𝑝̃ (Ω) ≤ 𝐷.
(49)
We need the following well-known results.
Theorem 6 (see [31, Theorem 6.2.1]). Let 𝑋 be a reflexive
Banach space and let 𝑓 : 𝑀 ⊂ 𝑋 → R be Gateaux
differentiable over the closed set 𝑀. Then, the following are
equivalent.
(i) 𝑓 is convex over 𝑀.
(ii) We have
−
> 1). We get from (37)
Step 2 (||𝑇𝛾 (𝑔)||𝐿𝑝𝑀
(Ω)
󵄩󵄩 𝜕𝑇 (𝑔) 󵄩󵄩𝑝𝑖−
−
󵄩󵄩
󵄩󵄩 𝛾
󵄩󵄩𝑝𝑀
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑇𝛾 (𝑔)󵄩󵄩 𝑝−
+
󵄩
𝑀
󵄩𝐿 (Ω) 󵄩󵄩 𝜕𝑥𝑖 󵄩󵄩󵄩󵄩 𝑝− ≤ 𝐶 (𝛾 + 1) .
󵄩
󵄩𝐿 𝑖 (Ω)
󵄩
𝑓 (𝑢) − 𝑓 (V) ≥ ⟨𝑓󸀠 (V) , 𝑢 − V⟩𝑋∗ ×𝑋
∀𝑢, V ∈ 𝑀,
(41)
where 𝑋∗ denotes the dual of the space 𝑋.
(50)
International Journal of Differential Equations
5
Lemma 9. The functional Λ is well-defined on 𝐸. In addition,
Λ is of class C1 (𝐸, R) and
(iii) The first Gateaux derivative is monotone, that is,
⟨𝑓󸀠 (𝑢) − 𝑓󸀠 (V) , 𝑢 − V⟩𝑋∗ ×𝑋 ≥ 0
∀𝑢, V ∈ 𝑀.
(51)
𝑁
(iv) The second Gateaux derivative of 𝑓 exists and it is
positive, that is,
⟨𝑓󸀠󸀠 (𝑢) ∘ V, V⟩𝑋∗ ×𝑋 ≥ 0
∀V ∈ 𝑀.
(52)
Theorem 7 (see [32, Theorem 1.2]). Suppose 𝑋 is a reflexive
Banach space with norm || ⋅ ||𝑋 , and let 𝑀 ⊂ 𝑋 be a weakly
closed subset of 𝑋. Suppose Ψ : 𝑀 ⊂ 𝑋 → R ∪ {∞} is
coercive and (sequentially) weakly lower semicontinuous on M
with respect to 𝑋, that is, suppose the following conditions are
fulfilled.
⟨Λ󸀠 (𝑢) , V⟩ = ∫ ∑𝑎𝑖 (𝑥,
Ω 𝑖=1
𝑚→∞
(53)
Then, Ψ is bounded from below and attains its infinimun in 𝑀.
3. Main Results
⃗
In the sequel, we denote 𝑊1,𝑝(⋅)
(Ω) = 𝐸 and ‖ ⋅ ‖𝑊1,𝑝(⋅)
⃗
(Ω) =
‖ ⋅ ‖𝐸 .
(57)
for all 𝑢, V ∈ 𝐸.
Due to Lemma 9, a standard calculus leads to the facts
that 𝐼 is well-defined on 𝐸 and 𝐼 ∈ C1 (𝐸, R) with the
derivative given by
𝑁
⟨𝐼󸀠 (𝑢) , V⟩ = ∫ ∑𝑎𝑖 (𝑥,
Ω 𝑖=1
(i) Ψ(𝑢) → ∞ as ‖𝑢‖𝑋 → ∞, 𝑢 ∈ 𝑀.
(ii) For any 𝑢 ∈ 𝑀, any subsequence (𝑢𝑚 ) in 𝑀 such that
𝑢𝑚 ⇀ 𝑢 weakly in 𝑋 there holds
Ψ (𝑢) ≤ lim inf Ψ (𝑢𝑚 ) .
𝜕𝑢 𝜕V
𝑑𝑥,
)
𝜕𝑥𝑖 𝜕𝑥𝑖
𝜕𝑢 𝜕V
)
𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖
(58)
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢V𝑑𝑥 − ∫ 𝑓 (𝑥) V𝑑𝑥
Ω
Ω
for all 𝑢, V ∈ 𝐸. Obviously, the weak solutions of (1) are the
critical points of 𝐼; so by means of Theorem 7, we intend to
prove the existence of critical points in order to deduce the
existence of weak solutions.
Theorem 10. Assume (4)–(8) and 𝑓 ∈ 𝐿∞ (Ω). Then, there
exists a unique weak solution of problem (1).
Let us start the proof by establishing some useful lemmas.
3.1. Weak Solutions. Let us define first the notion of weak
solution.
Lemma 11. If hypotheses (4)–(8) are fulfilled, then the functional 𝐼 is coercive.
Definition 8. Let 𝑢 : Ω → R be a measurable function, we
say that 𝑢 is a weak solution of problem (1) if 𝑢 belongs to
⃗
(Ω) and satisfies the following equation:
𝑊1,𝑝(⋅)
Proof. Let 𝑢 ∈ 𝐸 be such that ||𝑢||𝐸 → ∞. Using (7), we
deduce that
𝑁
∫ ∑𝑎𝑖 (𝑥,
Ω 𝑖=1
𝜕𝑢 𝜕V
)
𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖
𝑝𝑀 (𝑥)−2
+ ∫ |𝑢|
Ω
Λ (𝑢) ≥
(54)
𝑢V𝑑𝑥 − ∫ 𝑓 (𝑥) V𝑑𝑥 = 0,
⃗
for every V ∈ 𝑊1,𝑝(⋅)
(Ω).
We associate to problem (1) the energy functional 𝐼 : 𝐸 →
R, defined by
𝑁
1
+∫
|𝑢|𝑝𝑀 (𝑥) 𝑑𝑥 − ∫ 𝑓 (𝑥) 𝑢𝑑𝑥.
Ω 𝑝𝑀 (𝑥)
Ω
(55)
𝑁
Ω 𝑖=1
𝜕𝑢
) 𝑑𝑥.
𝜕𝑥𝑖
We recall the following result (see [15, Lemma 3.4]).
󵄨󵄨 𝜕𝑢 󵄨󵄨
󵄨
󵄨󵄨
≤ 1} ,
I1 = {𝑖 ∈ {1, . . . , 𝑁} : 󵄨󵄨󵄨
󵄨
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝐿𝑝𝑖 (⋅) (Ω)
󵄨󵄨 𝜕𝑢 󵄨󵄨
󵄨
󵄨󵄨
I2 = {𝑖 ∈ {1, . . . , 𝑁} : 󵄨󵄨󵄨
> 1} .
󵄨
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝐿𝑝𝑖 (⋅) (Ω)
(60)
We then have
To simplify our writing, we denote by Λ : 𝐸 → R the
functional
Λ (𝑢) = ∫ ∑𝐴 𝑖 (𝑥,
(59)
We make the following notations:
Ω
𝜕𝑢
) 𝑑𝑥
𝐼 (𝑢) = ∫ ∑𝐴 𝑖 (𝑥,
𝜕𝑥𝑖
Ω 𝑖=1
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)
1 𝑁
󵄨󵄨
󵄨󵄨
𝑑𝑥.
∑
∫
󵄨󵄨
󵄨󵄨
+
󵄨
󵄨󵄨
𝑝𝑀
𝜕𝑥
Ω
󵄨
𝑖
𝑖=1
(56)
Λ (𝑢) ≥
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)
1
󵄨󵄨
󵄨󵄨
𝑑𝑥
∑
∫
󵄨󵄨
󵄨
+
󵄨
𝑝𝑀 𝑖∈I Ω 󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
1
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)
1
󵄨
󵄨󵄨
+ + ∑ ∫ 󵄨󵄨󵄨
𝑑𝑥.
󵄨
𝑝𝑀 𝑖∈I Ω 󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
2
(61)
6
International Journal of Differential Equations
Using (19), (20), and (21), we have
Therefore,
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖+
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖−
1
1
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨
Λ (𝑢) ≥ + ∑ 󵄨󵄨
+ + ∑ 󵄨󵄨󵄨
󵄨
󵄨
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝 (⋅)
𝑝𝑀 𝑖∈I 󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅) 𝑝𝑀
𝑖∈I
𝑖
1
2
≥
𝐼 (𝑢) ≥
1
−
+
2𝑝𝑚 −1 𝑝𝑀
min (1,
1
−
𝑁𝑝𝑚 −1
𝑝−
) ‖𝑢‖𝐸𝑚
(66)
𝑁
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − + .
