# Existence of Solutions for NonhomogeneousA-Harmonic Equations with Variable Growth

Authors: Yongqiang Fu; Lifeng Guo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the following nonhomogeneousA-harmonic equations: d*A(x,du(x))+B(x,u(x))=0,x∈Ω,u(x)=0,x∈∂Ω, where Ω⊂ℝn is a bounded and convex Lipschitz domain, A(x,du(x)) and B(x,u(x)) satisfy some p(x)-growth conditions, respectively. We obtain the existence of weak solutions for the above equations in subspace 𝔎01,p(x)(Ω,Λl-1) of W01,p(x)(Ω,Λl-1).


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## Body

## 1. Introduction

Spaces of differential forms have been discussed in great details (see [1, 2] and the references therein). The theory of differential forms is an approach to multivariable calculus that is independent of coordinates and provides a better definition for integrals. Differential forms have played an important role in physical laws of thermodynamics, analytical mechanics, and physical theories, in particular Maxwell's theory, and the Yang-Mills theory, the theory of relativity, see for example [3–6].In recent years, the study ofA-harmonic equations for differential forms has developed rapidly. Many interesting results concerning A-harmonic equation have been established recently (see [7–11] and the references therein). In [12], spaces Lp(x)(Ω) and Wk,p(x)(Ω) are first introduced, and they used them to study the solutions of nonlinear Dirichlet boundary value problems with p(x)-growth conditions. In [13], spaces Lp(x)(Ω,Λl,ω) and W1,p(x)(Ω,Λl,ω) are first introduced and used to study the weak solutions of obstacle problems of A-harmonic equations with variable growth for differential forms.LetΩ⊂ℝn be a bounded and convex Lipschitz domain. It is our purpose to study the following systems:
(1.1)d*A(x,du(x))+B(x,u(x))=0,x∈Ω,u(x)=0,x∈∂Ω,
where u∈Λl-1(Ω), l=1,2,…,n, and A:Ω×Λl(Ω)→Λl(Ω), B:Ω×Λl-1(Ω)→Λl-1(Ω) satisfy the following conditions.(H1)
A(x,ξ) and B(x,ς) are measurable with respect to x for all ξ,ς and continuous with respect to ξ,ς, respectively, for a.e. x∈Ω.(H2)
|A(x,ξ)|+|B(x,ς)|≤C1|ξ|p(x)-1+C2|ς|p(x)-1+G(x), where G∈Lp′(x)(Ω) and C1,C2≥0 are constants.(H3)
〈A(x,ξ),ξ〉≥a|ξ|p(x)-|h(x)|, where a>0 is a constant and h∈L1(Ω).(H4)
〈B(x,ς),ς〉≥a¯|ς|p(x)-|h¯(x)|, where a¯≥0 is a constant and h¯∈L1(Ω).(H5)
For a.e.x0∈Ω, the mapping ξ→A(x0,ξ) satisfies
(1.2)∫D‍〈A(x0,ξ0+dv(x)),dv(x)〉dx≥γ∫D‍|dv(x)|p(x)dx,
for each ξ0∈Λl(Ω), D⊂Ω and v∈C01(Ω,Λl-1), where γ>0 is a constant. Here p′ is the conjugate function of p. Throughout this paper we suppose (unless declare specially)
(1.3)p∈Plog⁡(Ω),1<p*=essinfΩp(x)≤p(x)≤esssup⁡Ωp(x)=p*<∞.


## 2. Preliminaries

Lete1,e2,…,en be the standard orthogonal basis of ℝn. The space of all l-forms in ℝn is denoted by Λl(ℝn). The dual basis to e1,e2,…,en is denoted by e1,e2,…,en and referred to as the standard basis for 1-form Λ1(ℝn). The Grassman algebra Λ(ℝn)=⊕Λl(ℝn) is a graded algebra with respect to the exterior products. The standard ordered basis for Λ(ℝn) consists of the forms
(2.1)1,e1,e2,…,en,e1∧e2,…,en-1∧en,…,e1∧e2…∧en.Forα(x)=∑‍αI(x)eI∈Λl(ℝn) and β(x)=∑‍βI(x)eI∈Λl(ℝn), the inner product is obtained by 〈α,β〉=∑‍αI(x)βI(x) with summation over all l–tuples I=(i1,…il) and all integers l=0,1,…,n. The Hodge star operator (see [14]) ⋆:Λ(ℝn)→Λ(ℝn) is defined by the formulas
(2.2)⋆1=e1∧e2⋯∧en,α∧⋆β=β∧⋆α=〈α,β〉e1∧e2⋯∧en.
Hence, the norm of α is given by the formula |α|2=〈α,α〉=⋆(α∧⋆α)=∑‍αI(x)αI(x)∈Λ0(ℝn)=ℝ. Notice, the Hodge star operator is an isometric isomorphism operator on Λ(ℝn). Moreover,
(2.3)⋆:Λl(Rn)⟶Λn-l(Rn),⋆⋆=(-I)l(n-l):Λl(Rn)⟶Λl(Rn),
where I is the identity map.LetΩ⊂ℝn be a bounded domain. The coordinate functions x1,x2,…,xn in Ω are considered to be differential forms of degree 0. The 1-forms dx1,dx2,…,dxn are constant functions from Ω into Λl(ℝn). The value of dxi is simply ei, i=1,2,…,n. Therefore, every l-form u:Ω→Λl(ℝn) may be written uniquely as
(2.4)u(x)=∑I‍uI(x)dxI=∑1≤i1<⋯<il≤n‍ui1,…il(x)dxi1∧⋯∧dxil,
where the coefficients ui1,…il(x) are distributions from 𝒟′(Ω), dual to the space of smooth functions with compact support on Ω.We use𝒟′(Ω,Λl) to denote the space of all differential l-forms. For each form u(x)∈𝒟′(Ω,Λl), the exterior differential d:𝒟′(Ω,Λl)→𝒟′(Ω,Λl+1) is expressed by
(2.5)du(x)=∑k=1n‍∑1≤i1<⋯<il≤n‍∂ui1,…il(x)∂xkdxk∧dxi1∧⋯∧dxil.Foru∈𝒟′(Ω,Λl), the vector-valued differential form
(2.6)∇u=(∂u∂x1,∂u∂x2,…,∂u∂xn)
consists of differential forms ∂u/∂xi∈𝒟′(Ω,Λl), where the partial differentiation is applied to the coefficients of u.The formal adjoint operator, called the Hodge codifferential, is given by(2.7)d⋆=(-1)nl-1⋆d⋆:D′(Ω,Λl+1)⟶D′(Ω,Λl).
