Existence of Solutions for Nonhomogeneous A-Harmonic Equations with Variable Growth

Abstract

We study the following nonhomogeneous 𝐴-harmonic equations: 𝑑^∗𝐴(𝑥,𝑑𝑢(𝑥))+𝐵(𝑥,𝑢(𝑥))=0,𝑥∈Ω,𝑢(𝑥)=0,𝑥∈𝜕Ω, where Ω⊂ℝ^𝑛 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 𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙−1) of 𝑊₀^1,𝑝(𝑥)(Ω,Λ^𝑙−1).

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 of A-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 𝐿^𝑝(𝑥)(Ω) and 𝑊^{𝑘,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 𝐿^𝑝(𝑥)(Ω,Λ^𝑙,𝜔) and 𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙,𝜔) are first introduced and used to study the weak solutions of obstacle problems of A-harmonic equations with variable growth for differential forms.

Let Ω⊂ℝ^𝑛 be a bounded and convex Lipschitz domain. It is our purpose to study the following systems: 𝑑^∗𝐴((𝑥((𝑥𝑢(𝑥)𝑥,𝑑𝑢))+𝐵𝑥,𝑢))=0,𝑥∈Ω,=0,𝑥∈𝜕Ω, (1.1) where 𝑢∈Λ^𝑙−1(Ω), 𝑙=1,2,…,𝑛, and 𝐴∶Ω×Λ^𝑙(Ω)→Λ^𝑙(Ω), 𝐵∶Ω×Λ^𝑙−1(Ω)→Λ^𝑙−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∈𝐿^{𝑝′(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 _𝐷𝐴𝑥₀,𝜉₀+𝑑𝑣(𝑥),𝑑𝑣(𝑥)𝑑𝑥≥𝛾_𝐷|𝑑𝑣(𝑥)|^𝑝(𝑥)𝑑𝑥, (1.2) for each 𝜉₀∈Λ^𝑙(Ω), 𝐷⊂Ω and 𝑣∈𝐶¹₀(Ω,Λ^𝑙−1), where 𝛾>0 is a constant. Here 𝑝′ is the conjugate function of 𝑝. Throughout this paper we suppose (unless declare specially) 𝑝∈𝒫^log(Ω),1<𝑝_∗=essinf_Ω𝑝(𝑥)≤𝑝(𝑥)≤esssup_Ω𝑝(𝑥)=𝑝^∗<∞. (1.3)

2. Preliminaries

Let 𝑒₁,𝑒₂,…,𝑒_𝑛 be the standard orthogonal basis of ℝ^𝑛. The space of all 𝑙-forms in ℝ^𝑛 is denoted by Λ^𝑙(ℝ^𝑛). The dual basis to 𝑒₁,𝑒₂,…,𝑒_𝑛 is denoted by 𝑒¹,𝑒²,…,𝑒^𝑛 and referred to as the standard basis for 1-form Λ¹(ℝ^𝑛). The Grassman algebra Λ(ℝ^𝑛)=⊕Λ^𝑙(ℝ^𝑛) is a graded algebra with respect to the exterior products. The standard ordered basis for Λ(ℝ^𝑛) consists of the forms 1,𝑒¹,𝑒²,…,𝑒^𝑛,𝑒¹∧𝑒²,…,𝑒^𝑛−1∧𝑒^𝑛,…,𝑒¹∧𝑒²…∧𝑒^𝑛. (2.1)

For 𝛼(𝑥)=∑𝛼_𝐼(𝑥)𝑒^𝐼∈Λ^𝑙(ℝ^𝑛) and 𝛽(𝑥)=∑𝛽_𝐼(𝑥)𝑒^𝐼∈Λ^𝑙(ℝ^𝑛), the inner product is obtained by ⟨𝛼,𝛽⟩=∑𝛼_𝐼(𝑥)𝛽_𝐼(𝑥) with summation over all 𝑙–tuples 𝐼=(𝑖₁,…𝑖_𝑙) and all integers 𝑙=0,1,…,𝑛. The Hodge star operator (see [14]) ⋆∶Λ(ℝ^𝑛)→Λ(ℝ^𝑛) is defined by the formulas ⋆1=𝑒¹∧𝑒²⋯∧𝑒^𝑛,𝛼∧⋆𝛽=𝛽∧⋆𝛼=𝛼,𝛽𝑒¹∧𝑒²⋯∧𝑒^𝑛. (2.2) Hence, the norm of 𝛼 is given by the formula |𝛼|²=⟨𝛼,𝛼⟩=⋆(𝛼∧⋆𝛼)=∑𝛼_𝐼(𝑥)𝛼_𝐼(𝑥)∈Λ⁰(ℝ^𝑛)=ℝ. Notice, the Hodge star operator is an isometric isomorphism operator on Λ(ℝ^𝑛). Moreover, ⋆∶Λ^𝑙ℝ^𝑛⟶Λ^𝑛−𝑙ℝ^𝑛,⋆⋆=(−I)^{𝑙(𝑛−𝑙)}∶Λ^𝑙ℝ^𝑛⟶Λ^𝑙ℝ^𝑛, (2.3) where I is the identity map.

Let Ω⊂ℝ^𝑛 be a bounded domain. The coordinate functions 𝑥₁,𝑥₂,…,𝑥_𝑛 in Ω are considered to be differential forms of degree 0. The 1-forms 𝑑𝑥₁,𝑑𝑥₂,…,𝑑𝑥_𝑛 are constant functions from Ω into Λ^𝑙(ℝ^𝑛). The value of 𝑑𝑥_𝑖 is simply 𝑒^𝑖, 𝑖=1,2,…,𝑛. Therefore, every 𝑙-form 𝑢∶Ω→Λ^𝑙(ℝ^𝑛) may be written uniquely as 𝑢(𝑥)=_𝐼𝑢_𝐼(𝑥)𝑑𝑥_𝐼=_{1≤𝑖1<⋯<𝑖𝑙≤𝑛}𝑢_{𝑖1,…𝑖𝑙}(𝑥)𝑑𝑥_𝑖1∧⋯∧𝑑𝑥_𝑖𝑙, (2.4) where the coefficients 𝑢_{𝑖1,…𝑖𝑙}(𝑥) are distributions from 𝒟′(Ω), dual to the space of smooth functions with compact support on Ω.

We use 𝒟′(Ω,Λ^𝑙) to denote the space of all differential 𝑙-forms. For each form 𝑢(𝑥)∈𝒟′(Ω,Λ^𝑙), the exterior differential 𝑑∶𝒟′(Ω,Λ^𝑙)→𝒟′(Ω,Λ^𝑙+1) is expressed by 𝑑𝑢(𝑥)=^𝑛_{𝑘=11≤𝑖1<⋯<𝑖𝑙≤𝑛}𝜕𝑢_{𝑖1,…𝑖𝑙}(𝑥)𝜕𝑥_𝑘𝑑𝑥_𝑘∧𝑑𝑥_𝑖1∧⋯∧𝑑𝑥_𝑖𝑙. (2.5)

For 𝑢∈𝒟′(Ω,Λ^𝑙), the vector-valued differential form ∇𝑢=𝜕𝑢𝜕𝑥₁,𝜕𝑢𝜕𝑥₂,…,𝜕𝑢𝜕𝑥_𝑛 (2.6) consists of differential forms 𝜕𝑢/𝜕𝑥_𝑖∈𝒟′(Ω,Λ^𝑙), where the partial differentiation is applied to the coefficients of 𝑢.

The formal adjoint operator, called the Hodge codifferential, is given by 𝑑^⋆=(−1)^𝑛𝑙−1⋆𝑑⋆∶𝒟^′Ω,Λ^𝑙+1⟶𝒟^′Ω,Λ^𝑙. (2.7) By 𝐶^∞(Ω,Λ^𝑙) denote the space of infinitely differentiable 𝑙-forms on Ω and by 𝐶^∞₀(Ω,Λ^𝑙) denote the subspace of 𝐶^∞(Ω,Λ^𝑙) with compact support on Ω.

Let 𝒫(Ω) be the set of all Lebesgue measurable functions 𝑝∶Ω→(1,∞). For 𝑝∈𝒫(Ω), we put 𝑝_∗=essinf_Ω𝑝(𝑥) and 𝑝^∗=esssup_Ω𝑝(𝑥). Given 𝑝∈𝒫(Ω) we define the conjugate function 𝑝′∈𝒫(Ω) by 𝑝^′(𝑥)=𝑝(𝑥)𝑝(𝑥)−1,∀𝑥∈Ω. (2.8)

Definition 2.1 (see [15]). A Lebesgue measurable function 𝑝∶Ω→ℝ is called globally log-Hölder continuous in Ω if there exist 𝑝_∞∈ℝ and a constant 𝐶>0 such that 𝑝(𝑥)𝑦≤𝐶−𝑝,𝑝(𝑥)log𝑒+1/𝑥−𝑦−𝑝_∞≤𝐶log(𝑒+|𝑥|) (2.9) hold for all 𝑥,𝑦∈Ω. 𝑃^log(Ω) is defined by 𝒫^log(Ω)=(Ω)∶1𝑝∈𝒫𝑝isgloballylog-Höldercontinuous. (2.10)

For a differential 𝑙-form 𝑢(𝑥) on Ω, 𝑙=0,1,…,𝑛, define the functional 𝜌_𝑝(𝑥) by 𝜌_{𝑝(𝑥),Λ𝑙}(𝑢)=_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (2.11)

The space 𝐿^𝑝(𝑥)(Ω,Λ^𝑙)={𝑢∈Λ^𝑙(Ω)∶∃𝜆>0,𝜌_{𝑝(𝑥),Λ𝑙}(𝜆𝑢)<∞} is a reflexive Banach space endowed with the norm ‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}=inf𝜆>0∶𝜌_{𝑝(𝑥),Λ𝑙}𝑢𝜆≤1. (2.12)

The space 𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙)={𝑢∈Λ^𝑙(Ω)∶𝑢∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙)and𝑑𝑢∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙+1)} is a reflexive Banach space endowed with the norm ‖𝑢‖_{𝑊1,𝑝(𝑥)(Ω,Λ𝑙)}=‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}+‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙+1)}. (2.13)

Note that 𝐿^𝑝(𝑚)(Ω,Λ⁰) and 𝑊^1,𝑝(𝑚)(Ω,Λ⁰) are spaces of functions on Ω. In this paper, we denote them by 𝐿^𝑝(𝑚)(Ω) and 𝑊^1,𝑝(𝑚)(Ω).