𝑝𝑀
󵄨󵄨 𝜕𝑢 󵄨󵄨
1
󵄨󵄨
󵄨󵄨
∑
󵄨󵄨
󵄨󵄨
+
󵄨
󵄨
𝑝𝑀
𝜕𝑥
𝑖 󵄨𝑝𝑖 (⋅)
𝑖∈I 󵄨
𝑝𝑖−
2
𝑝−
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑚−
1
󵄨
󵄨󵄨
≥ + ∑ 󵄨󵄨󵄨
󵄨
𝑝𝑀 𝑖∈I 󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
(62)
Case 2 (|𝑢|𝑝𝑀 (⋅) < 1). Then |𝑢|𝑝𝑚 (⋅) − 1 < 0, and we get
𝑀
𝐼 (𝑢)
2
≥
𝑁 󵄨󵄨
󵄨
󵄨󵄨 𝜕𝑢 󵄨󵄨
1
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
󵄨
󵄨󵄨
− ∑ 󵄨󵄨󵄨
󵄨󵄨
󵄨 )
+ (∑󵄨󵄨󵄨
𝑝𝑀 𝑖=1󵄨 𝜕𝑥𝑖 󵄨󵄨𝑝𝑖 (⋅) 𝑖∈I 󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
≥
𝑁 󵄨󵄨
󵄨
1
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
− 𝑁) .
󵄨
+ (∑󵄨󵄨󵄨
𝑝𝑀 𝑖=1󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
−
𝑝𝑚
−
𝑝𝑚
−
𝑝𝑚
𝑁 󵄨󵄨
󵄨
−
1 [ 1
𝑝𝑚
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
]
( ∑󵄨
≥ +
󵄨 ) + |𝑢|𝑝𝑀
−
(⋅) − 𝑁 − 1
𝑝𝑀 𝑁𝑝𝑚 −1 𝑖=1󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
]
[
󵄩󵄩 󵄩󵄩
− 󵄩󵄩𝑓󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐿1 (Ω)
1
−
𝑝𝑚
−
By the generalized mean inequality or the Jensen’s inequality
applied to the convex function 𝑧 : R+ → R+ , 𝑧(𝑡) =
−
−
𝑡𝑝𝑚 , 𝑝𝑚
> 1, we get
𝑝𝑚
𝑁 󵄨󵄨
󵄨
−
1 [ 1
𝑝𝑚
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
]
≥ +
󵄨󵄨 ) + |𝑢|𝑝𝑀
− −1 (∑󵄨
󵄨
(⋅)
𝑝
󵄨
󵄨
𝑝𝑀 𝑁 𝑚
𝜕𝑥
󵄨
󵄨
𝑖
𝑝𝑖 (⋅)
𝑖=1
[
]
𝑁+1
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − +
𝑝𝑀
−
𝑝𝑚
𝑁 󵄨󵄨
𝑁 󵄨󵄨
󵄨
󵄨
1
󵄨 𝜕𝑢 󵄨󵄨󵄨
󵄨 𝜕𝑢 󵄨󵄨󵄨
≥ 𝑝− −1 (∑󵄨󵄨󵄨
∑󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 ) ,
󵄨
󵄨
󵄨
󵄨
𝑁 𝑚
𝑖=1 󵄨 𝜕𝑥𝑖 󵄨𝑝𝑖 (⋅)
𝑖=1 󵄨 𝜕𝑥𝑖 󵄨𝑝𝑖 (⋅)
−
𝑝𝑚
(63)
≥
thus,
−
𝑝𝑚
𝑁 󵄨󵄨
󵄨
1 [ 1
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
Λ (𝑢) ≥ +
(∑󵄨
󵄨 ) − 𝑁] .
−
𝑝𝑀 𝑁𝑝𝑚 −1 𝑖=1󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
]
[
(64)
1
−
+
2𝑝𝑚 −1 𝑝𝑀
−
𝑝𝑚
󵄨
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
min (1, 𝑝− −1 ) [∑󵄨󵄨
+ |𝑢|𝑝𝑀 (⋅) ]
󵄨󵄨
󵄨
󵄨
𝑁 𝑚
𝑖=1 󵄨 𝜕𝑥𝑖 󵄨𝑝𝑖 (⋅)
1
𝑁 󵄨󵄨
𝑁+1
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − + .
𝑝𝑀
(67)
So, we obtain
Case 1 (|𝑢|𝑝𝑀 (.) ≥ 1). We have
𝐼 (𝑢) ≥
−
𝑝𝑚
𝑁 󵄨󵄨
󵄨
1 [ 1
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
𝐼 (𝑢) ≥ +
(∑󵄨
󵄨 ) − 𝑁]
−
𝑝𝑀 𝑁𝑝𝑚 −1 𝑖=1󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
]
[
+
1
min (1,
1
−
𝑁𝑝𝑚 −1
(65)
−
𝑝𝑚
󵄨
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
⋅ [∑󵄨󵄨
+ |𝑢|𝑝𝑀 (⋅) ]
󵄨󵄨
󵄨
󵄨
𝑖=1 󵄨 𝜕𝑥𝑖 󵄨𝑝𝑖 (⋅)
𝑁 󵄨󵄨
𝑁
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − + .
𝑝𝑀
(68)
Proof. By [33, Corollary III.8], it is enough to show that 𝐼 is
lower semicontinuous. To this aim, fix 𝑢 ∈ 𝐸 and 𝜖 > 0.
Since for every 𝑖 ∈ {1, . . . , 𝑁}, 𝑎𝑖 (𝑥, ⋅) is monotone, Theorem 6
yields
𝐴 𝑖 (𝑥,
)
𝑝−
) ‖𝑢‖𝐸𝑚
Lemma 12. The functional 𝐼 is weakly lower semicontinuous.
−
−
+
2𝑝𝑚 −1 𝑝𝑀
1
−
𝑁𝑝𝑚 −1
Then, letting ‖𝑢‖𝐸 goes to infinity in (66) and (68), we
conclude that 𝐼(𝑢) reaches infinity. Thus, 𝐼 is coercive.
𝑝𝑚
𝑁 󵄨󵄨
󵄨
−
1 [ 1
𝑝𝑚
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
]
≥ +
(
)
+
∑
|𝑢|
󵄨
󵄨
−
󵄨
󵄨
𝑝
𝑀 (⋅)
𝑝𝑀 𝑁𝑝𝑚 −1 𝑖=1󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨𝑝𝑖 (⋅)
[
]
≥
min (1,
𝑁+1
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − + .
𝑝𝑀
−
1
𝑝𝑀
󵄩󵄩 󵄩󵄩
󵄩𝑓󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐿1 (Ω)
+ |𝑢|𝑝𝑀 (⋅) − 󵄩
𝑝𝑀
𝑁
󵄩 󵄩
− 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢‖𝐸 − +
𝑝𝑀
1
−
+
2𝑝𝑚 −1 𝑝𝑀
𝜕V
𝜕𝑢
𝜕𝑢
𝜕V
𝜕𝑢
) − 𝐴 𝑖 (𝑥,
) ≥ 𝑎𝑖 (𝑥,
)(
−
)
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥𝑖 𝜕𝑥𝑖
𝑁
󳨐⇒ ∑ ∫ 𝐴 𝑖 (𝑥,
𝑖=1 𝜔
𝑁
𝑁
𝜕V
𝜕𝑢
) 𝑑𝑥 ≥ ∑ ∫ 𝐴 𝑖 (𝑥,
) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥
𝜔
𝑖
𝑖=1
+ ∑ ∫ 𝑎𝑖 (𝑥,
𝑖=1 𝜔
𝜕𝑢
𝜕V
𝜕𝑢
)(
−
) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖 𝜕𝑥𝑖
International Journal of Differential Equations
𝑁
󳨐⇒ 𝐼 (V) ≥ 𝐼 (𝑢) + ∑ ∫ 𝑎𝑖 (𝑥,
𝑖=1 𝜔
+∫
𝜔
7
𝜕𝑢
𝜕V
𝜕𝑢
)(
−
) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖 𝜕𝑥𝑖
For the fourth term in the right-hand side of (71), we have
󵄨
󵄨
∫ 𝑓 (𝑥) (𝑢 − V) 𝑑𝑥 ≥ − ∫ 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 |𝑢 − V| 𝑑𝑥
Ω
Ω
1
(|V|𝑝𝑀 (𝑥) − |𝑢|𝑝𝑀 (𝑥) ) 𝑑𝑥
𝑝𝑀 (𝑥)
󵄩 󵄩
≥ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿∞ (Ω) ‖𝑢 − V‖𝐿1 (Ω)
≥ − 𝐶2 ‖𝑢 − V‖𝐸 .
+ ∫ 𝑓 (𝑥) (𝑢 − V) 𝑑𝑥.
𝜔
(69)
Since the map 𝑡 󳨃→ 𝑡𝑝𝑀 (𝑥) , 𝑡 > 0 is convex, again by
Theorem 6, we have
|V|𝑝𝑀 (𝑥) − |𝑢|𝑝𝑀 (𝑥) ≥ 𝑝𝑀 (𝑥) |𝑢|𝑝𝑀 (𝑥)−2 𝑢 (V − 𝑢) ,
The third term in the right-hand side of (71) gives by using
Hölder type inequality
∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢 (V − 𝑢) 𝑑𝑥
Ω
(70)
≥ − ∫ |𝑢|𝑝𝑀 (𝑥)−1 |𝑢 − V| 𝑑𝑥
Ω
then (69) becomes
𝑁
𝐼 (V) ≥ 𝐼 (𝑢) + ∑ ∫ 𝑎𝑖 (𝑥,
𝑖=1 Ω
󵄨
󵄨
≥ −𝐶󸀠 󵄨󵄨󵄨󵄨|𝑢|𝑝𝑀 (𝑥)−1 󵄨󵄨󵄨󵄨𝑃󸀠 (⋅) |𝑢 − V|𝑝𝑀 (⋅)
𝑀
𝜕𝑢
𝜕V
𝜕𝑢
)(
−
) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖 𝜕𝑥𝑖
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢 (V − 𝑢) 𝑑𝑥
(71)
Ω
Consider the second term in the right-hand side of (71). By
(5) and Hölder type inequality, we have
𝑖=1 Ω
Gathering these inequalities, it follows that
𝐼 (V) ≥ 𝐼 (𝑢) − 𝐶‖𝑢 − V‖𝐸 ≥ 𝐼 (𝑢) − 𝜖,
+ ∫ 𝑓 (𝑥) (𝑢 − V) 𝑑𝑥.
