# Entropy Solutions for Nonlinear Elliptic Anisotropic Homogeneous Neumann Problem

Authors: B. K. Bonzi; S. Ouaro; F. D. Y. Zongo
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We prove the existence and uniqueness of entropy solution for
nonlinear anisotropic elliptic equations with Neumann homogeneous
boundary value condition forL1-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.


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## Body

## 1. Introduction

LetΩ 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
(1)-∑i=1N‍∂∂xiai(x,∂∂xiu)+|u|pM(x)-2u=finΩ,∑i=1N‍ai(x,∂∂xiu)νi=0on∂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 anyΩ⊂ℝN, we set
(2)C+(Ω¯)={h∈C(Ω¯):infx∈Ωh(x)>1},
and we denote
(3)h+=supx∈Ωh(x),h-=infx∈Ωh(x).
For the exponents, p→(·):Ω¯→ℝN, p→(·)=(p1(·),…,pN(·)) with pi∈C+(Ω¯) for every i∈{1,…,N} and for all x∈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 fori=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 Ai:Ω×ℝ→ℝ, Ai=Ai(x,ξ), that is, ai(x,ξ)=(∂/∂ξ)Ai(x,ξ) such that the following equality and inequalities holds
(4)Ai(x,0)=0,
for almost every x∈Ω.There exists a positive constantC1 such that
(5)|ai(x,ξ)|≤C1(ji(x)+|ξ|pi(x)-1),
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 constantC2 such that
(6)(ai(x,ξ)-ai(x,η))·(ξ-η)≥{C2|ξ-η|pi(x)if|ξ-η|≥1,C2|ξ-η|pi-if|ξ-η|<1,
for almost every x∈Ω and for every ξ,η∈ℝ, with ξ≠η and
(7)|ξ|pi(x)≤ai(x,ξ)·ξ≤pi(x)Ai(x,ξ),
for almost every x∈Ω and for every ξ∈ℝ.We also assume that the variable exponentspi(·):Ω¯→[2,N) are continuous functions for all i=1,…,N such that
(8)p-(N-1)N(p--1)<pi-<p-(N-1)N-p-,∑i=1N‍1pi->1,pi+-pi--1pi-<p--Np-(N-1),
where 1/p-=(1/N)∑i=1N‍(1/pi-).We introduce the numbers(9)q=N(p--1)N-1,q*=NqN-q=N(p--1)N-p-.
A prototype example, that is, covered by our assumptions is the following anisotropic equation:SetAi(x,ξ)=(1/pi(x))|ξ|pi(x), ai(x,ξ)=|ξ|pi(x)-2ξ where pi(x)≥2. Then, we get the following equation. 						(10)-∑i=1N‍∂∂xi(|∂∂xiu|pi(x)-2∂∂xiu)+|u|pM(x)-2=f.
Actually, one of the topics from the field of PDEs that continuously gained interest is the one concerning the Sobolev space with variable exponents, W1,p(·)(Ω) or W01,p(·)(Ω) depending on the boundary condition (see [1–23]). In that context, problems involving the p(·)-Laplace operator
(11)Δp(x)u=div(|∇u|p(x)-2∇u)
or the more general operator
(12)diva(x,∇u)
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
(13)Δp→(x)u=∑i=1N‍∂xi(|∂xiu|pi(x)-2∂xiu)
or the more general variable exponent anisotropic operator
(14)∑i=1N‍∂xia(x,∂xiu).
When the homogeneous Dirichlet boundary condition is replaced by the Neumann boundary condition, one has to work with the anisotropic variable exponent Sobolev space W1,p→(·)(Ω) instead of W01,p→(·)(Ω). The main difficulty which 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 ℳp~ and ℳp~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 Section2, 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 functionp(·):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
(15)ρp(·)(u):=∫Ω‍|u|p(x)dx
is finite. If the exponent is bounded, that is, if p+<∞, then the expression
(16)|u|p(·):=inf{λ>0:ρp(·)(uλ)≤1}
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.Proposition 1 (generalized Hölder inequality, see [10]).
(i) For anyu∈Lp(·)(Ω) and v∈Lp′(·)(Ω), we have
(17)|∫Ω‍uvdx|≤(1p-+1p-′)|u|p(·)|v|p′(·).
(ii) Ifp1,p2∈𝒞+(Ω¯),  p1(x)≤p2(x) for any x∈Ω¯, then Lp2(x)(Ω)↪Lp1(x)(Ω) and the imbedding is continuous.Moreover, the applicationρp(·):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
(18)|u|p(·)<1⟹|u|p(·)p+≤ρp(·)(u)≤|u|p(·)p-,(19)|u|p(·)>1⟹|u|p(·)p-≤ρp(·)(u)≤|u|p(·)p+,(20)|u|p(·)<1(=1;>1)⟹ρp(·)(u)<1(=1;>1),(21)|u|p(·)⟶0(⟶∞)⟺ρp(·)(u)⟶0(⟶∞).
If, in addition, (un)n⊂Lp(·)(Ω), thenlim
n
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u
n
-
u
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p
(
·
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lim
n
→
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ρ
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(
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-
u
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u
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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(22)W1,p(·)(Ω):={u∈Lp(·)(Ω):|∇u|∈Lp(·)(Ω)},
which is a Banach space equipped with the norm
(23)∥u∥1,p(·):=|u|p(·)+|∇u|p(·).Now, we present a natural generalization of the variable exponent Sobolev spaceW1,p(·)(Ω) that will enable us to study the problem (1) with sufficient accuracy.Theanisotropic variable exponent Sobolev spaceW1,p→(·)(Ω) is defined as follows:
(24)W1,p→(·)(Ω)={u∈LpM(·)(Ω);∂u∂xi∈Lpi(·)(Ω),∀i∈{1,…,N}}.
Endowed with the norm
(25)∥u∥p→(·):=|u|pM(·)+∑i=1N‍|∂∂xiu|pi(·),
the space (W1,p→(·)(Ω),∥·∥p→(·)) is a reflexive Banach space (see [11, Theorems 2.1 and 2.2]).We have the following result.Theorem 2 (see [11, Corollary 2.1]).
LetΩ⊂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
(26)essinfx∈Ω(pM(x)-r(x))>0,
we have the compact embedding
(27)W1,p→(·)(Ω)↪Lr(·)(Ω).Next, we define(28)𝒯1,p→(·)(Ω)={u:Ω→ℝ;Tk(u)∈W1,p→(·)(Ω),∀k>0}.
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 spacesℳq(Ω)(1<q<∞) contain the measurable functions h:Ω→ℝ for which the distribution function
(29)λh(γ)=|{x∈Ω:|h(x)|>γ}|,γ≥0
satisfies an estimate of the form
(30)λh(γ)≤Cγ-q,forsomefiniteconstantC>0.
The space ℳq(Ω) is a Banach space under the norm
(31)∥h∥ℳq(Ω)*=supt>0t1/q(1t∫0t‍h*(s)ds),
where h* denotes the nonincreasing rearrangement of h:
(32)h*(t)=inf{γ>0:λh(γ)≤t}.
We will use the following pseudonorm
(33)∥h∥ℳq(Ω)=inf{C:λh(γ)≤Cγ-q,∀γ>0},
which is equivalent to the norm ∥h∥ℳq(Ω)* (see [27]).We need the following Lemma (see [28, Lemma  A.2]).Lemma 3.
Let1≤q<p<+∞. Then, for every measurable function u on Ω, we have(i)
((p-1)p/pp+1)∥u∥ℳp(Ω)p≤supλ>0{λpmeas[x∈Ω:|u(x)|>λ]}≤∥u∥ℳp(Ω)p.Moreover,(ii)
∫K‍|u|qdx≤(p/(p-q))(p/q)q/p∥u∥ℳp(Ω)q(meas(K))p-q/p, for every measurable subset K⊂Ω.
In particular,ℳp(Ω)⊂Llocq(Ω) with continuous injection and u∈ℳp(Ω) implies |u|q∈ℳp/q(Ω).The following result is due to Troisi (see [30]).Theorem 4.
Letp1,p2,…,pN∈[1,+∞);g∈W1,(p1,p2,…,pN)(Ω) and let
(34)q=p-*ifp-*<N,q∈[1,+∞)ifp-*≥N.