By C∞(Ω,Λl) denote the space of infinitely differentiable l-forms on Ω and by C0∞(Ω,Λl) denote the subspace of C∞(Ω,Λl) with compact support on Ω.Let𝒫(Ω) be the set of all Lebesgue measurable functions p:Ω→(1,∞). For p∈𝒫(Ω), we put p*=essinfΩp(x) and p*=esssupΩp(x). Given p∈𝒫(Ω) we define the conjugate function p′∈𝒫(Ω) by
(2.8)p′(x)=p(x)p(x)-1,∀x∈Ω.Definition 2.1 (see [15]).
A Lebesgue measurable functionp:Ω→ℝ is called globally log-Hölder continuous in Ω if there exist p∞∈ℝ and a constant C>0 such that
(2.9)|p(x)-p(y)|≤Clog⁡(e+1/|x-y|),|p(x)-p∞|≤Clog⁡(e+|x|)
hold for all x,y∈Ω. Plog(Ω) is defined by
(2.10)Plog⁡(Ω)={p∈P(Ω):1pisgloballylog-Ho¨ldercontinuous}.For a differentiall-form u(x) on Ω, l=0,1,…,n, define the functional ρp(x) by
(2.11)ρp(x),Λl(u)=∫Ω|u(x)|p(x)dx.The spaceLp(x)(Ω,Λl)={u∈Λl(Ω):∃λ>0,ρp(x),Λl(λu)<∞} is a reflexive Banach space endowed with the norm
(2.12)‖u‖Lp(x)(Ω,Λl)=inf⁡{λ>0:ρp(x),Λl(uλ)≤1}.The spaceW1,p(x)(Ω,Λl)={u∈Λl(Ω):u∈Lp(x)(Ω,Λl)anddu∈Lp(x)(Ω,Λl+1)} is a reflexive Banach space endowed with the norm
(2.13)‖u‖W1,p(x)(Ω,Λl)=‖u‖Lp(x)(Ω,Λl)+‖du‖Lp(x)(Ω,Λl+1).Note thatLp(m)(Ω,Λ0) and W1,p(m)(Ω,Λ0) are spaces of functions on Ω. In this paper, we denote them by Lp(m)(Ω) and W1,p(m)(Ω).Iwaniec and Lutoborski proved the following results in [2].LetΩ⊂ℝn be a bounded and convex domain. If u(x)∈Λl(ℝn) is defined for some x∈Ω, then the value of u(x) at the vectors ξ1,…,ξl∈ℝn is denoted by u(x)(ξ1,…,ξl). Then to each y∈Ω, there corresponds a linear operator Ky:Lloc1(Ω,∧l)→Lloc1(Ω,∧l-1) defined by
(2.14)Kyu(x)(ξ1,ξ2,…,ξl-1)=∫01‍tl-1u(tx+y-ty)(x-y,ξ1,ξ2,…,ξl-1)dt.
The homotopy operator T:Lloc1(Ω,∧l)→Lloc1(Ω,∧l-1) is defined by averaging Ky over all points y∈Ω(2.15)Tu(x)=∫Ω‍φ(y)Kyu(x)dy,
where φ∈C0∞(Ω) is normalized so that ∫Ω‍φ(y)dy=1. Then we have a pointwise estimate
(2.16)|Tu(x)|≤2nμ(Ω)∫Ω‍|u(y)||x-y|n-1dy,∀x∈Ω,
where
(2.17)μ(Ω)=(diam⁡Ω)n+1inf⁡{‖∇φ‖L∞(Ω)‖φ‖L1(Ω):φ∈C0∞(Ω)},
further infimum is attained at φ(x)=diam(x,∂Ω), and the decomposition
(2.18)u=dTu+Tdu
holds for u∈Lloc1(Ω,Λl).Definition 2.2.
Foru∈Lloc1(Ω,Λl), define the l-form uΩ∈𝒟′(Ω,Λl) by
(2.19)uΩ={1meas⁡(Ω)∫Ω‍u(x)dx,forl=0,dTu,forl=1,2,⋯,n,
and the Maximal operator is defined by
(2.20)(Mu)(x)=sup⁡r>01meas⁡(Br(x))∫Br(x)‍|u(y)|dy,
where Br(x)={y∈ℝn:|y-x|<r}.Lemma 2.3 (see [15]).
Letp(x) satisfies (1.3). Then the inequality
(2.21)‖(Mu)(x)‖Lp(x)(Rn)≤C(n,p)‖u(x)‖Lp(x)(Rn)
holds for every u∈Lp(x)(ℝn).Lemma 2.4 (see [15]).
LetΩ⊂ℝn be a bounded convex domain, x∈Ω and u∈Lloc1(ℝn). Then
(2.22)∫Ω‍|u(y)||x-y|n-1dy≤C(n)(diam⁡Ω)(Mu)(x).Lemma 2.5 (see [15]).
LetΨ be a Calderón-Zygmund operator with Calderón-Zygmund kernel K on ℝn×ℝn. Then Ψ is bounded on Lp(x)(ℝn). Further there exists a constant C = C(n,p) such that
(2.23)‖Ψu(x)‖Lp(x)(Rn)≤C(n,p)‖u(x)‖Lp(x)(Rn)
holds for every u∈Lp(x)(ℝn).Lemma 2.6.
Ifu∈Lp(x)(Ω,Λl), then
(2.24)‖Tu‖Lp(x)(Ω,Λl-1)≤C(n,p)μ(Ω)(diam⁡Ω)‖u‖Lp(x)(Ω,Λl).
Moreover, if u∈W1,p(x)(Ω,Λl), then
(2.25)‖uΩ‖Lp(x)(Ω,Λl)≤C(p)‖u‖Lp(x)(Ω,Λl)+C(n,p)μ(Ω)(diam⁡Ω)‖du‖Lp(x)(Ω,Λl+1).Proof.
First defineu(x)=0ifx∈ℝn∖Ω. From pointwise estimate (2.16) and Lemma 2.4,
(2.26)|Tu(x)|≤C(n)μ(Ω)(diam⁡Ω)M(|u|)(x),∀x∈Ω.
In view of Lemma 2.3, we have
(2.27)‖|Tu|‖Lp(x)(Ω)≤C(n,p)μ(Ω)(diam⁡Ω)‖|u|‖Lp(x)(Ω),
that is to say, (2.24) holds.
From the definition ofuΩ and (2.18), we have uΩ=u-Tdu. Therefore,
(2.28)‖uΩ‖Lp(x)(Ω,Λl)≤C(p)‖u‖Lp(x)(Ω,Λl)+C(n,p)‖Tdu‖Lp(x)(Ω,Λl).
Now in (2.24) replace u with du, we obtain (2.25).Lemma 2.7.
Letp(x) satisfies (1.3).(1)
C0∞(Ω,Λl) is dense in Lp(x)(Ω,Λl),(2)
Lp(x)(Ω,Λl) is separable.Proof.