𝑁
(74)
≥ −𝐶3 |𝑢 − V|𝑝𝑀 (⋅) .
Ω
∑ ∫ 𝑎𝑖 (𝑥,
(73)
𝜕𝑢
𝜕V
𝜕𝑢
)(
−
) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖 𝜕𝑥𝑖
(75)
for every V ∈ 𝐸 such that ‖𝑢 − V‖𝐸 < 𝜖/𝐶. Thus, 𝐼 is lower
semicontinuous.
Proof of Theorem 10. Consider the following
Step 1. Existence of weak solutions. The proof follows directly
from Lemmas 11 and 12 and Theorem 7.
Step 2. Uniqueness of weak solution. Let 𝑢, V ∈ 𝐸 be two weak
solutions of problem (1). Choosing a test function in (54), 𝜑 =
V−𝑢 for the weak solution 𝑢 and 𝜑 = 𝑢−V for the weak solution
V, we get
𝑁
󵄨󵄨
𝜕𝑢 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 𝜕V
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨
)󵄨󵄨 󵄨󵄨
−
≥ −∑ ∫ 󵄨󵄨󵄨𝑎𝑖 (𝑥,
󵄨 𝑑𝑥
󵄨
𝜕𝑥𝑖 󵄨󵄨 󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨
𝑖=1 Ω 󵄨
≥ − max {𝐶1 , . . . , 𝐶𝑁}
𝑁
∫ ∑𝑎𝑖 (𝑥,
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨 𝜕V
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨
󵄨
󵄨󵄨
⋅ ∑ ∫ (𝑗𝑖 + 󵄨󵄨󵄨
) 󵄨󵄨󵄨
−
󵄨󵄨
󵄨 𝑑𝑥
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨
𝑖=1 Ω
𝑁
Ω 𝑖=1
𝜕𝑢 𝜕 (V − 𝑢)
)
𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢 (V − 𝑢) 𝑑𝑥
𝑁
󵄨󵄨 𝜕V
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨
≥ −𝐾∑ ∫ 𝑗𝑖 (𝑥) 󵄨󵄨󵄨
−
󵄨 𝑑𝑥
󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨
𝑖=1 Ω
(76)
Ω
− ∫ 𝑓 (𝑥) (V − 𝑢) 𝑑𝑥 = 0,
Ω
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨 𝜕V
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
− 𝐾∑ ∫ 󵄨󵄨󵄨
−
󵄨󵄨
󵄨󵄨
󵄨 𝑑𝑥
󵄨
󵄨
󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨
𝑖=1 Ω 󵄨 𝜕𝑥𝑖 󵄨
𝑁
𝑁
∫ ∑𝑎𝑖 (𝑥,
Ω 𝑖=1
𝑁
󵄨
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨 󵄨 󵄨󵄨 𝜕V
≥ −𝐾󸀠 ∑󵄨󵄨󵄨𝑗𝑖 󵄨󵄨󵄨𝑝󸀠 (⋅) 󵄨󵄨󵄨
−
󵄨
𝑖
󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝 (⋅)
𝑖=1
𝑖
𝜕V 𝜕 (𝑢 − V)
)
𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
+ ∫ |V|𝑝𝑀 (𝑥)−2 V (𝑢 − V) 𝑑𝑥
(77)
Ω
󵄨
𝑁 󵄨󵄨󵄨󵄨
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 𝜕V
󵄨󵄨 󵄨󵄨
− 𝐾󸀠 ∑󵄨󵄨󵄨󵄨󵄨󵄨󵄨
−
󵄨󵄨
󵄨󵄨 󸀠 󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨󵄨
󵄨
󵄨
𝑝𝑖 (⋅)
󵄨𝑝𝑖 (⋅)
𝑖=1 󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨
− ∫ 𝑓 (𝑥) (𝑢 − V) 𝑑𝑥 = 0.
Ω
Summing up (76) and (77), we obtain
󵄨
󵄨󵄨󵄨󵄨
󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨󵄨
󵄨 󵄨
󵄨󵄨 }
≥ −𝐾󸀠 max {󵄨󵄨󵄨𝑗𝑖 󵄨󵄨󵄨𝑝󸀠 (⋅) , 󵄨󵄨󵄨󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 󸀠
𝑖
𝑖
󵄨󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
󵄨𝑝𝑖 (⋅)
𝑁
∫ ∑ (𝑎𝑖 (𝑥,
Ω 𝑖=1
𝑁 󵄨󵄨
𝑁 󵄨󵄨
𝜕𝑢 󵄨󵄨󵄨󵄨
𝜕V 󵄨󵄨󵄨󵄨
󵄨 𝜕V
󵄨 𝜕𝑢
⋅ ∑󵄨󵄨󵄨
−
≥ −𝐶1 ∑󵄨󵄨󵄨
−
󵄨󵄨
󵄨 .
󵄨
󵄨
𝜕𝑥𝑖 󵄨󵄨𝑝𝑖 (⋅)
𝜕𝑥𝑖 󵄨󵄨󵄨𝑝𝑖 (⋅)
𝑖=1 󵄨 𝜕𝑥𝑖
𝑖=1 󵄨 𝜕𝑥𝑖
+∫
Ω
(72)
𝜕𝑢
𝜕V
𝜕 (𝑢 − V)
) − 𝑎𝑖 (𝑥,
))
𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥𝑖
1
(|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) (𝑢 − V) 𝑑𝑥 = 0.
𝑝𝑀 (𝑥)
(78)
8
International Journal of Differential Equations
Thus, by the monotonicity of the functions 𝑎𝑖 (𝑥, ⋅) and 𝑡 󳨃→
|𝑡|𝑝𝑀 (𝑥)−2 𝑡, we deduce that 𝑢 = V almost everywhere.
3.2. Entropy Solutions. First of all, we define a space in which
we will look for entropy solutions. We define the space
⃗
T1,𝑝(⋅)
(Ω) as the set of every measurable function 𝑢 : Ω → R
⃗
which satisfies for every 𝑘 > 0, 𝑇𝑘 (𝑢) ∈ 𝑊1,𝑝(⋅)
(Ω).
⃗
1,𝑝(⋅)
(Ω). Then, there exists
Lemma 13 (see [34, 35]). Let 𝑢 ∈ T
a unique measurable function V𝑖 : Ω → R such that
then
𝑁
∑∫
𝛼𝑖 (⋅)
𝑖=1 {|(𝜕/𝜕𝑥𝑖 )𝑢| >𝑡}
Proof. Take 𝜑 = 0 in (80), we have
𝑁
∑ ∫ 𝑎𝑖 (𝑥,
𝑖=1 Ω
𝜕
𝜕
𝑇𝑡 (𝑢)) ⋅
𝑇 (𝑢) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖 𝑡
𝑝𝑀 (𝑥)−2
+ ∫ |𝑢|
Ω
𝜕𝑇𝑘 (𝑢)
𝜕𝑥𝑖
for 𝑎.𝑒. 𝑥 ∈ Ω, ∀𝑘 > 0, 𝑖 ∈ {1, . . . , 𝑁} ,
𝑁
𝑖=1 Ω
𝜕𝑢
𝜕
)
𝑇 (𝑢 − 𝜑) 𝑑𝑥
∑ ∫ 𝑎𝑖 (𝑥,
𝜕𝑥𝑖 𝜕𝑥𝑖 𝑘
𝑖=1 Ω
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥
Ω
𝑁
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
󵄨
󵄨
󵄩 󵄩
𝑇𝑡 (𝑢)󵄨󵄨󵄨
𝑑𝑥 ≤ 𝑡󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 ,
∑ ∫ 󵄨󵄨󵄨
󵄨
󵄨
𝜕𝑥
󵄨
𝑖
𝑖=1 Ω 󵄨
𝑁
󵄨󵄨 𝜕 󵄨󵄨𝑝𝑖 (𝑥)
󵄨
󵄨
𝜓󵄨󵄨󵄨
𝑑𝑥
∑ ∫ 𝑡𝑝𝑖 (𝑥)−1 󵄨󵄨󵄨
󵄨
󵄨
𝜕𝑥
Ω
󵄨
𝑖 󵄨
𝑖=1
for all 𝜑 ∈ 𝑊
𝑖=1 {|(𝜕/𝜕𝑥𝑖
𝑁
+ ∑∫
𝑖=1 {|𝑢|>𝑡}
Our main result in this section is the following.