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
(35)∥g∥Lq(Ω)≤C∏i=1N‍[∥g∥LpM(Ω)+∥∂g∂xi∥Lpi(Ω)]1/N,
where pM=max{p1,p2,…,pN} and 1/p-=(1/N)∑i=1N‍(1/pi).
We will use through the paper, the truncation functionTγ at height  (γ>0), that is
(36)Tγ(s)={sif|s|≤γ,γsign(s)if|s|>γ.We need the following lemma.Lemma 5.
Letg be a nonnegative function in W1,p→(·)(Ω). Assume p-<N and there exists a constant C>0 such that
(37)∫Ω‍|Tγ(g)|pM-dx+∑i=1N‍∫{|g|≤γ}‍|∂g∂xi|pi-dx≤C(γ+1),∀γ>0.
Then, there exists a constant D, depending on C, such that
(38)∥g∥ℳp~(Ω)≤D,
where p~=N(p--1)/(N-p-).Proof.
Consider the following
Step  1 (||Tγ(g)||LpM-(Ω)≤1). Then, obviously we have ∥g∥ℳp~(Ω)≤D, for some positive constant D. Indeed, since 1<p~≤p-≤pM-, according to Proposition 1 there exists a positive constant C such that
(39)∥Tγ(g)∥Lp~(Ω)≤C∥Tγ(g)∥LpM-(Ω)≤C.
It follows that there exists a positive constant D such that
(40)∥g∥ℳp~(Ω)≤D.
Step  2 (||Tγ(g)||LpM-(Ω)>1). We get from (37)
(41)∥Tγ(g)∥LpM-(Ω)pM-+∥∂Tγ(g)∂xi∥Lpi-(Ω)pi-≤C(γ+1).
Not also that
(42)(∥Tγ(g)∥LpM-(Ω)+∥∂Tγ(g)∂xi∥Lpi-(Ω))pi-≤2(pi--1)(∥Tγ(g)∥LpM-(Ω)pi-+∥∂Tγ(g)∂xi∥Lpi-(Ω)pi-)≤2(pi--1)(∥Tγ(g)∥LpM-(Ω)pM-+∥∂Tγ(g)∂xi∥Lpi-(Ω)pi-).
Therefore, by using (35), we obtain for γ>1,
(43)∥Tγ(g)∥Lq(Ω)≤C∏i=1N‍[2(pi--1)/Npi-γ1/Npi-]≤Dγ∑i=1N(1/Npi-)=Dγ1/p-.
It follows that
(44)∫{|g|>γ}‍|Tγ(g)|qdx≤Dγq/p-
which is equivalent to
(45)γqmeas({|g|>γ})≤Dγq/p-.
Therefore,
(46)meas({|g|>γ})≤Dγ-q(p--1)/p-.
Since, q=p-*=Np-/(N-p-) we get
(47)meas({|g|>γ})≤Dγ-N(p--1)/(N-p-)
which implies that ∥g∥ℳp~(Ω)≤D.
For0<γ≤1 we have
(48)meas({|g|>γ})≤meas(Ω)≤meas(Ω)γ-p~.
So,
(49)∥g∥ℳp~(Ω)≤D.
We need the following well-known results.Theorem 6 (see [31, Theorem 6.2.1]).
LetX 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(50)f(u)-f(v)≥〈f′(v),u-v〉X*×X∀u,v∈M,
where X* denotes the dual of the space X.(iii)
The first Gateaux derivative is monotone, that is,(51)〈f′(u)-f′(v),u-v〉X*×X≥0∀u,v∈M.(iv)
The second Gateaux derivative off exists and it is positive, that is,
(52)〈f′′(u)∘v,v〉X*×X≥0∀v∈M.Theorem 7 (see [32, Theorem 1.2]).
SupposeX 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.(i)
Ψ(u)→∞ as ∥u∥X→∞,u∈M.(ii)
For anyu∈M, any subsequence (um) in M such that um⇀u weakly in X there holds
(53)Ψ(u)≤liminfm→∞Ψ(um).Then, Ψ is bounded from below and attains its infinimun in M.


## 3. Main Results

In the sequel, we denoteW1,p→(·)(Ω)=E and ∥·∥W1,p→(·)(Ω)=∥·∥E.


### 3.1. Weak Solutions

Let us define first the notion of weak solution.Definition 8.
Letu:Ω→ℝ be a measurable function, we say that u is a weak solution of problem (1) if u belongs to W1,p→(·)(Ω) and satisfies the following equation:
(54)∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx+∫Ω‍|u|pM(x)-2uvdx-∫Ω‍f(x)vdx=0,
for every v∈W1,p→(·)(Ω).
We associate to problem (1) the energy functional I:E→ℝ, defined by
(55)I(u)=∫Ω‍∑i=1N‍Ai(x,∂u∂xi)dx+∫Ω‍1pM(x)|u|pM(x)dx-∫Ω‍f(x)udx.
To simplify our writing, we denote by Λ:E→ℝ the functional
(56)Λ(u)=∫Ω‍∑i=1N‍Ai(x,∂u∂xi)dx.
We recall the following result (see [15, Lemma 3.4]).Lemma 9.
The functionalΛ is well-defined on E. In addition, Λ is of class 𝒞1(E,ℝ) and
(57)〈Λ′(u),v〉=∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx,
for all u,v∈E.Due to Lemma9, a standard calculus leads to the facts that I is well-defined on E and I∈𝒞1(E,ℝ) with the derivative given by
(58)〈I′(u),v〉=∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx+∫Ω‍|u|pM(x)-2uvdx-∫Ω‍f(x)vdx
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.Lemma 11.
If hypotheses (4)–(8) are fulfilled, then the functional I is coercive.Proof.
Letu∈E be such that ||u||E→∞. Using (7), we deduce that
(59)Λ(u)≥1pM+∑i=1N‍∫Ω‍|∂u∂xi|pi(x)dx.
We make the following notations:
(60)ℐ1={i∈{1,…,N}:|∂u∂xi|Lpi(·)(Ω)≤1},ℐ2={i∈{1,…,N}:|∂u∂xi|Lpi(·)(Ω)>1}.
We then have
(61)Λ(u)≥1pM+∑i∈ℐ1‍∫Ω‍|∂u∂xi|pi(x)dx+1pM+∑i∈ℐ2‍∫Ω‍|∂u∂xi|pi(x)dx.
Using (19), (20), and (21), we have
(62)Λ(u)≥1pM+∑i∈ℐ1‍|∂u∂xi|pi(·)pi++1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pi-≥1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pi-≥1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pm-≥1pM+(∑i=1N‍|∂u∂xi|pi(·)pm--∑i∈ℐ1‍|∂u∂xi|pi(·)pm-)≥1pM+(∑i=1N‍|∂u∂xi|pi(·)pm--N).
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
(63)∑i=1N‍|∂u∂xi|pi(·)pm-≥1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-,
thus,
(64)Λ(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm--N].
Case  1 (|u|pM(.)≥1). We have
(65)I(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm--N]+1pM+|u|pM(·)pM--∥f∥L∞(Ω)∥u∥L1(Ω)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm-]-C∥f∥L∞(Ω)∥u∥E-NpM+≥12pm--1pM+min(1,1Npm--1)·[∑i=1N‍|∂u∂xi|pi(·)+|u|pM(·)]pm--C∥f∥L∞(Ω)∥u∥E-NpM+.
Therefore,
(66)I(u)≥12pm--1pM+min(1,1Npm--1)∥u∥Epm--C∥f∥L∞(Ω)∥u∥E-NpM+.
Case  2  (|u|pM(·)<1). Then |u|pM(·)pm--1<0, and we get
(67)I(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm--N-1]-∥f∥L∞(Ω)∥u∥L1(Ω)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm-]-C∥f∥L∞(Ω)∥u∥E-N+1pM+≥12pm--1pM+min(1,1Npm--1)[∑i=1N‍|∂u∂xi|pi(·)+|u|pM(·)]pm--C∥f∥L∞(Ω)∥u∥E-N+1pM+.
So, we obtain
(68)I(u)≥12pm--1pM+min(1,1Npm--1)∥u∥Epm--C∥f∥L∞(Ω)∥u∥E-N+1pM+.
Then, letting ∥u∥E goes to infinity in (66) and (68), we conclude that I(u) reaches infinity. Thus, I is coercive.Lemma 12.
