Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

Abstract

We study a kind of nonlinear elliptic boundary value problems with generalized 𝑝-Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

1. Introduction

The study on nonlinear boundary value problems with 𝑝-Laplacian operator, Δ_𝑝, is a hot topic since it has a close relationship with practical problems. Some significant work has been done by us, see [1–8], and so forth.

Specifically, in 2004, we studied the following nonlinear elliptic boundary value problem involving the generalized 𝑝-Laplacian operator: −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+|𝑢|^𝑝−2((𝑥(𝑥)−𝐶(𝑥)+||𝑢+𝑔𝑥,𝑢))=𝑓,a.e.inΩ𝜗,∇𝑢^{2(𝑝−2)/2}∇𝑢=0,a.e.onΓ. (1.1)

In Wei and Hou [4], we proved that under some conditions (1.1) has solutions in 𝐿²(Ω), where 2≤𝑝<+∞. In [5, 6], we extended our work to the following problem: −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+|𝑢|^𝑝−2((𝑥(𝑥)−𝐶(𝑥)+||𝑢+𝑔𝑥,𝑢))=𝑓,a.e.inΩ𝜗,∇𝑢^{2(𝑝−2)/2}∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (1.2) In Wei and Zhou [5], we established that (1.2) has solutions in 𝐿^𝑝(Ω), where 2≤𝑝<+∞, and in Wei [6] we proved that (1.2) has solutions in 𝐿^𝑠(Ω), where max(𝑁,2)≤𝑝≤𝑠<+∞. As the summary and extension of [5, 6], we studied the following nonlinear boundary value problem: −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+𝜀|𝑢|^𝑞−2((𝑥(𝑥)−𝐶(𝑥)+||𝑢+𝑔𝑥,𝑢))=𝑓,a.e.inΩ𝜗,∇𝑢^{2(𝑝−2)/2}∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (1.3)

It was shown by Wei and Agarwal [7] that (1.3) had a solution in 𝐿^𝑠(Ω), where 2𝑁/(𝑁+1)<𝑝≤𝑠<+∞, 1≤𝑞<+∞ if 𝑝≥𝑁 and 1≤𝑞≤𝑁𝑝/(𝑁−𝑝) if 𝑝<𝑁, for 𝑁≥1.

Clearly, if 𝐶(𝑥)≡0, then (1.1), (1.2), and (1.3) reduce the cases of involving 𝑝-Laplacian operators.

It is worth to mention that all of the work done in [4–7] is based on a perturbation result of the ranges of 𝑚-accretive mappings by Calvert and Gupta [9].

In this paper, we will study the following nonlinear elliptic boundary value problem: −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+𝜀|𝑢|^𝑞−2((𝑥)(𝑥(𝑥)−𝐶(𝑥)+||𝑢+𝑔𝑥,𝑢,∇𝑢))=𝑓,a.e.inΩ𝜗,∇𝑢^{2(𝑝−2)/2}∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (1.4)

Necessary details of (1.4) will be provided in Section 3.

We may notice that the principal part of the concerned equation is almost the same as those in (1.1), (1.2), and (1.3) while the nonlinear item 𝑔(𝑥,𝑢(𝑥)) is replaced by the item 𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥)), which is rather general. It seems that the difference is minor; however, the previous method cannot be employed. In this paper, we will use some perturbation results of the ranges for maximal monotone operators by Pascali and Sburlan [10] to prove that (1.4) has a unique solution in 𝑊^1,𝑝(Ω) and later we will prove that the unique solution is the zero point of a suitably defined maximal monotone operator. Finally, we will employ an iterative scheme to approximate strongly to the unique solution. Some new ideas of combining the knowledge of the ranges of nonlinear operators, zero point of nonlinear operators, iterative schemes, and the solution of nonlinear boundary value problems are demonstrated in this paper.

2. Preliminaries

Now, we list some of the knowledge we need in the sequel.

Let 𝑋 be a real Banach space with a strictly convex dual space 𝑋^∗. We use (⋅,⋅) to denote the generalized duality pairing between 𝑋 and 𝑋^∗. We use “→” to denote strong convergence. Let “𝑋↪𝑌” denote the space 𝑋 embedded continuously in space 𝑌. For any subset 𝐺 of 𝑋, we denote by int𝐺 its interior.

Function Φ is called a proper convex function on 𝑋 [11] if Φ is defined from 𝑋 to (−∞,+∞], not identically +∞ such that Φ((1−𝜆)𝑥+𝜆𝑦)≤(1−𝜆)Φ(𝑥)+𝜆Φ(𝑦), whenever 𝑥,𝑦∈𝑋 and 0≤𝜆≤1.

Function Φ∶𝑋→(−∞,+∞] is said to be lower semicontinuous on 𝑋 [11] if liminf_𝑦→𝑥Φ(𝑦)≥Φ(𝑥), for any 𝑥∈𝑋.

Given a proper convex function Φ on 𝑋 and a point 𝑥∈𝑋, we denote by 𝜕Φ(𝑥) the set of all 𝑥^∗∈𝑋^∗ such that Φ(𝑥)≤Φ(𝑦)+(𝑥−𝑦,𝑥^∗), for every 𝑦∈𝑋. Such elements 𝑥^∗ are called subgradients of Φ at 𝑥, and 𝜕Φ(𝑥) is called the subdifferential of Φ at 𝑥 [11].