𝑁
Theorem 16. Assume (4)–(8) and 𝑓 ∈ 𝐿1 (Ω). Then, there
exists a unique entropy solution 𝑢 to problem (1).
≤ ∑∫
Proof. The proof of this Theorem will be done in three steps.
≤ ∑∫
𝑖=1 {|𝑢|≤𝑡}
𝑁
𝑖=1
Step 1 (a priori estimates).
Lemma 17. Assume (4)–(8) and f ∈ L1 (Ω). Let u be an entropy
solution of (1). If there exists a positive constant M such that
𝑁
𝑖=1 {|𝑢|>𝑡}
𝑡𝑞𝑖 (𝑥) 𝑑𝑥 ≤ 𝑀,
∀𝑡 > 0,
(81)
(86)
𝑡𝑞𝑖 (𝑥) 𝑑𝑥
𝛼𝑖 (⋅)
𝑖=1 {|(𝜕/𝜕𝑥𝑖 )𝑢| >𝑡}∩{|𝑢|≤𝑡}
(Ω) ∩ 𝐿∞ (Ω).
∑∫
)𝑢|𝛼𝑖 (⋅) >𝑡}
𝑁
Ω
(85)
From the previous inequality, the definition of 𝛼𝑖 (⋅) and (81),
we have
≤ ∑∫
≤ ∫ 𝑓 (𝑥) 𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥,
⃗
1,𝑝(⋅)
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
1
󵄨
󵄨
󵄩 󵄩
𝑇𝑡 (𝑢)󵄨󵄨󵄨
𝑑𝑥 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
= ∑ ∫ 󵄨󵄨󵄨
󵄨
󵄨
𝑡
𝜕𝑥
Ω󵄨
󵄨
𝑖
𝑖=1
𝑁
𝑁
(80)
∀𝑡 > 0.
Therefore, defining 𝜓 := 𝑇𝑡 (𝑢)/𝑡, we have for all 𝑡 > 0,
∑∫
Ω
(84)
According to (7), we deduce that
(b) 𝜕𝑇𝑘 (𝑢𝑛 )/𝜕𝑥𝑖 → 𝜕𝑇𝑘 (𝑢)/𝜕𝑥𝑖 in 𝐿1 (Ω), for all 𝑘 > 0.
𝑁
𝜕
𝜕
𝑇 (𝑢)) ⋅
𝑇 (𝑢) 𝑑𝑥
𝜕𝑥𝑖 𝑡
𝜕𝑥𝑖 𝑡
≤ ∫ 𝑓 (𝑥) 𝑇𝑡 (𝑢) 𝑑𝑥.
(a) 𝑢𝑛 → 𝑢 a.e. in Ω,
Definition 15. A measurable function 𝑢 is an entropy solution
⃗
1,𝑝(⋅)
of (1) if 𝑢 ∈ TH (Ω) and for every 𝑘 > 0,
𝑢𝑇𝑡 (𝑢) 𝑑𝑥 ≤ ∫ 𝑓 (𝑥) 𝑇𝑡 (𝑢) 𝑑𝑥.
Ω
∑ ∫ 𝑎𝑖 (𝑥,
⃗
1,𝑝(⋅)
Definition 14. We define the space TH (Ω) as the set of
⃗
(Ω) such that there exists a sequence
function 𝑢 ∈ T1,𝑝(⋅)
⃗
1,𝑝(⋅)
(Ω) satisfying
(𝑢𝑛 )𝑛 ⊂ 𝑊
(83)
Since the second term in the previous inequality is nonnegative, it follows that
(79)
where 𝜒𝐴 denotes the characteristic function of a measurable
set 𝐴. The functions V𝑖 are called the weak partial gradients
of 𝑢 and are still denoted 𝜕𝑢/𝜕𝑥𝑖 . Moreover, if 𝑢 belongs to
⃗
(Ω), then V𝑖 ∈ 𝐿𝑝𝑖 (⋅) (Ω) and coincides with the standard
𝑊1,𝑝(⋅)
distributional gradient of 𝑢, that is, V𝑖 = 𝜕𝑢/𝜕𝑥𝑖 .
(82)
where 𝛼i (⋅) = pi (⋅)/(qi (⋅) + 1), for all i = 1, . . . , N.
V𝑖 𝜒{|𝑢|<𝑘}
=
󵄩 󵄩
𝑡𝑞𝑖 (𝑥) 𝑑𝑥 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 + 𝑀 ∀𝑡 > 0,
𝑡𝑞𝑖 (𝑥) 𝑑𝑥
𝑞𝑖 (𝑥)
𝑡
𝑡𝑞𝑖 (𝑥) 𝑑𝑥
󵄨𝛼 (𝑥) 𝑝𝑖 (𝑥)/𝛼𝑖 (𝑥)
󵄨󵄨
󵄨󵄨(𝜕/𝜕𝑥𝑖 ) 𝑢󵄨󵄨󵄨 𝑖
(
𝑑𝑥 + 𝑀
)
𝑡
󵄨
󵄨𝑝𝑖 (𝑥)
𝑞𝑖 (𝑥)−(𝑝𝑖 (𝑥)/𝛼𝑖 (𝑥)) 󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
𝑡
𝑑𝑥 + 𝑀
󵄨
󵄨
󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
{|(𝜕/𝜕𝑥𝑖 )𝑢|𝛼𝑖 (⋅) >𝑡; |𝑢|≤𝑡}
𝑁
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖 (𝑥)
1
󵄨󵄨
󵄨󵄨
𝑑𝑥
≤∑ ∫
󵄨󵄨
󵄨󵄨
𝛼𝑖 (⋅)
󵄨
󵄨
𝑡
𝜕𝑥
{|(𝜕/𝜕𝑥
)𝑢|
>𝑡;
|𝑢|≤𝑡}
󵄨
𝑖󵄨
𝑖
𝑖=1
󵄩 󵄩
+ 𝑀 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 + 𝑀.
(87)
International Journal of Differential Equations
9
Lemma 18. Assume (4)–(8) and f ∈ L1 (Ω). Let u be an entropy
solution of (1), then
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
1
󵄨󵄨
󵄨
𝑇ℎ (𝑢)󵄨󵄨󵄨
𝑑𝑥 ≤ 𝑀
∑∫
󵄨󵄨
󵄨󵄨
ℎ 𝑖=1 {|𝑢|≤ℎ} 󵄨󵄨 𝜕𝑥𝑖
Therefore,
−
ℎ𝑝𝑀 −1 meas {|𝑢| > ℎ} ≤ 𝐷 (𝑓, Ω)
𝑁
(88)
which implies
for every h > 0, with M a positive constant. Moreover, we have
󵄩
󵄩
󵄩󵄩 𝑝𝑀 (𝑥)−2 󵄩󵄩
󵄩 󵄩
󵄩󵄩|𝑢|
𝑢󵄩󵄩󵄩1 = 󵄩󵄩󵄩󵄩|𝑢|𝑝𝑀 (𝑥)−1 󵄩󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1
(89)
󵄩
and there exists a constant D > 0 which depends on f and Ω
such that
meas {|𝑢| > ℎ} ≤
𝐷
−
𝑃𝑀
−1
ℎ
,
∀ℎ > 0.
(90)
Proof. Taking 𝜑 = 0 in the entropy inequality (80) and using
(7), we obtain
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
󵄨󵄨
󵄨
󵄩 󵄩
𝑇ℎ (𝑢)󵄨󵄨󵄨
𝑑𝑥 ≤ ℎ󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 ≤ 𝑀ℎ,
∑∫
󵄨󵄨
󵄨
󵄨
𝜕𝑥
{|𝑢|≤ℎ}
󵄨 𝑖
󵄨
𝑖=1
meas {|𝑢| > ℎ} ≤
𝐷 (𝑓, Ω)
−
ℎ𝑝𝑀 −1
.
𝑁
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑑𝑥 ≤ 𝐶 (𝑘 + 1) ,
󵄨
󵄨
𝜕𝑥
{|𝑢|≤𝑘} 󵄨
𝑖󵄨
󵄨𝑃−
󵄨
∫ 󵄨󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥 + ∑ ∫
Ω
𝑖=1
(91)
Ω
Proof. Taking 𝜑 = 0 in the entropy inequality (80) and using
(7), we get
for all ℎ > 0. This yields
{|𝑢|>ℎ}
∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑘 (𝑢) 𝑑𝑥
Ω
(92)
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
󵄨󵄨
󵄨
󵄩 󵄩
+ ∑∫
𝑇𝑘 (𝑢)󵄨󵄨󵄨
𝑑𝑥 ≤ 𝑘󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
󵄨󵄨
󵄨
󵄨
𝜕𝑥
󵄨
𝑖
𝑖=1 {|𝑢|≤𝑘} 󵄨
𝑁
As 𝑢𝑇ℎ (𝑢)𝜒{|𝑢|>ℎ} = ℎ|𝑢|𝜒{|𝑢|>ℎ} , we get from the previous
inequality by using Fatou’s lemma
󵄩 󵄩
∫ |𝑢|𝑝𝑀 (𝑥)−2 |𝑢| 𝑑𝑥 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
Ω
(94)
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨
𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑑𝑥
∑∫
󵄨󵄨
󵄨
󵄨󵄨
𝑖=1 {|𝑢|≤𝑘} 󵄨 𝜕𝑥𝑖
𝑁
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨
𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑑𝑥
󵄨󵄨
󵄨󵄨
{|𝑢|≤𝑘,|𝜕𝑢/𝜕𝑥𝑖 |≤1} 󵄨󵄨 𝜕𝑥𝑖
𝑁
= ∑∫
𝑖=1
We deduce that
󵄨𝑝− −1
󵄨
∫ 󵄨󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥 ≤ 𝐷 (𝑓, Ω) .