The functionalI is weakly lower semicontinuous.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
(69)Ai(x,∂v∂xi)-Ai(x,∂u∂xi)≥ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)⟹∑i=1N‍∫ω‍Ai(x,∂v∂xi)dx≥∑i=1N‍∫ω‍Ai(x,∂u∂xi)dx+∑i=1N‍∫ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx⟹I(v)≥I(u)+∑i=1N‍∫ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx+∫ω‍1pM(x)(|v|pM(x)-|u|pM(x))dx+∫ω‍f(x)(u-v)dx.
Since the map t↦tpM(x),t>0 is convex, again by Theorem 6, we have
(70)|v|pM(x)-|u|pM(x)≥pM(x)|u|pM(x)-2u(v-u),
then (69) becomes
(71)I(v)≥I(u)+∑i=1N‍∫Ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx+∫Ω‍|u|pM(x)-2u(v-u)dx+∫Ω‍f(x)(u-v)dx.
Consider the second term in the right-hand side of (71). By (5) and Hölder type inequality, we have
(72)∑i=1N‍∫Ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx≥-∑i=1N‍∫Ω‍|ai(x,∂u∂xi)||∂v∂xi-∂u∂xi|dx≥-max{C1,…,CN}·∑i=1N‍∫Ω‍(ji+|∂u∂xi|pi(x)-1)|∂v∂xi-∂u∂xi|dx≥-K∑i=1N‍∫Ω‍ji(x)|∂v∂xi-∂u∂xi|dx-K∑i=1N‍∫Ω‍|∂u∂xi|pi(x)-1|∂v∂xi-∂u∂xi|dx≥-K′∑i=1N‍|ji|pi′(·)|∂v∂xi-∂u∂xi|pi(·)-K′∑i=1N‍||∂u∂xi|pi(x)-1|pi′(·)|∂v∂xi-∂u∂xi|pi(·)≥-K′maxi{|ji|pi′(·),||∂u∂xi|pi(x)-1|pi′(·)}·∑i=1N‍|∂v∂xi-∂u∂xi|pi(·)≥-C1∑i=1N‍|∂u∂xi-∂v∂xi|pi(·).
For the fourth term in the right-hand side of (71), we have
(73)∫Ω‍f(x)(u-v)dx≥-∫Ω‍|f(x)||u-v|dx≥∥f∥L∞(Ω)∥u-v∥L1(Ω)≥-C2∥u-v∥E.
The third term in the right-hand side of (71) gives by using Hölder type inequality
(74)∫Ω‍|u|pM(x)-2u(v-u)dx≥-∫Ω‍|u|pM(x)-1|u-v|dx≥-C′||u|pM(x)-1|PM′(·)|u-v|pM(·)≥-C3|u-v|pM(·).
Gathering these inequalities, it follows that
(75)I(v)≥I(u)-C∥u-v∥E≥I(u)-ϵ,
for every v∈E such that ∥u-v∥E<ϵ/C. Thus, I is lower semicontinuous.Proof of Theorem10.
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
(76)∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂(v-u)∂xidx+∫Ω‍|u|pM(x)-2u(v-u)dx-∫Ω‍f(x)(v-u)dx=0,(77)∫Ω‍∑i=1N‍ai(x,∂v∂xi)∂(u-v)∂xidx+∫Ω‍|v|pM(x)-2v(u-v)dx-∫Ω‍f(x)(u-v)dx=0.
Summing up (76) and (77), we obtain
(78)∫Ω‍∑i=1N‍(ai(x,∂u∂xi)-ai(x,∂v∂xi))∂(u-v)∂xidx+∫Ω‍1pM(x)(|u|pM(x)-2u-|v|pM(x)-2v)(u-v)dx=0.
Thus, by the monotonicity of the functions ai(x,·) and t↦|t|pM(x)-2t, 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 space𝒯1,p→(·)(Ω) as the set of every measurable function u:Ω→ℝ which satisfies for every k>0,Tk(u)∈W1,p→(·)(Ω).Lemma 13 (see [34, 35]).
Letu∈𝒯1,p→(·)(Ω). Then, there exists a unique measurable function vi:Ω→ℝ such that
(79)viχ{|u|<k}=∂Tk(u)∂xifora.e.x∈Ω,∀k>0,i∈{1,…,N},
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 W1,p→(·)(Ω), then vi∈Lpi(·)(Ω) and coincides with the standard distributional gradient of u, that is, vi=∂u/∂xi.Definition 14.
We define the space𝒯ℋ1,p→(·)(Ω) as the set of function u∈𝒯1,p→(·)(Ω) such that there exists a sequence (un)n⊂W1,p→(·)(Ω) satisfying(a)
un→u a.e. in Ω,(b)
∂Tk(un)/∂xi→∂Tk(u)/∂xi in L1(Ω), for all k>0.Definition 15.
A measurable functionu is an entropy solution of (1) if u∈𝒯ℋ1,p→(·)(Ω) and for every k>0,
(80)∑i=1N‍∫Ω‍ai(x,∂u∂xi)∂∂xiTk(u-φ)dx+∫Ω‍|u|pM(x)-2uTk(u-φ)dx≤∫Ω‍f(x)Tk(u-φ)dx,
for all φ∈W1,p→(·)(Ω)∩L∞(Ω).Our main result in this section is the following.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.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
(81)∑i=1N‍∫{|u|>t}‍tqi(x)dx≤M,∀t>0,
then
(82)∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}‍tqi(x)dx≤∥f∥1+M∀t>0,
where αi(·)=pi(·)/(qi(·)+1), for all i=1,…,N.Proof.
Takeφ=0 in (80), we have
(83)∑i=1N‍∫Ω‍ai(x,∂∂xiTt(u))·∂∂xiTt(u)dx+∫Ω‍|u|pM(x)-2uTt(u)dx≤∫Ω‍f(x)Tt(u)dx.
Since the second term in the previous inequality is nonnegative, it follows that
(84)∑i=1N‍∫Ω‍ai(x,∂∂xiTt(u))·∂∂xiTt(u)dx≤∫Ω‍f(x)Tt(u)dx.
According to (7), we deduce that
(85)∑i=1N‍∫Ω‍|∂∂xiTt(u)|pi(x)dx≤t∥f∥1,∀t>0.
Therefore, defining ψ:=Tt(u)/t, we have for all t>0,
(86)∑i=1N‍∫Ω‍tpi(x)-1|∂∂xiψ|pi(x)dx=∑i=1N‍1t∫Ω‍|∂∂xiTt(u)|pi(x)dx≤∥f∥1.
From the previous inequality, the definition of αi(·) and (81), we have
(87)∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}‍tqi(x)dx≤∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}∩{|u|≤t}‍tqi(x)dx+∑i=1N‍∫{|u|>t}‍tqi(x)dx≤∑i=1N‍∫{|u|≤t}‍tqi(x)(|(∂/∂xi)u|αi(x)t)pi(x)/αi(x)dx+M≤∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t;|u|≤t}‍tqi(x)-(pi(x)/αi(x))|∂u∂xi|pi(x)dx+M≤∑i=1N‍1t∫{|(∂/∂xi)u|αi(·)>t;|u|≤t}‍|∂u∂xi|pi(x)dx+M≤∥f∥1+M.Lemma 18.
Assume (4)–(8) and f∈L1(Ω). Let u be an entropy solution of (1), then
(88)1h∑i=1N‍∫{|u|≤h}‍|∂∂xiTh(u)|pi(x)dx≤M
for every h>0, with M a positive constant. Moreover, we have
(89)∥|u|pM(x)-2u∥1=∥|u|pM(x)-1∥1≤∥f∥1
and there exists a constant D>0 which depends on f and Ω such that
(90)meas{|u|>h}≤DhPM--1,∀h>0.Proof.
Takingφ=0 in the entropy inequality (80) and using (7), we obtain
(91)∑i=1N‍∫{|u|≤h}‍|∂∂xiTh(u)|pi(x)dx≤h∥f∥1≤Mh,∫Ω‍|u|pM(x)-2uTh(u)dx≤h∥f∥1,
for all h>0. This yields
(92)∫{|u|>h}‍|u|pM(x)-2uTh(u)dx≤h∥f∥1.
As uTh(u)χ{|u|>h}=h|u|χ{|u|>h}, we get from the previous inequality by using Fatou’s lemma
(93)∫Ω‍|u|pM(x)-2|u|dx≤∥f∥1.