A single-valued mapping 𝑇∶𝐷(𝑇)=𝑋→𝑋^∗ is said to be hemicontinuous [11] if 𝑤−lim_𝑡→0𝑇(𝑥+𝑡𝑦)=𝑇𝑥, for any 𝑥,𝑦∈𝑋.

A multivalued mapping 𝐴∶𝑋→2^𝑋∗ is said to be monotone [10] if its graph 𝐺(𝐴) is a monotone subset of 𝑋×𝑋^∗ in the sense that 𝑢₁−𝑢₂,𝑤₁−𝑤₂≥0, (2.1) for any [𝑢_𝑖,𝑤_𝑖]∈𝐺(𝐴),  𝑖=1,2. The mapping 𝐴 is said to be strictly monotone if the equality in (2.1) implies that 𝑢₁=𝑢₂. The monotone operator 𝐴 is said to be maximal monotone if 𝐺(𝐴) is maximal among all monotone subsets of 𝑋×𝑋^∗ in the sense of inclusion. The mapping 𝐴 is said to be coercive [10] if lim_𝑛→+∞(𝑥_𝑛,𝑥^∗_𝑛)/‖𝑥_𝑛‖=+∞ for all [𝑥_𝑛,𝑥^∗_𝑛]∈𝐺(𝐴) such that lim_𝑛→+∞‖𝑥_𝑛‖=+∞. A point 𝑥∈𝐷(𝐴) is said to be a zero point of 𝐴 if 0∈𝐴𝑥, and we denote by 𝐴⁻¹(0)={𝑥∈𝑋∶0∈𝐴𝑥} the set of zero points of 𝐴.

Lemma 2.1 (Adams [12]). Let Ω be a bounded conical domain in 𝑅^𝑁. If 𝑚𝑝>𝑁, then 𝑊^𝑚,𝑝(Ω)↪𝐶_𝐵(Ω); if 𝑚𝑝<𝑁 and 𝑞=𝑁𝑝/(𝑁−𝑚𝑝), then 𝑊^𝑚,𝑝(Ω)↪𝐿^𝑞(Ω); if 𝑚𝑝=𝑁 and 𝑝>1, then for 1≤𝑞<+∞,𝑊^𝑚,𝑝(Ω)↪𝐿^𝑞(Ω).

Lemma 2.2 (Pascali and Sburlan [10]). If 𝐵∶𝑋→2^𝑋∗ is an everywhere defined, monotone, and hemicontinuous operator, then 𝐵 is maximal monotone.

Lemma 2.3 (Pascali and Sburlan [10]). If Φ∶𝑋→(−∞,+∞] is a proper convex and lower semicontinuous function, then 𝜕Φ is maximal monotone from 𝑋 to 𝑋^∗.

Lemma 2.4 (Pascali and Sburlan [10]). If 𝐵₁ and 𝐵₂ are two maximal monotone operators in 𝑋 such that (int𝐷(𝐵₁))⋂𝐷(𝐵₂)≠∅, then 𝐵₁+𝐵₂ is maximal monotone.

Lemma 2.5 (Pascali and Sburlan [10]). If 𝐴∶𝑋→2^𝑋∗ is maximal monotone and coercive, then 𝑅(𝐴)=𝑋^∗.

Definition 2.6 (Kamimura and Takahashi [13]). Let 𝑋 be a real smooth Banach space. Then the Lyapunov functional 𝜑∶𝑋×𝑋→𝑅⁺ is defined as follows: 𝜑𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝑋, (2.2) where 𝐽∶𝑋→2^𝑋∗ is the duality mapping defined by 𝐽𝑥={𝑓∈𝑋^∗∶(𝑥,𝑓)=‖𝑥‖‖𝑓‖,‖𝑓‖=‖𝑥‖}, for 𝑥∈𝑋.

Lemma 2.7 (Kamimura and Takahashi [13]). Let 𝑋 be a real reflexive, strictly convex, and smooth Banach space, let 𝐶 be a nonempty closed and convex subset of 𝑋, and let 𝑥∈𝑋. Then there exists a unique element 𝑥₀∈𝐶 such that 𝜑𝑥₀,𝑥=min𝜑(𝑧,𝑥)∶𝑧∈𝐶. (2.3)

Define a mapping Π_𝐶 from 𝑋 onto 𝐶 by Π_𝐶𝑥=𝑥₀ for all 𝑥∈𝑋. Then Π_𝐶 is called the generalized projection mapping from 𝑋 onto 𝐶. It is easy to see that Π_𝐶 coincides with the metric projection 𝑃_𝐶 in a Hilbert space.

3. Main Results

3.1. Notations and Assumptions of (1.4)

In the following of this paper, unless otherwise stated, we will assume that 2𝑁/(𝑁+1)<𝑝<+∞,  1≤𝑞<+∞ if 𝑝≥𝑁, and 1≤𝑞≤𝑁𝑝/(𝑁−𝑝) if 𝑝<𝑁, for 𝑁≥1. Let 1/𝑝+1/𝑝^′=1. We use ‖⋅‖_𝑝,‖⋅‖_𝑝′, and ‖⋅‖_1,𝑝,Ω to denote the norm of spaces 𝐿^𝑝(Ω),𝐿^𝑝′(Ω), and 𝑊^1,𝑝(Ω), respectively.

