Study on the Existence and Uniqueness of Solution of Generalized Capillarity Problem

Abstract

By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence and uniqueness of the solution in 𝐿^𝑝(Ω) of the generalized Capillarity equation with nonlinear Neumann boundary value conditions, where 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁(≥1) denotes the dimension of 𝑅^𝑁, is studied. The equation discussed in this paper and the methods here are a continuation of and a complement to the previous corresponding results. To obtain the results, some new techniques are used in this paper.

1. Introduction and Preliminary

Since the 𝑝-Laplacian operator −Δ_𝑝 with 𝑝≠2 arises from a variety of physical phenomena, such as nonNewtonian fluids, reaction-diffusion problems, and petroleum extraction, it becomes a very popular topic in mathematical fields.

We began our study on this topic in 1995. We used a perturbation result of ranges for m-accretive mappings in Calvert and Gupta [1] to obtain a sufficient condition in Wei and He [2] so that the following zero boundary value problem, −Δ_𝑝((𝑥(𝑥)−𝑢+𝑔𝑥,𝑢))=𝑓,a.e.onΩ,𝜕𝑢𝜕𝑛=0,a.e.onΓ, (1.1) has solutions in 𝐿^𝑝(Ω), where 2≤𝑝<+∞. Later on, a series work of ours has been done from different angles on this kind of equations, cf. [3–7], and so forth.

Especially, in 2008, as a summary of the work done in [2–6], we use some new techniques to work for the following problem with so-called generalized 𝑝-Laplacian operator: −div𝐶(𝑥)+|∇𝑢|^{2(𝑝−2)/2}∇𝑢+𝜀|𝑢|^𝑞−2((𝑥(𝑥)−𝐶(𝑥)+||𝑢+𝑔𝑥,𝑢))=𝑓,a.e.inΩ,𝜗,∇𝑢^{2(𝑝−2)/2}∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ, (1.2) where 0≤𝐶(𝑥)∈𝐿^𝑝(Ω), 𝜀 is a nonnegative constant and 𝜗 denotes the exterior normal derivative of Γ. We showed in Wei and Agarwal [7] that (1.2) has solutions in 𝐿^𝑠(Ω) under some conditions, where 2𝑁/(𝑁+1)<𝑝≤𝑠<+∞,  1≤𝑞<+∞ if 𝑝≥𝑁, and 1≤𝑞≤𝑁𝑝/(𝑁−𝑝) if 𝑝<𝑁, for 𝑁≥1.

Capillarity equation is another important equation appeared in the capillarity phenomenon and we notice that in Chen and Luo [8], the authors studied the eigenvalue problem for the following generalized Capillarity equations: ||−div1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢=𝜆|𝑢|^𝑞−2𝑢+|𝑢|^𝑟−2𝑢,inΩ,𝑢=0,a.e.on𝜕Ω. (1.3)

Their work inspired us and one idea came to our mind. Can we borrow the main ideas dealing with the nonlinear elliptic boundary value problems with the generalized 𝑝-Laplacian operator to study the nonlinear generalized Capillarity equation with Neumann boundary conditions?

We will answer the question in this paper. By using the perturbation results of ranges for m-accretive mappings in Calvert and Gupta [1] again, we will study the following one: ||−div1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢+𝜆|𝑢|^𝑞−2𝑢+|𝑢|^𝑟−2𝑢((𝑥(𝑥)−||+𝑔𝑥,𝑢))=𝑓,a.e.inΩ,𝜗,1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (1.4)

More details on (1.4) will be given in Section 2. Our methods and techniques are different from those in Chen and Luo [8].

Now, we list some basic knowledge we need in sequel.

Let 𝑋 be a real Banach space with a strictly convex dual space 𝑋^∗. We use “→” and “𝑤−lim” to denote strong and weak convergence, respectively. For any subset 𝐺 of 𝑋, we denote by int𝐺 its interior and 𝐺 its closure, respectively. Let “𝑋↪↪𝑌” denote that space X is embedded compactly in space 𝑌 and “𝑋↪𝑌” denote that space 𝑋 is embedded continuously in space 𝑌. A mapping 𝑇∶𝐷(𝑇)=𝑋→𝑋^∗ is said to be hemicontinuous on 𝑋 if 𝑤−lim_𝑡→0𝑇(𝑥+𝑡𝑦)=𝑇𝑥, for any 𝑥,𝑦∈𝑋. Let 𝐽 denote the duality mapping from 𝑋 into 2^𝑋∗ defined by 𝐽(𝑥)=𝑓∈𝑋^∗∶𝑥,𝑓=‖𝑥‖⋅𝑓,𝑓=‖𝑥‖,𝑥∈𝑋, (1.5) where (⋅,⋅) denotes the generalized duality pairing between 𝑋 and 𝑋^∗. It is wellknown that 𝐽 is a single-valued mapping since 𝑋^∗ is strictly convex.

Let 𝐴∶𝑋→2^𝑋 be a given multivalued mapping. We say that 𝐴 is boundedly-inversely compact if for any pair of bounded subsets 𝐺 and 𝐺^′ of 𝑋, the subset 𝐺⋂𝐴⁻¹(𝐺^′) is relatively compact in 𝑋. The mapping 𝐴∶𝑋→2^𝑋 is said to be accretive if (𝑣₁−𝑣₂,𝐽(𝑢₁−𝑢₂))≥0, for any 𝑢_𝑖∈𝐷(𝐴) and 𝑣_𝑖∈𝐴𝑢_𝑖,  𝑖=1,2. The accretive mapping 𝐴 is said to be 𝑚-accretive if 𝑅(𝐼+𝜇𝐴)=𝑋, for some 𝜇>0.

Let 𝐵∶𝑋→2^𝑋∗ be a given multi-valued mapping. The graph of 𝐵,  𝐺(𝐵), is defined by 𝐺(𝐵)={[𝑢,𝑤]∣𝑢∈𝐷(𝐵),𝑤∈𝐵𝑢}. Then 𝐵∶𝑋→2^𝑋∗ is said to be monotone if 𝐺(𝐵) is a monotone subset of 𝑋×𝑋^∗ in the sense that 𝑢₁−𝑢₂,𝑤₁−𝑤₂≥0, (1.6) for any [𝑢_𝑖,𝑤_𝑖]∈𝐺(𝐵),  𝑖=1,2. The monotone operator 𝐵 is said to be maximal monotone if 𝐺(𝐵) is maximal among all monotone subsets of 𝑋×𝑋^∗ in the sense of inclusion. The mapping 𝐵 is said to be strictly monotone if the equality in (1.6) implies that 𝑢₁=𝑢₂. The mapping 𝐵 is said to be coercive if lim_𝑛→+∞((𝑥_𝑛,𝑥^∗_𝑛)/‖𝑥_𝑛‖)=∞ for all [𝑥_𝑛,𝑥^∗_𝑛]∈𝐺(𝐵) such that lim_𝑛→+∞‖𝑥_𝑛‖=+∞.

Definition 1.1. The duality mapping 𝐽∶𝑋→2^𝑋∗ is said to be satisfying Condition (𝐼) if there exists a function 𝜂∶𝑋→[0,+∞) such that ‖𝐽𝑢−𝐽𝑣‖≤𝜂(𝑢−𝑣),for∀𝑢,𝑣∈𝑋. (𝐼)

Definition 1.2. Let 𝐴∶𝑋→2^𝑋 be an accretive mapping and 𝐽∶𝑋→𝑋^∗ be a duality mapping. We say that 𝐴 satisfies Condition (∗) if, for any 𝑓∈𝑅(𝐴) and 𝑎∈𝐷(𝐴), there exists a constant 𝐶(𝑎,𝑓) such that 𝑣−𝑓,𝐽(𝑢−𝑎)≥𝐶𝑎,𝑓,forany𝑢∈𝐷(𝐴),𝑣∈𝐴𝑢. (∗)

Lemma 1.3 (Li and Guo [9]). Let Ω be a bounded conical domain in 𝑅^𝑁. Then we have the following results.(a)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).(b)If 𝑚𝑝>𝑁, then 𝑊^𝑚,𝑝(Ω)↪↪𝐶_𝐵(Ω); if 0<𝑚𝑝≤𝑁 and 𝑞₀=𝑁𝑝/(𝑁−𝑚𝑝), then 𝑊^𝑚,𝑝(Ω)↪↪𝐿^𝑞(Ω),  1≤𝑞<𝑞₀.