(1) For anyu(x)=∑I‍uI(x)dxI∈Lp(x)(Ω,Λl), since C0∞(Ω) is dense in Lp(x)(Ω) and uI(x)∈Lp(x)(Ω) for all I, we can find a sequence {uIk}k=1∞⊂C0∞(Ω) which converges to uI(x) in Lp(x)(Ω) for each I. Now let uk(x)=∑I‍uIkdxI, then the sequence {uk(x)}⊂C0∞(Ω,Λl) converges to u(x) in Lp(x)(Ω,Λl), since
(2.29)∫Ω‍|u(x)-uk(x)|p(x)dx=∫Ω‍((∑I‍|uI(x)-uIk(x)|2)1/2)p(x)dx≤∫Ω‍(∑I‍|uI(x)-uIk(x)|)p(x)dx≤2p*∑I‍∫Ω‍|uI(x)-uIk(x)|p(x)dx.
That is to say, C0∞(Ω,Λl) is dense in Lp(x)(Ω,Λl).
(2) Letu(x)=∑I‍uI(x)dxI∈Lp(x)(Ω,Λl). Since Lp(x)(Ω) is separable, there exists a countable dense subset K of Lp(x)(Ω). Then for any uI(x) above we can extract a sequence {uIk(x)} in K which converges to uI(x) in Lp(x)(Ω). Similar to (1), the sequence {uk:uk(x)=∑I‍uIk(x)dxI} converges to u(x) in Lp(x)(Ω,Λl). That is to say, Lp(x)(Ω,Λl) is separable.Let𝔎1,p(x)(Ω,Λl)={u(x)=ϑ(x)-ϑΩ(x):ϑ∈W1,p(x)(Ω,Λl)}. Note that u∈𝔎1,p(x)(Ω,Λl) if and only if uΩ=0.Lemma 2.8.
Letp(x) satisfies (1.3). Then 𝔎1,p(x)(Ω,Λl) is a closed subspace of W1,p(x)(Ω,Λl). In particular, it is a reflexive Banach space.Proof.
Set a sequence{uk(x)}⊂𝔎1,p(x)(Ω,Λl) convergent to u(x) in W1,p(x)(Ω,Λl), then (uk)Ω=0. By Lemma 2.6, the operator T is continuous on 𝔎1,p(x)(Ω,Λl). Therefore, uΩ=0, we have u(x)∈𝔎1,p(x)(Ω,Λl). That is to say, 𝔎1,p(x)(Ω,Λl) is a closed subspace of W1,p(x)(Ω,Λl).In [2], Iwaniec and Lutoborski obtained
(2.30)∂∂xi(Tu)=Aiu+Siu,
where
(2.31)|Aiu(x)|≤2nμ(Ω)diam⁡(Ω)∫Ω‍|u(z)||x-z|n-1dz,(2.32)Siu(x)(ξ)=∫Ω‍u(z)(Ki(z,x-z),ξ)dz,
where ξ=(ξ1,ξ2,…,ξl-1) and
(2.33)Ki(z,x-z)=ei|x-z|n∫0∞‍sn-1φ(z-sx-z|x-z|)ds-x-z|x-z|n+1∫0∞‍snφi(z-sx-z|x-z|)ds.
Further for each z∈Ω and h∈ℝn-{0}, Ki(z,h) satisfies the following properties:(i)
Ki(z,h)≤μ(Ω)|h|-n,(ii)
Ki(z,sh)=s-nKi(z,h),s>0,(iii)
∫|h|=1‍Ki(z,h)=0 for all z∈Ω.LetKi(z,h)=(Ki1,Ki2,…,Kin). Then Kiα satisfies the conditions of Calderón-Zygmund kernel on ℝn×ℝn for each α=1,2,…,n.Lemma 2.9.
Letu∈Lp(x)(Ω,Λl). Then
(2.34)‖|∇Tu|‖Lp(x)(Ω)≤C(n,p,Ω)‖u‖Lp(x)(Ω,Λl).Proof.
By Lemmas2.3 and 2.4, and (2.31),
(2.35)‖Aiu‖Lp(x)(Ω,Λl)≤C(n,p)μ(Ω)‖u‖Lp(x)(Ω,Λl).
Let
(2.36)Siu(x)=∑1≤j1<j2<⋯<jl-1≤n‍ωj1,j2,…,jl-1dxj1∧dxj2∧⋯∧dxjl-1,
we can write u(x) as
(2.37)u(x)=∑1≤α≤n,α≠j1,j2,…,jl-1‍∑1≤j1<j2<⋯<jl-1≤n‍uα,j1,j2,…,jl-1dxα∧dxj1∧⋯∧dxjl-1.
Hence,
(2.38)ωj1,j2,…,jl-1(x)=Siu(x)(ej1,ej2,…,ejl-1).
Taking ξ=(ej1,ej2,…,ejl-1) in (2.32), we obtain
(2.39)ωj1,j2,…,jl-1(x)=∫Ω‍∑1≤α≤n,α≠j1,…,jl-1‍Kiα(z,x-z)uα,j1,j2,…,jl-1(z)dz.
Now define u(x)=0ifx∈ℝn∖Ω. Since Kiα satisfies the conditions of Calderón-Zygmund kernel on ℝn×ℝn for each α, in view of Lemma 2.5,
(2.40)‖ωj1,j2,…,jl-1‖Lp(x)(Ω)≤C(n,p)∑1≤α≤n,α≠j1,…,jl-1‍‖uα,j1,j2,…,jl-1‖Lp(x)(Ω).
So that
(2.41)‖Siu‖Lp(x)(Ω,Λl)≤C(n,p)‖u‖Lp(x)(Ω,Λl).
By (2.30), (2.35), and (2.41), we have
(2.42)‖|∇Tu|‖Lp(x)(Ω)≤C(n,p,Ω)‖u‖Lp(x)(Ω,Λl).Now define another norm(2.43)‖|ω|‖K1,p(x)(Ω,Λl)=‖ω‖Lp(x)(Ω,Λl)+‖|∇ω|‖Lp(x)(Ω).Remark 2.10.
Replacingu with du in (2.34), we get by the definition of uΩ(2.44)‖|∇(u-uΩ)|‖Lp(x)(Ω)=‖|∇Tdu|‖Lp(x)(Ω)≤C(n,p)μ(Ω)‖du‖Lp(x)(Ω,Λl)=C(n,p)μ(Ω)‖d(u-uΩ)‖Lp(x)(Ω,Λl).
Therefore ∥|·|∥𝔎1,p(x)(Ω,Λl) is equivalent to ∥·∥𝔎1,p(x)(Ω,Λl).In this paper we also need the following two lemmas.Lemma 2.11 (see[15]).
Letp(x) satisfies (1.3). Then the embedding W01,p(x)(Ω)↪Lp(x)(Ω) is compact.Lemma 2.12 (see[15]).
Suppose thatp∈L∞(Ω). Let {uk}k=1∞ be bounded in Lp(x)(Ω). If uk→u a.e. on Ω, then uk⇀u weakly in Lp(x)(Ω).Remark 2.13.