In nonlinear boundary value problem (1.4), Ω is a bounded conical domain of an Euclidean space 𝑅^𝑁 with its boundary Γ∈𝐶¹ (see Wei and He [1]). We will assume that Green’s formula is available. 𝑓(𝑥)∈𝐿^𝑝′(Ω) is a given function. 0≤𝐶(𝑥)∈𝐿^𝑝(Ω),  𝜀 is a nonnegative constant and 𝜗 denotes the exterior normal derivative of Γ.

Let 𝜑∶Γ×𝑅→𝑅 be a given function such that, for each 𝑥∈Γ, 𝜑_𝑥=𝜑(𝑥,⋅)∶𝑅→𝑅 is a proper, convex, and lower semicontinuous function with 𝜑_𝑥(0)=0. Let 𝛽_𝑥 be the subdifferential of 𝜑_𝑥, that is, 𝛽_𝑥≡𝜕𝜑_𝑥. Suppose that 0∈𝛽_𝑥(0) and for each 𝑡∈𝑅, the function 𝑥∈Γ→(𝐼+𝜆𝛽_𝑥)⁻¹(𝑡)∈𝑅 is measurable for 𝜆>0.

Suppose that 𝑔∶Ω×𝑅^𝑁+1→𝑅 is a given function satisfying the following conditions, which can be found in Zeidler [14].(a)Carathéodory's conditions: 𝑥⟶𝑔(𝑥,𝑟)ismeasurableonΩ∀𝑟∈𝑅^𝑁+1,𝑟⟶𝑔(𝑥,𝑟)iscontinuouson𝑅^𝑁+1foralmostall𝑥∈Ω. (3.1)(b)Growth condition: 𝑔𝑥,𝑟₁,…,𝑟_𝑁+1≤ℎ₁(𝑥)+𝑏^𝑁+1_𝑖=1𝑟_𝑖^𝑝−1, (3.2) where (𝑟₁,𝑟₂,…,𝑟_𝑁+1)∈𝑅^𝑁+1,  ℎ₁(𝑥)∈𝐿^𝑝(Ω) and 𝑏 is a fixed positive constant.(c)Monotone condition: 𝑔 is monotone with respect to 𝑟₁, that is, 𝑔𝑥,𝑟₁,…,𝑟_𝑁+1−𝑔𝑥,𝑡₁,…,𝑡_𝑁+1𝑟₁−𝑡₁≥0, (3.3) for all 𝑥∈Ω and (𝑟₁,…,𝑟_𝑁+1),(𝑡₁,…,𝑡_𝑁+1)∈𝑅^𝑁+1.

3.2. Existence and Uniqueness of the Solution of (1.4)

Lemma 3.1. Define the mapping 𝐵_𝑝,𝑞∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ by 𝑣,𝐵_𝑝,𝑞𝑢=_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝑣𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)𝑑𝑥 (3.4) for any 𝑢,𝑣∈𝑊^1,𝑝(Ω). Then, 𝐵_𝑝,𝑞 is everywhere defined, strictly monotone, hemicontinuous, and coercive.

Moreover, Lemma 2.2 implies that 𝐵_𝑝,𝑞 is maximal monotone.

Proof. From Lemma  3.2 by Wei and Agarwal [7], we know that 𝐵_𝑝,𝑞 is everywhere defined, monotone, hemicontinuous, and coercive. Then we only need to show that 𝐵_𝑝,𝑞 is strictly monotone.
In fact, for any 𝑢,𝑣∈𝑊^1,𝑝(Ω), 𝑢−𝑣,𝐵_𝑝,𝑞𝑢−𝐵_𝑝,𝑞𝑣=_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢−𝐶(𝑥)+|∇𝑣|^{2(𝑝−2)/2}∇𝑣,∇𝑢−∇𝑣𝑑𝑥+𝜀_Ω|𝑢|^𝑞−2𝑢−|𝑣|^𝑞−2𝑣)≥(𝑢−𝑣𝑑𝑥_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}|∇𝑢|²−𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}||−𝐶(𝑥)+||∇𝑢||∇𝑣∇𝑣^{2(𝑝−2)/2}|∇𝑣||∇𝑢|+𝐶(𝑥)+|∇𝑣|^{2(𝑝−2)/2}|∇𝑣|²𝑑𝑥+𝜀_Ω|𝑢|^𝑞−|𝑢|^𝑞−1|𝑣|−|𝑣|^𝑞−1|𝑢|+|𝑣|^𝑞=𝑑𝑥_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}|∇𝑢|−𝐶(𝑥)+|∇𝑣|^{2(𝑝−2)/2}|∇𝑣|(|∇𝑢|−|∇𝑣|)𝑑𝑥+𝜀_Ω|𝑢|^𝑞−1−|𝑣|^𝑞−1(|𝑢|−|𝑣|)𝑑𝑥. (3.5)
Define ℎ(𝑡)=(𝑘+𝑡²)^(𝑝−2)/2𝑡, for all 𝑡>0, where 𝑘 is a nonnegative constant.
Case 1. If 𝑝≥2, then ℎ^′(𝑡)=(𝑝−2)(𝑘+𝑡²)^𝑝/2−2𝑡²+(𝑘+𝑡²)^𝑝/2−1>0, for all 𝑡>0. That is, ℎ∶𝑅→𝑅 is strictly increasing. Thus we can see from (3.5) that if (𝑢−𝑣,𝐵_𝑝,𝑞𝑢−𝐵_𝑝,𝑞𝑣)=0, then 𝑢(𝑥)=𝑣(𝑥) and 𝜕𝑢/𝜕𝑥_𝑖=𝜕𝑣/𝜕𝑥_𝑖(𝑖=1,2,…,𝑁), a.e. in Ω, which implies that 𝑢(𝑥)=𝑣(𝑥) in 𝑊^1,𝑝(Ω). Therefore, 𝐵_𝑝,𝑞 is strictly monotone.
Case 2. If 2𝑁/(𝑁+1)<𝑝<2, then for all 𝑡>0, ℎ^′(𝑡)=𝑝−2𝑘+𝑡^2𝑝/2−2𝑡²+𝑘+𝑡^2𝑝/2−1=𝑝−2𝑘+𝑡^2𝑝/2−1−𝑝−2𝑘+𝑡^2𝑝/2−2𝑘+𝑘+𝑡^2𝑝/2−1=𝑝−1𝑘+𝑡^2𝑝/2−1−𝑝−2𝑘+𝑡^2𝑝/2−2𝑘=𝑝−1𝑘+𝑡^2𝑝/2−1+2−𝑝𝑘+𝑡^2𝑝/2−2𝑘>0, (3.6) which implies that ℎ∶𝑅→𝑅 is also strictly increasing. In the same way as Case 1, we know that 𝐵_𝑝,𝑞 is strictly monotone.
This completes the proof.