Iwaniec and Lutoborski proved the following results in [2].

Let Ω⊂ℝ^𝑛 be a bounded and convex domain. If 𝑢(𝑥)∈Λ^𝑙(ℝ^𝑛) is defined for some 𝑥∈Ω, then the value of 𝑢(𝑥) at the vectors 𝜉₁,…,𝜉_𝑙∈ℝ^𝑛 is denoted by 𝑢(𝑥)(𝜉₁,…,𝜉_𝑙). Then to each 𝑦∈Ω, there corresponds a linear operator 𝐾_𝑦∶𝐿¹_loc(Ω,∧^𝑙)→𝐿¹_loc(Ω,∧^𝑙−1) defined by 𝐾_𝑦𝑢(𝑥)𝜉₁,𝜉₂,…,𝜉_𝑙−1=¹₀𝑡^𝑙−1𝑢𝑡𝑥+𝑦−𝑡𝑦𝑥−𝑦,𝜉₁,𝜉₂,…,𝜉_𝑙−1𝑑𝑡. (2.14) The homotopy operator 𝑇∶𝐿¹_loc(Ω,∧^𝑙)→𝐿¹_loc(Ω,∧^𝑙−1) is defined by averaging 𝐾_𝑦 over all points 𝑦∈Ω𝑇𝑢(𝑥)=_Ω𝜑𝑦𝐾_𝑦𝑢(𝑥)𝑑𝑦, (2.15) where 𝜑∈𝐶^∞₀(Ω) is normalized so that ∫_Ω𝜑(𝑦)𝑑𝑦=1. Then we have a pointwise estimate |𝑇𝑢(𝑥)|≤2^𝑛𝜇(Ω)_Ω𝑢𝑦𝑥−𝑦^𝑛−1𝑑𝑦,∀𝑥∈Ω, (2.16) where 𝜇(Ω)=(diamΩ)^𝑛+1inf∇𝜑_𝐿∞(Ω)𝜑_𝐿1(Ω)∶𝜑∈𝐶^∞₀(Ω), (2.17) further infimum is attained at 𝜑(𝑥)=diam(𝑥,𝜕Ω), and the decomposition 𝑢=𝑑𝑇𝑢+𝑇𝑑𝑢 (2.18) holds for 𝑢∈𝐿¹_loc(Ω,Λ^𝑙).

Definition 2.2. For 𝑢∈𝐿¹_loc(Ω,Λ^𝑙), define the 𝑙-form 𝑢_Ω∈𝒟′(Ω,Λ^𝑙) by 𝑢_Ω=1meas(Ω)_Ω𝑢(𝑥)𝑑𝑥,for𝑙=0,𝑑𝑇𝑢,for𝑙=1,2,⋯,𝑛, (2.19) and the Maximal operator is defined by (𝑀𝑢)(𝑥)=sup_𝑟>01meas𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑢𝑦𝑑𝑦, (2.20) where 𝐵_𝑟(𝑥)={𝑦∈ℝ^𝑛∶|𝑦−𝑥|<𝑟}.

Lemma 2.3 (see [15]). Let 𝑝(𝑥) satisfies (1.3). Then the inequality ‖(𝑀𝑢)(𝑥)‖_{𝐿𝑝(𝑥)(ℝ𝑛)}≤𝐶𝑛,𝑝‖𝑢(𝑥)‖_{𝐿𝑝(𝑥)(ℝ𝑛)} (2.21) holds for every 𝑢∈𝐿^𝑝(𝑥)(ℝⁿ).

Lemma 2.4 (see [15]). Let Ω⊂ℝ^𝑛 be a bounded convex domain, 𝑥∈Ω and 𝑢∈𝐿¹_loc(ℝ^𝑛). Then _Ω𝑢𝑦𝑥−𝑦^𝑛−1d𝑦≤𝐶(𝑛)(diamΩ)(𝑀𝑢)(𝑥). (2.22)

Lemma 2.5 (see [15]). Let Ψ be a Calderón-Zygmund operator with Calderón-Zygmund kernel 𝐾 on ℝ^𝑛×ℝ^𝑛. Then Ψ is bounded on 𝐿^𝑝(𝑥)(ℝ^𝑛). Further there exists a constant C = C(n,p) such that ‖Ψ𝑢(𝑥)‖_{𝐿𝑝(𝑥)(ℝ𝑛)}≤𝐶𝑛,𝑝‖𝑢(𝑥)‖_{𝐿𝑝(𝑥)(ℝ𝑛)} (2.23) holds for every 𝑢∈𝐿^𝑝(𝑥)(ℝ^𝑛).

Lemma 2.6. If 𝑢∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙), then ‖𝑇𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙−1)}≤𝐶𝑛,𝑝𝜇(Ω)(diamΩ)‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.24) Moreover, if 𝑢∈𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙), then 𝑢_{Ω𝐿𝑝(𝑥)(Ω,Λ𝑙)}≤𝐶𝑝‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}+𝐶𝑛,𝑝𝜇(Ω)(diamΩ)‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙+1)}. (2.25)

Proof. First define 𝑢(𝑥)=0if𝑥∈ℝ^𝑛⧵Ω. From pointwise estimate (2.16) and Lemma 2.4, |𝑇𝑢(𝑥)|≤𝐶(𝑛)𝜇(Ω)(diamΩ)𝑀(|𝑢|)(𝑥),∀𝑥∈Ω. (2.26) In view of Lemma 2.3, we have ‖|𝑇𝑢|‖_{𝐿𝑝(𝑥)(Ω)}≤𝐶𝑛,𝑝𝜇(Ω)(diamΩ)‖|𝑢|‖_{𝐿𝑝(𝑥)(Ω)}, (2.27) that is to say, (2.24) holds.
From the definition of 𝑢_Ω and (2.18), we have 𝑢_Ω=𝑢−𝑇𝑑𝑢. Therefore, 𝑢_{Ω𝐿𝑝(𝑥)(Ω,Λ𝑙)}≤𝐶𝑝‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}+𝐶𝑛,𝑝‖𝑇𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.28) Now in (2.24) replace 𝑢 with 𝑑𝑢, we obtain (2.25).

Lemma 2.7. Let 𝑝(𝑥) satisfies (1.3).(1)𝐶^∞₀(Ω,Λ^𝑙) is dense in 𝐿^𝑝(𝑥)(Ω,Λ^𝑙),(2)𝐿^𝑝(𝑥)(Ω,Λ^𝑙) is separable.

Proof. (1) For any 𝑢(𝑥)=∑_𝐼𝑢_𝐼(𝑥)𝑑𝑥_𝐼∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙), since 𝐶^∞₀(Ω) is dense in 𝐿^𝑝(𝑥)(Ω) and 𝑢_𝐼(𝑥)∈𝐿^𝑝(𝑥)(Ω) for all 𝐼, we can find a sequence {𝑢_𝐼𝑘}^∞_𝑘=1⊂𝐶^∞₀(Ω) which converges to 𝑢_𝐼(𝑥) in 𝐿^𝑝(𝑥)(Ω) for each 𝐼. Now let 𝑢_𝑘(𝑥)=∑_𝐼𝑢_𝐼𝑘𝑑𝑥_𝐼, then the sequence {𝑢_𝑘(𝑥)}⊂𝐶^∞₀(Ω,Λ^𝑙) converges to 𝑢(𝑥) in 𝐿^𝑝(𝑥)(Ω,Λ^𝑙), since _Ω𝑢(𝑥)−𝑢_𝑘(𝑥)^𝑝(𝑥)𝑑𝑥=_Ω𝐼𝑢_I(𝑥)−𝑢_𝐼𝑘(𝑥)^{21/2𝑝(𝑥)}≤𝑑𝑥_Ω𝐼𝑢_𝐼(𝑥)−𝑢_𝐼𝑘(𝑥)^𝑝(𝑥)𝑑𝑥≤2^𝑝∗_𝐼Ω𝑢_𝐼(𝑥)−𝑢_𝐼𝑘(𝑥)^𝑝(𝑥)𝑑𝑥. (2.29) That is to say, 𝐶^∞₀(Ω,Λ^𝑙) is dense in 𝐿^𝑝(𝑥)(Ω,Λ^𝑙).
(2) Let 𝑢(𝑥)=∑_𝐼𝑢_𝐼(𝑥)𝑑𝑥_𝐼∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙). Since 𝐿^𝑝(𝑥)(Ω) is separable, there exists a countable dense subset 𝐾 of 𝐿^𝑝(𝑥)(Ω). Then for any 𝑢_𝐼(𝑥) above we can extract a sequence {𝑢_𝐼𝑘(𝑥)} in 𝐾 which converges to 𝑢_𝐼(𝑥) in 𝐿^𝑝(𝑥)(Ω). Similar to (1), the sequence {𝑢_𝑘∶𝑢_𝑘(𝑥)=∑_𝐼𝑢_𝐼𝑘(𝑥)𝑑𝑥_𝐼} converges to 𝑢(𝑥) in 𝐿^𝑝(𝑥)(Ω,Λ^𝑙). That is to say, 𝐿^𝑝(𝑥)(Ω,Λ^𝑙) is separable.