Ω
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨
+ ∑∫
𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑑𝑥
󵄨󵄨
󵄨
󵄨󵄨
𝜕𝑥
𝑖
𝑖=1 {|𝑢|≤𝑘,|𝜕𝑢/𝜕𝑥𝑖 |>1} 󵄨
𝑁
(95)
Indeed,
𝑁
𝑖=1
𝑁
󵄨𝑝− −1
󵄨󵄨
󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
{|𝑇 (𝑢)|≤1}
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
󵄨󵄨
󵄨
𝑇𝑘 (𝑢)󵄨󵄨󵄨
𝑑𝑥,
󵄨󵄨
󵄨󵄨
{|𝑢|≤𝑘} 󵄨󵄨 𝜕𝑥𝑖
≤ 𝑁 meas (Ω) + ∑ ∫
≤∫
𝑖=1
ℎ
󵄨𝑝− −1
󵄨󵄨
󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
{|𝑇 (𝑢)|>1}
󵄨󵄨 𝜕
󵄨󵄨𝑝𝑖 (𝑥)
󵄨󵄨
󵄨
𝑇𝑘 (𝑢)󵄨󵄨󵄨
𝑑𝑥
󵄨󵄨
󵄨󵄨
{|𝑢|≤𝑘,|𝜕𝑢/𝜕𝑥𝑖 |>1} 󵄨󵄨 𝜕𝑥𝑖
≤ 𝑁 meas (Ω) + ∑ ∫
󵄨𝑝− −1
󵄨
∫ 󵄨󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
Ω
+∫
(96)
ℎ
󵄨 𝑝−
󵄨
∫ 󵄨󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
Ω
󵄨𝑝−
󵄨 𝑝−
󵄨󵄨
󵄨󵄨
󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥 + ∫
󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
{|𝑇 (𝑢)|≤1}
{|𝑇 (𝑢)|>1}
󵄨𝑝 (𝑥)−1
󵄨
≤ meas (Ω) + ∫ 󵄨󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀
𝑑𝑥
Ω
≤∫
󵄩 󵄩
≤ meas (Ω) + 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
󵄨𝑝 (𝑥)
󵄨
≤ meas (Ω) + ∫ 󵄨󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥
Ω
𝑘
From aforementioned, we get
󵄨𝑝− −1
󵄨󵄨
∫
󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀 𝑑𝑥 ≤ 𝐷 (𝑓, Ω) .
{|𝑢|>ℎ}
(101)
Note that
(93)
Now, since |𝑇ℎ (𝑢)| ≤ |𝑢| we have
󵄨𝑝 (𝑥)−1
󵄨
󵄩 󵄩
𝑑𝑥 ≤ ∫ |𝑢|𝑝𝑀 (𝑥)−1 𝑑𝑥 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
∫ 󵄨󵄨󵄨𝑇ℎ (𝑢)󵄨󵄨󵄨 𝑀
Ω
Ω
(100)
∀𝑘 > 0.
󵄩 󵄩
∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇ℎ (𝑢) 𝑑𝑥 ≤ ℎ󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 ,
󵄩 󵄩
|𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇ℎ (𝑢) 𝑑𝑥 ≤ ℎ󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
(99)
Lemma 19. If u is an entropy solution of (1) then there exists
a constant C > 0 such that
𝑁
∫
(98)
(97)
𝑘
≤ meas (Ω) + ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑘 (𝑢) 𝑑𝑥.
Ω
(102)
10
International Journal of Differential Equations
Lemma 22. Assume (4)–(8) and f ∈ L1 (Ω). Let u be an
entropy solution of (1), then
Therefore, we deduce according to (101) that
𝑁
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨󵄨
󵄨󵄨𝑃𝑀−
󵄨󵄨
∫ 󵄨󵄨𝑇𝑘 (𝑢)󵄨󵄨 𝑑𝑥 + ∑ ∫
󵄨󵄨
󵄨󵄨 𝑑𝑥
󵄨
󵄨
Ω
𝑖=1 {|𝑢|≤𝑘} 󵄨 𝜕𝑥𝑖 󵄨
󵄩 󵄩
≤ (𝑁 + 1) meas (Ω) + 𝑘󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 , ∀𝑘 > 0.
(103)
∀𝑖 = 1, . . . , 𝑁,
(104)
where F = {h < |u| ≤ h + t}, h > 0, t > 0.
Proof. Taking 𝜑 = 𝑇ℎ (𝑢) as a test function in the entropy
inequality (80), we get
𝑁
𝜕
𝜕
𝑢) ⋅
𝑇𝑡 (𝑢 − 𝑇ℎ (𝑢)) 𝑑𝑥
∑ ∫ 𝑎𝑖 (𝑥,
𝜕𝑥
𝜕𝑥
Ω
𝑖
𝑖
𝑖=1
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (𝑢)) 𝑑𝑥
𝑁
∑𝑎𝑖 (𝑥,
∫
{|𝑢−𝑇ℎ (V)|≤𝑡} 𝑖=1
⋅
(105)
≤ ∫ 𝑓 (𝑥) 𝑇𝑡 (𝑢 − 𝑇ℎ (𝑢)) 𝑑𝑥.
𝜕
𝑢)
𝜕𝑥𝑖
𝜕
(𝑢 − 𝑇ℎ (V)) 𝑑𝑥
𝜕𝑥𝑖
𝑁
+∫
Ω
∑𝑎𝑖 (𝑥,
{|V−𝑇ℎ (𝑢)|≤𝑡} 𝑖=1
It follows by using (7) that
⋅
(106)
𝜕
V)
𝜕𝑥𝑖
𝜕
(V − 𝑇ℎ (𝑢)) 𝑑𝑥
𝜕𝑥𝑖
(112)
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (V)) 𝑑𝑥
Ω
Therefore,
󵄨󵄨 𝜕 󵄨󵄨𝑝𝑖 (𝑥)−1
󵄨
󵄨
𝜌𝑝󸀠 (⋅) (󵄨󵄨󵄨
𝑢󵄨󵄨
𝜒𝐹 ) ≤ 𝐶,
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
(111)
Step 2 (uniqueness of entropy solution). The proof of the
uniqueness of entropy solutions follows the same techniques
by Ouaro [20] (see also [35]). Indeed, let ℎ > 0 and 𝑢, V be
two entropy solutions of (1). We write the entropy inequality
(54) corresponding to the solution 𝑢, with 𝑇ℎ (V) as test
function, and to the solution V, with 𝑇ℎ (𝑢) as test function.
Upon addition, we get
Ω
󵄨󵄨 𝜕 󵄨󵄨𝑝𝑖 (𝑥)
󵄨
󵄨
󵄩 󵄩
𝑢󵄨󵄨󵄨
𝑑𝑥 ≤ 𝑡󵄩󵄩󵄩𝑓󵄩󵄩󵄩1 .
∫ 󵄨󵄨󵄨
󵄨
󵄨
𝜕𝑥
𝐹󵄨
𝑖 󵄨
∀ℎ ≥ 1, ∀𝑖 = 1, . . . , 𝑁,
where D󸀠 is a positive constant which depends on f and p−M .
Lemma 20. If u is an entropy solution of (1) then
󵄨󵄨 𝜕 󵄨󵄨𝑝𝑖 (𝑥)−1
󵄨
󵄨
𝜌𝑝󸀠 (⋅) (󵄨󵄨󵄨
𝑢󵄨󵄨
𝜒𝐹 ) ≤ 𝐶,
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
󵄨󵄨 𝜕 󵄨󵄨
𝐷󸀠
󵄨
󵄨
meas {󵄨󵄨󵄨
𝑢󵄨󵄨󵄨 > ℎ} ≤
,
− 󸀠
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
ℎ1/(𝑃𝑀 )
+ ∫ |V|𝑝𝑀 (𝑥)−2 V𝑇𝑡 (V − 𝑇ℎ (𝑢)) 𝑑𝑥
Ω
∀𝑖 = 1, . . . , 𝑁.
(107)
Ω
Define
Lemma 21. If u is an entropy solution of (1) then
󵄨 󵄨
lim ∫ 󵄨󵄨𝑓󵄨󵄨 𝜒{|𝑢|>ℎ−𝑡} 𝑑𝑥 = 0,
ℎ → +∞ Ω 󵄨 󵄨
≤ ∫ 𝑓 (𝑥) (𝑇𝑡 (𝑢 − 𝑇ℎ (V)) + 𝑇𝑡 (V − 𝑇ℎ (𝑢))) 𝑑𝑥.