Lemma 3.2 (see Wei and Agarwal [7]). The mapping Φ_𝑝∶𝑊^1,𝑝(Ω)→𝑅 defined by Φ_𝑝(𝑢)=∫_Γ𝜑_𝑥(𝑢|_Γ(𝑥))𝑑Γ(𝑥), for any 𝑢∈𝑊^1,𝑝(Ω), is proper, convex, and lower semicontinuous on 𝑊^1,𝑝(Ω).

Therefore, 𝜕Φ_𝑝 is maximal monotone in view of Lemma 2.3.

Lemma 3.3. Define the mapping 𝐹∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ by (𝑣,𝐹𝑢(𝑥))=_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝑣(𝑥)𝑑𝑥, (3.7) for all 𝑢,𝑣∈𝑊^1,𝑝(Ω); then 𝐹 is everywhere defined, monotone, and hemicontinuous on 𝑊^1,𝑝(Ω).

Moreover, Lemma 2.2 implies that 𝐹 is maximal monotone.

Proof. We split our proof into four steps.
Step 1. For 𝑢∈𝑊^1,𝑝(Ω),  𝑥→𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥)) is measurable on Ω.
From the facts that 𝑢(𝑥),𝜕𝑢/𝜕𝑥_𝑖∈𝐿^𝑝(Ω),  𝑖=1,2,…,𝑁, we know that 𝑥→(𝑢(𝑥),𝜕𝑢/𝜕𝑥₁,…,𝜕𝑢/𝜕𝑥_𝑁) is measurable on Ω. Combining with the fact that 𝑔 satisfies Carathéodory’s conditions, we know that 𝑥→𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥)) is measurable on Ω.
Step 2. 𝐹 is everywhere defined.
From Lemma 2.1, we know that 𝑊^1,𝑝(Ω)↪𝐶_𝐵(Ω), when 𝑝>𝑁, and 𝑊^1,𝑝(Ω)↪𝐿^𝑝′(Ω), when 𝑝=𝑁. Thus, for all 𝑤∈𝑊^1,𝑝(Ω), ‖𝑤‖_𝑝′≤𝑘‖𝑤‖_1,𝑝,Ω, (3.8) where 𝑘>0 is a constant.
When 𝑝<𝑁, we know from Lemma 2.1 that 𝑊^1,𝑝(Ω)↪𝐿^{𝑁𝑝/(𝑁−𝑝)}(Ω). Since 2𝑁/(𝑁+1)<𝑝<+∞, then 𝑁𝑝/(𝑁−𝑝)>𝑝^′ and 𝐿^{𝑁𝑝/(𝑁−𝑝)}(Ω)↪𝐿^𝑝′(Ω), which implies that (3.8) is still true.
Now, for 𝑢,𝑣∈𝑊^1,𝑝(Ω), we have from (3.8) that |(𝑣,𝐹𝑢)|≤_Ωℎ₁(𝑥)|𝑣|𝑑𝑥+𝑏_Ω|𝑢|^𝑝−1|𝑣|𝑑𝑥+𝑏^𝑁_𝑖=1Ω𝜕𝑢𝜕𝑥_𝑖^𝑝−1|𝑣|≤ℎ𝑑𝑥₁(𝑥)_𝑝‖𝑣‖_𝑝′+𝑏‖𝑢‖^𝑝/𝑝′_𝑝‖𝑣‖_𝑝+𝑏^𝑁_𝑖=1𝜕𝑢𝜕𝑥_𝑖^𝑝/𝑝′_𝑝‖𝑣‖_𝑝≤ℎ₁(𝑥)_𝑝‖𝑣‖_1,𝑝,Ω+𝑏(𝑁+1)‖𝑢‖^𝑝/𝑝′_1,𝑝,Ω‖𝑣‖_1,𝑝,Ω, (3.9) which implies that 𝐹 is everywhere defined.
Step 3. 𝐹 is monotone.
Since 𝑔(𝑥,𝑟₁,…,𝑟_𝑁+1) is monotone with respect to 𝑟₁, then 𝐹 is monotone.
Step 4. 𝐹 is hemicontinuous.
In fact, it suffices to show that, for any 𝑢,𝑣,𝑤∈𝑊^1,𝑝(Ω) and 𝑡∈[0,1],  (𝑤,𝐹(𝑢+𝑡𝑣)−𝐹𝑢)→0, as 𝑡→0.
Noticing the facts that 𝑔 is measurable on Ω and 𝑔 satisfies Carathéodory’s conditions, by using Lebesque’s dominated convergence theorem, we have 0≤lim_𝑡→0()≤|(𝑤,𝐹𝑢+𝑡𝑣−𝐹𝑢)|_Ωlim_𝑡→0𝑔(𝑥,𝑢(𝑥)+𝑡𝑣(𝑥),∇𝑢+𝑡∇𝑣)−𝑔(𝑥,𝑢,∇𝑢)|𝑤|𝑑𝑥=0, (3.10) and hence 𝐹 is hemicontinuous.
This completes the proof.