Let 𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙)={𝑢(𝑥)=𝜗(𝑥)−𝜗_Ω(𝑥)∶𝜗∈𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙)}. Note that 𝑢∈𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙) if and only if 𝑢_Ω=0.

Lemma 2.8. Let 𝑝(𝑥) satisfies (1.3). Then 𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙) is a closed subspace of 𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙). In particular, it is a reflexive Banach space.

Proof. Set a sequence {𝑢_𝑘(𝑥)}⊂𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙) convergent to 𝑢(𝑥) in 𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙), then (𝑢_𝑘)_Ω=0. By Lemma 2.6, the operator 𝑇 is continuous on 𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙). Therefore, 𝑢_Ω=0, we have 𝑢(𝑥)∈𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙). That is to say, 𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙) is a closed subspace of 𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙).

In [2], Iwaniec and Lutoborski obtained 𝜕𝜕𝑥_𝑖(𝑇𝑢)=𝐴_𝑖𝑢+𝑆_𝑖𝑢, (2.30) where 𝐴_𝑖𝑢(𝑥≤2)^𝑛𝜇(Ω)diam(Ω)_Ω|𝑢(𝑧)||𝑥−𝑧|^𝑛−1𝑆𝑑𝑧,_𝑖𝑢(𝑥)(𝜉)=_Ω𝑢(𝑧)𝐾_𝑖(𝑧,𝑥−𝑧),𝜉𝑑𝑧, (2.31)(2.32) where 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑙−1) and 𝐾_𝑖()=𝑒𝑧,𝑥−𝑧^𝑖|𝑥−𝑧|^𝑛∞₀𝑠^𝑛−1𝜑𝑧−𝑠𝑥−𝑧||−𝑥−𝑧𝑑𝑠𝑥−𝑧|𝑥−𝑧|^𝑛+1∞₀𝑠^𝑛𝜑_𝑖𝑧−𝑠𝑥−𝑧|𝑥−𝑧|𝑑𝑠. (2.33) Further for each 𝑧∈Ω and ℎ∈ℝ^𝑛−{0}, 𝐾_𝑖(𝑧,ℎ) satisfies the following properties:(i)𝐾_𝑖(𝑧,ℎ)≤𝜇(Ω)|ℎ|^−𝑛,(ii)𝐾_𝑖(𝑧,𝑠ℎ)=𝑠^−𝑛𝐾_𝑖(𝑧,ℎ),𝑠>0,(iii)∫_|ℎ|=1𝐾_𝑖(𝑧,ℎ)=0 for all 𝑧∈Ω.

Let 𝐾_𝑖(𝑧,ℎ)=(𝐾_𝑖1,𝐾_𝑖2,…,𝐾_𝑖𝑛). Then 𝐾_𝑖𝛼 satisfies the conditions of Calderón-Zygmund kernel on ℝ^𝑛×ℝ^𝑛 for each 𝛼=1,2,…,𝑛.

Lemma 2.9. Let 𝑢∈𝐿^𝑝(𝑥)(Ω,Λ^𝑙). Then ‖|∇𝑇𝑢|‖_{𝐿𝑝(𝑥)(Ω)}≤𝐶𝑛,𝑝,Ω‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.34)

Proof. By Lemmas 2.3 and 2.4, and (2.31), 𝐴_𝑖𝑢_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}≤𝐶𝑛,𝑝𝜇(Ω)‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.35) Let 𝑆_𝑖𝑢(𝑥)=_{1≤𝑗1<𝑗2<⋯<𝑗𝑙−1≤𝑛}𝜔_{𝑗1,𝑗2,…,𝑗𝑙−1}𝑑𝑥_𝑗1∧𝑑𝑥_𝑗2∧⋯∧𝑑𝑥_𝑗𝑙−1, (2.36) we can write 𝑢(𝑥) as 𝑢(𝑥)=_{1≤𝛼≤𝑛,𝛼≠𝑗1,𝑗2,…,𝑗𝑙−11≤𝑗1<𝑗2<⋯<𝑗𝑙−1≤𝑛}𝑢_{𝛼,𝑗1,𝑗2,…,𝑗𝑙−1}𝑑𝑥_𝛼∧𝑑𝑥_𝑗1∧⋯∧𝑑𝑥_𝑗𝑙−1. (2.37) Hence, 𝜔_{𝑗1,𝑗2,…,𝑗𝑙−1}(𝑥)=𝑆_𝑖𝑢(𝑥)𝑒_𝑗1,𝑒_𝑗2,…,𝑒_𝑗𝑙−1. (2.38) Taking 𝜉=(𝑒_𝑗1,𝑒_𝑗2,…,𝑒_𝑗𝑙−1) in (2.32), we obtain 𝜔_{𝑗1,𝑗2,…,𝑗𝑙−1}(𝑥)=_{Ω1≤𝛼≤𝑛,𝛼≠𝑗1,…,𝑗𝑙−1}𝐾_𝑖𝛼(𝑧,𝑥−𝑧)𝑢_{𝛼,𝑗1,𝑗2,…,𝑗𝑙−1}(𝑧)𝑑𝑧. (2.39) Now define 𝑢(𝑥)=0if𝑥∈ℝ^𝑛⧵Ω. Since 𝐾_𝑖𝛼 satisfies the conditions of Calderón-Zygmund kernel on ℝ^𝑛×ℝ^𝑛 for each 𝛼, in view of Lemma 2.5, 𝜔_{𝑗1,𝑗2,…,𝑗𝑙−1𝐿𝑝(𝑥)(Ω)}≤𝐶𝑛,𝑝_{1≤𝛼≤𝑛,𝛼≠𝑗1,…,𝑗𝑙−1}𝑢_{𝛼,𝑗1,𝑗2,…,𝑗𝑙−1𝐿𝑝(𝑥)(Ω)}. (2.40) So that 𝑆_𝑖𝑢_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}≤𝐶𝑛,𝑝‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.41) By (2.30), (2.35), and (2.41), we have ‖|∇𝑇𝑢|‖_{𝐿𝑝(𝑥)(Ω)}≤𝐶𝑛,𝑝,Ω‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.42)

Now define another norm ‖|𝜔|‖_𝔎^1,𝑝(𝑥)_(Ω,Λ𝑙)=‖𝜔‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}+‖|∇𝜔|‖_{𝐿𝑝(𝑥)(Ω)}. (2.43)

Remark 2.10. Replacing 𝑢 with 𝑑𝑢 in (2.34), we get by the definition of 𝑢_Ω∇𝑢−𝑢_{Ω𝐿𝑝(𝑥)(Ω)}=‖|∇𝑇𝑑𝑢|‖_{𝐿𝑝(𝑥)(Ω)}≤𝐶𝑛,𝑝𝜇(Ω)‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}=𝐶𝑛,𝑝𝜇(Ω)𝑑𝑢−𝑢_{Ω𝐿𝑝(𝑥)(Ω,Λ𝑙)}. (2.44) 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]). Let 𝑝(𝑥) satisfies (1.3). Then the embedding 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝑝(𝑥)(Ω) is compact.

Lemma 2.12 (see[15]). Suppose that 𝑝∈𝐿^∞(Ω). Let {𝑢_𝑘}^∞_𝑘=1 be bounded in 𝐿^𝑝(𝑥)(Ω). If 𝑢_𝑘→𝑢 a.e. on Ω, then 𝑢_𝑘⇀𝑢 weakly in 𝐿^𝑝(𝑥)(Ω).

Remark 2.13. Let 𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙) be the completion of 𝐶^∞₀(Ω,∧^𝑙) in 𝔎^1,𝑝(𝑥)(Ω,Λ^𝑙). Then from Remark 2.10 and Lemma 2.11, the embedding 𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙)↪𝐿^𝑝(𝑥)(Ω,Λ^𝑙) is compact.

Remark 2.14. Suppose 𝑝(𝑥) satisfies (1.3), Lemma 2.12 also holds on space 𝐿^𝑝(𝑥)(Ω,∧^𝑙).

3. Weak Solutions of Dirichlet Problems for the 𝐴-Harmonic Equations with Variable Growth

Theorem 3.1. Under conditions (H1)–(H5), the Dirichlet problem (1.1) has at least one weak solution in 𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙), that is to say, there exists at least one 𝑢=𝜗−𝜗_Ω∈𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙) satisfying _Ω𝐴(𝑥,𝑑𝑢(𝑥)),𝑑𝜑(𝑥)+𝐵(𝑥,𝑢(𝑥)),𝜑(𝑥)𝑑𝑥=0, (3.1) for all 𝜑∈𝑊₀^1,𝑝(𝑥)(Ω,Λ^𝑙−1). Here, 𝜗∈𝑊^1,𝑝(𝑥)(Ω,Λ^𝑙−1) and 𝑝(𝑥) satisfies (1.3).