Now, since |Th(u)|≤|u| we have
(94)∫Ω‍|Th(u)|pM(x)-1dx≤∫Ω‍|u|pM(x)-1dx≤∥f∥1.
We deduce that
(95)∫Ω‍|Th(u)|pM--1dx≤D(f,Ω).
Indeed,
(96)∫Ω‍|Th(u)|pM--1dx≤∫{|Th(u)|≤1}‍|Th(u)|pM--1dx+∫{|Th(u)|>1}‍|Th(u)|pM--1dx≤meas(Ω)+∫Ω‍|Th(u)|pM(x)-1dx≤meas(Ω)+∥f∥1.
From aforementioned, we get
(97)∫{|u|>h}‍|Th(u)|pM--1dx≤D(f,Ω).
Therefore,
(98)hpM--1meas{|u|>h}≤D(f,Ω)
which implies
(99)meas{|u|>h}≤D(f,Ω)hpM--1.Lemma 19.
Ifu is an entropy solution of (1) then there exists a constant C>0 such that
(100)∫Ω‍|Tk(u)|PM-dx+∑i=1N‍∫{|u|≤k}‍|∂u∂xi|pi-dx≤C(k+1),∀k>0.Proof.
Takingφ=0 in the entropy inequality (80) and using (7), we get
(101)∫Ω‍|u|pM(x)-2uTk(u)dx+∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi(x)dx≤k∥f∥1.
Note that
(102)∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi-dx=∑i=1N‍∫{|u|≤k,|∂u/∂xi|≤1}‍|∂∂xiTk(u)|pi-dx+∑i=1N‍∫{|u|≤k,|∂u/∂xi|>1}‍|∂∂xiTk(u)|pi-dx≤Nmeas(Ω)+∑i=1N‍∫{|u|≤k,|∂u/∂xi|>1}‍|∂∂xiTk(u)|pi(x)dx≤Nmeas(Ω)+∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi(x)dx,∫Ω‍|Tk(u)|pM-dx≤∫{|Tk(u)|≤1}‍|Tk(u)|pM-dx+∫{|Tk(u)|>1}‍|Tk(u)|pM-dx≤meas(Ω)+∫Ω‍|Tk(u)|pM(x)dx≤meas(Ω)+∫Ω‍|u|pM(x)-2uTk(u)dx.
Therefore, we deduce according to (101) that
(103)∫Ω‍|Tk(u)|PM-dx+∑i=1N‍∫{|u|≤k}‍|∂u∂xi|pi-dx≤(N+1)meas(Ω)+k∥f∥1,∀k>0.Lemma 20.
Ifu is an entropy solution of (1) then
(104)ρp′(·)(|∂∂xiu|pi(x)-1χF)≤C,∀i=1,…,N,
where F={h<|u|≤h+t}, h>0, t>0.Proof.
Takingφ=Th(u) as a test function in the entropy inequality (80), we get
(105)∑i=1N‍∫Ω‍ai(x,∂∂xiu)·∂∂xiTt(u-Th(u))dx+∫Ω‍|u|pM(x)-2uTt(u-Th(u))dx≤∫Ω‍f(x)Tt(u-Th(u))dx.
It follows by using (7) that
(106)∫F‍|∂∂xiu|pi(x)dx≤t∥f∥1.
Therefore,
(107)ρp′(·)(|∂∂xiu|pi(x)-1χF)≤C,∀i=1,…,N.Lemma 21.
Ifu is an entropy solution of (1) then
(108)limh→+∞∫Ω‍|f|χ{|u|>h-t}dx=0,
where h>0, t>0.Proof.
By Lemma18, we deduce that
(109)limh→+∞|f|χ{|u|>h-t}=0
and as f∈L1(Ω), it follows by using the Lebesgue dominated convergence theorem that
(110)limh→+∞∫Ω‍|f|χ{|u|>h-t}dx=0.
The proof of the following lemma can be found in [1].Lemma 22.
Assume (4)–(8) and f∈L1(Ω). Let u be an entropy solution of (1), then
(111)meas{|∂∂xiu|>h}≤D′h1/(PM-)′,∀h≥1,∀i=1,…,N,
where D′ is a positive constant which depends on f and pM-.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
(112)∫{|u-Th(v)|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx+∫{|v-Th(u)|≤t}‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(v-Th(u))dx+∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx+∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx≤∫Ω‍f(x)(Tt(u-Th(v))+Tt(v-Th(u)))dx.
Define
(113)E1:={|u-v|≤t,|v|≤h},E2:=E1∩{|u|≤h},E3:=E1∩{|u|>h}.
We start with the first integral in (112). By (7), we have
(114)∫{|u-Thv|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx≥∫E2‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-v)dx-∫E3‍∑i=1N‍ai(x,∂∂xiu)·∂∂xivdx.
Using (5) and Proposition 1, we estimate the last integral in (114) as follows:
(115)|∫E3‍∑i=1N‍ai(x,∂∂xiu)·∂∂xivdx|≤C1∫E3‍∑i=1N‍(ji(x)+|∂∂xiu|pi(x)-1)|∂∂xiv|dx≤C∑i=1N‍(|j|pi′(·)+||∂∂xiu|pi(x)-1|pi′(·),{h<|u|≤h+t})·|∂∂xiv|pi(·),{h-t<|v|≤h},
where
(116)||∂∂xiu|pi(x)-1|pi′(·),{h<|u|≤h+t}=∥|∂∂xiu|pi(x)-1∥Lpi′(·)({h<|u|≤h+t}).
For all i=1,…,N, the quantity (|ji|pi′(·)+||(∂/∂xi)u|pi(x)-1|pi′(·),{h<|u|≤h+t}) is finite according to relations (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
(117)∫{|u-Thv|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx≥Ih+∫E2‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-v)dx,
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
(118)∫{|v-Th(u)|≤t}‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(v-Th(u))dx≥Jh-∫E2‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(u-v)dx,
where Jh converges to zero as h tends to infinity.For the two other terms in the left-hand side of (112), we denote
(119)Kh=∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx+∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx.
We have |u|pM(x)-2uTt(u-Th(v))→|u|pM(x)-2uTt(u-v) a.e. as h goes to infinity and
(120)|u|pM(x)-2uTt(u-Th(v))∣≤t|u|pM(x)-2u∈L1(Ω).
Then, by the Lebesgue dominated convergence theorem, we obtain
(121)∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx⟶∫Ω‍|u|pM(x)-2uTt(u-v)dx,ash⟶∞.
In the same way, we get
(122)∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx⟶∫Ω‍|v|pM(x)-2vTt(v-u)dx,ash⟶∞.
Therefore,
(123)limh→∞Kh=∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)dx.Furthermore, consider the right-hand side of inequality (112). We have
(124)limh→∞∫Ω‍f(x)(Tt(u-Th(v))+Tt(v-Th(u)))dx=0.
Indeed,
(125)f(x)(Tt(u-Th(v))+Tt(v-Th(u)))⟶f(x)(Tt(u-v)+Tt(v-u))=0a.e.inΩash⟶∞,|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:
(126)∑i=1N‍∫{|u-v|≤t}‍(ai(x,∂∂xiu)-ai(x,∂∂xiv))·∂∂xi(u-v)+∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)≤0.
Using (6) and as t↦|t|pM(x)-2t is monotone, we deduce from (126) that
(127)∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)dx≤0.
Since pM->1, the following relation is true for any ξ,η∈ℝ, ξ≠η (cf. [12])
(128)(|ξ|pM(x)-2ξ-|η|pM(x)-2η)(ξ-η)>0.
Therefore, from (127), we get that (|u|pM(x)-2u-|v|pM(x)-2v)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,
(129)(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)=0.
Therefore,
(130)(|u|pM(x)-2u-|v|pM(x)-2v)(u-v)=0,∀x∈Ω∖⋃t∈ℕ*‍Ωt.
Now, using (128) and (130), we obtain
(131)u1=u2a.e.inΩ.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
(132)∥fn∥1≤∥f∥1,∀n∈ℕ*.
We consider the problem
(133)-∑i=1N‍∂∂xiai(x,∂∂xiun)+|un|pM(x)-2un=fninΩ,∑i=1N‍ai(x,∂∂xiun)νi=0on∂Ω.