Lemma 1.4 (Pascali and Sburlan [10]). If 𝐵∶𝑋→2^𝑋∗ is an everywhere defined, monotone and hemicontinuous operator, then 𝐵 is maximal monotone. If 𝐵∶𝑋→2^𝑋∗ is maximal monotone and coercive, then 𝑅(𝐵)=𝑋^∗.

Lemma 1.5 (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 1.6 (Pascali and Sburlan [10]). If 𝐴 and 𝐵 are two maximal monotone operators in 𝑋 such that (int𝐷(𝐴))⋂𝐷(𝐵) ≠∅, then 𝐴+𝐵 is maximal monotone.

Proposition 1.7 (Calvert and Gupta [1]). Let 𝑋=𝐿^𝑝(Ω) and Ω be a bounded domain in 𝑅^𝑁. For 2≤𝑝<+∞, the duality mapping 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω) defined by 𝐽_𝑝𝑢=|𝑢|^𝑝−1sgn𝑢‖𝑢‖_𝑝^2−𝑝, for 𝑢∈𝐿^𝑝(Ω), satisfies Condition (𝐼); for 2𝑁/(𝑁+1)<𝑝≤2 and 𝑁≥1, the duality mapping 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω) defined by 𝐽_𝑝𝑢=|𝑢|^𝑝−1𝑠𝑔𝑛𝑢, for 𝑢∈𝐿^𝑝(Ω), satisfies Condition (𝐼), where (1/𝑝)+(1/𝑝^′)=1.

Lemma 1.8 (see Calvert and Gupta [1]). Let Ω be a bounded domain in 𝑅^𝑁 and 𝑔∶Ω×𝑅→𝑅 be a function satisfying Caratheodory's conditions such that (i)𝑔(𝑥,⋅) is monotonically increasing on 𝑅; (ii)the mapping 𝑢∈𝐿^𝑝(Ω)→𝑔(𝑥,𝑢(𝑥))∈𝐿^𝑝(Ω) is well defined, where 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁≥1.
Let 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω),  (1/𝑝)+(1/𝑝^′)=1 be the duality mapping defined by 𝐽_𝑝|𝑢|𝑢=^𝑝−1sgn𝑢,if2𝑁|𝑢|𝑁+1<𝑝≤2,_𝑝−1sgn𝑢‖𝑢‖_𝑝^2−𝑝,if2≤𝑝<+∞, (1.7) for 𝑢∈𝐿^𝑝(Ω). Then the mapping 𝐵∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) defined by (𝐵𝑢)(𝑥)=𝑔(𝑥,𝑢(𝑥)), for any 𝑥∈Ω satisfies Condition (∗).

Theorem 1.9 (Calvert and Gupta [1]). Let 𝑋 be a real Banach space with a strictly convex dual 𝑋^∗. Let 𝐽∶𝑋→𝑋^∗ be a duality mapping on 𝑋 satisfying Condition (I). Let 𝐴,𝐵₁∶𝑋→2^𝑋 be accretive mappings such that (i)either both 𝐴,𝐵₁ satisfy Condition (∗) or 𝐷(𝐴)⊂𝐷(𝐵₁) and 𝐵₁ satisfies Condition (∗),(ii)𝐴+𝐵₁ is m-accretive and boundedly inversely compact.
If 𝐵₂∶𝑋→𝑋 be a bounded continuous mapping such that, for any 𝑦∈𝑋, there is a constant 𝐶(𝑦) satisfying (𝐵₂(𝑢+𝑦),𝐽𝑢)≥−𝐶(𝑦) for any 𝑢∈𝑋. Then. (a)[𝑅(𝐴)+𝑅(𝐵₁)]⊂𝑅(𝐴+𝐵₁+𝐵₂).(b)int[𝑅(𝐴)+𝑅(𝐵₁)]⊂int𝑅(𝐴+𝐵₁+𝐵₂).

2. The Main Results

2.1. Notations and Assumptions of (1.4)

In the next of this paper, we assume 2𝑁/(𝑁+1)<𝑝<+∞,  1≤𝑞,  𝑟<+∞ if 𝑝≥𝑁, and 1≤𝑞,  𝑟≤𝑁𝑝/(𝑁−𝑝) if 𝑝<𝑁, where 𝑁≥1. We use ‖⋅‖_𝑝,  ‖⋅‖_𝑞,  ‖⋅‖_𝑟, and ‖⋅‖_1,𝑝,Ω to denote the norms in 𝐿^𝑝(Ω),  𝐿^𝑞(Ω),  𝐿^𝑟(Ω) and 𝑊^1,𝑝(Ω). Let (1/𝑝)+(1/𝑝^′)=1,  (1/𝑞)+(1/𝑞^′)=1, and (1/𝑟)+(1/𝑟^′)=1.

In (1.4), Ω is a bounded conical domain of a Euclidean space 𝑅^𝑁 with its boundary Γ∈𝐶¹, (c.f. [4]). We suppose that the Green's Formula is available. Let |⋅| denote the Euclidean norm in 𝑅^𝑁,  ⟨⋅,⋅⟩ the Euclidean inner-product and 𝜗 the exterior normal derivative of Γ.  𝜆 is a nonnegative constant.

Let 𝜑∶Γ×𝑅→𝑅 be a given function such that, for each 𝑥∈Γ,(i)𝜑_𝑥=𝜑(𝑥,⋅)∶𝑅→𝑅 is a proper, convex, lower-semicontinuous function with 𝜑_𝑥(0)=0.(ii)𝛽_𝑥=𝜕𝜑_𝑥 (: subdifferential of 𝜑_𝑥) is maximal monotone mapping on 𝑅 with 0∈𝛽_𝑥(0) and for each 𝑡∈𝑅, the function 𝑥∈Γ→(𝐼+𝜇𝛽_𝑥)⁻¹(𝑡)∈𝑅 is measurable for 𝜇>0.

Let 𝑔∶Ω×𝑅→𝑅 be a given function satisfying Caratheodory's conditions such that for 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁≥1, the mapping 𝑢∈𝐿^𝑝(Ω)→𝑔(𝑥,𝑢(𝑥))∈𝐿^𝑝(Ω) is defined. Further, suppose that there is a function 𝑇(𝑥)∈𝐿^𝑝(Ω) such that 𝑔(𝑥,𝑡)𝑡≥0, for |𝑡|≥𝑇(𝑥), 𝑥∈Ω.