In view of Lemma 2.4, we can easily obtain the following result.

Lemma 3.4. 𝐵_𝑝,𝑞+𝐹+𝜕Φ_𝑝∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ is maximal monotone.

Lemma 3.5. Define the mapping 𝑆∶𝐿^𝑝(Γ)→𝐿^𝑝′(Γ) by 𝑆𝑢=𝛽_𝑥(𝑢(𝑥)), for any 𝑢∈𝐿^𝑝(Γ). Define the mapping 𝐾∶𝑊^1,𝑝(Ω)→𝐿^𝑝(Γ) by 𝐾𝑣=𝑣|_Γ for any 𝑣∈𝑊^1,𝑝(Ω). Then 𝐾^∗𝑆𝐾=𝜕Φ_𝑝, where Φ_𝑝 is the same as that in Lemma 3.2.

Proof. We will prove the result under the additional condition |𝛽_𝑥(𝑢)|≤𝑎|𝑢|^𝑝/𝑝′+𝑏(𝑥), where 𝑏(𝑥)∈𝐿^𝑝′(Γ) and 𝑎∈𝑅. Refer to the result of Brezis [15] for the general case.
It is obvious that 𝑆 is continuous. For all 𝑢(𝑥),𝑣(𝑥)∈𝐿^𝑝(Γ), since 𝛽_𝑥 is monotone, then (𝑢−𝑣,𝑆𝑢−𝑆𝑣)=∫_Γ(𝛽_𝑥(𝑢(𝑥))−𝛽_𝑥(𝑣(𝑥)))(𝑢(𝑥)−𝑣(𝑥))𝑑Γ(𝑥)≥0, which implies that 𝑆 is monotone. Thus, 𝑆∶𝐿^𝑝(Γ)→𝐿^𝑝′(Γ) is maximal monotone in view of Lemma 2.2.
Define Ψ∶𝐿^𝑝(Γ)→𝑅 by Ψ(𝑢)=∫_Γ𝜑_𝑥(𝑢(𝑥))𝑑Γ(𝑥), for all 𝑢(𝑥)∈𝐿^𝑝(Γ); then it is easy to see that Ψ is a proper, convex, and lower semicontinuous function on 𝐿^𝑝(Γ), which implies that 𝜕Ψ∶𝐿^𝑝(Γ)→𝐿^𝑝′(Γ) is maximal monotone. Since Ψ(𝑢)−Ψ(𝑣)=∫_Γ[𝜑_𝑥(𝑢(𝑥))−𝜑_𝑥(𝑣(𝑥))]𝑑Γ(𝑥)≥∫_Γ𝛽_𝑥(𝑣(𝑥))(𝑢(𝑥)−𝑣(𝑥))𝑑Γ(𝑥)=(𝑆𝑣,𝑢−𝑣), for all 𝑢(𝑥),𝑣(𝑥)∈𝐿^𝑝(Γ), then 𝐵𝑣∈𝜕Ψ(𝑣). So 𝑆=𝜕Ψ.
Now clearly, 𝐾^∗𝑆𝐾∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ is maximal monotone since both 𝐾,𝑆 are continuous. Finally, for any 𝑢,𝑣∈𝑊^1,𝑝(Ω), we have Φ_𝑝(𝑣)−Φ_𝑝(𝑢)()()==Ψ𝐾𝑣−Ψ𝐾𝑢_Γ𝜑_𝑥𝑣|_Γ(𝑥)−𝜑_𝑥𝑢|_Γ(𝑥)(𝑥)≥𝑑Γ_Γ𝛽_𝑥𝑢|_Γ(𝑥)𝑣|_Γ(𝑥)−𝑢|_Γ(𝑥)(𝑥)=()=𝐾𝑑Γ𝑆𝐾𝑢,𝐾𝑣−𝐾𝑢^∗𝑆𝐾𝑢,𝑣−𝑢. (3.11) Hence we get 𝐾^∗𝑆𝐾⊂𝜕Φ_𝑝 and so 𝐾^∗𝑆𝐾=𝜕Φ_𝑝.
This completes the proof.

Lemma 3.6. One has 𝑅(𝐵_𝑝,𝑞+𝐹+𝜕Φ_𝑝)=(𝑊^1,𝑝(Ω))^∗.