Let𝔎01,p(x)(Ω,Λl) be the completion of C0∞(Ω,∧l) in 𝔎1,p(x)(Ω,Λl). Then from Remark 2.10 and Lemma 2.11, the embedding 𝔎01,p(x)(Ω,Λl)↪Lp(x)(Ω,Λl) is compact.Remark 2.14.
Supposep(x) satisfies (1.3), Lemma 2.12 also holds on space Lp(x)(Ω,∧l).


## 3. Weak Solutions of Dirichlet Problems for theA-Harmonic Equations with Variable Growth

Theorem 3.1.
Under conditions (H1)–(H5), the Dirichlet problem (1.1) has at least one weak solution in 𝔎01,p(x)(Ω,Λl), that is to say, there exists at least one u=ϑ-ϑΩ∈𝔎01,p(x)(Ω,Λl) satisfying
(3.1)∫Ω‍〈A(x,du(x)),dφ(x)〉+〈B(x,u(x)),φ(x)〉dx=0,
for all φ∈W01,p(x)(Ω,Λl-1). Here, ϑ∈W1,p(x)(Ω,Λl-1) and p(x) satisfies (1.3).LetV=W01,p(x)(Ω,Λl-1) and 𝔎0=𝔎01,p(x)(Ω,Λl). For u∈V, define 𝔄:V→V* in the following way: for each φ∈V(3.2)(Au,φ)=∫Ω‍〈A(x,du(x)),dφ(x)〉+〈B(x,u(x)),φ(x)〉dx.
Now we need only to show that there exists u∈𝔎0 such that (𝔄u,φ)=0 for all φ=∑‍φI(x)dxI∈V.Lemma 3.2.
𝔄 is strong-weakly continuous on V.Proof.
Let{uk:uk(x)=∑I‍ukI(x)dxI}⊂V be a sequence strongly convergent to an element u(x)=∑‍uI(x)dxI∈V in V. Let duk(x)=∑J‍ωkJ(x)dxJ and du(x)=∑J‍ωJ(x)dxJ. Then (h1)
∥uk∥V≤C for some constant C,(h2)
{ωkJ(x)} is a sequence strongly convergent to ωJ(x) in Lp(x)(Ω) for each J.
In view of (H2) and(h1), we know that A(x,duk)=∑‍AkJ(x)dxJ and B(x,uk)=∑‍BkI(x)dxI are uniformly bounded in Lp′(x)(Ω,Λl) and Lp′(x)(Ω,Λl-1), respectively. Hence, AkJ(x) and BkI(x) are uniformly bounded in Lp′(x)(Ω). On the other hand, by (h2), there exists a subsequence of {ωkJ(x)} (still denoted by {ωkJ(x)}) such that
(3.3)lim⁡k→∞ωkJ(x)=ωJ(x),a.e.x∈Ω,foreachJ.
Then there exists a subsequence of {uk(x)} (still denoted by {uk(x)}) such that
(3.4)lim⁡k→∞uk(x)=u(x),lim⁡k→∞duk(x)=du(x),a.e.x∈Ω.
In view of (H1), we obtain
(3.5)lim⁡k→∞A(x,duk)=A(x,du),a.e.x∈Ω,lim⁡k→∞B(x,uk)=B(x,u),a.e.x∈Ω.
Let dφ(x)=∑‍ψJ(x)dxJ, A(x,du)=∑‍AJ(x)dxJ, and B(x,u)=∑‍BI(x)dxI, then ψJ(x)∈Lp(x)(Ω), in the meantime
(3.6)lim⁡k→∞AkJ(x)=AJ(x),a.e.x∈Ω,(3.7)lim⁡k→∞BkI(x)=BI(x),a.e.x∈Ω,
for each J and I.
Now by Lemma2.12, we can show that ∫Ω‍AkJ(x)ψJ(x)dx→∫Ω‍AJ(x)ψJ(x)dx and ∫Ω‍BkI(x)φI(x)dx→∫Ω‍BI(x)φI(x)dx as k→∞. Therefore,
(3.8)(Auk,φ)=∫Ω‍〈A(x,duk),dφ〉+〈B(x,uk),φ〉dx⟶∫Ω‍〈A(x,du),dφ〉+〈B(x,u),φ〉dx=(Au,φ),
that is to say, 𝔄 is strong-weakly continuous on V.Lemma 3.3.
𝔄 is coercive on 𝔎0, that is,
(3.9)lim⁡‖u‖K→∞(Au,u)‖u‖K=+∞,∀u∈K0.Proof.
By (H3) and (H4),(3.10)(Au,u)=∫Ω‍〈A(x,du),du〉+〈B(x,u),u〉dx≥∫Ω‍(a|du|p(x)-|h(x)|+a¯|u|p(x)-|h¯(x)|)dx≥∫Ω‍a|du|p(x)dx-C(h,h¯).
By dϑΩ=0 and Lemma 2.6, we have
(3.11)‖u‖Lp(x)(Ω,Λl-1)=‖Tdϑ‖Lp(x)(Ω,Λl-1)≤2nC(n,p)μ(Ω)(diam⁡Ω)‖du‖Lp(x)(Ω,Λl),
for all u=ϑ-ϑΩ∈𝔎0. Then ∥du∥Lp(x)(Ω,Λl)→∞, as ∥u∥𝔎→∞. Taking
(3.12)δ=12‖du‖Lp(x)(Ω,Λl)>1,
we have
(3.13)∫Ω‍|du|p(x)dx‖du‖Lp(x)(Ω,Λl)=∫Ω‍(|du|‖du‖Lp(x)(Ω,Λl)-δ)p(x)(‖du‖Lp(x)(Ω,Λl)-δ)p(x)‖du‖Lp(x)(Ω,Λl)dx≥(‖du‖Lp(x)(Ω,Λl)-δ)p*‖du‖Lp(x)(Ω,Λl)=(12)p*‖du‖Lp(x)(Ω,Λl)p*-1.
Therefore,
(3.14)∫Ω‍|du|p(x)dx‖du‖Lp(x)(Ω,Λl)⟶∞as‖du‖Lp(x)(Ω,Λl)⟶∞.
Then it is immediate to obtain that
(3.15)(Au,u)‖u‖K⟶∞as‖u‖K⟶∞.
That is to say, 𝔄 is coercive on 𝔎0.Lemma 3.4 (see[16]).
Supposeg=A(x) is a mapping from ℝm into itself such that
(3.16)lim⁡|x|→∞A(x)⋅x|x|=∞.
Then the range of A is the whole of ℝm.Lemma 3.5.
There exists a sequence{uk}⊂𝔎0 and u0∈𝔎0, such that
(3.17)(Auk,uk-u0)⟶0ask⟶∞.Proof .