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

Definition 2.1 (Calvert and Gupta [1]). Define 𝑔₊(𝑥)=liminf_𝑡→+∞𝑔(𝑥,𝑡) and 𝑔₋(𝑥)=limsup_{𝑡→−∞}𝑔(𝑥,𝑡).
Further, define a function 𝑔₁∶Ω×𝑅→𝑅 by 𝑔₁(𝑥,𝑡)=inf_𝑠≥𝑡𝑔()∧((𝑥(𝑥),[(𝑥)(𝑥,𝑥,𝑠𝑡−𝑇)),∀𝑡≥𝑇0,∀𝑡∈−𝑇,𝑇)]sup_𝑠≤𝑡𝑔(𝑥,𝑠)∨(𝑡+𝑇(𝑥)),∀𝑡≤−𝑇(𝑥). (2.1) Then for all 𝑥∈Ω,  𝑔₁(𝑥,𝑡) is increasing in 𝑡 and lim_𝑡→±∞𝑔₁(𝑥,𝑡)=𝑔_±(𝑥). Moreover, 𝑔₁∶Ω×𝑅→𝑅 satisfies Caratheodory's conditions and the functions 𝑔_±(𝑥) are measurable on Ω. And, if 𝑔₂(𝑥,𝑡)=𝑔(𝑥,𝑡)−𝑔₁(𝑥,𝑡) then 𝑔₂(𝑥,𝑡)𝑡≥0, for |𝑡|≥𝑇(𝑥), 𝑥∈Ω.

Proposition 2.2 (see Calvert and Gupta [1]). For 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁≥1, define the mapping 𝐵₁∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) by (𝐵₁𝑢)(𝑥)=𝑔₁(𝑥,𝑢(𝑥)), for all 𝑢∈𝐿^𝑝(Ω) and 𝑥∈Ω, then 𝐵₁ is a bounded, continuous, and m-accretive mapping.

Moreover, Lemma 1.8 implies that 𝐵₁ satisfies Condition (∗).

Define 𝐵₂∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) by (𝐵₂𝑢)(𝑥)=𝑔₂(𝑥,𝑢(𝑥)), where 𝑔₂(𝑥,𝑡)=𝑔(𝑥,𝑡)−𝑔₁(𝑥,𝑡), then 𝐵₂ satisfies the inequality: 𝐵₂𝑢+𝑦,𝐽_𝑝𝑢≥−𝐶𝑦, (2.2) for any 𝑢,𝑦∈𝐿^𝑝(Ω), where 𝐶(𝑦) is a constant depending on 𝑦 and 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω) denotes the duality mapping, where (1/𝑝)+(1/𝑝^′)=1.

Lemma 2.3 (Wei and Agarwal [7]). The mapping Φ_𝑝∶𝑊^1,𝑝(Ω)→𝑅 defined by Φ_𝑝(𝑢)=_Γ𝜑_𝑥𝑢|_Γ(𝑥)𝑑Γ(𝑥), (2.3) for any 𝑢∈𝑊^1,𝑝(Ω), is a proper, convex, and lower-semicontinuous mapping on 𝑊^1,𝑝(Ω).

Moreover, Lemma 1.5 implies that 𝜕Φ_𝑝, the subdifferential of Φ_𝑝, is maximal monotone.

Lemma 2.4 (Wei and He [2]). Let 𝑋₀ denote the closed subspace of all constant functions in 𝑊^1,𝑝(Ω). Let 𝑋 be the quotient space 𝑊^1,𝑝(Ω)/𝑋₀. For 𝑢∈𝑊^1,𝑝(Ω), define the mapping 𝑃∶𝑊^1,𝑝(Ω)→𝑋₀ by 𝑃𝑢=(1/meas(Ω))∫_Ω𝑢𝑑𝑥. Then, there is a constant 𝐶>0, such that for all 𝑢∈𝑊^1,𝑝(Ω), ‖𝑢−𝑃𝑢‖_𝑝≤𝐶‖∇𝑢‖_{(𝐿𝑝(Ω))}^𝑁. (2.4)

Lemma 2.5. Define the mapping 𝐵_{𝑝,𝑞,𝑟}∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ by 𝑣,𝐵_{𝑝,𝑞,𝑟}𝑢=_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢,∇𝑣𝑑𝑥+𝜆_Ω|𝑢(𝑥)|^𝑞−2𝑢(𝑥)𝑣(𝑥)𝑑𝑥+𝜆_Ω|𝑢(𝑥)|^𝑟−2𝑢(𝑥)𝑣(𝑥)𝑑𝑥, (2.5) for any 𝑢,𝑣∈𝑊^1,𝑝(Ω). Then 𝐵_{𝑝,𝑞,𝑟} is everywhere defined, strictly monotone, hemicontinuous, and coercive.