Let 𝑉=𝑊₀^1,𝑝(𝑥)(Ω,Λ^𝑙−1) and 𝔎₀=𝔎₀^1,𝑝(𝑥)(Ω,Λ^𝑙). For 𝑢∈𝑉, define 𝔄∶𝑉→𝑉^∗ in the following way: for each 𝜑∈𝑉𝔄𝑢,𝜑=_Ω𝐴(𝑥,𝑑𝑢(𝑥)),𝑑𝜑(𝑥)+𝐵(𝑥,𝑢(𝑥)),𝜑(𝑥)𝑑𝑥. (3.2) Now we need only to show that there exists 𝑢∈𝔎₀ such that (𝔄𝑢,𝜑)=0 for all 𝜑=∑𝜑_𝐼(𝑥)𝑑𝑥_𝐼∈𝑉.

Lemma 3.2. 𝔄 is strong-weakly continuous on 𝑉.

Proof. Let {𝑢_𝑘∶𝑢_𝑘(𝑥)=∑_𝐼𝑢_𝑘𝐼(𝑥)𝑑𝑥_𝐼}⊂𝑉 be a sequence strongly convergent to an element 𝑢(𝑥)=∑𝑢_𝐼(𝑥)𝑑𝑥_𝐼∈𝑉 in 𝑉. Let 𝑑𝑢_𝑘(𝑥)=∑_𝐽𝜔_𝑘𝐽(𝑥)𝑑𝑥_𝐽 and 𝑑𝑢(𝑥)=∑_𝐽𝜔_𝐽(𝑥)𝑑𝑥_𝐽. Then (ℎ₁)‖𝑢_𝑘‖_𝑉≤𝐶 for some constant 𝐶, (ℎ₂){𝜔_𝑘𝐽(𝑥)} is a sequence strongly convergent to 𝜔_𝐽(𝑥) in 𝐿^𝑝(𝑥)(Ω) for each 𝐽.
In view of (H2) and (ℎ₁), we know that 𝐴(𝑥,𝑑𝑢_𝑘)=∑𝐴_𝑘𝐽(𝑥)𝑑𝑥_𝐽 and 𝐵(𝑥,𝑢_𝑘)=∑𝐵_𝑘𝐼(𝑥)𝑑𝑥_𝐼 are uniformly bounded in 𝐿^{𝑝′(𝑥)}(Ω,Λ^𝑙) and 𝐿^{𝑝′(𝑥)}(Ω,Λ^𝑙−1), respectively. Hence, 𝐴_𝑘𝐽(𝑥) and 𝐵_𝑘𝐼(𝑥) are uniformly bounded in 𝐿^{𝑝′(𝑥)}(Ω). On the other hand, by (ℎ₂), there exists a subsequence of {𝜔_𝑘𝐽(𝑥)} (still denoted by {𝜔_𝑘𝐽(𝑥)}) such that lim_𝑘→∞𝜔_𝑘𝐽(𝑥)=𝜔_𝐽(𝑥),a.e.𝑥∈Ω,foreach𝐽. (3.3) Then there exists a subsequence of {𝑢_𝑘(𝑥)} (still denoted by {𝑢_𝑘(𝑥)}) such that lim_𝑘→∞𝑢_𝑘(𝑥)=𝑢(𝑥),lim_𝑘→∞𝑑𝑢_𝑘(𝑥)=𝑑𝑢(𝑥),a.e.𝑥∈Ω. (3.4) In view of (H1), we obtain lim_𝑘→∞𝐴𝑥,𝑑𝑢_𝑘=𝐴(𝑥,𝑑𝑢),a.e.𝑥∈Ω,lim_𝑘→∞𝐵𝑥,𝑢_𝑘=𝐵(𝑥,𝑢),a.e.𝑥∈Ω. (3.5) Let 𝑑𝜑(𝑥)=∑𝜓_𝐽(𝑥)𝑑𝑥_𝐽, 𝐴(𝑥,𝑑𝑢)=∑𝐴_𝐽(𝑥)𝑑𝑥_𝐽, and 𝐵(𝑥,𝑢)=∑𝐵_𝐼(𝑥)𝑑𝑥_𝐼, then 𝜓_𝐽(𝑥)∈𝐿^𝑝(𝑥)(Ω), in the meantime lim_𝑘→∞𝐴_𝑘𝐽(𝑥)=𝐴_𝐽(𝑥),a.e.𝑥∈Ω,lim_𝑘→∞𝐵_𝑘𝐼(𝑥)=𝐵_𝐼(𝑥),a.e.𝑥∈Ω, (3.6)(3.7) for each 𝐽 and 𝐼.
Now by Lemma 2.12, we can show that ∫_Ω𝐴_𝑘𝐽(𝑥)𝜓_𝐽(𝑥)𝑑𝑥→∫_Ω𝐴_𝐽(𝑥)𝜓_𝐽(𝑥)𝑑𝑥 and ∫_Ω𝐵_𝑘𝐼(𝑥)𝜑_𝐼(𝑥)𝑑𝑥→∫_Ω𝐵_𝐼(𝑥)𝜑_𝐼(𝑥)𝑑𝑥 as 𝑘→∞. Therefore, 𝔄𝑢_𝑘,𝜑=_Ω⟨𝐴𝑥,𝑑𝑢_𝑘,𝑑𝜑⟩+⟨𝐵𝑥,𝑢_𝑘,𝜑⟩𝑑𝑥⟶_Ω()()=,⟨𝐴𝑥,𝑑𝑢,𝑑𝜑⟩+⟨𝐵𝑥,𝑢,𝜑⟩𝑑𝑥𝔄𝑢,𝜑 (3.8) that is to say, 𝔄 is strong-weakly continuous on 𝑉.

Lemma 3.3. 𝔄 is coercive on 𝔎₀, that is, lim_{‖𝑢‖𝔎→∞}(𝔄𝑢,𝑢)‖𝑢‖_𝔎=+∞,∀𝑢∈𝔎₀. (3.9)

Proof. By (H3) and (H4), (𝔄𝑢,𝑢)=_Ω⟨𝐴(𝑥,𝑑𝑢),𝑑𝑢⟩+⟨𝐵(𝑥,𝑢),𝑢⟩𝑑𝑥≥_Ω𝑎|𝑑𝑢|^𝑝(𝑥)−|ℎ(𝑥)|+𝑎|𝑢|^𝑝(𝑥)−ℎ(𝑥≥)𝑑𝑥_Ω𝑎|𝑑𝑢|^𝑝(𝑥)𝑑𝑥−𝐶ℎ,ℎ. (3.10) By 𝑑𝜗_Ω=0 and Lemma 2.6, we have ‖𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙−1)}=‖𝑇𝑑𝜗‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙−1)}≤2^𝑛𝐶𝑛,𝑝𝜇(Ω)(diamΩ)‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}, (3.11) for all 𝑢=𝜗−𝜗_Ω∈𝔎₀. Then ‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}→∞, as ‖𝑢‖_𝔎→∞. Taking 1𝛿=2‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}>1, (3.12) we have ∫_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)}≥‖‖𝑑𝑥𝑑𝑢_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}−𝛿^𝑝∗‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}=12^𝑝∗‖𝑑𝑢‖^𝑝∗_𝐿⁻¹_{𝑝(𝑥)Ω,Λ𝑙}. (3.13) Therefore, ∫_Ω|𝑑𝑢|^𝑝(𝑥)𝑑𝑥‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}⟶∞as‖𝑑𝑢‖_{𝐿𝑝(𝑥)(Ω,Λ𝑙)}⟶∞. (3.14) Then it is immediate to obtain that (𝔄𝑢,𝑢)‖𝑢‖_𝔎⟶∞as‖𝑢‖_𝔎⟶∞. (3.15) That is to say, 𝔄 is coercive on 𝔎₀.

Lemma 3.4 (see[16]). Suppose 𝑔=𝐴(𝑥) is a mapping from ℝ^𝑚 into itself such that lim_|𝑥|→∞𝐴(𝑥)⋅𝑥|𝑥|=∞. (3.16) Then the range of 𝐴 is the whole of ℝ^𝑚.

Lemma 3.5. There exists a sequence {𝑢_𝑘}⊂𝔎₀ and 𝑢₀∈𝔎₀, such that 𝔄𝑢_𝑘,𝑢_𝑘−𝑢₀⟶0as𝑘⟶∞. (3.17)

Proof . By Lemmas 2.7 and 2.8, we can choose a Schauder basis {𝜔_𝑠} of 𝔎₀ such that the union of subspace finitely generated from 𝜔_𝑠 is dense in 𝔎₀. Let 𝔎^𝑘₀ be the subspace of 𝔎₀ generated by 𝜔₁,𝜔₂,…,𝜔_𝑘. Since 𝔎^𝑘₀ is topologically isomorphic to ℝ^𝑘. By Lemmas 3.3, and 3.4, there exists 𝑢_𝑘∈𝔎^𝑘₀ such that 𝔄𝑢_𝑘,𝜔=0∀𝜔∈𝔎^𝑘₀. (3.18) By Lemma 3.3 again, we know that ‖𝑢_𝑘‖_𝔎≤𝐶, where 𝐶 is independent of 𝑘. Since 𝔎₀ is reflexive, by Remark 2.14 and (H1), we can extract a subsequence of {𝑢_𝑘} (still denoted by {𝑢_𝑘}) such that 𝑢_𝑘⇀𝑢₀weaklyin𝔎₀,𝔄𝑢_𝑘⇀𝜉weakly^∗in𝔎^∗₀,(𝜉,𝜔)=0, (3.19) 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 (𝔄𝑢_𝑘,𝑢_𝑘−𝑢₀), we have 𝔄𝑢_𝑘,𝑢_𝑘−𝑢₀=𝔄𝑢_𝑘,𝑢_𝑘−𝔄𝑢_𝑘,𝑢₀=−𝔄𝑢_𝑘,𝑢₀⟶0as𝑘⟶∞. (3.20) This completes the proof of Lemma 3.5.