By Lemmas2.7 and 2.8, we can choose a Schauder basis {ωs} of 𝔎0 such that the union of subspace finitely generated from ωs is dense in 𝔎0. Let 𝔎0k be the subspace of 𝔎0 generated by ω1,ω2,…,ωk. Since 𝔎0k is topologically isomorphic to ℝk. By Lemmas 3.3, and 3.4, there exists uk∈𝔎0k such that
(3.18)(Auk,ω)=0∀ω∈K0k.
By Lemma 3.3 again, we know that ∥uk∥𝔎≤C, where C is independent of k. Since 𝔎0 is reflexive, by Remark 2.14 and (H1), we can extract a subsequence of {uk} (still denoted by {uk}) such that
(3.19)uk⇀u0weaklyinK0,Auk⇀ξweakly*inK0*,(ξ,ω)=0,
where ω is in a dense subset of 𝔎0. For fixed ξ, by the continuity of (ξ,·), we get (ξ,ω)=0 for all ω∈𝔎0. For (𝔄uk,uk-u0), we have
(3.20)(Auk,uk-u0)=(Auk,uk)-(Auk,u0)=-(Auk,u0)⟶0ask⟶∞.
This completes the proof of Lemma 3.5.Setvk=uk-u0=∑‍vkIdxI. Then
(3.21)vk⇀0weaklyinK0ask⟶∞.
Consider (𝔄uk,uk-u0) once more, then
(3.22)(Auk,uk-u0)=∫Ω‍〈A(x,du0+dvk),dvk〉+〈B(x,u0+vk),vk〉dx⟶0,
as k→∞. By Remark 2.13, we get
(3.23)vk⟶0stronglyinLp(x)(Ω,Λl-1).
In view of (3.23) and (H2), it is immediate that
(3.24)∫Ω‍〈B(x,u0+vk),vk〉dx⟶0ask⟶∞,
that is to say,
(3.25)∫Ω‍〈A(x,du0+dvk),dvk〉dx⟶0ask⟶∞.Now if we can prove that there exists a subsequence of{vk} which is strongly convergent in 𝔎0, then from the strong-weakly continuity of 𝔄, we get 𝔄uk⇀𝔄u0=ξ weakly in 𝔎0 as k→∞ and u0 will be a weak solution of (1.1). We need the following lemmas.Definition 3.6.
LetΩ be an open subset of ℝn provided with the Lebesgue measure. The mapping f:Ω×ℝN→ℝ is said Carathéodory function if for almost all x∈Ω, f(x,·) is continuous on ℝN, for all ξ∈ℝN is measurable on Ω.Lemma 3.7 (see[17]).
A mappingf:Ω×ℝN→ℝ is a Carathéodory function if and only if for all compact sets K⊂Ω and all ɛ>0, there exists a compact subset Kɛ⊂K such that meas(K-Kɛ)<ɛ for with the restriction of f to Kɛ×ℝN is continuous.Lemma 3.8 (see[15]).
Let{fk} be a sequence of bounded function in L1(ℝn). For each ɛ>0 there exists (Aɛ,δ,N) (where Aɛ is measurable and meas(Aɛ)<ɛ,δ>0,N is an infinite subset of natural numbers set ℕ) such that for each k∈N,
(3.26)∫B‍|fk(x)|dx<ɛ,
where B and Aɛ are disjoint and meas(B)<δ.Definition 3.9.
Foru∈C01(ℝn), define
(3.27)(M*u)(x)=(Mu)(x)+∑α=1n(M∂u∂xα)(x).Lemma 3.10 (see[18]).
Ifu∈C0∞(ℝn), then M*u∈C0(ℝn) and for all x∈ℝn,
(3.28)|u(x)|+∑α=1n|∂u∂xα(x)|≤(M*u)(x).
Furthermore, if p>1, then
(3.29)‖M*u‖Lp(Rn)≤C(n,p)‖u‖W1,p(Rn),
and if p=1, then
(3.30)meas⁡({x∈Rn:(M*u)(x)>λ})≤C(n)λ‖u‖W1,1(Rn),
for all λ>0.Lemma 3.11 (see[19]).
Letu∈C0∞(ℝn) and λ>0. Set
(3.31)Hλ={x∈Rn:(M*u)(x)<λ}.
Then for all x,y∈Hλ, we have
(3.32)|u(y)-u(x)|≤C(n)λ|y-x|.Lemma 3.12 (see[16]).
LetX be a metric space, E be a subspace of X, and k be a positive number. Then any k-Lipchitz mapping from E into ℝ can be extended to a k-Lipchitz mapping from X into ℝ.Proof of Theorem3.1.
We need only to show that there exists subsequence of{vk} which is strongly convergent in 𝔎0.
For each measurable setS⊂Ω, define
(3.33)F(v,S)=∫S‍〈A(x,du0+dv),dv〉dx,
where v∈𝔎0. Similar to the proof of Lemma 3.2, F(·,S) is strongly continuous on 𝔎0. Since C0∞(Ω,Λl-1) is dense in 𝔎0, there exists hk⊂C0∞(Ω,Λl-1) such that
(3.34)‖hk-vk‖K<1k,|F(hk,Ω)-F(vk,Ω)|<1k.
So we can suppose that {vk}⊂C0∞(Ω,Λl-1) is bounded in 𝔎0.
Next define(3.35)vk(x)=0whenx∈Rn∖Ω.
In this way, we extend the domain of vk to ℝn and suppvk⊂Ω.
Letβ:ℝ+→ℝ+ be a continuous increasing function satisfying β(0)=0 and for each measurable set D⊂Ω,
(3.36)∫D‍(|G(x)|p′(x)+|h(x)|+(C1+1)|du0|p(x))dx≤β(meas⁡(D)),
where C1 is the constant in (H2).
Let{ɛj} be a positive decreasing sequence with ɛj→0 as j→∞. For ɛ1, by Lemma 3.8, we get a subsequence {k1} of {k}, a set Aɛ1⊂Ω satisfying meas(Aɛ1)<ɛ1, and a real number δ1>0 such that
(3.37)∫B‍(M*vk1I)p(x)dx<ɛ1,
for each k1,I and B⊂Ω∖Aɛ1 satisfying meas(B)<δ1. By Lemma 3.10, we can choose λ>1 so large that for all I and k1,
(3.38)meas⁡({x∈Rn:(M*vk1I)(x)≥λ})≤min⁡{ɛ1,δ1}.
For eachI and k1, define
(3.39)Hk1Iλ={x∈Rn:(M*vk1I)(x)<λ},Hk1λ=⋂I‍Hk1Iλ.
In view of Lemma 3.11, we have
(3.40)|vk1I(y)-vk1I(x)||y-x|≤C(n)λ∀x,y∈Hk1λandI.