Proof. Step 1. 𝐵_{𝑝,𝑞,𝑟} is everywhere defined.
From Lemma 1.3, we know that 𝑊^1,𝑝(Ω)↪𝐶_𝐵(Ω), when 𝑝>𝑁. And, 𝑊^1,𝑝(Ω)↪𝐿^𝑞(Ω),  𝑊^1,𝑝(Ω)↪𝐿^𝑟(Ω), when 𝑝≤𝑁. Thus, for all  𝑣∈𝑊^1,𝑝(Ω),  ‖𝑣‖_𝑞≤𝑘₁‖𝑣‖_1,𝑝,Ω,  ‖𝑣‖_𝑟≤𝑘₂‖𝑣‖_1,𝑝,Ω, where 𝑘₁,𝑘₂ are positive constants.
For 𝑢,𝑣∈𝑊^1,𝑝(Ω), we have 𝑣,𝐵_{𝑝,𝑞,𝑟}𝑢≤2_Ω|∇𝑢|^𝑝−1|∇𝑣|𝑑𝑥+𝜆_Ω|𝑢|^𝑞−1|𝑣|𝑑𝑥+𝜆_Ω|𝑢|^𝑟−1|𝑣|𝑑𝑥≤2‖|∇𝑢|‖^𝑝/𝑝′_𝑝‖|∇𝑣|‖_𝑝+𝜆‖𝑣‖_𝑞‖𝑢‖^𝑞/𝑞′_𝑞+𝜆‖𝑣‖_𝑟‖𝑢‖^𝑟/𝑟′_𝑟≤2‖𝑢‖^𝑝/𝑝′_1,𝑝,Ω‖𝑣‖_1,𝑝,Ω+𝑘^′₁𝜆‖𝑣‖_1,𝑝,Ω‖𝑢‖^𝑞/𝑞′_1,𝑝,Ω+𝑘^′₂𝜆‖𝑣‖_1,𝑝,Ω‖𝑢‖^𝑟/𝑟′_1,𝑝,Ω, (2.6) where 𝑘^′₁ and 𝑘^′₂ are positive constants. Thus 𝐵_{𝑝,𝑞,𝑟} is everywhere defined.
Step 2. 𝐵_{𝑝,𝑞,𝑟} is strictly monotone.
For 𝑢,𝑣∈𝑊^1,𝑝(Ω), we have 𝑢−𝑣,𝐵_{𝑝,𝑞,𝑟}𝑢−𝐵_{𝑝,𝑞,𝑟}𝑣=_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2||∇𝑢−1+∇𝑣^𝑝1+|∇𝑣|^2𝑝|∇𝑣|^𝑝−2∇𝑣,∇𝑢−∇𝑣𝑑𝑥+𝜆_Ω|𝑢|^𝑞−2𝑢−|𝑣|^𝑞−2𝑣(𝑢−𝑣)𝑑𝑥+𝜆_Ω|𝑢|^𝑟−2𝑢−|𝑣|^𝑟−2𝑣()=𝑢−𝑣𝑑𝑥_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2−||∇𝑢∇𝑣1+∇𝑣^𝑝1+|∇𝑣|^2𝑝|∇𝑣|^𝑝−2||∇𝑢∇𝑣+1+∇𝑣^𝑝1+|∇𝑣|^2𝑝|∇𝑣|^𝑝𝑑𝑥+𝜆_Ω|𝑢|^𝑞−2𝑢−|𝑣|^𝑞−2𝑣(𝑢−𝑣)𝑑𝑥+𝜆_Ω|𝑢|^𝑟−2𝑢−|𝑣|^𝑟−2𝑣()≥𝑢−𝑣𝑑𝑥_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−1−||1+∇𝑣^𝑝1+|∇𝑣|^2𝑝|∇𝑣|^𝑝−1(|∇𝑢|−|∇𝑣|)𝑑𝑥+𝜆_Ω|𝑢|^𝑞−1−|𝑣|^𝑞−1(|𝑢|−|𝑣|)𝑑𝑥+𝜆_Ω|𝑢|^𝑟−1−|𝑣|^𝑟−1(|𝑢|−|𝑣|)𝑑𝑥. (2.7)
If, we let ℎ(𝑡)=(1+(𝑡/√1+𝑡²))𝑡^{(𝑝−1)/𝑝}, for 𝑡≥0. Then we know that ℎ^′(𝑡)=𝑡^{(𝑝−1)/𝑝}1+𝑡^23/2+𝑡^−(1/𝑝)𝑡1+√1+𝑡²𝑝−1𝑝≥0, (2.8) since 𝑡≥0. And, ℎ^′(𝑡)=0 if and only if 𝑡=0. Then ℎ(𝑡) is strictly monotone. Thus we can easily know that 𝐵_{𝑝,𝑞,𝑟} is strictly monotone.
Step 3.  𝐵_{𝑝,𝑞,𝑟} is hemicontinuous.
In fact, it suffices to show that, for any 𝑢,𝑣,𝑤∈𝑊^1,𝑝(Ω) and 𝑡∈[0,1],  (𝑤,𝐵_{𝑝,𝑞,𝑟}(𝑢+𝑡𝑣)−𝐵_{𝑝,𝑞,𝑟}𝑢)→0, as 𝑡→0.
By Lebesque's dominated convergence theorem, it follows that 0≤lim_𝑡→0𝑤,𝐵_{𝑝,𝑞,𝑟}(𝑢+𝑡𝑣)−𝐵_{𝑝,𝑞,𝑟}𝑢≤_Ωlim_𝑡→0||1+∇𝑢+𝑡∇𝑣^𝑝1+|∇𝑢+𝑡∇𝑣|^2𝑝|∇𝑢+𝑡∇𝑣|^𝑝−2()−||∇𝑢+𝑡∇𝑣1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢|∇𝑤|𝑑𝑥+𝜆_Ωlim_𝑡→0|𝑢+𝑡𝑣|^𝑞−2(𝑢+𝑡𝑣)−|𝑢|^𝑞−2𝑢|𝑤|𝑑𝑥+𝜆_Ωlim_𝑡→0|𝑢+𝑡𝑣|^𝑟−2(𝑢+𝑡𝑣)−|𝑢|^𝑟−2𝑢|𝑤|𝑑𝑥=0, (2.9) and hence 𝐵_{𝑝,𝑞,𝑟} is hemicontinuous.
Step 4.  𝐵_{𝑝,𝑞,𝑟} is coercive.
Now, for 𝑢∈𝑊^1,𝑝(Ω), Lemma 2.4 implies that ‖𝑢‖_1,𝑝,Ω→∞ is equivalent to ‖𝑢−(1/meas(Ω))∫_Ω𝑢𝑑𝑥‖_1,𝑝,Ω→∞ and hence we have the following result: 𝑢,𝐵_{𝑝,𝑞,𝑟}𝑢‖𝑢‖_1,𝑝,Ω=∫_Ω1+|∇𝑢|^𝑝/1+|∇𝑢|^2𝑝|∇𝑢|^𝑝𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑞𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑟𝑑𝑥‖𝑢‖_1,𝑝,Ω=∫_Ω|∇𝑢|^𝑝+1+|∇𝑢|^2𝑝𝑑𝑥−∫_Ω1/1+|∇𝑢|^2𝑝𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑞𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑟𝑑𝑥‖𝑢‖_1,𝑝,Ω≥2∫_Ω|∇𝑢|^𝑝𝑑𝑥−∫_Ω1/1+|∇𝑢|^2𝑝𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑞𝑑𝑥‖𝑢‖_1,𝑝,Ω∫+𝜆^Ω|𝑢|^𝑟𝑑𝑥‖𝑢‖_1,𝑝,Ω→+∞, (2.10) as ‖𝑢‖_1,𝑝,Ω→+∞, which implies that 𝐵_{𝑝,𝑞,𝑟} is coercive.
This completes the proof.

Remark 2.6. Lemma 2.5 is a key result for later use.

Definition 2.7. Define a mapping 𝐴_𝑝∶𝐿^𝑝(Ω)→2^𝐿𝑝(Ω) as follows: 𝐷𝐴_𝑝=𝑢∈𝐿^𝑝(Ω)∣thereexistsan𝑓∈𝐿^𝑝(Ω),suchthat𝑓∈𝐵_{𝑝,𝑞,𝑟}𝑢+𝜕Φ_𝑝(𝑢). (2.11) For 𝑢∈𝐷(𝐴_𝑝), let 𝐴_𝑝𝑢={𝑓∈𝐿^𝑝(Ω)∣𝑓∈𝐵_{𝑝,𝑞,𝑟}𝑢+𝜕Φ_𝑝(𝑢)}.

Proposition 2.8. The mapping 𝐴_𝑝∶𝐿^𝑝(Ω)→2^𝐿𝑝(Ω) is m-accretive.