Set 𝑣_𝑘=𝑢_𝑘−𝑢₀=∑𝑣_𝑘𝐼𝑑𝑥_𝐼. Then 𝑣_𝑘⇀0weaklyin𝔎₀as𝑘⟶∞. (3.21) Consider (𝔄𝑢_𝑘,𝑢_𝑘−𝑢₀) once more, then 𝔄𝑢_𝑘,𝑢_𝑘−𝑢₀=_Ω𝐴𝑥,𝑑𝑢₀+𝑑𝑣_𝑘,𝑑𝑣_𝑘+𝐵𝑥,𝑢₀+𝑣_𝑘,𝑣_𝑘𝑑𝑥⟶0, (3.22) as 𝑘→∞. By Remark 2.13, we get 𝑣_𝑘⟶0stronglyin𝐿^𝑝(𝑥)Ω,Λ^𝑙−1. (3.23) In view of (3.23) and (H2), it is immediate that _Ω𝐵𝑥,𝑢₀+𝑣_𝑘,𝑣_𝑘𝑑𝑥⟶0as𝑘⟶∞, (3.24) that is to say, _Ω𝐴𝑥,𝑑𝑢₀+𝑑𝑣_𝑘,𝑑𝑣_𝑘𝑑𝑥⟶0as𝑘⟶∞. (3.25)

Now if we can prove that there exists a subsequence of {𝑣_𝑘} which is strongly convergent in 𝔎₀, then from the strong-weakly continuity of 𝔄, we get 𝔄𝑢_𝑘⇀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. Let Ω be an open subset of ℝ^𝑛 provided with the Lebesgue measure. The mapping 𝑓∶Ω×ℝ^𝑁→ℝ is said Carathéodory function if for almost all 𝑥∈Ω, 𝑓(𝑥,⋅) is continuous on ℝ^𝑁, for all 𝜉∈ℝ^𝑁 is measurable on Ω.

Lemma 3.7 (see[17]). A mapping 𝑓∶Ω×ℝ^𝑁→ℝ is a Carathéodory function if and only if for all compact sets 𝐾⊂Ω and all 𝜀>0, there exists a compact subset 𝐾_𝜀⊂𝐾 such that meas(𝐾−𝐾_𝜀)<𝜀 for with the restriction of 𝑓 to 𝐾_𝜀×ℝ^𝑁 is continuous.

Lemma 3.8 (see[15]). Let {𝑓_𝑘} be a sequence of bounded function in 𝐿¹(ℝ^𝑛). For each 𝜀>0 there exists (𝐴_𝜀,𝛿,𝑁) (where 𝐴_𝜀 is measurable and meas(𝐴_𝜀)<𝜀,𝛿>0,𝑁 is an infinite subset of natural numbers set ℕ) such that for each 𝑘∈𝑁, _𝐵𝑓_𝑘(𝑥)𝑑𝑥<𝜀, (3.26) where 𝐵 and 𝐴_𝜀 are disjoint and meas(𝐵)<𝛿.

Definition 3.9. For 𝑢∈𝐶¹₀(ℝ^𝑛), define 𝑀^∗𝑢(𝑥)=(𝑀𝑢)(𝑥)+^𝑛_𝛼=1𝑀𝜕𝑢𝜕𝑥_𝛼(𝑥). (3.27)

Lemma 3.10 (see[18]). If 𝑢∈𝐶^∞₀(ℝ^𝑛), then 𝑀^∗𝑢∈𝐶⁰(ℝ^𝑛) and for all 𝑥∈ℝ^𝑛, |𝑢(𝑥)|+^𝑛_𝛼=1𝜕𝑢𝜕𝑥_𝛼(𝑥)≤𝑀^∗𝑢(𝑥). (3.28) Furthermore, if 𝑝>1, then 𝑀^∗𝑢_{𝐿𝑝(ℝ𝑛)}≤𝐶𝑛,𝑝‖𝑢‖_{𝑊1,𝑝(ℝ𝑛)}, (3.29) and if 𝑝=1, then meas𝑥∈ℝ^𝑛∶𝑀^∗𝑢(𝑥)≤𝐶(𝑛)>𝜆𝜆‖𝑢‖_{𝑊1,1(ℝ𝑛)}, (3.30) for all 𝜆>0.

Lemma 3.11 (see[19]). Let 𝑢∈𝐶^∞₀(ℝ^𝑛) and 𝜆>0. Set 𝐻^𝜆=𝑥∈ℝ^𝑛∶𝑀^∗𝑢(𝑥)<𝜆. (3.31) Then for all 𝑥,𝑦∈𝐻^𝜆, we have 𝑢𝑦−𝑢(𝑥)≤𝐶(𝑛)𝜆𝑦−𝑥. (3.32)