Form Lemma 3.12, there exists a Lipschitz function gk1I which extends vk1I outside Hk1λ and Lipschitz constant of gk1I is no more than C(n)λ. As Hk1λ is an open set, we have gk1I=vk1I and ∇gk1I(x)=∇vk1I(x) for all x∈Hk1λ, and ∥∇gk1I∥L∞(ℝn)≤C(n)λ. We can further suppose that
(3.41)‖gk1I‖L∞(Rn)≤‖vk1I‖L∞(Hk1λ)≤λ,‖gk1I‖W1,∞(Ω)≤C(n)λ.
By the uniformly boundedness of {∥gk1I∥W1,∞(Ω)}, there exists a subsequence of {gk1I} (still denoted by {gk1I}) such that
(3.42)gk1I⇀ωIweakly*inW1,∞(Ω)ask1⟶∞∀I.
Set ω=∑I‍ωIdxI and gk1=∑I‍gk1IdxI. We have
(3.43)F(vk1,Ω)=F(gk1,Ω∖Aɛ1)-F(gk1,(Ω∖Aɛ1)∖Hk1λ)+F(vk1,Aɛ1∪(Ω∖Hk1λ)).
Next we estimateF(vk1,Ω) in four steps.
(1)The estimate ofF(gk1,(Ω∖Aɛ1)∖Hk1λ)  andF(vk1,Aɛ1∪(Ω∖Hk1λ)). Since
(3.44)meas⁡((Ω∖Aɛ1)∖Hk1λ)≤∑I‍meas⁡((Ω∖Aɛ1)∖Hk1Iλ)≤Cnl-1min⁡{ɛ1,δ1},
where Cnl-1=n(n-1)⋯(n-l+2)/(l-1)(l-2)⋯1, from (H2), (H3), and the choose of Aɛ1, we have
(3.45)|F(gk1,(Ω∖Aε1)∖Hk1λ)|gk1≤∫(Ω∖Aε1)∖Hk1λ‍(C1|du0+dgk1|p(x)-1|dgk1|+|G(x)||dgk1|)dxgk1≤∫(Ω∖Aε1)∖Hk1λ‍(C12p*-1(|du0|p(x)+|dgk1|p(x))+C1|dgk1|p(x)+|G(x)|p′(x)+|dgk1|p(x))dxgk1≤2p*-1β(meas⁡((Ω∖Aε1)∖Hk1λ))+2p*(C1+1)∫(Ω∖Aε1)∖Hk1λ‍|dgk1|p(x)dxgk1≤2p*-1β(meas⁡((Ω∖Aε1)∖Hk1λ))+2p*(C1+1)∫(Ω∖Aε1)∖Hk1λ‍(∑I‍|∇gk1I|)p(x)dxgk1≤2p*-1β(Cnl-1ε1)+2p*C(C1,n,l)∫(Ω∖Aε1)∖Hk1λ‍λp(x)dxgk1≤2p*-1β(Cnl-1ε1)+2p*C(C1,n,l)∑I‍∫(Ω∖Aε1)∖Hk1Iλ‍(M*vk1I)p(x)dxgk1≤2p*-1β(Cnl-1ε1)+2p*C(C1,n,l)ε1≤O(ε1),(3.46)F(vk1,Aɛ1∪(Ω∖Hk1λ))vk1=∫Aɛ1∪(Ω∖Hk1λ)‍〈A(x,du0+dvk1),du0+dvk1〉-〈A(x,du0+dvk1),du0〉dxvk1≥∫Aɛ1∪(Ω∖Hk1λ)‍(a|du0+dvk1|p(x)-h(x))-(C1|du0+dvk1|p(x)-1|du0|+|G(x)||du0|)dxvk1≥∫Aɛ1∪(Ω∖Hk1λ)‍((a2-(p*-1)-C1μ2p*-1)|dvk1|p(x)-|h(x)|-|G(x)|p′(x))vk1vk1-(-a2-(p*-1)+C1μ2p*-1+C1C(μ)+1)|du0|p(x)dxvk1≥(a2-(p*-1)-C1μ2p*-1)∫Aɛ1∪(Ω∖Hk1λ)‍|dvk1|p(x)dx-C(a,p,C1,μ)β(meas⁡(Aɛ1∪(Ω∖Hk1λ)))vk1≥a2-p*∫Aɛ1∪(Ω∖Hk1λ)‍|dvk1|p(x)dx-O(ɛ1),
where μ>0 is small enough.
From (3.43)–(3.46), we get
(3.47)F(vk1,Ω)≥F(gk1,Ω∖Aɛ1)+a2-p*∫Aɛ1∪(Ω∖Hk1λ)‍|dvk1|p(x)dx-O(ɛ1).
(2)The estimate ofF(gk1,Ω∖Aɛ1). Set fk1I=gk1I-ωI, where ωI is defined by (3.42). Then
(3.48)fk1I⇀0weakly*inW1,∞(Ω)ask1⟶∞∀I,‖fk1I‖L∞(Ω)≤2λ,‖dfk1I‖L∞(Ω,Λl)≤2C(n)λ.
LetG=⋃I‍GI with GI={x∈Ω:ωI(x)≠0}. According to Acerbi and Fusco [19], we have meas(G)≤(Cnl-1+1)ɛ1 where Cnl-1=n(n-1)···(n-l+2)/(l-1)(l-2)···1, and set fk1=∑I‍fk1IdxI, then
(3.49)F(gk1,Ω∖Aɛ1)=F(fk1,(Ω∖Aɛ1)∖G)+F(vk1,(Ω∖Aɛ1)∩Hk1λ∩G)+F(gk1,(Ω∖Aɛ1)∩(G∖Hk1λ)).
Define
(3.50)Ω1ɛ1,k1=Aɛ1∪(Ω∖Hk1λ),Ω2ɛ1=(Ω∖Aɛ1)∖G,Ω3ɛ1,k1=(Ω∖Aɛ1)∩Hk1λ∩G,Ω4ɛ1,k1=(Ω∖Aɛ1)∩(G∖Hk1λ).
Similar to the proof of (3.46), we get
(3.51)F(vk1,Ω3ɛ1,k1)≥a2-p*∫Ω3ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1).
Since on Ω4ɛ1,k1 we have
(3.52)∫Ω4ɛ1,k1‍|dgk1|p(x)dx≤C(n,p)(Cnl-1+1)ɛ1,
then similar to the proof of (3.45), we get
(3.53)|F(gk1,Ω4ɛ1,k1)|≤O(ɛ1).
By (3.49)–(3.53), we have
(3.54)F(gk1,Ω∖Aɛ1)≥F(fk1,Ω2ɛ1)+a2-p*∫Ω3ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1).
Thus, we have
(3.55)F(vk1,Ω)≥F(fk1,Ω2ɛ1)+a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1),
where Ω5ɛ1,k1=Ω1ɛ1,k1∪Ω3ɛ1,k1.
Choose an open setΩ′⊂Ω which contains Ω2ɛ1 such that
(3.56)|F(fk1,Ω′)-F(fk1,Ω2ɛ1)|<ɛ1.