Proof. Step 1.    𝐴_𝑝 is accretive.
Case 1.  If 𝑝≥2, the duality mapping 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω) is defined by 𝐽_𝑝𝑢=|𝑢|^𝑝−1sgn𝑢‖𝑢‖_𝑝^2−𝑝 for 𝑢∈𝐿^𝑝(Ω). It then suffices to prove that for any 𝑢_𝑖∈𝐷(𝐴_𝑝) and 𝑣_𝑖∈𝐴_𝑝𝑢_𝑖,𝑖=1,2, 𝑣₁−𝑣₂,𝐽_𝑝𝑢₁−𝑢₂≥0. (2.12) To this, we are left to prove that both 𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑢₁−𝑢_2𝑝^2−𝑝,𝐵_{𝑝,𝑞,𝑟}𝑢₁−𝐵_{𝑝,𝑞,𝑟}𝑢₂𝑢≥0,₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑢₁−𝑢_2𝑝^2−𝑝,𝜕Φ_𝑝𝑢₁−𝜕Φ_𝑝𝑢₂≥0 (2.13) are available.
Now take for a constant 𝑘>0, 𝜒_𝑘∶𝑅→𝑅 is defined by 𝜒_𝑘(𝑡)=|(𝑡∧𝑘)∨(−𝑘)|^𝑝−1sgn𝑡. Then 𝜒_𝑘 is monotone, Lipschitz with 𝜒_𝑘(0)=0 and 𝜒^′_𝑘 is continuous except at finitely many points on 𝑅. This gives that 𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑢₁−𝑢_2𝑝^2−𝑝,𝜕Φ_𝑝𝑢₁−𝜕Φ_𝑝𝑢₂=lim_𝑘→+∞𝑢₁−𝑢_2𝑝^2−𝑝𝜒_𝑘𝑢₁−𝑢₂,𝜕Φ_𝑝𝑢₁−𝜕Φ_𝑝𝑢₂≥0. (2.14) Also, 𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑢₁−𝑢_2𝑝^2−𝑝,𝐵_{𝑝,𝑞,𝑟}𝑢₁−𝐵_{𝑝,𝑞,𝑟}𝑢₂=𝑢₁−𝑢_2𝑝^2−𝑝×lim_{𝑘→+∞Ω}1+∇𝑢₁^𝑝1+∇𝑢₁^2𝑝∇𝑢₁^𝑝−2∇𝑢₁−1+∇𝑢₂^𝑝1+∇𝑢₂^2𝑝∇𝑢₂^𝑝−2∇𝑢₂,∇𝑢₁−∇𝑢₂×𝜒^′_𝑘𝑢₁−𝑢₂𝑑𝑥+𝜆𝑢₁−𝑢_2𝑝^2−𝑝_Ω𝑢₁^𝑞−2𝑢₁−𝑢₂^𝑞−2𝑢₂𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑢𝑑𝑥+𝜆₁−𝑢_2𝑝^2−𝑝_Ω𝑢₁^𝑟−2𝑢₁−𝑢₂^𝑟−2𝑢₂𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂𝑑𝑥≥0, (2.15) the last inequality is available since 𝜒_𝑘 is monotone and 𝜒_𝑘(0)=0.
Case 2.  If 2𝑁/(𝑁+1)<𝑝<2, the duality mapping 𝐽_𝑝∶𝐿^𝑝(Ω)→𝐿^𝑝′(Ω) is defined by 𝐽_𝑝𝑢=|𝑢|^𝑝−1sgn𝑢, for 𝑢∈𝐿^𝑝(Ω). It then suffices to prove that for any 𝑢_𝑖∈𝐷(𝐴_𝑝) and 𝑣_𝑖∈𝐴_𝑝𝑢_𝑖,𝑖=1,2, 𝑣₁−𝑣₂,𝐽_𝑝𝑢₁−𝑢₂≥0. (2.16) To this, we define the function 𝜒_𝑛∶𝑅→𝑅 by 𝜒_𝑛(𝑡)=|𝑡|^𝑝−1|𝑡|≥1sgn𝑡,if𝑛,1𝑛^𝑝−2|𝑡|≤1𝑡,if𝑛. (2.17) Then 𝜒_𝑛 is monotone, Lipschitz with 𝜒_𝑛(0)=0 and 𝜒^′_𝑛 is continuous except at finitely many points on 𝑅. So (𝜒_𝑛(𝑢₁−𝑢₂),𝜕Φ_𝑝(𝑢₁)−𝜕Φ_𝑝(𝑢₂))≥0.
Then, for 𝑢_𝑖∈𝐷(𝐴_𝑝),  𝑣_𝑖∈𝐴_𝑝𝑢_𝑖,𝑖=1,2, we have𝑣₁−𝑣₂,𝐽_𝑝𝑢₁−𝑢₂=𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂,𝐵_{𝑝,𝑞,𝑟}𝑢₁−𝐵_{𝑝,𝑞,𝑟}𝑢₂+𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂,𝜕Φ_𝑝𝑢₁−𝜕Φ_𝑝𝑢₂=𝑢₁−𝑢₂^𝑝−1sgn𝑢₁−𝑢₂,𝐵_{𝑝,𝑞,𝑟}𝑢₁−𝐵_{𝑝,𝑞,𝑟}𝑢₂+lim_𝑛→∞𝜒_𝑛𝑢₁−𝑢₂,𝜕Φ_𝑝𝑢₁−𝜕Φ_𝑝𝑢₂≥0. (2.18)
Step 2.  𝑅(𝐼+𝜇𝐴_𝑝)=𝐿^𝑝(Ω), for every 𝜇>0.
First, define the mapping 𝐼_𝑝∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ by 𝐼_𝑝𝑢=𝑢 and (𝑣,𝐼_𝑝𝑢)_{(𝑊1,𝑝(Ω))∗×𝑊1,𝑝(Ω)}=(𝑣,𝑢)_𝐿²_(Ω) for 𝑢,𝑣∈𝑊^1,𝑝(Ω), where (⋅,⋅)_𝐿²_(Ω) denotes the inner product of 𝐿²(Ω). Then 𝐼_𝑝 is maximal monotone [7].
Secondly, for any 𝜇>0, define the mapping 𝑇_𝜇∶𝑊^1,𝑝(Ω)→2^{(𝑊1,𝑝(Ω))∗} by 𝑇_𝜇𝑢=𝐼_𝑝𝑢+𝜇𝐵_{𝑝,𝑞,𝑟}𝑢+𝜇𝜕Φ_𝑝(𝑢), for 𝑢∈𝑊^1,𝑝(Ω). Then similar to that in [7], by using Lemmas 1.4, 1.6, 2.3, and 2.5, we know that 𝑇_𝜇 is maximal monotone and coercive, so that 𝑅(𝑇_𝜇)=(𝑊^1,𝑝(Ω))^∗, for any 𝜇>0.
Therefore, for any 𝑓∈𝐿^𝑝(Ω), there exists 𝑢∈𝑊^1,𝑝(Ω), such that 𝑓=𝑇_𝜇𝑢=𝑢+𝜇𝐵_{𝑝,𝑞,𝑟}𝑢+𝜇𝜕Φ_𝑝(𝑢). (2.19) From the definition of 𝐴_𝑝, it follows that 𝑅(𝐼+𝜇𝐴_𝑝)=𝐿^𝑝(Ω), for all 𝜇>0.
This completes the proof.

Lemma 2.9. The mapping 𝐴_𝑝∶𝐿^𝑝(Ω)→2^𝐿𝑝(Ω) has a compact resolvent for 2𝑁/(𝑁+1)<𝑝<2 and 𝑁≥1.

Proof. Since 𝐴_𝑝 is 𝑚-accretive by Proposition 2.8, it suffices to prove that if 𝑢+𝜇𝐴_𝑝𝑢=𝑓 (𝜇>0) and {𝑓} is bounded in 𝐿^𝑝(Ω), then {𝑢} is relatively compact in 𝐿^𝑝(Ω). Now define functions 𝜒_𝑛,𝜉_𝑛∶𝑅→𝑅 by 𝜒_𝑛(𝑡)=|𝑡|^𝑝−1|𝑡|≥1sgn𝑡,if𝑛1𝑛^𝑝−2|𝑡|≤1𝑡,if𝑛,𝜉_𝑛(𝑡)=|𝑡|^2−(2/𝑝)|𝑡|≥1sgn𝑡,if𝑛,1𝑛^1−(2/𝑝)|𝑡|≤1𝑡,if𝑛. (2.20) Noticing that 𝜒^′_𝑛(𝑡)=(𝑝−1)×(𝑝^′/2)^𝑝×(𝜉^′_𝑛(𝑡))^𝑝,  for|𝑡|≥1/𝑛,  where(1/𝑝)+(1/𝑝^′)=1 and 𝜒^′_𝑛(𝑡)=(𝜉^′_𝑛(𝑡))^𝑝,  for|𝑡|≤1/𝑛. We know that (𝜒_𝑛(𝑢),𝜕Φ_𝑝(𝑢))≥0 for 𝑢∈𝑊^1,𝑝(Ω) since 𝜒_𝑛 is monotone, Lipschitz with 𝜒_𝑛(0)=0, and 𝜒^′_𝑛 is continuous except at finitely many points on 𝑅. Then |𝑢|^𝑝−1sgn𝑢,𝐴_𝑝𝑢=lim_𝑛→∞𝜒_𝑛(𝑢),𝐴_𝑝𝑢≥lim_𝑛→∞𝜒_𝑛(𝑢),𝐵_{𝑝,𝑞,𝑟}𝑢=lim_𝑛→∞Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝𝜒^′_𝑛(𝑢)𝑑𝑥+𝜆lim_𝑛→∞Ω|𝑢|^𝑞−2𝑢𝜒_𝑛(𝑢)𝑑𝑥+𝜆lim_𝑛→∞Ω|𝑢|^𝑟−2𝑢𝜒_𝑛(𝑢)𝑑𝑥≥lim_𝑛→∞Ω|∇𝑢|^𝑝𝜒^′_𝑛(𝑢)𝑑𝑥≥const⋅lim_𝑛→∞Ωgrad𝜉_𝑛(𝑢)^𝑝𝑑𝑥≥const_Ωgrad|𝑢|^2−(2/𝑝)sgn𝑢^𝑝𝑑𝑥. (2.21)
We now have from 𝑓=𝑢+𝜇𝐴_𝑝𝑢 that 𝑓_𝑝|𝑢|^2−(2/𝑝)sgn𝑢^{𝑝2/2(𝑝−1)𝑝′}_{𝑝2/2𝑝−1}≥|𝑢|^𝑝−1sgn𝑢,𝑓=|𝑢|^𝑝−1sgn𝑢,𝑢+𝜇|𝑢|^𝑝−1sgn𝑢,𝐴_𝑝𝑢≥|𝑢|^2−(2/𝑝)sgn𝑢^𝑝2_𝑝^/2(𝑝−1)_2/2𝑝−1+𝜇⋅const⋅grad|𝑢|^2−(2/𝑝)sgn𝑢^𝑝_𝑝, (2.22) which gives that |𝑢|^2−(2/𝑝)sgn𝑢_𝑝^{𝑝/2(𝑝−1)}≤|𝑢|^2−(2/𝑝)sgn𝑢_𝑝^{𝑝/2(𝑝−1)}_{2/2(𝑝−1)}𝑓_𝑝≤const, (2.23) in view of the fact that 𝑝<𝑝²/2(𝑝−1) when 2𝑁/(𝑁+1)<𝑝<2 for 𝑁≥1. Again from (2.22), we have ‖grad(|𝑢|^2−(2/𝑝)sgn𝑢)‖_𝑝≤const. Hence {𝑓} bounded in 𝐿^𝑝(Ω) implies that {|𝑢|^2−(2/𝑝)sgn𝑢} is bounded in 𝑊^1,𝑝(Ω).
We notice that 𝑊^1,𝑝(Ω)↪↪𝐿^{𝑝2/2(𝑝−1)}(Ω) when 𝑁≥2 and 𝑊^1,𝑝(Ω)↪↪𝐶_𝐵(Ω) when 𝑁=1 by Lemma 1.3, hence{|𝑢|^2−(2/𝑝)sgn𝑢} is relatively compact in 𝐿^{𝑝2/2(𝑝−1)}(Ω). This gives that {𝑢} is relatively compact in 𝐿^𝑝(Ω) since the Nemytskii mapping 𝑢∈𝐿^{𝑝2/2(𝑝−1)}(Ω)→|𝑢|^{𝑝/2(𝑝−1)}sgn𝑢∈𝐿^𝑝(Ω) is continuous.
This completes the proof.