It follows from Theorem 10 that problem (133) admits a unique weak solution un∈W1,p→(·)(Ω) which satisfies
(134)∑i=1N‍∫Ω‍ai(x,∂un∂xi)∂∂xiφdx+∫Ω‍|un|pM(x)-2unφdx=∫Ω‍fn(x)φdx,
for all φ∈W1,p→(·)(Ω).Our interest is to prove that these approximated solutionsun 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.
Ifun is a weak solution of (126) then there exist some constants C1,C2>0 such that(i)
∥un∥ℳp~(Ω)≤C1,(ii)
∥∂un/∂xi∥ℳpi-q/p-(Ω)≤C2, for  all  i=1,…,N.Proof.
(i) is a consequence of Lemmas19 and 5 by using Tk(un) for all k>0 as a test function in (134).
(ii) We first useTγ(un) for all γ>0 as a test function in (134) to get
(135)∑i=1N‍∫{|u|≤γ}‍|∂u∂xi|pi-dx≤C(γ+1).
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,
(136)λ|∂un/∂xi|(α)≤meas{x∈Ω:|∂un∂xi|>α,|un|≤γ}+meas{x∈Ω:|∂un∂xi|>α,|un|>γ}≤∫{|∂un/∂xi|>α,|un|≤γ}‍(1α|∂un∂xi|)pi-dx+λ|un|(γ)≤1αpi-∫{|un|≤γ}‍|∂un∂xi|pi-dx+λ|un|(γ).
Using (135) and (i), we get
(137)λ|∂un/∂xi|(α)≤C(γαpi-+γ-p~),
from which we deduce (ii).
By lemmas3 and 23, it follows that (un)n∈ℕ* is uniformly bounded in Ls0(Ω) for some 1≤s0<p~, and in the same way, (|∂un/∂xi|)n∈ℕ* is uniformly bounded in Lsi(Ω) for some 1≤si<p~i. From this, we get that the sequence (un)n∈ℕ* is uniformly bounded in W1,s(Ω), where s=min(s0,s1,…,sN). Consequently, we can extract a subsequence, still denoted (un) satisfying
(138)un⟶ua.e.inΩ,inLs(Ω),un⇀uinW1,s(Ω),|∂un∂xi-∂u∂xi|⇀ℋi(x)inLs(Ω),∀i=1,…,N.
By the same way as in the proof of [16, Lemma 3.5] (see also [27]), we prove that
(139)ℋi(x)=0a.e.x∈Ω∀i=1,…,N.
We deduce from (139) that
(140)ai(x,∂un∂xi)⟶ai(x,∂u∂xi)a.e.inΩ,inL1(Ω),∀i=1,…,N.
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].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). Let un∈W1,p→(·)(Ω) be the solution of (133). The following assertions hold.(i)
For alli=1,…,N, ∂un/∂xi converges in measure to the weak partial gradient of u.(ii)
For alli=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∈W1,p→(·)(Ω)∩L∞(Ω). For any k>0, choose Tk(un-φ) as a test function in (134), we get
(141)∑i=1N‍∫Ω‍ai(x,∂un∂xi)∂∂xiTk(un-φ)dx+∫Ω‍|un|pM(x)-2unTk(un-φ)dx=∫Ω‍fn(x)Tk(un-φ)dx.
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
(142)∫Ω‍|un|pM(x)-2unTk(un-φ)dx=∫Ω‍(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)dx+∫Ω‍|φ|pM(x)-2φTk(un-φ)dx. The quantity (|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ) is nonnegative and since for all x∈Ω,s↦|s|pM(x)-2s is continuous; we get
(143)(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)⟶(|u|pM(x)-2u-|φ|pM(x)-2φ)Tk(u-φ)a.e.inΩ.
Then, it follows by Fatou’s Lemma that
(144)liminfn→+∞∫Ω‍(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)dx≥∫Ω‍(|u|pM(x)-2u-|φ|pM(x)-2φ)Tk(u-φ)dx.Let us show that|φ|pM(x)-2φ∈L1(Ω).We have(145)∫Ω‍||φ|pM(x)-2φ|dx=∫Ω‍|φ|pM(x)-1dx≤∫Ω‍(∥φ∥∞)pM(x)-1dx.
If ∥φ∥∞≤1, then ∫Ω‍||φ|pM(x)-2φ|dx≤meas(Ω)<+∞.If∥φ∥∞>1, then
(146)∫Ω‍||φ|pM(x)-2φ|dx≤∫Ω‍(∥φ∥∞)pM+-1dx=(∥φ∥∞)pM+-1meas(Ω)<+∞.
Hence, |φ|pM(x)-2φ∈L1(Ω).SinceTk(un-φ) converges weakly-* to Tk(u-φ) in L∞(Ω) and |φ|pM(x)-2φ∈L1(Ω), it follows that
(147)limn→+∞∫Ω‍|φ|pM(x)-2φTk(un-φ)dx=∫Ω‍|φ|pM(x)-2φTk(u-φ)dx.
For the first term of (141), we write it as follows:
(148)∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiundx-∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiφdx.
The first term of (148) is nonnegative by (7), then by Fatou’s Lemma and (138), we get
(149)∑i=1N‍∫{|u-φ|≤k}‍ai(x,∂u∂xi)∂∂xiudx≤liminfn→∞∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiundx.
According to Proposition 25, the second term of (148) converges to
(150)∑i=1N‍∫{|u-φ|≤k}‍ai(x,∂u∂xi)∂∂xiφdx.
Combining the previous convergence results, we obtain
(151)∑i=1N‍∫Ω‍ai(x,∂u∂xi)∂∂xiTk(u-φ)dx+∫Ω‍|u|pM(x)-2uTk(u-φ)dx≤∫Ω‍f(x)Tk(u-φ)dx.


## 3.1. Weak Solutions

Let us define first the notion of weak solution.Definition 8.
Letu:Ω→ℝ be a measurable function, we say that u is a weak solution of problem (1) if u belongs to W1,p→(·)(Ω) and satisfies the following equation:
(54)∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx+∫Ω‍|u|pM(x)-2uvdx-∫Ω‍f(x)vdx=0,
for every v∈W1,p→(·)(Ω).
We associate to problem (1) the energy functional I:E→ℝ, defined by
(55)I(u)=∫Ω‍∑i=1N‍Ai(x,∂u∂xi)dx+∫Ω‍1pM(x)|u|pM(x)dx-∫Ω‍f(x)udx.
To simplify our writing, we denote by Λ:E→ℝ the functional
(56)Λ(u)=∫Ω‍∑i=1N‍Ai(x,∂u∂xi)dx.
We recall the following result (see [15, Lemma 3.4]).Lemma 9.
The functionalΛ is well-defined on E. In addition, Λ is of class 𝒞1(E,ℝ) and
(57)〈Λ′(u),v〉=∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx,
for all u,v∈E.Due to Lemma9, a standard calculus leads to the facts that I is well-defined on E and I∈𝒞1(E,ℝ) with the derivative given by
(58)〈I′(u),v〉=∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂v∂xidx+∫Ω‍|u|pM(x)-2uvdx-∫Ω‍f(x)vdx
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.Lemma 11.
If hypotheses (4)–(8) are fulfilled, then the functional I is coercive.Proof.
Letu∈E be such that ||u||E→∞. Using (7), we deduce that
(59)Λ(u)≥1pM+∑i=1N‍∫Ω‍|∂u∂xi|pi(x)dx.
We make the following notations:
(60)ℐ1={i∈{1,…,N}:|∂u∂xi|Lpi(·)(Ω)≤1},ℐ2={i∈{1,…,N}:|∂u∂xi|Lpi(·)(Ω)>1}.
We then have
(61)Λ(u)≥1pM+∑i∈ℐ1‍∫Ω‍|∂u∂xi|pi(x)dx+1pM+∑i∈ℐ2‍∫Ω‍|∂u∂xi|pi(x)dx.
Using (19), (20), and (21), we have
(62)Λ(u)≥1pM+∑i∈ℐ1‍|∂u∂xi|pi(·)pi++1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pi-≥1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pi-≥1pM+∑i∈ℐ2‍|∂u∂xi|pi(·)pm-≥1pM+(∑i=1N‍|∂u∂xi|pi(·)pm--∑i∈ℐ1‍|∂u∂xi|pi(·)pm-)≥1pM+(∑i=1N‍|∂u∂xi|pi(·)pm--N).