Lemma 3.12 (see[16]). Let 𝑋 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 Theorem 3.1. We need only to show that there exists subsequence of {𝑣_𝑘} which is strongly convergent in 𝔎₀.
For each measurable set 𝑆⊂Ω, define 𝐹(𝑣,𝑆)=_𝑆𝐴𝑥,𝑑𝑢₀+𝑑𝑣,𝑑𝑣𝑑𝑥, (3.33) where 𝑣∈𝔎₀. Similar to the proof of Lemma 3.2, 𝐹(⋅,𝑆) is strongly continuous on 𝔎₀. Since 𝐶^∞₀(Ω,Λ^𝑙−1) is dense in 𝔎₀, there exists ℎ_𝑘⊂𝐶^∞₀(Ω,Λ^𝑙−1) such that ℎ_𝑘−𝑣_𝑘𝔎<1𝑘,𝐹ℎ_𝑘,Ω−𝐹𝑣_𝑘<1,Ω𝑘. (3.34) So we can suppose that {𝑣_𝑘}⊂𝐶^∞₀(Ω,Λ^𝑙−1) is bounded in 𝔎₀.
Next define 𝑣_𝑘(𝑥)=0when𝑥∈ℝ^𝑛⧵Ω. (3.35) In this way, we extend the domain of 𝑣_𝑘 to ℝ^𝑛 and supp𝑣_𝑘⊂Ω.
Let 𝛽∶ℝ⁺→ℝ⁺ be a continuous increasing function satisfying 𝛽(0)=0 and for each measurable set 𝐷⊂Ω, _𝐷|𝐺(𝑥)|^{𝑝′(𝑥)}+|ℎ(𝑥)|+𝐶₁+1𝑑𝑢₀^𝑝(𝑥)𝑑𝑥≤𝛽(meas(𝐷)), (3.36) where 𝐶₁ is the constant in (H2).
Let {𝜀_𝑗} be a positive decreasing sequence with 𝜀_𝑗→0 as 𝑗→∞. For 𝜀₁, by Lemma 3.8, we get a subsequence {𝑘₁} of {𝑘}, a set 𝐴_𝜀1⊂Ω satisfying meas(𝐴_𝜀1)<𝜀₁, and a real number 𝛿₁>0 such that _𝐵𝑀^∗𝑣_𝑘1𝐼^𝑝(𝑥)𝑑𝑥<𝜀₁, (3.37) for each 𝑘₁,𝐼 and 𝐵⊂Ω⧵𝐴_𝜀1 satisfying meas(𝐵)<𝛿₁. By Lemma 3.10, we can choose 𝜆>1 so large that for all 𝐼 and 𝑘₁, meas𝑥∈ℝ^𝑛∶𝑀^∗𝑣_𝑘1𝐼(𝑥)≥𝜆≤min𝜀₁,𝛿₁. (3.38)
For each 𝐼 and 𝑘₁, define 𝐻^𝜆_𝑘1𝐼=𝑥∈ℝ^𝑛∶𝑀^∗𝑣_𝑘1𝐼(𝑥)<𝜆,𝐻^𝜆_𝑘1=_𝐼𝐻^𝜆_𝑘1𝐼. (3.39) In view of Lemma 3.11, we have 𝑣_𝑘1𝐼𝑦−𝑣_𝑘1𝐼(𝑥)(𝑛)𝑦−𝑥≤𝐶𝜆∀𝑥,𝑦∈𝐻^𝜆_𝑘1and𝐼. (3.40) Form Lemma 3.12, there exists a Lipschitz function 𝑔_𝑘1𝐼 which extends 𝑣_𝑘1𝐼 outside 𝐻^𝜆_𝑘1 and Lipschitz constant of 𝑔_𝑘1𝐼 is no more than 𝐶(𝑛)𝜆. As 𝐻^𝜆_𝑘1 is an open set, we have 𝑔_𝑘1𝐼=𝑣_𝑘1𝐼 and ∇𝑔_𝑘1𝐼(𝑥)=∇𝑣_𝑘1𝐼(𝑥) for all 𝑥∈𝐻^𝜆_𝑘1, and ‖∇𝑔_𝑘1𝐼‖_{𝐿∞(ℝ𝑛)}≤𝐶(𝑛)𝜆. We can further suppose that 𝑔_{𝑘1𝐼𝐿∞(ℝ𝑛)}≤𝑣_{𝑘1𝐼𝐿∞(𝐻𝜆𝑘1)}≤𝜆,𝑔_{𝑘1𝐼𝑊1,∞(Ω)}≤𝐶(𝑛)𝜆. (3.41) By the uniformly boundedness of {‖𝑔_𝑘1𝐼‖_{𝑊1,∞(Ω)}}, there exists a subsequence of {𝑔_𝑘1𝐼} (still denoted by {𝑔_𝑘1𝐼}) such that 𝑔_𝑘1𝐼⇀𝜔_𝐼weakly^∗in𝑊^1,∞(Ω)as𝑘₁⟶∞∀𝐼. (3.42) Set 𝜔=∑_𝐼𝜔_𝐼𝑑𝑥_𝐼 and 𝑔_𝑘1=∑_𝐼𝑔_𝑘1𝐼𝑑𝑥_𝐼. We have 𝐹𝑣_𝑘1,Ω=𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1−𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1+𝐹𝑣_𝑘1,𝐴_𝜀1∪Ω⧵𝐻^𝜆_𝑘1. (3.43)
Next we estimate 𝐹(𝑣_𝑘1,Ω) in four steps.
(1) The estimate of 𝐹(𝑔_𝑘1,(Ω⧵𝐴_𝜀1)⧵𝐻^𝜆_𝑘1)  and 𝐹(𝑣_𝑘1,𝐴_𝜀1∪(Ω⧵𝐻^𝜆_𝑘1)). Since measΩ⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1≤_𝐼measΩ⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1𝐼≤C_𝑛^𝑙−1min𝜀₁,𝛿₁, (3.44) where C_𝑛^𝑙−1=𝑛(𝑛−1)⋯(𝑛−𝑙+2)/(𝑙−1)(𝑙−2)⋯1, from (H2), (H3), and the choose of 𝐴_𝜀1, we have 𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1≤_{(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1}𝐶₁𝑑𝑢₀+𝑑𝑔_𝑘1^{𝑝(𝑥)−1}𝑑𝑔_𝑘1+|𝐺(𝑥)|𝑑𝑔_𝑘1≤𝑑𝑥_{(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1}𝐶₁2^𝑝∗−1𝑑𝑢₀^𝑝(𝑥)+𝑑𝑔_𝑘1^𝑝(𝑥)+𝐶₁𝑑𝑔_𝑘1^𝑝(𝑥)+|𝐺(𝑥)|^{𝑝′(𝑥)}+𝑑𝑔_𝑘1^𝑝(𝑥)𝑑𝑥≤2^𝑝∗−1𝛽measΩ⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1+2^𝑝∗𝐶₁+1_{(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1}𝑑𝑔_𝑘1^𝑝(𝑥)𝑑𝑥≤2^𝑝∗−1𝛽measΩ⧵𝐴_𝜀1⧵𝐻^𝜆_𝑘1+2^𝑝∗𝐶₁+1_{(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1𝐼}∇𝑔_𝑘1𝐼^𝑝(𝑥)𝑑𝑥≤2^𝑝∗−1𝛽C_𝑛^𝑙−1𝜀₁+2^𝑝∗𝐶𝐶₁,𝑛,𝑙_{(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1}𝜆^𝑝(𝑥)𝑑𝑥≤2^𝑝∗−1𝛽C_𝑛^𝑙−1𝜀₁+2^𝑝∗𝐶𝐶₁,𝑛,𝑙_{𝐼(Ω⧵𝐴𝜀1)⧵𝐻𝜆𝑘1𝐼}𝑀^∗𝑣_𝑘1𝐼^𝑝(𝑥)𝑑𝑥≤2^𝑝∗−1𝛽C_𝑛^𝑙−1𝜀₁+2^𝑝∗𝐶𝐶₁,𝑛,𝑙𝜀₁≤𝑂𝜀₁,𝐹𝑣_𝑘1,𝐴_𝜀1∪Ω⧵𝐻^𝜆_𝑘1=_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝐴𝑥,𝑑𝑢₀+𝑑𝑣_𝑘1,𝑑𝑢₀+𝑑𝑣_𝑘1−𝐴𝑥,𝑑𝑢₀+𝑑𝑣_𝑘1,𝑑𝑢₀≥𝑑𝑥_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝑎𝑑𝑢₀+𝑑𝑣_𝑘1^𝑝(𝑥)−ℎ(𝑥)−𝐶₁𝑑𝑢₀+𝑑𝑣_𝑘1^{𝑝(𝑥)−1}𝑑𝑢₀+|𝐺(𝑥)|𝑑𝑢₀≥𝑑𝑥_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝑎2^{−(𝑝∗−1)}−𝐶₁𝜇2^𝑝∗−1𝑑𝑣_𝑘1^𝑝(𝑥)−|ℎ(𝑥)|−|𝐺(𝑥)|^{𝑝′(𝑥)}−−𝑎2^{−(𝑝∗−1)}+𝐶₁𝜇2^𝑝∗−1+𝐶₁𝐶𝜇+1𝑑𝑢₀^𝑝(𝑥)≥𝑑𝑥𝑎2^{−(𝑝∗−1)}−𝐶₁𝜇2^𝑝∗−1_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝐶𝑎,𝑝,𝐶₁,𝜇𝛽meas𝐴_𝜀1∪Ω⧵𝐻^𝜆_𝑘1≥𝑎2^−𝑝∗_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁, (3.45)(3.46) where 𝜇>0 is small enough.
From (3.43)–(3.46), we get 𝐹𝑣_𝑘1,Ω≥𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1+𝑎2^−𝑝∗_{𝐴𝜀1∪(Ω⧵𝐻𝜆𝑘1)}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁. (3.47)
(2) The estimate of 𝐹(𝑔_𝑘1,Ω⧵𝐴_𝜀1). Set 𝑓_𝑘1𝐼=𝑔_𝑘1𝐼−𝜔_𝐼, where 𝜔_𝐼 is defined by (3.42). Then 𝑓_𝑘1𝐼⇀0weakly^∗in𝑊^1,∞(Ω)as𝑘₁𝑓⟶∞∀𝐼,_{𝑘1𝐼𝐿∞(Ω)}≤2𝜆,𝑑𝑓_{𝑘1𝐼𝐿∞(Ω,Λ𝑙)}≤2𝐶(𝑛)𝜆. (3.48)
Let 𝐺=⋃_𝐼𝐺_𝐼 with 𝐺_𝐼={𝑥∈Ω∶𝜔_𝐼(𝑥)≠0}. According to Acerbi and Fusco [19], we have meas(𝐺)≤(C_𝑛^𝑙−1+1)𝜀₁ where C_𝑛^𝑙−1=𝑛(𝑛−1)⋅⋅⋅(𝑛−𝑙+2)/(𝑙−1)(𝑙−2)⋅⋅⋅1, and set 𝑓_𝑘1=∑_𝐼𝑓_𝑘1𝐼𝑑𝑥_𝐼, then 𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1=𝐹𝑓_𝑘1,Ω⧵𝐴_𝜀1𝑣⧵𝐺+𝐹_𝑘1,Ω⧵𝐴_𝜀1∩𝐻^𝜆_𝑘1𝑔∩𝐺+𝐹_𝑘1,Ω⧵𝐴_𝜀1∩𝐺⧵𝐻^𝜆_𝑘1. (3.49) Define Ω^𝜀1,𝑘1₁=𝐴_𝜀1∪Ω⧵𝐻^𝜆_𝑘1,Ω^𝜀1₂=Ω⧵𝐴_𝜀1Ω⧵𝐺,_𝜀1,𝑘13=Ω⧵𝐴_𝜀1∩𝐻^𝜆_𝑘1∩𝐺,Ω^𝜀1,𝑘1₄=Ω⧵𝐴_𝜀1∩𝐺⧵𝐻^𝜆_𝑘1. (3.50) Similar to the proof of (3.46), we get 𝐹𝑣_𝑘1,Ω^𝜀1,𝑘1₃≥𝑎2^−𝑝∗_{Ω𝜀113,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁. (3.51) Since on Ω^𝜀1,𝑘1₄ we have _{Ω𝜀114,𝑘}𝑑𝑔_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐶𝑛,𝑝C_𝑛^𝑙−1+1𝜀₁, (3.52) then similar to the proof of (3.45), we get 𝐹𝑔_𝑘1,Ω^𝜀1,𝑘1₄≤𝑂𝜀₁. (3.53) By (3.49)–(3.53), we have 𝐹𝑔_𝑘1,Ω⧵𝐴_𝜀1≥𝐹𝑓_𝑘1,Ω^𝜀1₂+𝑎2^−𝑝∗_{Ω𝜀113,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁. (3.54) Thus, we have 𝐹𝑣_𝑘1,Ω≥𝐹𝑓_𝑘1,Ω^𝜀1₂+𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁, (3.55) where Ω^𝜀1,𝑘1₅=Ω^𝜀1,𝑘1₁∪Ω^𝜀1,𝑘1₃.
Choose an open set Ω′⊂Ω which contains Ω^𝜀1₂ such that 𝐹𝑓_𝑘1,Ω^′−𝐹𝑓_𝑘1,Ω^𝜀1₂<𝜀₁. (3.56) From (3.55), we get 𝐹𝑣_𝑘1,Ω≥𝐹𝑓_𝑘1,Ω^′+𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁. (3.57)
Approximate Ω′ by hypercubes with edges parallel to coordinate axes, that is, construct 𝐻_𝑗⊂Ω^′,measΩ^′⧵𝐻_𝑗𝐻⟶0as𝑗⟶∞,_𝑗=^ℎ𝑗_𝑠=1𝐷_𝑗,𝑠,meas𝐷_𝑗,𝑠=1/2^𝑛𝑗,1≤𝑠≤ℎ_𝑗. (3.58)
Let 𝑗>0 be large enough such that for all 𝑘₁>0, we have𝐹𝑓_𝑘1,Ω^′−𝐹𝑓_𝑘1,𝐻_𝑗<𝜀₁,_{Ω′⧵𝐻𝑗}𝑑𝑓_𝑘1^𝑝(𝑥)𝑑𝑥<𝜀₁,measΩ^′⧵𝐻_𝑗<min𝜀₁,𝛿₁. (3.59) Thus, 𝐹𝑣_𝑘1,Ω≥𝐹𝑓_𝑘1,𝐻_𝑗+𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁−2𝜀₁. (3.60)
(3) The estimate of 𝐹(𝑓_𝑘1,𝐻_𝑗). Let 𝛼>0 be large enough such that for 𝐸={𝑥∈Ω′∶𝜂(𝑥)≤𝛼}. Then measΩ^′<𝜀⧵𝐸¹𝑁,_Ω′⧵𝐸𝜂(𝑥)𝑑𝑥<𝜀₁, (3.61) where ‖𝑑𝑓_𝑘1‖_{𝐿∞(Ω,Λ𝑙)}≤2C_𝑛^𝑙−1𝐶(𝑛)𝜆=𝑁 and 𝜂(𝑥)=|𝐺(𝑥)|^{𝑝′(𝑥)}+2^𝑝∗−1(𝐶₁+1)|𝑑𝑢₀|^𝑝(𝑥).