Remark 2.10. Since Φ_𝑝(𝑢+𝛼)=Φ_𝑝(𝑢), for any 𝑢∈𝑊^1,𝑝(Ω) and 𝛼∈𝐶^∞₀(Ω), we have 𝑓∈𝐴_𝑝𝑢 implies that 𝑓=𝐵_{𝑝,𝑞,𝑟}𝑢 in the sense of distributions.

Proposition 2.11. For 𝑓∈𝐿^𝑝(Ω), if there exists 𝑢∈𝐿^𝑝(Ω) such that 𝑓∈𝐴_𝑝𝑢, then 𝑢 is the unique solution of (1.4).

Proof. First, we show that ||−div1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢+𝜆|𝑢|^𝑞−2𝑢+𝜆|𝑢|^𝑟−2𝑢=𝑓(𝑥),a.e.𝑥∈Ω (2.24) is available.
Now 𝑓∈𝐴_𝑝𝑢 implies that 𝑓=𝐵_{𝑝,𝑞,𝑟}𝑢+𝜕Φ_𝑝(𝑢). For all 𝜑∈𝐶^∞₀(Ω), by Remark 2.10, we have 𝜑,𝑓=𝜑,𝐵_{𝑝,𝑞,𝑟}𝑢+𝜕Φ_𝑝(𝑢=)𝜑,𝐵_{𝑝,𝑞,𝑟}𝑢=_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢,∇𝜑𝑑𝑥+𝜆_Ω|𝑢|^𝑞−2𝑢𝜑𝑑𝑥+𝜆_Ω|𝑢|^𝑟−2=𝑢𝜑𝑑𝑥_Ω||−div1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢𝜑𝑑𝑥+𝜆_Ω|𝑢|^𝑞−2𝑢𝜑𝑑𝑥+𝜆_Ω|𝑢|^𝑟−2𝑢𝜑𝑑𝑥, (2.25) which implies that (2.24) is true.
Secondly, we show that −||𝜗,1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.𝑥∈Γ. (2.26) We will prove (2.26) 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 [11] for the general case.
Now, from (2.24), 𝑓∈𝐴_𝑝𝑢 implies that 𝑓(𝑥)=−div[(1+|∇𝑢|^𝑝/1+|∇𝑢|^2𝑝)|∇𝑢|^𝑝−2∇𝑢]+𝜆|𝑢(𝑥)|^𝑞−2𝑢(𝑥)+𝜆|𝑢|^𝑟−2𝑢∈𝐿^𝑝(Ω). By using Green's Formula, we have that for any 𝑣∈𝑊^1,𝑝(Ω), _Γ||𝜗,1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢𝑣|_Γ(𝑥)=𝑑Γ_Ω||div1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢𝑣𝑑𝑥+_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢,∇𝑣𝑑𝑥. (2.27)
Then −⟨𝜗,(1+|∇𝑢|^𝑝/1+|∇𝑢|^2𝑝)|∇𝑢|^𝑝−2∇𝑢⟩∈𝑊^{−(1/𝑝),𝑝′}(Γ)=(𝑊^1/𝑝,𝑝(Γ))^∗, where 𝑊^1/𝑝,𝑝(Γ) is the space of traces of 𝑊^1,𝑝(Ω).
Now let the mapping 𝐵∶𝐿^𝑝(Γ)→𝐿^𝑝′(Γ) be defined by 𝐵𝑢=𝑔(𝑥), for any 𝑢∈𝐿^𝑝(Γ), where 𝑔(𝑥)=𝛽_𝑥(𝑢(𝑥)) a.e. on Γ. Clearly, 𝐵=𝜕Ψ where Ψ(𝑢)=∫_Γ𝜑_𝑥(𝑢(𝑥))𝑑Γ(𝑥) is a proper, convex, and lower-semicontinuous function on 𝐿^𝑝(Γ). Now define the mapping 𝐾∶𝑊^1,𝑝(Ω)→𝐿^𝑝(Γ) by 𝐾(𝑣)=𝑣|_Γ for any 𝑣∈𝑊^1,𝑝(Ω). Then 𝐾^∗𝐵𝐾∶𝑊^1,𝑝(Ω)→(𝑊^1,𝑝(Ω))^∗ is maximal monotone since both 𝐾,𝐵 are continuous. Finally, for any 𝑢,𝑣∈𝑊^1,𝑝(Ω), we have Ψ(𝐾𝑣)−Ψ(𝐾𝑢)=_Γ𝜑_𝑥𝑣|_Γ(𝑥)−𝜑_𝑥𝑢|_Γ(𝑥)(𝑥)≥𝑑Γ_Γ𝛽_𝑥𝑢|_Γ(𝑥)𝑣|_Γ(𝑥)−𝑢|_Γ(𝑥(𝑥)=()=𝐾)𝑑Γ𝐵𝐾𝑢,𝐾𝑣−𝐾𝑢^∗𝐵𝐾𝑢,𝑣−𝑢. (2.28) Hence we get 𝐾^∗𝐵𝐾⊂𝜕Φ_𝑝 and so 𝐾^∗𝐵𝐾=𝜕Φ_𝑝. Therefore, we have −||𝜗,1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢∈𝛽_𝑥(𝑢(𝑥)),a.e.onΓ. (2.29)
Finally, we will show that 𝑢 is unique.
If 𝑓∈𝐴_𝑝𝑢 and 𝑓∈𝐴_𝑝𝑣, where 𝑢,𝑣∈𝐷(𝐴_𝑝). Then 0≤𝑢−𝑣,𝐵_{𝑝,𝑞,𝑟}𝑢−𝐵_{𝑝,𝑞,𝑟}𝑣=𝑢−𝑣,𝜕Φ_𝑝(𝑣)−𝜕Φ_𝑝(𝑢)≤0, (2.30) since 𝐵_{𝑝,𝑞,𝑟} is strictly monotone and 𝜕Φ_𝑝 is maximal monotone, which implies that 𝑢(𝑥)=𝑣(𝑥).
This completes the proof.