From (3.55), we get
(3.57)F(vk1,Ω)≥F(fk1,Ω′)+a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1).
ApproximateΩ′ by hypercubes with edges parallel to coordinate axes, that is, construct
(3.58)Hj⊂Ω′,meas⁡(Ω′∖Hj)⟶0asj⟶∞,Hj=⋃s=1hjDj,s,meas⁡(Dj,s)=1/2nj,1≤s≤hj.
Letj>0 be large enough such that for all k1>0, we have(3.59)|F(fk1,Ω′)-F(fk1,Hj)|<ɛ1,∫Ω′∖Hj‍|dfk1|p(x)dx<ɛ1,meas⁡(Ω′∖Hj)<min⁡{ɛ1,δ1}.
Thus,
(3.60)F(vk1,Ω)≥F(fk1,Hj)+a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1)-2ɛ1.
(3)The estimate ofF(fk1,Hj). Let α>0 be large enough such that for E={x∈Ω′:η(x)≤α}. Then
(3.61)meas⁡(Ω′∖E)<ɛ1N,∫Ω′∖E‍η(x)dx<ɛ1,
where ∥dfk1∥L∞(Ω,Λl)≤2Cnl-1C(n)λ=N and η(x)=|G(x)|p′(x)+2p*-1(C1+1)|du0|p(x).
Forx∈Ω,ξ∈Λl(Ω), define
(3.62)ψ(x,ξ)=〈A(x,du0(x)+ξ),ξ〉.
By Lemma 3.7 and (H1), there exists a compact subset K⊂Hj such that ψ(x,ξ) is continuous on K×Λl(Ω) and meas(Hj∖K)<ɛ1/(α+N). Hence, ψ(x,ξ) is uniformly continuous on bounded subsets of K×Λl(Ω).
Divide eachDj,s into 2nm hypercubes Qt,j,sm with edge length 2-jm,1≤t≤2nm. For all j,s,m,t, take xt,j,sm∈Qt,j,sm∩K∩E (if this set is empty, take xt,j,sm∈Qt,j,sm) such that
(3.63)η(xt,j,sm)meas⁡(Qt,j,sm)≤∫Qt,j,sm‍η(x)dx.
Then
(3.64)F(fk1,Hj)Hj=F(fk1,Hj∩K∩E)+F(fk1,Hj∖E)+F(fk1,(Hj∩E)∖K)Hj≥F(fk1,Hj∩K∩E)-∫Hj∖E‍η(x)dx-∫(Hj∩E)∖K‍η(x)dxHjHj-2p*(C1+1)(∫Hj∖E‍|dfk1|p(x)dx+∫(Hj∩E)∖K‍|dfk1|p(x)dx)Hj=F(fk1,Hj∩K∩E)-O(ɛ1)Hj=bk1m,j+ck1m,j+dk1m,j-O(ɛ1),
where
(3.65)bk1m,j=∑t,s‍∫Qt,j,sm∩K∩E‍(ψ(x,dfk1(x))-ψ(xt,j,sm,dfk1(x)))dx,ck1m,j=∑t,s‍∫Qt,j,sm‍ψ(xt,j,sm,dfk1(x))dx,dk1m,j=-∑t,s‍∫Qt,j,sm∖(K∩E)‍ψ(xt,j,sm,dfk1(x))dx.
By (3.25), we have
(3.66)lim⁡k1→∞F(vk1,Ω)=0.
Note that ifQt,j,sm∩K∩E is an empty set, then
(3.67)∫Qt,j,sm∩K∩E‍[ψ(x,dfk1(x))-ψ(xt,j,sm,dfk1(x))]dx=0.
Now we only consider Qt,j,sm which satisfies Qt,j,sm∩K∩E≠ϕ. Since du0(x) is uniformly continuous on Hj, then by the uniform continuity of ψ on bounded subsets of K×Λl(Ω), we obtain that for x∈Qt,j,sm, there exists a constant L>0 such that
(3.68)|ψ(x,dfk1(x))-ψ(xt,j,sm,dfk1(x))|ψψψ=|〈A(x,du0(x)+dfk1(x))-A(xt,j,sm,du0(xt,j,sm)+dfk1(x)),dfk1(x)〉|ψψψ<1meas⁡(Hj)ɛ1
holds for all m>L and each k1. Therefore, |bk1m,j|<ɛ1 for all k1. 							(3.69)|dk1m,j|≤∑t,s‍∫Qt,j,sm∖(K∩E)‍|ψ(xt,j,sm,dfk1(x))|dx=∑t,s‍∫Qt,j,sm∖(K∩E)‍〈A(xt,j,sm,du0(xt,j,sm)+dfk1(x)),dfk1(x)〉dx≤∑t,s‍∫Qt,j,sm∖(K∩E)‍C1|du0(xt,j,sm)+dfk1(x)|p(x)-1|dfk(x)|+|G(xt,j,sm)||dfk1(x)|dx≤∑t,s‍∫Qt,j,sm∖(K∩E)‍(η(xt,j,sm)+2p*(C1+1)N)dx≤∫(Hj∩E)∖K‍(η(xt,j,sm)+2p*(C1+1)N)dx+C(C1,p)∑t,s‍∫Qt,j,sm∖E‍(η(xt,j,sm)+N)dx≤C(α,N,C1,p)meas⁡((Hj∩E)∖K)+C(C1,p)∫Hj∖E‍[η(x)+N]dx≤C(α,N,C1,p)ɛ1≤O(ɛ1).
Now we suppose thatm is large enough that |bk1m,j|<ɛ1 for each k1>0 and there exists k¯1>0 such that F(vk1Ω)<ɛ1 for k1>k¯1. Therefore, from (3.25), (3.60), and (3.64), we have
(3.70)ɛ1≥F(vk1,Ω)≥ck1m,j+a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1)-3ɛ1-C(C1,p)ɛ1=ck1m,j+a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx-O(ɛ1).
(4)The estimate ofck1m,j. By fk1I⇀0  weakly* in W1,∞(Ω) as k1→∞, we obtain ∥fk1I∥L∞(Ω)→0ask1→∞ for each I. Then
(3.71)Rt,s,jk1,m=‖|fk1|‖L∞(Qt,s,jm)⟶0ask1⟶∞forfixedm.
Define a hypercubeEt,s,jk1,m contained in Qt,s,jm with edge length 1/2jm-2Rt,s,jk1,m such that dist(∂Qt,s,jm,Et,s,jk1,m)=Rt,s,jk1,m.
Next define(3.72)φk1(x)=0,x∈∂Qt,s,jm,φk1(x)=fk1(x),x∈Et,s,jm.
Since φk1I is a Lipschitz mapping on set Et,s,jm∪∂Qt,s,jm and its Lipschitz constant is no more than 2C(n)λ, by Lemma 3.12, φk1I can be extended to the whole Qt,s,jm, where it is also a Lipschitz mapping with the same Lipchistz constant. We still denote the extension by φk1I and suppose that it is defined on the whole Hj. Then by [20]
(3.73)∇φk1I-∇fk1I⟶0a.e.on⁡Hj.