𝐸1 := {|𝑢 − V| ≤ 𝑡, |V| ≤ ℎ} ,
(108)
𝐸2 := 𝐸1 ∩ {|𝑢| ≤ ℎ} ,
(113)
𝐸3 := 𝐸1 ∩ {|𝑢| > ℎ} .
where h > 0, t > 0.
Proof. By Lemma 18, we deduce that
󵄨 󵄨
lim 󵄨󵄨𝑓󵄨󵄨󵄨 𝜒{|𝑢|>ℎ−𝑡} = 0
ℎ → +∞ 󵄨
We start with the first integral in (112). By (7), we have
(109)
and as 𝑓 ∈ 𝐿1 (Ω), it follows by using the Lebesgue dominated
convergence theorem that
󵄨 󵄨
lim ∫ 󵄨󵄨󵄨𝑓󵄨󵄨󵄨 𝜒{|𝑢|>ℎ−𝑡} 𝑑𝑥 = 0.
ℎ → +∞ Ω
The proof of the following lemma can be found in [1].
(110)
∫
𝑁
∑𝑎𝑖 (𝑥,
{|𝑢−𝑇ℎ V|≤𝑡} 𝑖=1
𝑁
𝜕
𝜕
𝑢) ⋅
(𝑢 − 𝑇ℎ (V)) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
≥ ∫ ∑𝑎𝑖 (𝑥,
𝐸2 𝑖=1
𝑁
𝜕
𝜕
𝑢) ⋅
(𝑢 − V) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
− ∫ ∑𝑎𝑖 (𝑥,
𝐸3 𝑖=1
𝜕
𝜕
𝑢) ⋅
V𝑑𝑥.
𝜕𝑥𝑖
𝜕𝑥𝑖
(114)
International Journal of Differential Equations
11
Using (5) and Proposition 1, we estimate the last integral in
(114) as follows:
󵄨󵄨
󵄨󵄨
𝑁
󵄨󵄨
󵄨
󵄨󵄨∫ ∑𝑎𝑖 (𝑥, 𝜕 𝑢) ⋅ 𝜕 V𝑑𝑥󵄨󵄨󵄨
󵄨󵄨 𝐸
󵄨󵄨
𝜕𝑥𝑖
𝜕𝑥𝑖
󵄨󵄨 3 𝑖=1
󵄨󵄨
Then, by the Lebesgue dominated convergence theorem, we
obtain
∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (V)) 𝑑𝑥
Ω
(121)
𝑝𝑀 (𝑥)−2
󳨀→ ∫ |𝑢|
𝑁
󵄨󵄨 𝜕 󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨 𝜕 󵄨󵄨
󵄨
󵄨
󵄨
󵄨
≤ 𝐶1 ∫ ∑ (𝑗𝑖 (𝑥) + 󵄨󵄨󵄨
𝑢󵄨󵄨󵄨
) 󵄨󵄨󵄨
V󵄨󵄨󵄨 𝑑𝑥
󵄨
󵄨
󵄨
𝜕𝑥
𝜕𝑥
𝐸3 𝑖=1
󵄨 𝑖 󵄨
󵄨 𝑖 󵄨󵄨
Ω
(115)
󵄨
󵄨󵄨󵄨󵄨
󵄨󵄨 𝜕 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨󵄨
󵄨 󵄨
󵄨󵄨
≤ 𝐶∑ (󵄨󵄨󵄨𝑗󵄨󵄨󵄨𝑝󸀠 (⋅) + 󵄨󵄨󵄨󵄨󵄨󵄨󵄨
𝑢󵄨󵄨
)
󵄨󵄨 󸀠
𝑖
󵄨󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
󵄨𝑝𝑖 (⋅),{ℎ<|𝑢|≤ℎ+𝑡}
𝑖=1
𝑁
∫ |V|𝑝𝑀 (𝑥)−2 V𝑇𝑡 (V − 𝑇ℎ (𝑢)) 𝑑𝑥
Ω
(122)
󳨀→ ∫ |V|𝑝𝑀 (𝑥)−2 V𝑇𝑡 (V − 𝑢) 𝑑𝑥,
Ω
as ℎ 󳨀→ ∞.
Therefore,
where
󵄨󵄨󵄨󵄨
󵄨
󵄨󵄨󵄨󵄨 𝜕 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨󵄨
󵄨󵄨󵄨󵄨
󵄨󵄨
𝑢
󵄨
󵄨󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
󵄨󵄨 󸀠
󵄨
󵄨𝑝𝑖 (⋅),{ℎ<|𝑢|≤ℎ+𝑡}
lim 𝐾ℎ = ∫ (|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) 𝑇𝑡 (𝑢 − V) 𝑑𝑥.
ℎ→∞
For all 𝑖
𝑖
||(𝜕/𝜕𝑥𝑖 )𝑢|
|𝑝󸀠 (⋅),{ℎ<|𝑢|≤ℎ+𝑡} ) is finite according to relations
𝑖
(18), (19) and Lemma 20. The quantity |(𝜕/𝜕𝑥𝑖 )V|𝑝𝑖 (⋅),{ℎ−𝑡<|V|≤ℎ}
converges to zero as ℎ goes to infinity according to Lemma 21.
Then, the last expression in (115) converges to zero as ℎ tends
to infinity. Therefore, from (114), we obtain
𝑁
∑𝑎𝑖 (𝑥,
{|𝑢−𝑇ℎ V|≤𝑡} 𝑖=1
𝜕
𝜕
𝑢) ⋅
(𝑢 − 𝑇ℎ (V)) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
𝑁
≥ 𝐼ℎ + ∫ ∑𝑎𝑖 (𝑥,
𝐸2 𝑖=1
𝑁
𝑁
≥ 𝐽ℎ − ∫ ∑𝑎𝑖 (𝑥,
𝐸2 𝑖=1
𝐾ℎ = ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (V)) 𝑑𝑥
(119)
𝑝𝑀 (𝑥)−2
Ω
a.e. in Ω
V𝑇𝑡 (V − 𝑇ℎ (𝑢)) 𝑑𝑥.
𝑝𝑀 (𝑥)−2
We have |𝑢|
𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (V)) → |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − V) a.e.
as ℎ goes to infinity and
|𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑡 (𝑢 − 𝑇ℎ (V)) |≤ 𝑡|𝑢|𝑝𝑀 (𝑥)−2 𝑢 ∈ 𝐿1 (Ω) . (120)
as ℎ 󳨀→ ∞,
(125)
󵄨󵄨
󵄨
󵄨󵄨𝑓 (𝑥) (𝑇𝑡 (𝑢 − 𝑇ℎ (V)) + 𝑇𝑡 (V − 𝑇ℎ (𝑢)))󵄨󵄨󵄨
󵄨
󵄨
≤ 2𝑡 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 ∈ 𝐿1 (Ω) ,
so that we are able to apply the Lebesgue dominated convergence theorem. Then, we deduce from relations (112)–(124)
after passing to the limit as ℎ → ∞ in (112) the following:
𝑁
(118)
where 𝐽ℎ converges to zero as ℎ tends to infinity.
For the two other terms in the left-hand side of (112), we
denote
+ ∫ |V|
𝑓 (𝑥) (𝑇𝑡 (𝑢 − 𝑇ℎ (V)) + 𝑇𝑡 (V − 𝑇ℎ (𝑢)))
(117)
𝜕
𝜕
V) ⋅
(𝑢 − V) 𝑑𝑥,
𝜕𝑥𝑖
𝜕𝑥𝑖
Ω
(124)
Indeed,
𝜕
𝜕
𝑢) ⋅
(𝑢 − V) 𝑑𝑥,
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕
𝜕
V) ⋅
(V − 𝑇ℎ (𝑢)) 𝑑𝑥
𝜕𝑥𝑖
𝜕𝑥𝑖
{|V−𝑇ℎ (𝑢)|≤𝑡} 𝑖=1
lim ∫ 𝑓 (𝑥) (𝑇𝑡 (𝑢 − 𝑇ℎ (V)) + 𝑇𝑡 (V − 𝑇ℎ (𝑢))) 𝑑𝑥 = 0.
ℎ→∞ Ω
󳨀→ 𝑓 (𝑥) (𝑇𝑡 (𝑢 − V) + 𝑇𝑡 (V − 𝑢)) = 0
where 𝐼ℎ converges to zero as ℎ tends to infinity. We may
adopt the same procedure to treat the second term in (112)
to obtain
∑𝑎𝑖 (𝑥,
(123)
Furthermore, consider the right-hand side of inequality
(112). We have
1, . . . , 𝑁, the quantity (|𝑗𝑖 |𝑝󸀠 (⋅) +
=
𝑝𝑖 (𝑥)−1
Ω
(116)
󵄩󵄩󵄨󵄨
󵄩
󵄩󵄨 𝜕 󵄨󵄨󵄨󵄨𝑝𝑖 (𝑥)−1 󵄩󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩󵄩󵄨󵄨󵄨
𝑢󵄨󵄨
.
󵄩󵄩 𝑝𝑖󸀠 (⋅)
󵄩󵄩󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
󵄩𝐿 ({ℎ<|𝑢|≤ℎ+𝑡})
∫
as ℎ 󳨀→ ∞.