Remark 2.12. If 𝛽_𝑥≡0 for any 𝑥∈Γ, then 𝜕Φ_𝑝(𝑢)≡0, for all 𝑢∈𝑊^1,𝑝(Ω).

Proposition 2.13. If 𝛽_𝑥≡0 for any 𝑥∈Γ, then {𝑓∈𝐿^𝑝(Ω)∣∫_Ω𝑓𝑑𝑥=0}⊂𝑅(𝐴_𝑝).

Proof. We can easily know that 𝑅(𝐵_{𝑝,𝑞,𝑟})=(𝑊^1,𝑝(Ω))^∗ in view of Lemmas 1.4 and 2.5. Note that for any 𝑓∈𝐿^𝑝(Ω) with ∫_Ω𝑓𝑑𝑥=0, the linear function 𝑢∈𝑊^1,𝑝(Ω)→∫_Ω𝑓𝑢𝑑𝑥 is an element of (𝑊^1,𝑝(Ω))^∗. So there exists a 𝑢∈𝑊^1,𝑝(Ω) such that _Ω𝑓𝑣𝑑𝑥=_Ω||1+∇𝑢^𝑝1+|∇𝑢|^2𝑝|∇𝑢|^𝑝−2∇𝑢,∇𝑣𝑑𝑥+𝜆_Ω|𝑢|^𝑞−2𝑢𝑣𝑑𝑥+𝜆_Ω|𝑢|^𝑟−2𝑢𝑣𝑑𝑥, (2.31) for any 𝑣∈𝑊^1,𝑝(Ω). So 𝑓=𝐴_𝑝𝑢 in view of Remark 2.12.
This completes the proof.

Definition 2.14 (see [1, 7]). For 𝑡∈𝑅,  𝑥∈Γ, let 𝛽⁰_𝑥(𝑡)∈𝛽_𝑥(𝑡) be the element with least absolute value if 𝛽_𝑥(𝑡)≠∅ and 𝛽⁰_𝑥(𝑡)=±∞, where 𝑡>0 or <0, respectively, in case 𝛽_𝑥(𝑡)=∅. Finally, let 𝛽_±(𝑥)=lim_𝑡→±∞𝛽⁰_𝑥(𝑡) (in the extended sense) for 𝑥∈Γ. 𝛽_±(𝑥) define measurable functions on Γ, in view of our assumptions on 𝛽_𝑥.

Proposition 2.15. Let 𝑓∈𝐿^𝑝(Ω) such that _Γ𝛽₋(𝑥)𝑑Γ(𝑥)<_Ω𝑓𝑑𝑥<_Γ𝛽₊(𝑥)𝑑Γ(𝑥). (2.32) Then 𝑓∈Int𝑅(𝐴_𝑝).

Proof. Let 𝑓∈𝐿^𝑝(Ω) and satisfy (2.32), by Proposition 2.8, there exists 𝑢_𝑛∈𝐿^𝑝(Ω) such that, for each 𝑛≥1,  𝑓=(1/𝑛)𝑢_𝑛+𝐴_𝑝𝑢_𝑛. In the same reason as that in [1], we only need to prove that ‖𝑢_𝑛‖_𝑝≤const, for all 𝑛≥1.
Indeed, suppose to the contrary that 1≤‖𝑢_𝑛‖_𝑝→∞, as 𝑛→∞. Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖_𝑝. Let 𝜓∶𝑅→𝑅 be defined by 𝜓(𝑡)=|𝑡|^𝑝,  𝜕𝜓∶𝑅→𝑅 be its subdifferential and for 𝜇>0,  𝜕𝜓_𝜇∶𝑅→𝑅 denote the Yosida-approximation of 𝜕𝜓.  Let 𝜃_𝜇∶𝑅→𝑅 denote the indefinite integral of [(𝜕𝜓_𝜇)^′]^1/𝑝 with 𝜃_𝜇(0)=0 so that (𝜃^′_𝜇)^𝑝=(𝜕𝜓_𝜇)^′. In view of Calvert and Gupta [1], we have 𝜕𝜓_𝜇𝑣_𝑛,𝜕Φ_𝑝𝑢_𝑛≥_Γ𝛽_𝑥1+𝜇𝜕𝜓⁻¹𝑢_𝑛|_Γ(𝑥)×𝜕𝜓_𝜇𝑣_𝑛|_Γ(𝑥)𝑑Γ(𝑥)≥0. (2.33)
Now multiplying the equation 𝑓=(1/𝑛)𝑢_𝑛+𝐴_𝑝𝑢_𝑛 by 𝜕𝜓_𝜇(𝑣_𝑛), we get that 𝜕𝜓_𝜇𝑣_𝑛,𝑓=𝜕𝜓_𝜇𝑣_𝑛,1𝑛𝑢_𝑛+𝜕𝜓_𝜇𝑣_𝑛,𝐵_{𝑝,𝑞,𝑟}𝑢_𝑛+𝜕𝜓_𝜇𝑣_𝑛,𝜕Φ_𝑝𝑢_𝑛. (2.34) Since 𝜕𝜓_𝜇(0)=0, it follows that (𝜕𝜓_𝜇(𝑣_𝑛),𝑢_𝑛)≥0. Also, we can know that 𝜕𝜓_𝜇𝑣_𝑛,𝐵_{𝑝,𝑞,𝑟}𝑢_𝑛=_Ω1+∇𝑢_𝑛^𝑝1+∇𝑢_𝑛^2𝑝∇𝑢_𝑛^𝑝−2∇𝑢_𝑛,∇𝑣_𝑛𝜕𝜓_𝜇^′𝑣_𝑛𝑑𝑥+𝜆_Ω𝑢_𝑛^𝑞−2𝑢_𝑛𝜕𝜓_𝜇𝑣_𝑛𝑑𝑥+𝜆_Ω𝑢_𝑛^𝑟−2𝑢_𝑛𝜕𝜓_𝜇𝑣_𝑛≥𝑑𝑥_Ω|∇𝑢|^𝑝𝑢_𝑛𝑝𝜕𝜓_𝜇^′𝑣_𝑛𝑑𝑥=𝑢_𝑛𝑝^𝑝−1_Ωgrad𝜃_𝜇𝑣_𝑛^𝑝𝑑𝑥. (2.35)
Then we get from (2.33) that 𝑢_𝑛𝑝^𝑝−1_Ωgrad𝜃_𝜇𝑣_𝑛^𝑝𝑑𝑥+_Γ𝛽_𝑥1+𝜇𝜕𝜓⁻¹𝑢_𝑛|_Γ(𝑥)×𝜕𝜓_𝜇𝑣_𝑛|_Γ(𝑥)(𝑥)≤𝑑Γ𝜕𝜓_𝜇𝑣_𝑛,𝑓. (2.36) Since |𝜕𝜓_𝜇(𝑡)|≤|𝜕𝜓(𝑡)| for any 𝑡∈𝑅 and 𝜇>0, we see from ‖𝑣_𝑛‖_𝑝=1 for 𝑛≥1, that ‖𝜕𝜓_𝜇(𝑣_𝑛)‖_𝑝′≤𝐶, for 𝜇>0, where 𝐶 is a constant which does not depend on 𝑛 or 𝜇 and (1/𝑝)+(1/𝑝^′)=1.
From (2.36), we have grad𝜃_Ω𝜇𝑣_𝑛^𝑝𝐶𝑑𝑥≤𝑢_𝑛𝑝^𝑝−1,for𝜇>0,𝑛≥1. (2.37) Now we easily know that (𝜃^′_𝜇)^𝑝=(𝜕𝜓_𝜇)^′→(𝜕𝜓)^′ as 𝜇→0  a.e. on 𝑅.
Letting 𝜇→0, we see from Fatou's lemma and (2.37) that grad𝑣_Ω𝑛^2−(2/𝑝)sgn𝑣_𝑛^𝑝𝐶𝑑𝑥≤𝑢_𝑛𝑝^𝑝−1. (2.38)
From (2.38), we know that |𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛→𝑘 (a constant) in 𝐿^𝑝(Ω), as 𝑛→+∞.
Next we will show that 𝑘≠0 is in 𝐿^𝑝(Ω) from two aspects.(i)If 𝑝≥2, since ‖|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛‖_𝑝=‖𝑣_𝑛‖^2−(2/𝑝)_2𝑝−2≥‖𝑣_𝑛‖_𝑝^2−(2/𝑝)=1, it follows that 𝑘≠0 in 𝐿^𝑝(Ω),(ii)if 2𝑁/(𝑁+1)<𝑝<2,  ‖|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛‖_𝑝=‖𝑣_𝑛‖^2−(2/𝑝)_2𝑝−2≥‖𝑣_𝑛‖_𝑝^2−(2/𝑝)=1, then {|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛} is bounded in 𝑊^1,𝑝(Ω). By Lemma 1.3, 𝑊^1,𝑝(Ω)↪↪𝐶_𝐵(Ω) when 𝑁=1 and 𝑊^1,𝑝(Ω)↪↪𝐿^{𝑝2/2(𝑝−1)}(Ω), when 𝑁≥2. So {|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛} is relatively compact in 𝐿^{𝑝2/2(𝑝−1)}(Ω). Then there exists a subsequence of {|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛}, for simplicity, we denote it by {|𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛}, satisfying |𝑣_𝑛|^2−(2/𝑝)sgn𝑣_𝑛→𝑔 in 𝐿^{𝑝2/2(𝑝−1)}(Ω). Noticing that 𝑝≤𝑝²/2(𝑝−1) when 2𝑁/(𝑁+1)<𝑝<2, it follows that 𝑘=𝑔  a.e. on Ω. Now, 1=𝑣_𝑛^𝑝_𝑝=_Ω𝑣_𝑛^2−(2/𝑝)sgn𝑣_𝑛^{𝑝2/2(𝑝−1)}𝑑𝑥≤const_Ω𝑣_𝑛^2−(2/𝑝)sgn𝑣_𝑛−𝑔^{𝑝2/2(𝑝−1)}𝑑𝑥+const𝑔^𝑝2_𝑝^/2(𝑝−1)_2/2𝑝−1, (2.39)it follows that 𝑔≠0 in 𝐿^𝑝(Ω) and then 𝑘≠0 in 𝐿^𝑝(Ω). Assume, now, 𝑘>0, we see from (2.36) that _Γ𝛽_𝑥1+𝜇𝜕𝜓⁻¹𝑢_𝑛|_Γ(𝑥)×𝜕𝜓_𝜇𝑣_𝑛|_Γ(𝑥)𝑑Γ(𝑥)≤𝜕𝜓_𝜇𝑣_𝑛,𝑓. (2.40) Choosing a subsequence so that 𝑢_𝑛|_Γ(𝑥)→+∞ a.e. on Γ, we see letting 𝑛→+∞ that ∫_Γ𝛽₊(𝑥)𝑑Γ(𝑥)≤∫_Ω𝑓(𝑥)𝑑𝑥, which is a contradiction with (2.32). Similarly, if 𝑘<0, it also leads to a contradiction. Thus 𝑓∈int𝑅(𝐴_𝑝).
This completes the proof.