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
(63)∑i=1N‍|∂u∂xi|pi(·)pm-≥1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-,
thus,
(64)Λ(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm--N].
Case  1 (|u|pM(.)≥1). We have
(65)I(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm--N]+1pM+|u|pM(·)pM--∥f∥L∞(Ω)∥u∥L1(Ω)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm-]-C∥f∥L∞(Ω)∥u∥E-NpM+≥12pm--1pM+min(1,1Npm--1)·[∑i=1N‍|∂u∂xi|pi(·)+|u|pM(·)]pm--C∥f∥L∞(Ω)∥u∥E-NpM+.
Therefore,
(66)I(u)≥12pm--1pM+min(1,1Npm--1)∥u∥Epm--C∥f∥L∞(Ω)∥u∥E-NpM+.
Case  2  (|u|pM(·)<1). Then |u|pM(·)pm--1<0, and we get
(67)I(u)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm--N-1]-∥f∥L∞(Ω)∥u∥L1(Ω)≥1pM+[1Npm--1(∑i=1N‍|∂u∂xi|pi(·))pm-+|u|pM(·)pm-]-C∥f∥L∞(Ω)∥u∥E-N+1pM+≥12pm--1pM+min(1,1Npm--1)[∑i=1N‍|∂u∂xi|pi(·)+|u|pM(·)]pm--C∥f∥L∞(Ω)∥u∥E-N+1pM+.
So, we obtain
(68)I(u)≥12pm--1pM+min(1,1Npm--1)∥u∥Epm--C∥f∥L∞(Ω)∥u∥E-N+1pM+.
Then, letting ∥u∥E goes to infinity in (66) and (68), we conclude that I(u) reaches infinity. Thus, I is coercive.Lemma 12.
The functionalI is weakly lower semicontinuous.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
(69)Ai(x,∂v∂xi)-Ai(x,∂u∂xi)≥ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)⟹∑i=1N‍∫ω‍Ai(x,∂v∂xi)dx≥∑i=1N‍∫ω‍Ai(x,∂u∂xi)dx+∑i=1N‍∫ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx⟹I(v)≥I(u)+∑i=1N‍∫ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx+∫ω‍1pM(x)(|v|pM(x)-|u|pM(x))dx+∫ω‍f(x)(u-v)dx.
Since the map t↦tpM(x),t>0 is convex, again by Theorem 6, we have
(70)|v|pM(x)-|u|pM(x)≥pM(x)|u|pM(x)-2u(v-u),
then (69) becomes
(71)I(v)≥I(u)+∑i=1N‍∫Ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx+∫Ω‍|u|pM(x)-2u(v-u)dx+∫Ω‍f(x)(u-v)dx.
Consider the second term in the right-hand side of (71). By (5) and Hölder type inequality, we have
(72)∑i=1N‍∫Ω‍ai(x,∂u∂xi)(∂v∂xi-∂u∂xi)dx≥-∑i=1N‍∫Ω‍|ai(x,∂u∂xi)||∂v∂xi-∂u∂xi|dx≥-max{C1,…,CN}·∑i=1N‍∫Ω‍(ji+|∂u∂xi|pi(x)-1)|∂v∂xi-∂u∂xi|dx≥-K∑i=1N‍∫Ω‍ji(x)|∂v∂xi-∂u∂xi|dx-K∑i=1N‍∫Ω‍|∂u∂xi|pi(x)-1|∂v∂xi-∂u∂xi|dx≥-K′∑i=1N‍|ji|pi′(·)|∂v∂xi-∂u∂xi|pi(·)-K′∑i=1N‍||∂u∂xi|pi(x)-1|pi′(·)|∂v∂xi-∂u∂xi|pi(·)≥-K′maxi{|ji|pi′(·),||∂u∂xi|pi(x)-1|pi′(·)}·∑i=1N‍|∂v∂xi-∂u∂xi|pi(·)≥-C1∑i=1N‍|∂u∂xi-∂v∂xi|pi(·).
For the fourth term in the right-hand side of (71), we have
(73)∫Ω‍f(x)(u-v)dx≥-∫Ω‍|f(x)||u-v|dx≥∥f∥L∞(Ω)∥u-v∥L1(Ω)≥-C2∥u-v∥E.
The third term in the right-hand side of (71) gives by using Hölder type inequality
(74)∫Ω‍|u|pM(x)-2u(v-u)dx≥-∫Ω‍|u|pM(x)-1|u-v|dx≥-C′||u|pM(x)-1|PM′(·)|u-v|pM(·)≥-C3|u-v|pM(·).
Gathering these inequalities, it follows that
(75)I(v)≥I(u)-C∥u-v∥E≥I(u)-ϵ,
for every v∈E such that ∥u-v∥E<ϵ/C. Thus, I is lower semicontinuous.Proof of Theorem10.
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
(76)∫Ω‍∑i=1N‍ai(x,∂u∂xi)∂(v-u)∂xidx+∫Ω‍|u|pM(x)-2u(v-u)dx-∫Ω‍f(x)(v-u)dx=0,(77)∫Ω‍∑i=1N‍ai(x,∂v∂xi)∂(u-v)∂xidx+∫Ω‍|v|pM(x)-2v(u-v)dx-∫Ω‍f(x)(u-v)dx=0.
Summing up (76) and (77), we obtain
(78)∫Ω‍∑i=1N‍(ai(x,∂u∂xi)-ai(x,∂v∂xi))∂(u-v)∂xidx+∫Ω‍1pM(x)(|u|pM(x)-2u-|v|pM(x)-2v)(u-v)dx=0.
Thus, by the monotonicity of the functions ai(x,·) and t↦|t|pM(x)-2t, 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 space𝒯1,p→(·)(Ω) as the set of every measurable function u:Ω→ℝ which satisfies for every k>0,Tk(u)∈W1,p→(·)(Ω).Lemma 13 (see [34, 35]).
Letu∈𝒯1,p→(·)(Ω). Then, there exists a unique measurable function vi:Ω→ℝ such that
(79)viχ{|u|<k}=∂Tk(u)∂xifora.e.x∈Ω,∀k>0,i∈{1,…,N},
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 W1,p→(·)(Ω), then vi∈Lpi(·)(Ω) and coincides with the standard distributional gradient of u, that is, vi=∂u/∂xi.Definition 14.
We define the space𝒯ℋ1,p→(·)(Ω) as the set of function u∈𝒯1,p→(·)(Ω) such that there exists a sequence (un)n⊂W1,p→(·)(Ω) satisfying(a)
un→u a.e. in Ω,(b)
∂Tk(un)/∂xi→∂Tk(u)/∂xi in L1(Ω), for all k>0.Definition 15.
A measurable functionu is an entropy solution of (1) if u∈𝒯ℋ1,p→(·)(Ω) and for every k>0,
(80)∑i=1N‍∫Ω‍ai(x,∂u∂xi)∂∂xiTk(u-φ)dx+∫Ω‍|u|pM(x)-2uTk(u-φ)dx≤∫Ω‍f(x)Tk(u-φ)dx,
for all φ∈W1,p→(·)(Ω)∩L∞(Ω).Our main result in this section is the following.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.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
(81)∑i=1N‍∫{|u|>t}‍tqi(x)dx≤M,∀t>0,
then
(82)∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}‍tqi(x)dx≤∥f∥1+M∀t>0,
where αi(·)=pi(·)/(qi(·)+1), for all i=1,…,N.Proof.
Takeφ=0 in (80), we have
(83)∑i=1N‍∫Ω‍ai(x,∂∂xiTt(u))·∂∂xiTt(u)dx+∫Ω‍|u|pM(x)-2uTt(u)dx≤∫Ω‍f(x)Tt(u)dx.
Since the second term in the previous inequality is nonnegative, it follows that
(84)∑i=1N‍∫Ω‍ai(x,∂∂xiTt(u))·∂∂xiTt(u)dx≤∫Ω‍f(x)Tt(u)dx.
According to (7), we deduce that
(85)∑i=1N‍∫Ω‍|∂∂xiTt(u)|pi(x)dx≤t∥f∥1,∀t>0.
Therefore, defining ψ:=Tt(u)/t, we have for all t>0,
(86)∑i=1N‍∫Ω‍tpi(x)-1|∂∂xiψ|pi(x)dx=∑i=1N‍1t∫Ω‍|∂∂xiTt(u)|pi(x)dx≤∥f∥1.