Proof. From Lemma 3.5, we know that for all 𝑢∈𝑊^1,𝑝(Ω), 𝑢,𝜕Φ_𝑝(𝑢)=_Γ𝛽_𝑥𝑢|_Γ(𝑥)𝑢|_Γ(𝑥)𝑑Γ(𝑥). (3.12) Since 0∈𝛽_𝑥(0), then for all 𝑢∈𝑊^1,𝑝(Ω),  (𝑢,𝜕Φ_𝑝(𝑢))≥0.
Since 𝐹 is monotone, then (𝑢,𝐹𝑢−𝐹0)≥0. Moreover, in view of (3.8), we have |(𝑢,𝐹0)|‖𝑢‖_1,𝑝,Ω≤∫_Ωℎ₁(𝑥)|𝑢(𝑥)|𝑑𝑥‖𝑢‖_1,𝑝,Ω≤ℎ₁(𝑥)_𝑝‖𝑢‖_𝑝′‖𝑢‖_1,𝑝,Ω≤𝑘ℎ₁(𝑥)_𝑝<+∞. (3.13) Then by using Lemma 3.1, we have 𝑢,𝐵_𝑝,𝑞𝑢+𝐹𝑢+𝜕Φ_𝑝(𝑢)‖𝑢‖_1,𝑝,Ω≥𝑢,𝐵_𝑝,𝑞𝑢‖𝑢‖_1,𝑝,Ω+(𝑢,𝐹0)‖𝑢‖_1,𝑝,Ω⟶+∞, (3.14) as ‖𝑢‖_1,𝑝,Ω→+∞, which implies that 𝐵_𝑝,𝑞+𝐹+𝜕Φ_𝑝 is coercive. Then Lemmas 3.4 and 2.5 ensure that the result is true.
This completes the proof.

Theorem 3.7. For 𝑓∈𝐿^𝑝′(Ω), nonlinear boundary value problem (1.4) has a unique solution 𝑢∈𝑊^1,𝑝(Ω).

Proof. From Lemma 3.6, we know that for 𝑓∈𝐿^𝑝′(Ω), there exists 𝑢∈𝑊^1,𝑝(Ω) such that 𝑓=𝐵_𝑝,𝑞𝑢+𝜕Φ_𝑝(𝑢)+𝐹𝑢. (3.15)
Now, we will show that this 𝑢 is unique.
Otherwise, there exists 𝑣∈𝑊^1,𝑝(Ω) satisfying (3.15). Then notice the facts that 𝐵_𝑝,𝑞,𝜕Φ_𝑝 and 𝐹 are all monotone, we have 0≤𝑢−𝑣,𝐵_𝑝,𝑞𝑢−𝐵_𝑝,𝑞𝑣=𝑢−𝑣,𝑓−𝐹𝑢−𝜕Φ_𝑝(𝑢)−𝑓+𝐹𝑣+𝜕Φ_𝑝(𝑣)=(𝑢−𝑣,𝐹𝑣−𝐹𝑢)+𝑢−𝑣,𝜕Φ_𝑝(𝑣)−𝜕Φ_𝑝(𝑢)≤0. (3.16) Since 𝐵_𝑝,𝑞 is strictly monotone, then 𝑢=𝑣.
Next, we will show that this 𝑢 is the solution of (1.4).
Since Φ_𝑝(𝑢+𝜑)=Φ_𝑝(𝑢) for any 𝑢∈𝑊^1,𝑝(Ω) and 𝜑∈𝐶^∞₀(Ω), where 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁≥1, we have (𝜑,𝜕Φ(𝑢))=0. Since 𝑓−𝐹𝑢=𝐵_𝑝,𝑞𝑢+𝜕Φ_𝑝(𝑢), then for all 𝜑∈𝐶^∞₀(Ω), _Ω((𝑥)(𝑥=𝑓−𝑔𝑥,𝑢,∇𝑢))𝜑𝑑𝑥_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝜑𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝜑(𝑥)𝑑𝑥=−_Ωdiv𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝜑𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝜑𝑑𝑥, (3.17) which implies that the result −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+𝜀|𝑢|^𝑞−2𝑢+𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))=𝑓(𝑥)a.e.𝑥∈Ω (3.18) is true.
From (3.18), we know that 𝑓(𝑥)=−div[(𝐶(𝑥)+|∇𝑢|²)^(𝑝−2)/2∇𝑢]+𝜀|𝑢|^𝑞−2𝑢+𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))∈𝐿^𝑝′(Ω). By using Green's formula, we have that for any 𝑣∈𝑊^1,𝑝(Ω), 𝜗,𝐶_Γ(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝑣|_Γ(𝑥)=𝑑Γ_Ωdiv𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝑣𝑑𝑥+_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝑣𝑑𝑥. (3.19)
Then −⟨𝜗,(𝐶(𝑥)+|∇𝑢|²)^(𝑝−2)/2∇𝑢⟩∈𝑊^{−1/𝑝,𝑝′}(Γ)=(𝑊^1/𝑝,𝑝(Γ))^∗, where 𝑊^1/𝑝,𝑝(Γ) is the space of the traces of 𝑊^1,𝑝(Ω).
Combining with the results of Lemma 3.5, (3.18), and (3.19), we have 𝜗,𝐶_Γ(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝑣|_Γ(𝑥)=|𝑢(𝑥𝑑Γ𝑣,𝜀)|^𝑞−2𝑢(𝑥)+𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))−𝑓+𝑣,𝐵_𝑝𝑢−𝜀|𝑢(𝑥)|^𝑞−2𝑢(𝑥)=𝑣,𝐵_𝑝𝑢+𝐹𝑢−𝑓=𝑣,−𝜕Φ_𝑝(𝑢)=𝑣,−𝐾^∗𝑆𝐾𝑢=−(𝐾𝑣,𝑆𝐾𝑢)=−_Γ𝛽_𝑥𝑢|_Γ(𝑥)𝑣|_Γ(𝑥)𝑑Γ(𝑥). (3.20)
Then −𝜗,𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (3.21)
From (3.19) and (3.21) we know that 𝑢 is the solution of (1.4).
This completes the proof.