Proposition 2.16. 𝐴_𝑝+𝐵₁∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) is m-accretive and has a compact resolvent.

Proof. Using a theorem in Corduneanu [12], we know that 𝐴_𝑝+𝐵₁∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) is 𝑚-accretive.
To show that 𝐴_𝑝+𝐵₁∶𝐿^𝑝(Ω)→𝐿^𝑝(Ω) has a compact resolvent, we only need to prove that if 𝑤∈𝐴_𝑝𝑢+𝐵₁𝑢 with {𝑤} and {𝑢} being bounded in 𝐿^𝑝(Ω), then {𝑢} is relatively compact in 𝐿^𝑝(Ω). Now we discuss it from two aspects.(i)If 𝑝≥2, since _Ω|∇𝑢|^𝑝𝑑𝑥≤𝑢,𝐵_{𝑝,𝑞,𝑟}𝑢=𝑢,𝐴_𝑝𝑢−𝑢,𝜕Φ_𝑝(𝑢)≤𝑢,𝐴_𝑝𝑢+𝑢,𝐵₁𝑢=(𝑢,𝑤)≤‖𝑢‖_𝑝‖𝑢‖_𝑝′≤const, (2.41) it follows that {𝑢} is bounded in 𝑊^1,𝑝(Ω), where (1/𝑝)+(1/𝑝^′)=1. Then {𝑢} is relatively compact in 𝐿^𝑝(Ω) since 𝑊^1,𝑝(Ω)↪↪𝐿^𝑝(Ω);(ii)if 2𝑁/(𝑁+1)<𝑝<2, since 𝑤∈𝐴_𝑝𝑢+𝐵₁𝑢 with {𝑤} and {𝑢} being bounded in 𝐿^𝑝(Ω), we have 𝑤−𝐵₁𝑢∈𝐴_𝑝𝑢 with {𝑤−𝐵₁𝑢} and {𝑢} being bounded in 𝐿^𝑝(Ω) which gives that {𝑢} is relatively compact in 𝐿^𝑝(Ω) since 𝐴_𝑝 is 𝑚-accretive by Proposition 2.8 and has a compact resolvent by Lemma 2.9.
This completes the proof.

Theorem 2.17. Let 𝑓∈𝐿^𝑝(Ω) be such that _Γ𝛽₋(𝑥)𝑑Γ(𝑥)+_Ω𝑔₋(𝑥)𝑑𝑥<_Ω𝑓(𝑥)𝑑𝑥<_Γ𝛽₊(𝑥)𝑑Γ(𝑥)+_Ω𝑔₊(𝑥)𝑑𝑥, (2.42) then (1.4) has a unique solution in 𝐿^𝑝(Ω), where 2𝑁/(𝑁+1)<𝑝<+∞ and 𝑁≥1.

Proof. We want to use Theorem 1.9 to finish our proof. From Propositions 1.7, 2.2, 2.8, and 2.16, we can see that all of the conditions in Theorem 1.9 are satisfied. It then suffices to show that 𝑓∈int[𝑅(𝐴_𝑝)+𝑅(𝐵₁)] which ensures that 𝑓∈𝑅(𝐴_𝑝+𝐵₁+𝐵₂). Thus Proposition 2.11 tells us (1.4) has a unique solution in 𝐿^𝑝(Ω).
Using the similar methods as those in [2, 4, 7], by dividing it into two cases and using Propositions 2.13 and 2.15, respectively, we know that 𝑓∈int[𝑅(𝐴_𝑝)+𝑅(𝐵₁)].
This completes the proof.

Remark 2.18. Compared to the work done in [1–7], not only the existence of the solution of (1.4) is obtained but also the uniqueness of the solution is obtained.

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 ).