For 𝑥∈Ω,𝜉∈Λ^𝑙(Ω), define 𝜓(𝑥,𝜉)=𝐴𝑥,𝑑𝑢₀(𝑥)+𝜉,𝜉. (3.62) By Lemma 3.7 and (H1), there exists a compact subset 𝐾⊂𝐻_𝑗 such that 𝜓(𝑥,𝜉) is continuous on 𝐾×Λ^𝑙(Ω) and meas(𝐻_𝑗⧵𝐾)<𝜀₁/(𝛼+𝑁). Hence, 𝜓(𝑥,𝜉) is uniformly continuous on bounded subsets of 𝐾×Λ^𝑙(Ω).
Divide each 𝐷_𝑗,𝑠 into 2^𝑛𝑚 hypercubes 𝑄^𝑚_{𝑡,𝑗,𝑠} with edge length 2^−𝑗𝑚,1≤𝑡≤2^𝑛𝑚. For all 𝑗,𝑠,𝑚,𝑡, take 𝑥^𝑚_{𝑡,𝑗,𝑠}∈𝑄^𝑚_{𝑡,𝑗,𝑠}∩𝐾∩𝐸 (if this set is empty, take 𝑥^𝑚_{𝑡,𝑗,𝑠}∈𝑄^𝑚_{𝑡,𝑗,𝑠}) such that 𝜂𝑥^𝑚_{𝑡,𝑗,𝑠}meas𝑄^𝑚_{𝑡,𝑗,𝑠}≤_{𝑄𝑚𝑡,𝑗,𝑠}𝜂(𝑥)𝑑𝑥. (3.63) Then 𝐹𝑓_𝑘1,𝐻_𝑗=𝐹𝑓_𝑘1,𝐻_𝑗∩𝐾∩𝐸+𝐹𝑓_𝑘1,𝐻_𝑗⧵𝐸+𝐹𝑓_𝑘1,𝐻_𝑗𝑓∩𝐸⧵𝐾≥𝐹_𝑘1,𝐻_𝑗∩𝐾∩𝐸−_{𝐻𝑗⧵𝐸}𝜂(𝑥)𝑑𝑥−_{(𝐻𝑗∩𝐸)⧵𝐾}𝜂(𝑥)𝑑𝑥−2^𝑝∗𝐶₁+1_{𝐻𝑗⧵𝐸}𝑑𝑓_𝑘1^𝑝(𝑥)𝑑𝑥+_{(𝐻𝑗∩𝐸)⧵𝐾}𝑑𝑓_𝑘1^𝑝(𝑥)𝑓𝑑𝑥=𝐹_𝑘1,𝐻_𝑗∩𝐾∩𝐸−𝑂𝜀₁=𝑏_𝑘^𝑚,𝑗₁+𝑐_𝑘^𝑚,𝑗₁+𝑑_𝑘^𝑚,𝑗₁−𝑂𝜀₁, (3.64) where 𝑏_𝑘^𝑚,𝑗₁=_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠∩𝐾∩𝐸}𝜓𝑥,𝑑𝑓_𝑘1(𝑥)−𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)𝑐𝑑𝑥,_{𝑘𝑚,𝑗1}=_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠}𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥𝑑)𝑑𝑥,_{𝑘𝑚,𝑗1}=−_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠⧵(𝐾∩𝐸)}𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)𝑑𝑥. (3.65)
By (3.25), we have lim_𝑘1→∞𝐹𝑣_𝑘1,Ω=0. (3.66)
Note that if 𝑄^𝑚_{𝑡,𝑗,𝑠}∩𝐾∩𝐸 is an empty set, then _{𝑄𝑚𝑡,𝑗,𝑠∩𝐾∩𝐸}𝜓𝑥,𝑑𝑓_𝑘1(𝑥)−𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)𝑑𝑥=0. (3.67) Now we only consider 𝑄^𝑚_{𝑡,𝑗,𝑠} which satisfies 𝑄^𝑚_{𝑡,𝑗,𝑠}∩𝐾∩𝐸≠𝜙. Since 𝑑𝑢₀(𝑥) is uniformly continuous on 𝐻_𝑗, then by the uniform continuity of 𝜓 on bounded subsets of 𝐾×Λ^𝑙(Ω), we obtain that for 𝑥∈𝑄^𝑚_{𝑡,𝑗,𝑠}, there exists a constant 𝐿>0 such that 𝜓𝑥,𝑑𝑓_𝑘1(𝑥)−𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥=𝐴)𝑥,𝑑𝑢₀(𝑥)+𝑑𝑓_𝑘1(𝑥)−𝐴𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑢₀𝑥^𝑚_{𝑡,𝑗,𝑠}+𝑑𝑓_𝑘1(𝑥),𝑑𝑓_𝑘1(𝑥<1)meas𝐻_𝑗𝜀₁ (3.68) holds for all 𝑚>𝐿 and each 𝑘₁. Therefore, |𝑏_𝑘^𝑚,𝑗₁|<𝜀₁ for all 𝑘₁. 𝑑_𝑘^𝑚,𝑗₁≤_{𝑡,𝑠𝑄𝑚𝑡,𝑗,s⧵(𝐾∩𝐸)}𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)𝑑𝑥=_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠⧵(𝐾∩𝐸)}𝐴𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑢₀𝑥^𝑚_{𝑡,𝑗,𝑠}+𝑑𝑓_𝑘1(𝑥),𝑑𝑓_𝑘1(𝑥≤)𝑑𝑥_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠⧵(𝐾∩𝐸)}𝐶₁𝑑𝑢₀𝑥^𝑚_{𝑡,𝑗,𝑠}+𝑑𝑓_𝑘1(𝑥)^{𝑝(𝑥)−1}𝑑𝑓_𝑘(𝑥)+𝐺𝑥^𝑚_{𝑡,𝑗,𝑠}𝑑𝑓_𝑘1(𝑥)𝑑𝑥≤_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠⧵(𝐾∩𝐸)}𝜂𝑥^𝑚_{𝑡,𝑗,𝑠}+2^𝑝∗𝐶₁𝑁≤+1𝑑𝑥_{(𝐻𝑗∩𝐸)⧵𝐾}𝜂𝑥^𝑚_{𝑡,𝑗,𝑠}+2^𝑝∗𝐶₁+1𝑁𝑑𝑥+𝐶𝐶₁,𝑝_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠⧵𝐸}𝜂𝑥^𝑚_{𝑡,𝑗,𝑠}+𝑁𝑑𝑥≤𝐶𝛼,𝑁,𝐶₁,𝑝meas𝐻_𝑗∩𝐸⧵𝐾+𝐶𝐶₁,𝑝_{𝐻𝑗⧵𝐸}𝜂(𝑥)+𝑁𝑑𝑥≤𝐶𝛼,𝑁,𝐶₁,𝑝𝜀₁≤𝑂𝜀₁. (3.69)
Now we suppose that 𝑚 is large enough that |𝑏_𝑘^𝑚,𝑗₁|<𝜀₁ for each 𝑘₁>0 and there exists 𝑘₁>0 such that 𝐹(𝑣_𝑘1Ω)<𝜀₁ for 𝑘₁>𝑘₁. Therefore, from (3.25), (3.60), and (3.64), we have 𝜀₁≥𝐹𝑣_𝑘1,Ω≥𝑐_𝑘^𝑚,𝑗₁+𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁−3𝜀₁−𝐶𝐶₁,𝑝𝜀₁=𝑐_𝑘^𝑚,𝑗₁+𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥−𝑂𝜀₁. (3.70)
(4) The estimate of 𝑐_𝑘^𝑚,𝑗₁. By 𝑓_𝑘1𝐼⇀0  weakly^∗ in 𝑊^1,∞(Ω) as 𝑘₁→∞, we obtain ‖𝑓_𝑘1𝐼‖_𝐿∞(Ω)→0as𝑘₁→∞ for each 𝐼. Then 𝑅^𝑘1,𝑚_{𝑡,𝑠,𝑗}=|𝑓_𝑘1|_{𝐿∞(𝑄𝑚𝑡,𝑠,𝑗)}⟶0as𝑘₁⟶∞forfixed𝑚. (3.71)
Define a hypercube 𝐸^𝑘1,𝑚_{𝑡,𝑠,𝑗} contained in 𝑄^𝑚_{𝑡,𝑠,𝑗} with edge length 1/2^𝑗𝑚−2𝑅^𝑘1,𝑚_{𝑡,𝑠,𝑗} such that dist(𝜕𝑄^𝑚_{𝑡,𝑠,𝑗},𝐸^𝑘1,𝑚_{𝑡,𝑠,𝑗})=𝑅^𝑘1,𝑚_{𝑡,𝑠,𝑗}.
Next define 𝜑_𝑘1(𝑥)=0,𝑥∈𝜕𝑄^𝑚_{𝑡,𝑠,𝑗},𝜑_𝑘1(𝑥)=𝑓_𝑘1(𝑥),𝑥∈𝐸^𝑚_{𝑡,𝑠,𝑗}. (3.72) Since 𝜑_𝑘1𝐼 is a Lipschitz mapping on set 𝐸^𝑚_{𝑡,𝑠,𝑗}∪𝜕𝑄^𝑚_{𝑡,𝑠,𝑗} and its Lipschitz constant is no more than 2𝐶(𝑛)𝜆, by Lemma 3.12, 𝜑_𝑘1𝐼 can be extended to the whole 𝑄^𝑚_{𝑡,𝑠,𝑗}, where it is also a Lipschitz mapping with the same Lipchistz constant. We still denote the extension by 𝜑_𝑘1𝐼 and suppose that it is defined on the whole 𝐻_𝑗. Then by [20] ∇𝜑_𝑘1𝐼−∇𝑓_𝑘1𝐼⟶0a.e.on𝐻_𝑗. (3.73) Thus, there exists a 𝑘₁>𝑘₁ such that for all 𝑘₁>𝑘₁, we have _𝐻𝑗𝑑𝜑_𝑘1−𝑑𝑓_𝑘1^𝑝(𝑥)𝜀𝑑𝑥≤¹2_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠},𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)−𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝜑_𝑘1(𝑥)≤𝜀𝑑𝑥¹2. (3.74) In view of (H5), we obtain that 𝑐_𝑘^𝑚,𝑗₁=_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠}𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑓_𝑘1(𝑥)𝑑𝑥≥_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠}𝜓𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝜑_𝑘1(𝑥𝜀)𝑑𝑥−¹2=_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠}𝐴𝑥^𝑚_{𝑡,𝑗,𝑠},𝑑𝑢₀𝑥^𝑚_{𝑡,𝑗,𝑠}+𝑑𝜑_𝑘1(𝑥),𝑑𝜑_𝑘1(𝑥𝜀)𝑑𝑥−¹2≥𝛾_{𝑡,𝑠𝑄𝑚𝑡,𝑗,𝑠}𝑑𝜑_𝑘1^𝑝(𝑥)𝜀𝑑𝑥−¹2≥𝛾2^𝑝∗−1_𝐻𝑗𝑑𝑓_𝑘1^𝑝(𝑥)𝜀𝑑𝑥−𝛾+1₁2. (3.75) Thus in (3.70) for 𝑘₁>𝑘₁, we obtain the estimate of 𝐹(𝑣_𝑘1,Ω) from the four steps above 𝜀₁≥𝐹𝑣_𝑘1,Ω≥𝑎2^−𝑝∗_{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝛾𝑑𝑥+2^𝑝∗−1_𝐻𝑗𝑑𝑓_𝑘1^𝑝(𝑥)𝜀𝑑𝑥−𝛾+1₁2−𝑂𝜀₁. (3.76) Let 𝐾(𝜀₁)=(𝛾+1)𝜀₁/(2+𝑜(𝜀₁))/min{𝑎2^−𝑝∗,𝛾/2^𝑝∗−1}. Then _{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥+_𝐻𝑗𝑑𝑓_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐾𝜀₁,for𝑘₁>𝑘₁. (3.77) Form (3.59) and (3.77), we deduce that _{Ω𝜀115,𝑘}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐾𝜀₁,_Ω′𝑑𝑓_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐾𝜀₁+𝜀₁. (3.78) According to the definition of Ω^𝜀1₂, we have _Ω2𝜀1𝑑𝑔_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐾𝜀₁+𝜀₁. (3.79) Since 𝑑𝑔_𝑘1(𝑥)=𝑑𝑣_𝑘1(𝑥) for each 𝑥∈𝐻^𝜆_𝑘1, we get _{Ω2𝜀1∩𝐻𝜆𝑘1}𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥≤𝐾𝜀₁+𝜀₁. (3.80) By the definitions of Ω^𝜀1₂ and Ω^𝜀1,𝑘1₅, it is immediate that Ω^𝜀1₂∩𝐻^𝜆_𝑘1∪Ω^𝜀1,𝑘1₅=Ω, (3.81) which implies that _Ω𝑑𝑣_𝑘1^𝑝(𝑥)𝑑𝑥≤2𝐾𝜀₁+𝜀₁≤𝑂𝜀₁. (3.82)
For 𝜀₂>0 and the sequence {𝑣_𝑘1}, repeating the above arguments we can extract a subsequence {𝑣_𝑘2} of {𝑣_𝑘1} such that _Ω𝑑𝑣_𝑘2^𝑝(𝑥)𝑑𝑥≤𝑂𝜀₂, (3.83) whenever 𝑘₂>𝑘₂ for some 𝑘₂. If {𝑣_𝑘𝑛} has been obtained, repeating the above process, we can extract a subsequence {𝑘_𝑛+1} of {𝑘_𝑛} such that _Ω𝑑𝑣_𝑘𝑛+1^𝑝(𝑥)𝑑𝑥≤𝑂𝜀_𝑛+1, (3.84) whenever 𝑘_𝑛+1>𝑘_𝑛+1 for some 𝑘_𝑛+1. Finally, by a diagonal argument we get a subsequence {𝑣_𝑘𝑖}^∞_𝑖=1 which satisfies _Ω𝑑𝑣_𝑘𝑖^𝑝(𝑥)𝑑𝑥⟶0as𝑖⟶∞. (3.85) Therefore, 𝑑𝑣_{𝑘𝑖𝐿𝑝(𝑥)(Ω,Λ𝑙)}⟶0as𝑖⟶∞, (3.86) and by (3.23), {𝑣_𝑘𝑖}^∞_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.