3.3. Iterative Construction of the Solution of (1.4)

Lemma 3.8 (Wei and Zhou [16]). Suppose that 𝑋 is a real smooth and uniformly convex Banach space and 𝐴∶𝑋→2^𝑋∗ is a maximal monotone operator with 𝐴⁻¹(0)≠∅. Let {𝑥_𝑛} be a sequence generated by the following iterative scheme: 𝑥₁∈𝑋,𝑟₁𝑥>0,_𝑛+1=𝐽⁻¹𝛽_𝑛𝐽𝑥₁+1−𝛽_𝑛𝐽𝐽+𝑟_𝑛𝐴⁻¹𝐽𝑥_𝑛,∀𝑛≥1. (3.22)
If {𝑟_𝑛}⊂(0,+∞) with lim_𝑛→∞𝑟_𝑛=+∞,  {𝛽_𝑛}⊂[0,1] with ∑^∞_𝑛=1𝛽_𝑛=+∞, and 𝛽_𝑛→0 as 𝑛→∞, then {𝑥_𝑛} converges strongly to Π_𝐴⁻¹₍₀₎(𝑥₁), where Π_𝐴⁻¹₍₀₎ is the generalized projection operator from 𝑋 onto 𝐴⁻¹(0).

Definition 3.9. Define the mapping 𝐶_𝑝,𝑞∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ by 𝑣,𝐶_𝑝,𝑞𝑢=_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝑣𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)−𝑑𝑥_Ω𝑓(𝑥)𝑣(𝑥)𝑑𝑥+_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝑣(𝑥)𝑑𝑥+𝑣,𝜕Φ_𝑝(𝑢) (3.23) for any 𝑢,𝑣∈𝑊^1,𝑝(Ω), where 𝑓∈𝐿^𝑝′(Ω) is the same as that in (1.4).

Similarly to Lemma 3.4, we know that 𝐶_𝑝,𝑞 is also a maximal monotone operator. Moreover, we can easily get the following result.

Lemma 3.10. 𝑢∈𝑊^1,𝑝(Ω) is the solution of (1.4) if and only if 𝑢∈𝑊^1,𝑝(Ω) is the zero point of 𝐶_𝑝,𝑞.

Proof. Let 𝑢(𝑥) be the solution of (1.4), then for all 𝑣∈𝑊^1,𝑝(Ω), by using Green’s formula and Lemma 3.5, we have 𝑣,𝐶_𝑝,𝑞𝑢=_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝑣𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)−𝑑𝑥_Ω𝑓(𝑥)𝑣(𝑥)𝑑𝑥+_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝑣(𝑥)𝑑𝑥+𝑣,𝜕Φ_𝑝(𝑢)=−_Ωdiv𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}+∇𝑢𝑣𝑑𝑥_Γ𝜗,𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝑣|_Γ𝑑Γ(𝑥)+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)𝑑𝑥−_Ω𝑓(𝑥)𝑣(𝑥)𝑑𝑥+_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝑣(𝑥)𝑑𝑥+𝑣,𝐾^∗=𝑆𝐾𝑢_Γ𝜗,𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢𝑣|_Γ𝑑Γ(𝑥)+𝑣,𝐾^∗𝑆𝐾𝑢=−_Γ𝛽_𝑥𝑢|_Γ𝑣|_Γ𝑑Γ(𝑥)+_Γ𝛽_𝑥𝑢|_Γ𝑣|_Γ𝑑Γ(𝑥)=0. (3.24)
Thus 𝑢∈𝐶⁻¹_𝑝,𝑞(0).
If 𝑢∈𝐶⁻¹_𝑝,𝑞(0), then for all 𝜑∈𝐶^∞₀(Ω), 0=_Ω𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢,∇𝜑𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝜑(𝑥)−𝑑𝑥_Ω𝑓𝜑𝑑𝑥+_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝜑𝑑𝑥, (3.25)
which implies the result −div[(𝐶(𝑥)+|∇𝑢|²)^(𝑝−2)/2∇𝑢]+𝜀|𝑢(𝑥)|^𝑞−2𝑢(𝑥)+𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))=𝑓(𝑥), a.e.𝑥∈Ω is true.
Copying the last part of Theorem 3.7, we can obtain (3.21), which implies that 𝑢 is the solution of (1.4).
This completes the proof.

Remark 3.11. From Lemma 3.10 we can see that 𝐶⁻¹_𝑝,𝑞(0)≠∅. It is a good example to show that the assumption that 𝐴⁻¹(0)≠∅ in Lemma 3.8 is valid.

Lemma 3.12 (Takahashi [17]). Let 𝑋 be a Banach space and 𝐽 a duality mapping defined on 𝑋. If 𝐽 is single valued, then 𝑋 is smooth.