In the same way, we get
󵄨󵄨 𝜕 󵄨󵄨
󵄨
󵄨
⋅ 󵄨󵄨󵄨
V󵄨󵄨
,
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨𝑝 (⋅),{ℎ−𝑡<|V|≤ℎ}
𝑖
∫
𝑢𝑇𝑡 (𝑢 − V) 𝑑𝑥,
∑∫
𝑖=1 {|𝑢−V|≤𝑡}
(𝑎𝑖 (𝑥,
𝜕
𝜕
𝜕
𝑢) − 𝑎𝑖 (𝑥,
V)) ⋅
(𝑢 − V)
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥𝑖
+ ∫ (|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) 𝑇𝑡 (𝑢 − V) ≤ 0.
Ω
(126)
Using (6) and as 𝑡 󳨃→ |𝑡|𝑝𝑀 (𝑥)−2 𝑡 is monotone, we deduce from
(126) that
∫ (|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) 𝑇𝑡 (𝑢 − V) 𝑑𝑥 ≤ 0.
Ω
(127)
−
Since 𝑝𝑀
> 1, the following relation is true for any 𝜉, 𝜂 ∈ R,
𝜉 ≠ 𝜂 (cf. [12])
󵄨 󵄨𝑝 (𝑥)−2
󵄨 󵄨𝑝 (𝑥)−2
𝜉 − 󵄨󵄨󵄨𝜂󵄨󵄨󵄨 𝑀
𝜂) (𝜉 − 𝜂) > 0.
(󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝑀
(128)
12
International Journal of Differential Equations
Therefore, from (127), we get that (|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V)
𝑇𝑡 (𝑢 − V) = 0 a.e. in Ω, which means that for all 𝑡 ∈ R+ , there
exists Ω𝑡 ⊂ Ω with meas (Ω𝑡 ) = 0 such that for all 𝑥 ∈ Ω \ Ω𝑡 ,
(|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) 𝑇𝑡 (𝑢 − V) = 0.
(129)
Therefore,
(|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − |V|𝑝𝑀 (𝑥)−2 V) (𝑢 − V) = 0,
∀𝑥 ∈ Ω \ ⋃ Ω𝑡 .
Then, let 𝜆 |𝜕𝑢𝑛 /𝜕𝑥𝑖 | (𝛼) = meas{𝑥 ∈ Ω : |𝜕𝑢𝑛 /𝜕𝑥𝑖 | > 𝛼} for all
𝑖 = 1, . . . , 𝑁, we have for any 𝛼 > 1, 𝛾 > 0,
󵄨󵄨 𝜕𝑢 󵄨󵄨
󵄨 󵄨
󵄨
󵄨
𝜆 |𝜕𝑢𝑛 /𝜕𝑥𝑖 | (𝛼) ≤ meas {𝑥 ∈ Ω : 󵄨󵄨󵄨 𝑛 󵄨󵄨󵄨 > 𝛼, 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 ≤ 𝛾}
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨
󵄨 󵄨
+ meas {𝑥 ∈ Ω : 󵄨󵄨󵄨 𝑛 󵄨󵄨󵄨 > 𝛼, 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 > 𝛾}
󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
(130)
1 󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨 𝑝𝑖
( 󵄨󵄨󵄨 𝑛 󵄨󵄨󵄨) 𝑑𝑥 + 𝜆 |𝑢𝑛 | (𝛾)
{|𝜕𝑢𝑛 /𝜕𝑥𝑖 |>𝛼,|𝑢𝑛 |≤𝛾} 𝛼 󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨
−
≤ ∫
𝑡∈N∗
Now, using (128) and (130), we obtain
𝑢1 = 𝑢2
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖
1
󵄨󵄨 𝑛 󵄨󵄨
∫
󵄨
󵄨 𝑑𝑥 + 𝜆 |𝑢𝑛 | (𝛾) .
−
𝛼𝑝𝑖 {|𝑢𝑛 |≤𝛾} 󵄨󵄨󵄨 𝜕𝑥𝑖 󵄨󵄨󵄨
−
a.e. in Ω.
≤
(131)
Step 3 (Existence of entropy solutions). Let (𝑓𝑛 )𝑛∈N∗ be a
sequence of bounded functions, strongly converging to 𝑓 ∈
𝐿1 (Ω) and such that
󵄩󵄩󵄩𝑓𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 , ∀𝑛 ∈ N∗ .
(132)
󵄩 󵄩1 󵄩 󵄩1
Using (135) and (i), we get
𝜆 |𝜕𝑢𝑛 /𝜕𝑥𝑖 | (𝛼) ≤ 𝐶 (
We consider the problem
𝑁
𝜕
𝜕
󵄨 󵄨𝑝 (𝑥)−2
𝑎𝑖 (𝑥,
𝑢𝑛 ) + 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
𝑢𝑛 = 𝑓𝑛
𝜕𝑥𝑖
𝑖=1 𝜕𝑥𝑖
−∑
𝑁
𝜕
𝑢 )𝜈 = 0
∑𝑎𝑖 (𝑥,
𝜕𝑥𝑖 𝑛 𝑖
𝑖=1
in Ω,
(133)
on 𝜕Ω.
It follows from Theorem 10 that problem (133) admits a
⃗
(Ω) which satisfies
unique weak solution 𝑢𝑛 ∈ 𝑊1,𝑝(⋅)
𝑁
∑ ∫ 𝑎𝑖 (𝑥,
𝑖=1 Ω
𝜕𝑢𝑛 𝜕
󵄨 󵄨𝑝 (𝑥)−2
)
𝜑𝑑𝑥 + ∫ 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
𝑢𝑛 𝜑𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖
Ω
𝛾
̃
−𝑝
),
− + 𝛾
𝛼𝑝𝑖
(137)
from which we deduce (ii).
By lemmas 3 and 23, it follows that (𝑢𝑛 )𝑛∈N∗ is uniformly
̃ , and in the same
bounded in 𝐿𝑠0 (Ω) for some 1 ≤ 𝑠0 < 𝑝
way, (|𝜕𝑢𝑛 /𝜕𝑥𝑖 |)𝑛∈N∗ is uniformly bounded in 𝐿𝑠𝑖 (Ω) for some
̃ 𝑖 . From this, we get that the sequence (𝑢𝑛 )𝑛∈N∗ is
1 ≤ 𝑠𝑖 < 𝑝
uniformly bounded in 𝑊1,𝑠 (Ω), where 𝑠 = min(𝑠0 , 𝑠1 , . . . , 𝑠𝑁).
Consequently, we can extract a subsequence, still denoted
(𝑢𝑛 ) satisfying
𝑢𝑛 󳨀→ 𝑢 a.e. in Ω, in 𝐿𝑠 (Ω) ,
𝑢𝑛 ⇀ 𝑢
(134)
= ∫ 𝑓𝑛 (𝑥) 𝜑𝑑𝑥,
Ω
⃗
(Ω).
for all 𝜑 ∈ 𝑊1,𝑝(⋅)
Our interest is to prove that these approximated solutions
𝑢𝑛 tend, as 𝑛 goes to infinity, to a measurable function 𝑢 which
is an entropy solution of the problem (1). We announce the
following important lemma, useful to get some convergence
results.
Lemma 23. If un is a weak solution of (126) then there exist
some constants C1 , C2 > 0 such that
(i) ‖𝑢𝑛 ‖M𝑝̃ (Ω) ≤ 𝐶1 ,
󵄨󵄨 𝜕𝑢
󵄨
󵄨󵄨 𝑛 𝜕𝑢 󵄨󵄨󵄨
−
󵄨󵄨
󵄨 ⇀ H𝑖 (𝑥)
󵄨󵄨 𝜕𝑥𝑖 𝜕𝑥𝑖 󵄨󵄨󵄨
Proof. (i) is a consequence of Lemmas 19 and 5 by using
𝑇𝑘 (𝑢𝑛 ) for all 𝑘 > 0 as a test function in (134).
(ii) We first use 𝑇𝛾 (𝑢𝑛 ) for all 𝛾 > 0 as a test function in
(134) to get
𝑁
(135)
in 𝑊1,𝑠 (Ω) ,
(138)
in 𝐿𝑠 (Ω) , ∀𝑖 = 1, . . . , 𝑁.
By the same way as in the proof of [16, Lemma 3.5] (see also
[27]), we prove that
H𝑖 (𝑥) = 0 a.e. 𝑥 ∈ Ω ∀𝑖 = 1, . . . , 𝑁.
(139)
We deduce from (139) that
𝑎𝑖 (𝑥,
(ii) ‖𝜕𝑢𝑛 /𝜕𝑥𝑖 ‖M𝑝𝑖− 𝑞/𝑝 (Ω) ≤ 𝐶2 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 = 1, . . . , 𝑁.
󵄨󵄨 𝜕𝑢 󵄨󵄨𝑝𝑖−
󵄨󵄨
󵄨󵄨
∑∫
󵄨󵄨
󵄨󵄨 𝑑𝑥 ≤ 𝐶 (𝛾 + 1) .
󵄨
󵄨
𝜕𝑥
𝑖󵄨
𝑖=1 {|𝑢|≤𝛾} 󵄨
(136)
𝜕𝑢𝑛
)
𝜕𝑥𝑖
󳨀→ 𝑎𝑖 (𝑥,
𝜕𝑢
) a.e. in Ω, in 𝐿1 (Ω) , ∀𝑖 = 1, . . . , 𝑁.