From the previous inequality, the definition of αi(·) and (81), we have
(87)∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}‍tqi(x)dx≤∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t}∩{|u|≤t}‍tqi(x)dx+∑i=1N‍∫{|u|>t}‍tqi(x)dx≤∑i=1N‍∫{|u|≤t}‍tqi(x)(|(∂/∂xi)u|αi(x)t)pi(x)/αi(x)dx+M≤∑i=1N‍∫{|(∂/∂xi)u|αi(·)>t;|u|≤t}‍tqi(x)-(pi(x)/αi(x))|∂u∂xi|pi(x)dx+M≤∑i=1N‍1t∫{|(∂/∂xi)u|αi(·)>t;|u|≤t}‍|∂u∂xi|pi(x)dx+M≤∥f∥1+M.Lemma 18.
Assume (4)–(8) and f∈L1(Ω). Let u be an entropy solution of (1), then
(88)1h∑i=1N‍∫{|u|≤h}‍|∂∂xiTh(u)|pi(x)dx≤M
for every h>0, with M a positive constant. Moreover, we have
(89)∥|u|pM(x)-2u∥1=∥|u|pM(x)-1∥1≤∥f∥1
and there exists a constant D>0 which depends on f and Ω such that
(90)meas{|u|>h}≤DhPM--1,∀h>0.Proof.
Takingφ=0 in the entropy inequality (80) and using (7), we obtain
(91)∑i=1N‍∫{|u|≤h}‍|∂∂xiTh(u)|pi(x)dx≤h∥f∥1≤Mh,∫Ω‍|u|pM(x)-2uTh(u)dx≤h∥f∥1,
for all h>0. This yields
(92)∫{|u|>h}‍|u|pM(x)-2uTh(u)dx≤h∥f∥1.
As uTh(u)χ{|u|>h}=h|u|χ{|u|>h}, we get from the previous inequality by using Fatou’s lemma
(93)∫Ω‍|u|pM(x)-2|u|dx≤∥f∥1.
Now, since |Th(u)|≤|u| we have
(94)∫Ω‍|Th(u)|pM(x)-1dx≤∫Ω‍|u|pM(x)-1dx≤∥f∥1.
We deduce that
(95)∫Ω‍|Th(u)|pM--1dx≤D(f,Ω).
Indeed,
(96)∫Ω‍|Th(u)|pM--1dx≤∫{|Th(u)|≤1}‍|Th(u)|pM--1dx+∫{|Th(u)|>1}‍|Th(u)|pM--1dx≤meas(Ω)+∫Ω‍|Th(u)|pM(x)-1dx≤meas(Ω)+∥f∥1.
From aforementioned, we get
(97)∫{|u|>h}‍|Th(u)|pM--1dx≤D(f,Ω).
Therefore,
(98)hpM--1meas{|u|>h}≤D(f,Ω)
which implies
(99)meas{|u|>h}≤D(f,Ω)hpM--1.Lemma 19.
Ifu is an entropy solution of (1) then there exists a constant C>0 such that
(100)∫Ω‍|Tk(u)|PM-dx+∑i=1N‍∫{|u|≤k}‍|∂u∂xi|pi-dx≤C(k+1),∀k>0.Proof.
Takingφ=0 in the entropy inequality (80) and using (7), we get
(101)∫Ω‍|u|pM(x)-2uTk(u)dx+∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi(x)dx≤k∥f∥1.
Note that
(102)∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi-dx=∑i=1N‍∫{|u|≤k,|∂u/∂xi|≤1}‍|∂∂xiTk(u)|pi-dx+∑i=1N‍∫{|u|≤k,|∂u/∂xi|>1}‍|∂∂xiTk(u)|pi-dx≤Nmeas(Ω)+∑i=1N‍∫{|u|≤k,|∂u/∂xi|>1}‍|∂∂xiTk(u)|pi(x)dx≤Nmeas(Ω)+∑i=1N‍∫{|u|≤k}‍|∂∂xiTk(u)|pi(x)dx,∫Ω‍|Tk(u)|pM-dx≤∫{|Tk(u)|≤1}‍|Tk(u)|pM-dx+∫{|Tk(u)|>1}‍|Tk(u)|pM-dx≤meas(Ω)+∫Ω‍|Tk(u)|pM(x)dx≤meas(Ω)+∫Ω‍|u|pM(x)-2uTk(u)dx.
Therefore, we deduce according to (101) that
(103)∫Ω‍|Tk(u)|PM-dx+∑i=1N‍∫{|u|≤k}‍|∂u∂xi|pi-dx≤(N+1)meas(Ω)+k∥f∥1,∀k>0.Lemma 20.
Ifu is an entropy solution of (1) then
(104)ρp′(·)(|∂∂xiu|pi(x)-1χF)≤C,∀i=1,…,N,
where F={h<|u|≤h+t}, h>0, t>0.Proof.
Takingφ=Th(u) as a test function in the entropy inequality (80), we get
(105)∑i=1N‍∫Ω‍ai(x,∂∂xiu)·∂∂xiTt(u-Th(u))dx+∫Ω‍|u|pM(x)-2uTt(u-Th(u))dx≤∫Ω‍f(x)Tt(u-Th(u))dx.
It follows by using (7) that
(106)∫F‍|∂∂xiu|pi(x)dx≤t∥f∥1.
Therefore,
(107)ρp′(·)(|∂∂xiu|pi(x)-1χF)≤C,∀i=1,…,N.Lemma 21.
Ifu is an entropy solution of (1) then
(108)limh→+∞∫Ω‍|f|χ{|u|>h-t}dx=0,
where h>0, t>0.Proof.
By Lemma18, we deduce that
(109)limh→+∞|f|χ{|u|>h-t}=0
and as f∈L1(Ω), it follows by using the Lebesgue dominated convergence theorem that
(110)limh→+∞∫Ω‍|f|χ{|u|>h-t}dx=0.
The proof of the following lemma can be found in [1].Lemma 22.
Assume (4)–(8) and f∈L1(Ω). Let u be an entropy solution of (1), then
(111)meas{|∂∂xiu|>h}≤D′h1/(PM-)′,∀h≥1,∀i=1,…,N,
where D′ is a positive constant which depends on f and pM-.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
(112)∫{|u-Th(v)|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx+∫{|v-Th(u)|≤t}‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(v-Th(u))dx+∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx+∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx≤∫Ω‍f(x)(Tt(u-Th(v))+Tt(v-Th(u)))dx.
Define
(113)E1:={|u-v|≤t,|v|≤h},E2:=E1∩{|u|≤h},E3:=E1∩{|u|>h}.
We start with the first integral in (112). By (7), we have
(114)∫{|u-Thv|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx≥∫E2‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-v)dx-∫E3‍∑i=1N‍ai(x,∂∂xiu)·∂∂xivdx.
Using (5) and Proposition 1, we estimate the last integral in (114) as follows:
(115)|∫E3‍∑i=1N‍ai(x,∂∂xiu)·∂∂xivdx|≤C1∫E3‍∑i=1N‍(ji(x)+|∂∂xiu|pi(x)-1)|∂∂xiv|dx≤C∑i=1N‍(|j|pi′(·)+||∂∂xiu|pi(x)-1|pi′(·),{h<|u|≤h+t})·|∂∂xiv|pi(·),{h-t<|v|≤h},
where
(116)||∂∂xiu|pi(x)-1|pi′(·),{h<|u|≤h+t}=∥|∂∂xiu|pi(x)-1∥Lpi′(·)({h<|u|≤h+t}).
For all i=1,…,N, the quantity (|ji|pi′(·)+||(∂/∂xi)u|pi(x)-1|pi′(·),{h<|u|≤h+t}) is finite according to relations (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
(117)∫{|u-Thv|≤t}‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-Th(v))dx≥Ih+∫E2‍∑i=1N‍ai(x,∂∂xiu)·∂∂xi(u-v)dx,
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
(118)∫{|v-Th(u)|≤t}‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(v-Th(u))dx≥Jh-∫E2‍∑i=1N‍ai(x,∂∂xiv)·∂∂xi(u-v)dx,
where Jh converges to zero as h tends to infinity.For the two other terms in the left-hand side of (112), we denote
(119)Kh=∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx+∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx.