Thus, there exists a k1¯¯>k1¯ such that for all k1>k1¯¯, we have
(3.74)∫Hj‍|dφk1-dfk1|p(x)dx≤ɛ12,∑t,s‍|∫Qt,j,sm‍ψ(xt,j,sm,dfk1(x))-ψ(xt,j,sm,dφk1(x))dx|≤ɛ12.
In view of (H5), we obtain that
(3.75)ck1m,j=∑t,s‍∫Qt,j,sm‍ψ(xt,j,sm,dfk1(x))dx≥∑t,s‍∫Qt,j,sm‍ψ(xt,j,sm,dφk1(x))dx-ɛ12=∑t,s‍∫Qt,j,sm‍〈A(xt,j,sm,du0(xt,j,sm)+dφk1(x)),dφk1(x)〉dx-ɛ12≥γ∑t,s‍∫Qt,j,sm‍|dφk1|p(x)dx-ɛ12≥γ2p*-1∫Hj‍|dfk1|p(x)dx-(γ+1)ɛ12.
Thus in (3.70) for k1>k1¯¯, we obtain the estimate of F(vk1,Ω) from the four steps above
(3.76)ɛ1≥F(vk1,Ω)≥a2-p*∫Ω5ɛ1,k1‍|dvk1|p(x)dx+γ2p*-1∫Hj‍|dfk1|p(x)dx-(γ+1)ɛ12-O(ɛ1).
Let K(ɛ1)=(γ+1)ɛ1/(2+o(ɛ1))/min{a2-p*,γ/2p*-1}. Then
(3.77)∫Ω5ɛ1,k1‍|dvk1|p(x)dx+∫Hj‍|dfk1|p(x)dx≤K(ɛ1),fork1>k1¯¯.
Form (3.59) and (3.77), we deduce that
(3.78)∫Ω5ɛ1,k1‍|dvk1|p(x)dx≤K(ɛ1),∫Ω′‍|dfk1|p(x)dx≤K(ɛ1)+ɛ1.
According to the definition of Ω2ɛ1, we have
(3.79)∫Ωɛ12‍|dgk1|p(x)dx≤K(ɛ1)+ɛ1.
Since dgk1(x)=dvk1(x) for each x∈Hk1λ, we get
(3.80)∫Ωɛ12∩Hk1λ‍|dvk1|p(x)dx≤K(ɛ1)+ɛ1.
By the definitions of Ω2ɛ1 and Ω5ɛ1,k1, it is immediate that
(3.81)(Ω2ɛ1∩Hk1λ)∪Ω5ɛ1,k1=Ω,
which implies that
(3.82)∫Ω‍|dvk1|p(x)dx≤2K(ɛ1)+ɛ1≤O(ɛ1).
Forɛ2>0 and the sequence {vk1}, repeating the above arguments we can extract a subsequence {vk2} of {vk1} such that
(3.83)∫Ω‍|dvk2|p(x)dx≤O(ɛ2),
whenever k2>k2¯¯ for some k2¯¯. If {vkn} has been obtained, repeating the above process, we can extract a subsequence {kn+1} of {kn} such that
(3.84)∫Ω‍|dvkn+1|p(x)dx≤O(ɛn+1),
whenever kn+1>k¯¯n+1 for some k¯¯n+1. Finally, by a diagonal argument we get a subsequence {vki}i=1∞ which satisfies
(3.85)∫Ω|dvki|p(x)dx⟶0asi⟶∞.
Therefore,
(3.86)‖dvki‖Lp(x)(Ω,Λl)⟶0asi⟶∞,
and by (3.23), {vki}i=1∞ strongly converges to zero in 𝔎0 as i→∞. This completes the proof of Theorem 3.1.


## 4. Applications

In this section, we explore applications of our results developed in this paper.LetΩ⊂ℝn be a bounded and convex Lipschitz domain. Suppose that maps A:Ω×Λl(Ω)→Λl(Ω) and B:Ω×Λl-1(Ω)→Λl-1(Ω), where l=1,2,…,n.Example 4.1.
Ifp(x) satisfies (1.3), let l=1, A(x,ξ)=ξ|ξ|p(x)-2 and B(x,ς)=ς|ς|p(x)-2-f(x), where f(x)∈Lp′(x)(Ω). Then A,B satisfy the required conditions, and (1.1) reduce to the following p(x)-Laplacian equations:
(4.1)-div⁡(|∇u|p(x)-2∇u)+|u|p(x)-2u=f(x),x∈Ω,u(x)=0,x∈∂Ω.
Now by Theorem 3.1, we deduce that the p(x)-Laplacian equations (4.1) have at least one weak solution in 𝔎1,p(x)(Ω) with u=0 on ∂Ω.Example 4.2.
Ifl=1, A(x,ξ)=∑i,j‍Aij(x)ξjdxi, B(x,ς)=B(x)ς-f(x), where f(x)∈L2(Ω), and Aij(x), B(x) satisfy the following conditions:
(4.2)Aij(x)=Aji(x),∧|ξ|2≥∑i,j=1n‍Aij(x)ξiξj≥λ|ξ|2,λ≤B(x)≤∧,
for some constants λ,∧>0. Then A,B satisfy the required conditions, and (1.1) reduce to the following Divergence form equations:
(4.3)∑i,j=1n‍∇j(Aij(x)∇iu(x))+B(x)u(x)=f(x),x∈Ω,(4.4)u(x)=0,x∈∂Ω,
where ∇i=(∂/∂xi). Now by Theorem 3.1, we deduce that the divergence form (4.3) have at least one weak solution u(x) in 𝔎1,2(Ω) with u=0 on ∂Ω. The comparison principles, the maximum principles, and the existence of weak solutions for divergence form equation (4.3) can be found in [21].Example 4.3.
Ifp(x) satisfies (1.3), let A(x,ξ)=ξ|ξ|p(x)-2 and B(x,ς)=ς|ς|p(x)-2-f(x), where f(x)∈Lp′(x)(Ω,∧l-1). Then A,B satisfy the required conditions, and (1.1) reduce to the following p(x)-harmonic equations for differential forms:
(4.5)d*(du|du|p(x)-2)+u|u|p(x)-2=f(x),x∈Ω,(4.6)u(x)=0,x∈∂Ω.
Now by Theorem 3.1, we deduce that (4.5) have at least one weak solution u(x) in 𝔎01,p(x)(Ω,∧l-1). If p(x) is a constant q and 1<q<∞, the equation (4.5) is called nonhomogeneous q-harmonic equation. In [2], Iwaniec and Lutoborski studied the Lq theory of weak solution for homogeneous q-harmonic equations.



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*Source: 421571-2012-07-25.xml*