𝜕𝑥𝑖
(140)
In order to pass to the limit in relation (134), we need also
the following convergence results which can be proved by the
same way as in [1].
International Journal of Differential Equations
13
Proposition 24. Assume (4)–(8), 𝑓 ∈ 𝐿1 (Ω) and (132). Let
⃗
𝑢𝑛 ∈ 𝑊1,𝑝(⋅)
(Ω) be the solution of (133). The sequence (𝑢𝑛 )𝑛∈N
is Cauchy in measure. In particular, there exists a measurable
function 𝑢 and a subsequence still denoted by 𝑢𝑛 such that
𝑢𝑛 → 𝑢 in measure.
Proposition 25. Assume (4)–(8), 𝑓 ∈ 𝐿1 (Ω) and (132).
⃗
(Ω) be the solution of (133). The following
Let 𝑢𝑛 ∈ 𝑊1,𝑝(⋅)
assertions hold.
(i) For all 𝑖 = 1, . . . , 𝑁, 𝜕𝑢𝑛 /𝜕𝑥𝑖 converges in measure to
the weak partial gradient of 𝑢.
(ii) For all 𝑖 = 1, . . . , 𝑁 and all 𝑘 > 0, 𝑎𝑖 (𝑥, 𝜕𝑇𝑘 (𝑢𝑛 )/𝜕𝑥𝑖 )
converges to 𝑎𝑖 (𝑥, 𝜕𝑇𝑘 (𝑢𝑛 )/𝜕𝑥𝑖 ) in 𝐿1 (Ω) strongly and
󸀠
in 𝐿𝑝𝑖 (⋅) (Ω) weakly.
We can now pass to the limit in (134). To this end, let 𝜑 ∈
1,𝑝(⋅)
𝑊 ⃗ (Ω) ∩ 𝐿∞ (Ω). For any 𝑘 > 0, choose 𝑇𝑘 (𝑢𝑛 − 𝜑) as a test
function in (134), we get
Let us show that |𝜑|𝑝𝑀 (𝑥)−2 𝜑 ∈ 𝐿1 (Ω).
We have
󵄨󵄨󵄨 󵄨𝑝 (𝑥)−2 󵄨󵄨
󵄨 󵄨𝑝 (𝑥)−1
𝜑󵄨󵄨󵄨 𝑑𝑥 = ∫ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝑑𝑥
∫ 󵄨󵄨󵄨󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
󵄨
Ω󵄨
Ω
󵄩 󵄩 𝑝 (𝑥)−1
≤ ∫ (󵄩󵄩󵄩𝜑󵄩󵄩󵄩∞ ) 𝑀
𝑑𝑥.
Ω
If ‖𝜑‖∞ ≤ 1, then ∫Ω ||𝜑|𝑝𝑀 (𝑥)−2 𝜑|𝑑𝑥 ≤ meas(Ω) < +∞.
If ‖𝜑‖∞ > 1, then
󵄨󵄨󵄨 󵄨𝑝 (𝑥)−2 󵄨󵄨
󵄩 󵄩 𝑝+ −1
𝜑󵄨󵄨󵄨 𝑑𝑥 ≤ ∫ (󵄩󵄩󵄩𝜑󵄩󵄩󵄩∞ ) 𝑀 𝑑𝑥
∫ 󵄨󵄨󵄨󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
󵄨
Ω󵄨
Ω
󵄩 󵄩 𝑝+ −1
= (󵄩󵄩󵄩𝜑󵄩󵄩󵄩∞ ) 𝑀 meas (Ω) < +∞.
󵄨 󵄨𝑝 (𝑥)−2
lim ∫ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥
𝜕𝑢
𝜕
𝑇 (𝑢 − 𝜑) 𝑑𝑥
∑ ∫ 𝑎𝑖 (𝑥, 𝑛 )
𝜕𝑥𝑖 𝜕𝑥𝑖 𝑘 𝑛
𝑖=1 Ω
𝑛 → +∞ Ω
󵄨 󵄨𝑝 (𝑥)−2
+ ∫ 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
𝑢𝑛 𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥
Ω
󵄨 󵄨𝑝 (𝑥)−2
= ∫ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥.
Ω
(141)
𝑁
Ω
∑∫
𝑖=1 {|𝑢𝑛 −𝜑|≤𝑘}
For the right-hand side of (141), the convergence is obvious
since 𝑓𝑛 converges strongly to 𝑓 in 𝐿1 (Ω), and 𝑇𝑘 (𝑢𝑛 − 𝜑)
converges weakly-∗ to 𝑇𝑘 (𝑢 − 𝜑) in 𝐿∞ (Ω) and a.e in Ω.
For the second term of (141), we have
󵄨 󵄨𝑝 (𝑥)−2
𝑢𝑛 𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥
∫ 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
Ω
󵄨 󵄨𝑝 (𝑥)−2
+ ∫ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥.
Ω
𝜕𝑢𝑛 𝜕
)
𝑢 𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖 𝑛
𝑁
(148)
𝜕𝑢
𝜕
𝑎𝑖 (𝑥, 𝑛 )
𝜑𝑑𝑥.
− ∑∫
𝜕𝑥
𝜕𝑥
{|𝑢
−𝜑|≤𝑘}
𝑖
𝑖
𝑛
𝑖=1
𝑁
∑∫
𝑖=1 {|𝑢−𝜑|≤𝑘}
𝑎𝑖 (𝑥,
𝜕𝑢
𝜕
)
𝑢𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖
𝑁
The quantity (|𝑢𝑛 |𝑝𝑀 (𝑥)−2 𝑢𝑛 − |𝜑|𝑝𝑀 (𝑥)−2 𝜑)𝑇𝑘 (𝑢𝑛 − 𝜑) is
nonnegative and since for all 𝑥 ∈ Ω, 𝑠 󳨃→ |𝑠|𝑝𝑀 (𝑥)−2 𝑠 is
continuous; we get
According to Proposition 25, the second term of (148) converges to
𝑁
∑∫
a.e. in Ω.
(143)
𝑖=1 {|𝑢−𝜑|≤𝑘}
∑ ∫ 𝑎𝑖 (𝑥,
󵄨 󵄨𝑝 (𝑥)−2
󵄨 󵄨𝑝 (𝑥)−2
𝑢𝑛 − 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑) 𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥
lim inf ∫ (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
𝑛 → +∞ Ω
𝑎𝑖 (𝑥,
𝜕𝑢
𝜕
)
𝜑𝑑𝑥.
𝜕𝑥𝑖 𝜕𝑥𝑖
𝑖=1 Ω
𝜕𝑢
𝜕
)
𝑇 (𝑢 − 𝜑) 𝑑𝑥
𝜕𝑥𝑖 𝜕𝑥𝑖 𝑘
+ ∫ |𝑢|𝑝𝑀 (𝑥)−2 𝑢𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥
Ω
󵄨 󵄨𝑝 (𝑥)−2
≥ ∫ (|𝑢|𝑝𝑀 (𝑥)−2 𝑢 − 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑) 𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥.
(144)
(150)
Combining the previous convergence results, we obtain
𝑁
Then, it follows by Fatou’s Lemma that
(149)
𝜕𝑢
𝜕
𝑎𝑖 (𝑥, 𝑛 )
𝑢𝑛 𝑑𝑥.
≤ lim inf ∑ ∫
𝑛→∞
𝜕𝑥
𝜕𝑥
{|𝑢
−𝜑|≤𝑘}
𝑖
𝑖
𝑛
𝑖=1
󵄨 󵄨𝑝 (𝑥)−2
󵄨 󵄨𝑝 (𝑥)−2
𝑢𝑛 − 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑) 𝑇𝑘 (𝑢𝑛 − 𝜑)
(󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
Ω
𝑎𝑖 (𝑥,
The first term of (148) is nonnegative by (7), then by Fatou’s
Lemma and (138), we get
󵄨 󵄨𝑝 (𝑥)−2
󵄨 󵄨𝑝 (𝑥)−2
= ∫ (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑀
𝑢𝑛 − 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑) 𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥 (142)
Ω
󳨀→ (|𝑢|
(147)
For the first term of (141), we write it as follows:
= ∫ 𝑓𝑛 (𝑥) 𝑇𝑘 (𝑢𝑛 − 𝜑) 𝑑𝑥.
󵄨 󵄨𝑝 (𝑥)−2
𝑢 − 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 𝑀
𝜑) 𝑇𝑘 (𝑢 − 𝜑)
(146)
Hence, |𝜑|𝑝𝑀 (𝑥)−2 𝜑 ∈ 𝐿1 (Ω).
Since 𝑇𝑘 (𝑢𝑛 −𝜑) converges weakly-∗ to 𝑇𝑘 (𝑢−𝜑) in 𝐿∞ (Ω)
and |𝜑|𝑝𝑀 (𝑥)−2 𝜑 ∈ 𝐿1 (Ω), it follows that
𝑁
𝑝𝑀 (𝑥)−2
(145)
≤ ∫ 𝑓 (𝑥) 𝑇𝑘 (𝑢 − 𝜑) 𝑑𝑥.
Ω
(151)
14
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Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014