We have |u|pM(x)-2uTt(u-Th(v))→|u|pM(x)-2uTt(u-v) a.e. as h goes to infinity and
(120)|u|pM(x)-2uTt(u-Th(v))∣≤t|u|pM(x)-2u∈L1(Ω).
Then, by the Lebesgue dominated convergence theorem, we obtain
(121)∫Ω‍|u|pM(x)-2uTt(u-Th(v))dx⟶∫Ω‍|u|pM(x)-2uTt(u-v)dx,ash⟶∞.
In the same way, we get
(122)∫Ω‍|v|pM(x)-2vTt(v-Th(u))dx⟶∫Ω‍|v|pM(x)-2vTt(v-u)dx,ash⟶∞.
Therefore,
(123)limh→∞Kh=∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)dx.Furthermore, consider the right-hand side of inequality (112). We have
(124)limh→∞∫Ω‍f(x)(Tt(u-Th(v))+Tt(v-Th(u)))dx=0.
Indeed,
(125)f(x)(Tt(u-Th(v))+Tt(v-Th(u)))⟶f(x)(Tt(u-v)+Tt(v-u))=0a.e.inΩash⟶∞,|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:
(126)∑i=1N‍∫{|u-v|≤t}‍(ai(x,∂∂xiu)-ai(x,∂∂xiv))·∂∂xi(u-v)+∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)≤0.
Using (6) and as t↦|t|pM(x)-2t is monotone, we deduce from (126) that
(127)∫Ω‍(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)dx≤0.
Since pM->1, the following relation is true for any ξ,η∈ℝ, ξ≠η (cf. [12])
(128)(|ξ|pM(x)-2ξ-|η|pM(x)-2η)(ξ-η)>0.
Therefore, from (127), we get that (|u|pM(x)-2u-|v|pM(x)-2v)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,
(129)(|u|pM(x)-2u-|v|pM(x)-2v)Tt(u-v)=0.
Therefore,
(130)(|u|pM(x)-2u-|v|pM(x)-2v)(u-v)=0,∀x∈Ω∖⋃t∈ℕ*‍Ωt.
Now, using (128) and (130), we obtain
(131)u1=u2a.e.inΩ.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
(132)∥fn∥1≤∥f∥1,∀n∈ℕ*.
We consider the problem
(133)-∑i=1N‍∂∂xiai(x,∂∂xiun)+|un|pM(x)-2un=fninΩ,∑i=1N‍ai(x,∂∂xiun)νi=0on∂Ω.
It follows from Theorem 10 that problem (133) admits a unique weak solution un∈W1,p→(·)(Ω) which satisfies
(134)∑i=1N‍∫Ω‍ai(x,∂un∂xi)∂∂xiφdx+∫Ω‍|un|pM(x)-2unφdx=∫Ω‍fn(x)φdx,
for all φ∈W1,p→(·)(Ω).Our interest is to prove that these approximated solutionsun 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.
Ifun is a weak solution of (126) then there exist some constants C1,C2>0 such that(i)
∥un∥ℳp~(Ω)≤C1,(ii)
∥∂un/∂xi∥ℳpi-q/p-(Ω)≤C2, for  all  i=1,…,N.Proof.
(i) is a consequence of Lemmas19 and 5 by using Tk(un) for all k>0 as a test function in (134).
(ii) We first useTγ(un) for all γ>0 as a test function in (134) to get
(135)∑i=1N‍∫{|u|≤γ}‍|∂u∂xi|pi-dx≤C(γ+1).
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,
(136)λ|∂un/∂xi|(α)≤meas{x∈Ω:|∂un∂xi|>α,|un|≤γ}+meas{x∈Ω:|∂un∂xi|>α,|un|>γ}≤∫{|∂un/∂xi|>α,|un|≤γ}‍(1α|∂un∂xi|)pi-dx+λ|un|(γ)≤1αpi-∫{|un|≤γ}‍|∂un∂xi|pi-dx+λ|un|(γ).
Using (135) and (i), we get
(137)λ|∂un/∂xi|(α)≤C(γαpi-+γ-p~),
from which we deduce (ii).
By lemmas3 and 23, it follows that (un)n∈ℕ* is uniformly bounded in Ls0(Ω) for some 1≤s0<p~, and in the same way, (|∂un/∂xi|)n∈ℕ* is uniformly bounded in Lsi(Ω) for some 1≤si<p~i. From this, we get that the sequence (un)n∈ℕ* is uniformly bounded in W1,s(Ω), where s=min(s0,s1,…,sN). Consequently, we can extract a subsequence, still denoted (un) satisfying
(138)un⟶ua.e.inΩ,inLs(Ω),un⇀uinW1,s(Ω),|∂un∂xi-∂u∂xi|⇀ℋi(x)inLs(Ω),∀i=1,…,N.
By the same way as in the proof of [16, Lemma 3.5] (see also [27]), we prove that
(139)ℋi(x)=0a.e.x∈Ω∀i=1,…,N.
We deduce from (139) that
(140)ai(x,∂un∂xi)⟶ai(x,∂u∂xi)a.e.inΩ,inL1(Ω),∀i=1,…,N.
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].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). Let un∈W1,p→(·)(Ω) be the solution of (133). The following assertions hold.(i)
For alli=1,…,N, ∂un/∂xi converges in measure to the weak partial gradient of u.(ii)
For alli=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∈W1,p→(·)(Ω)∩L∞(Ω). For any k>0, choose Tk(un-φ) as a test function in (134), we get
(141)∑i=1N‍∫Ω‍ai(x,∂un∂xi)∂∂xiTk(un-φ)dx+∫Ω‍|un|pM(x)-2unTk(un-φ)dx=∫Ω‍fn(x)Tk(un-φ)dx.
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
(142)∫Ω‍|un|pM(x)-2unTk(un-φ)dx=∫Ω‍(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)dx+∫Ω‍|φ|pM(x)-2φTk(un-φ)dx. The quantity (|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ) is nonnegative and since for all x∈Ω,s↦|s|pM(x)-2s is continuous; we get
(143)(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)⟶(|u|pM(x)-2u-|φ|pM(x)-2φ)Tk(u-φ)a.e.inΩ.
Then, it follows by Fatou’s Lemma that
(144)liminfn→+∞∫Ω‍(|un|pM(x)-2un-|φ|pM(x)-2φ)Tk(un-φ)dx≥∫Ω‍(|u|pM(x)-2u-|φ|pM(x)-2φ)Tk(u-φ)dx.Let us show that|φ|pM(x)-2φ∈L1(Ω).We have(145)∫Ω‍||φ|pM(x)-2φ|dx=∫Ω‍|φ|pM(x)-1dx≤∫Ω‍(∥φ∥∞)pM(x)-1dx.
If ∥φ∥∞≤1, then ∫Ω‍||φ|pM(x)-2φ|dx≤meas(Ω)<+∞.If∥φ∥∞>1, then
(146)∫Ω‍||φ|pM(x)-2φ|dx≤∫Ω‍(∥φ∥∞)pM+-1dx=(∥φ∥∞)pM+-1meas(Ω)<+∞.
Hence, |φ|pM(x)-2φ∈L1(Ω).SinceTk(un-φ) converges weakly-* to Tk(u-φ) in L∞(Ω) and |φ|pM(x)-2φ∈L1(Ω), it follows that
(147)limn→+∞∫Ω‍|φ|pM(x)-2φTk(un-φ)dx=∫Ω‍|φ|pM(x)-2φTk(u-φ)dx.
For the first term of (141), we write it as follows:
(148)∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiundx-∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiφdx.
The first term of (148) is nonnegative by (7), then by Fatou’s Lemma and (138), we get
(149)∑i=1N‍∫{|u-φ|≤k}‍ai(x,∂u∂xi)∂∂xiudx≤liminfn→∞∑i=1N‍∫{|un-φ|≤k}‍ai(x,∂un∂xi)∂∂xiundx.
According to Proposition 25, the second term of (148) converges to
(150)∑i=1N‍∫{|u-φ|≤k}‍ai(x,∂u∂xi)∂∂xiφdx.
Combining the previous convergence results, we obtain
(151)∑i=1N‍∫Ω‍ai(x,∂u∂xi)∂∂xiTk(u-φ)dx+∫Ω‍|u|pM(x)-2uTk(u-φ)dx≤∫Ω‍f(x)Tk(u-φ)dx.



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*Source: 476781-2013-04-03.xml*