Based on the facts of Lemmas 3.8, 3.10, and 3.12, we can construct an iterative sequence to approximate strongly to the solution of (1.4).

Theorem 3.13. Let {𝑢_𝑛(𝑥)} be a sequence generated by the following iterative scheme: 𝑢₁(𝑥)∈𝑊^1,𝑝(Ω),𝑟₁𝑢>0,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1(𝑥)=𝐽⁻¹𝛽_𝑛𝐽𝑢₁(𝑥)+1−𝛽_𝑛𝐽𝐽+𝑟_𝑛𝐶_𝑝,𝑞⁻¹𝐽𝑢_𝑛(𝑥),∀𝑛≥1. (3.26)
If {𝑟_𝑛}⊂(0,+∞) with lim_𝑛→∞𝑟_𝑛=+∞,  {𝛽_𝑛}⊂[0,1] with ∑^∞_𝑛=1𝛽_𝑛=+∞, and 𝛽_𝑛→0 as 𝑛→∞, then {𝑢_𝑛(𝑥)} converges strongly to Π_𝐶⁻¹_𝑝,𝑞(0)(𝑢₁(𝑥)).

Remark 3.14. Theorem 3.13 not only tells us that the sequence {𝑢_𝑛(𝑥)} generated by (3.26) converges strongly to the solution of (1.4), but also tells us that the unique solution of (1.4) is the generalized projection of the initial function 𝑢₁(𝑥) onto 𝐶⁻¹_𝑝,𝑞(0).

Remark 3.15. Compared to the work done in [8], we may find that different techniques are employed to show the existence and uniqueness of the solution of the desired equation because the nonlinear item 𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥)) is involved.

Remark 3.16. We can get the following special cases in our paper.

Corollary 3.17. For 𝑓∈𝐿²(Ω), the following equation has a unique solution in 𝐻¹(Ω): −Δ𝑢+𝜀|𝑢|^𝑞−2((𝑥)(𝑥(𝑥)−𝑢+𝑔𝑥,𝑢,∇𝑢))=𝑓,𝑎.𝑒.𝑖𝑛Ω,𝜕𝑢𝜕𝜗∈𝛽_𝑥(𝑢(𝑥)),𝑎.𝑒.𝑜𝑛Γ. (3.27)

Corollary 3.18. Let {𝑢_𝑛(𝑥)} be a sequence generated by 𝑢₁(𝑥)∈𝐻¹(Ω),𝑟₁𝑢>0,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1(𝑥)=𝛽_𝑛𝑢₁(𝑥)+1−𝛽_𝑛𝐼+𝑟_𝑛𝐶_2,𝑞⁻¹𝑢_𝑛(𝑥),∀𝑛≥1, (3.28) where 𝐶_2,𝑞 is a special case of 𝐶_𝑝,𝑞 defined in Definition 3.9 if 𝑝≡2.
If {𝑟_𝑛}⊂(0,+∞) with lim_𝑛→∞𝑟_𝑛=+∞,  {𝛽_𝑛}⊂[0,1] with ∑^∞_𝑛=1𝛽_𝑛=+∞, and 𝛽_𝑛→0 as 𝑛→∞, then {𝑢_𝑛(𝑥)} converges strongly to 𝑃_𝐶⁻¹_2,𝑞(0)(𝑢₁(𝑥)), where 𝑃_𝐶⁻¹_2,𝑞(0)(𝑢₁(𝑥)) denotes the metric projection from 𝐻¹(Ω) onto 𝐶⁻¹_2,𝑞(0). And,  𝑃_𝐶⁻¹_2,𝑞(0)(𝑢₁(𝑥)) is the unique solution of (3.27).

Corollary 3.19. For 𝑓∈𝐿^𝑝′(Ω), the following equation has a unique solution in 𝑊^1,𝑝(Ω):−Δ_𝑝𝑢+𝜀|𝑢|^𝑞−2((𝑥)(𝑥(𝑥)−||𝑢+𝑔𝑥,𝑢,∇𝑢))=𝑓,𝑎.𝑒.𝑖𝑛Ω,𝜗,∇𝑢^𝑝−2∇𝑢∈𝛽_𝑥(𝑢(𝑥)),𝑎.𝑒.𝑜𝑛Γ. (3.29)

Corollary 3.20. Let {𝑢_𝑛(𝑥)} be a sequence generated by (3.26), where 𝐶_𝑝,𝑞∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ is defined by𝑣,𝐶_𝑝,𝑞𝑢=_Ω|∇𝑢|^𝑝−2∇𝑢,∇𝑣𝑑𝑥+𝜀_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)−𝑑𝑥_Ω𝑓(𝑥)𝑣(𝑥)𝑑𝑥+_Ω𝑔(𝑥,𝑢(𝑥),∇𝑢(𝑥))𝑣(𝑥)𝑑𝑥+𝑣,𝜕Φ_𝑝(𝑢), (3.30) for 𝑢(𝑥),𝑣(𝑥)∈𝑊^1,𝑝(Ω).

Then under the conditions of Theorem 3.13, we know that {𝑢_𝑛(𝑥)} converges strongly to both the unique solution of (3.29) and the zero point of 𝐶_𝑝,𝑞.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11071053), the Natural Science Foundation of Hebei Province (Grant no. A2010001482), the Key Project of Science and Research of Hebei Education Department (Grant no. ZH2012080), and the Youth Project of Science and Research of Hebei Education Department (Grant no. Q2012054).