Let Ω⊂ℝ^𝑛 be a bounded and convex Lipschitz domain. Suppose that maps 𝐴∶Ω×Λ^𝑙(Ω)→Λ^𝑙(Ω) and 𝐵∶Ω×Λ^𝑙−1(Ω)→Λ^𝑙−1(Ω), where 𝑙=1,2,…,𝑛.

Example 4.1. If 𝑝(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)∈𝐿^{𝑝′(𝑥)}(Ω). Then 𝐴,𝐵 satisfy the required conditions, and (1.1) reduce to the following 𝑝(𝑥)-Laplacian equations: −div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢+|𝑢|^{𝑝(𝑥)−2}(𝑥)𝑢(𝑥)𝑢=𝑓,𝑥∈Ω,=0,𝑥∈𝜕Ω. (4.1) Now by Theorem 3.1, we deduce that the 𝑝(𝑥)-Laplacian equations (4.1) have at least one weak solution in 𝔎^1,𝑝(𝑥)(Ω) with 𝑢=0 on 𝜕Ω.

Example 4.2. If 𝑙=1, 𝐴(𝑥,𝜉)=∑_𝑖,𝑗𝐴_𝑖𝑗(𝑥)𝜉_𝑗𝑑𝑥_𝑖, 𝐵(𝑥,𝜍)=𝐵(𝑥)𝜍−𝑓(𝑥), where 𝑓(𝑥)∈𝐿²(Ω), and 𝐴_𝑖𝑗(𝑥), 𝐵(𝑥) satisfy the following conditions: 𝐴_𝑖𝑗(𝑥)=𝐴_𝑗𝑖(𝑥),∧𝜉²≥^𝑛_𝑖,𝑗=1𝐴_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝜆𝜉²,𝜆≤𝐵(𝑥)≤∧, (4.2) for some constants 𝜆,∧>0. Then 𝐴,𝐵 satisfy the required conditions, and (1.1) reduce to the following Divergence form equations: ^𝑛_𝑖,𝑗=1∇_𝑗𝐴_𝑖𝑗(𝑥)∇_𝑖𝑢(𝑥(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥))+𝐵=𝑓,𝑥∈Ω,=0,𝑥∈𝜕Ω, (4.3)(4.4) where ∇_𝑖=(𝜕/𝜕𝑥_𝑖). Now by Theorem 3.1, we deduce that the divergence form (4.3) have at least one weak solution 𝑢(𝑥) in 𝔎^1,2(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. If 𝑝(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)∈𝐿^{𝑝′(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: 𝑑^∗𝑑𝑢|𝑑𝑢|^{𝑝(𝑥)−2}+𝑢|𝑢|^{𝑝(𝑥)−2}(𝑥)𝑢(𝑥)=𝑓,𝑥∈Ω,=0,𝑥∈𝜕Ω. (4.5)(4.6) Now by Theorem 3.1, we deduce that (4.5) have at least one weak solution 𝑢(𝑥) in 𝔎₀^1,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 𝐿^𝑞 theory of weak solution for homogeneous 𝑞-harmonic equations.