𝐿^∞-Solutions for Some Nonlinear Degenerate Elliptic Equations

Abstract

We are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations −∑^𝑛_𝑗=1𝐷_𝑗[𝜔₂(𝑥)𝒜_𝑗(𝑥,𝑢,∇𝑢)]+𝜔₁(𝑥)𝑔(𝑥,𝑢(𝑥))+𝐻(𝑥,𝑢,∇𝑢)𝜔₂(𝑥)=𝑓(𝑥),onΩ in the setting of the weighted Sobolev spaces W₀^1,𝑝(Ω,𝜔₁,𝜔₂).

1. Introduction

In this paper we prove the existence of (weak) solutions in the weighted Sobolev spaces W₀^1,𝑝(Ω,𝜔₁,𝜔₂) for the homogeneous Dirichlet problem: (𝑥)(𝑥)𝑢(𝑥)𝐿𝑢=𝑓,onΩ,=0,on𝜕Ω, (𝑃) where 𝐿 is the partial differential operator: 𝐿𝑢(𝑥)=−div𝜔₂(𝑥)𝒜(𝑥,𝑢,∇𝑢)+𝑔(𝑥,𝑢)𝜔₁(𝑥)+𝐻(𝑥,𝑢,∇𝑢)𝜔₂(𝑥), (1.1) where Ω is a bounded open set in ℝ^𝑁 (𝑁≥2), 𝜔₁ and 𝜔₂ are two weight functions, and the functions 𝒜∶Ω×ℝ×ℝ^𝑁→ℝ^𝑁, 𝑔∶Ω×ℝ→ℝ, and 𝐻∶Ω×ℝ×ℝ^𝑁→ℝ are Carathéodory functions.

By a weight, we will mean a locally integrable function 𝜔 on ℝ^𝑁 such that 𝜔(𝑥)>0 for a.e. 𝑥∈ℝ^𝑁. Every weight 𝜔_𝑖 (𝑖=1,2) gives rise to a measure on the measurable subsets on ℝ^𝑁 through integration. This measure will be denoted by 𝜇_𝑖. Thus, 𝜇_𝑖(𝐸)=∫_𝐸𝜔_𝑖(𝑥)𝑑𝑥 (𝑖=1,2) for measurable sets 𝐸⊂ℝ^𝑁.

In general, the Sobolev spaces W^𝑘,𝑝(Ω) without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. For degenerate partial differential equations, that is, equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1–4]).

A class of weights, which is particularly well understood, is the class of 𝐴_𝑝-weights (or Muckenhoupt class) that was introduced by Muckenhoupt (see [5]). These classes have found many useful applications in harmonic analysis (see [6, 7]). Another reason for studying 𝐴_𝑝-weights is the fact that powers of distance to submanifolds of ℝ^𝑁 often belong to 𝐴_𝑝 (see [8]). There are, in fact, many interesting examples of weights (see [4] for 𝑝-admissible weights).

Equations like (1.1) have been studied by many authors in the nondegenerate case (i.e., with 𝜔(𝑥)≡1) (see, e.g., [9] and the references therein). The degenerate case with different conditions has been studied by many authors. In [2] Drábek et al. proved that under certain condition, the Dirichlet problem associated with the equation −div(𝑎(𝑥,𝑢,∇𝑢))=ℎ, ℎ∈[𝑊₀^1,𝑝(Ω,𝜔)]^∗ has at least one solution 𝑢∈𝑊₀^1,𝑝(Ω,𝜔), and in [1] the author proved the existence of solution when the nonlinear term 𝐻(𝑥,𝜂,𝜉) is equal to zero.

Firstly, we prove an 𝐿^∞ estimate for the bounded solutions of (𝑃): we assume that 𝑓/𝜔₁∈𝐿^𝑞(Ω,𝜔₁), with 𝑟₂/(𝑟₂−1)<𝑞<∞ (where 𝑟₂>1 as in Theorem 2.5), and we prove that any 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) that solves (𝑃) satisfies ‖𝑢‖_𝐿^∞_(Ω)≤𝐶, where 𝐶 depends only on the data, that is, Ω, 𝑁,𝑝,𝑞,𝛼₁,𝛼₂,𝐶₀,𝐶₁ and ‖𝑓/𝜔₁‖_{𝐿𝑞(Ω,𝜔1)}. After that, we prove the existence of solution for problem (𝑃) if 𝑓/𝜔₁∈𝐿^𝑞(Ω,𝜔₁), with 𝑝^′𝑟₂/(𝑟₂−1)<𝑞<∞.

Note that, in the proof of our main result, many ideas have been adapted from [9–11].

The following theorem will be proved in Section 3.

Theorem 1.1. Let 𝜔₁ and 𝜔₂ be 𝐴_𝑝-weights, 1<𝑝<∞, with 𝜔₁≤𝜔₂. Suppose the following. (H1)𝑥↦𝒜(𝑥,𝜂,𝜉) is measurable in Ω for all (𝜂,𝜉)∈ℝ×ℝ^𝑁: (𝜂,𝜉)↦𝒜(𝑥,𝜂,𝜉) is continuous in ℝ×ℝ^𝑁 for almost all 𝑥∈Ω. (H2)[𝒜(𝑥,𝜂,𝜉)−𝒜(𝑥,𝜂′,𝜉′)]⋅(𝜉−𝜉′)>0, whenever 𝜉,𝜉′∈ℝ^𝑁, 𝜉≠𝜉′.(H3)𝒜(𝑥,𝜂,𝜉)⋅𝜉≥𝛼₁|𝜉|^𝑝, with 1<𝑝<∞, where 𝛼₁>0.(H4)|𝒜(𝑥,𝜂,𝜉)|≤𝐾₂(𝑥)+ℎ₁(𝑥)|𝜂|^𝑝/𝑝′+ℎ₂(𝑥)|𝜉|^𝑝/𝑝′, where 𝐾₂,ℎ₁, and ℎ₂ are positive functions, with ℎ₁ and ℎ₂∈𝐿^∞(Ω), and 𝐾₂∈𝐿^𝑝′(Ω,𝜔₂) (1/𝑝+1/𝑝′=1).(H5)𝑥↦𝑔(𝑥,𝜂) is measurable in Ω for all 𝜂∈ℝ∶𝜂↦𝑔(𝑥,𝜂) is continuous in ℝ for almost all 𝑥∈Ω.(H6)|𝑔(𝑥,𝜂)|≤𝐾₁(𝑥)+ℎ₃(𝑥)|𝜂|^𝑝/𝑝′, where 𝐾₁ and ℎ₃ are positive functions, with ℎ₃∈𝐿^∞(Ω) and 𝐾₁∈𝐿^𝑝′(Ω,𝜔₁).(H7)𝑔(𝑥,𝜂)𝜂≥𝛼₀|𝜂|^𝑝, for all 𝜂∈ℝ, where 𝛼₀>0.(H8)𝑥↦𝐻(𝑥,𝜂,𝜉) is measurable in Ω for all (𝜂,𝜉)∈ℝ×ℝ^𝑁: (𝜂,𝜉)↦𝐻(𝑥,𝜂,𝜉) is continuous in ℝ×ℝ^𝑁 for almost all 𝑥∈Ω.(H9)|𝐻(𝑥,𝜂,𝜉)|≤𝐶₀+𝐶₁|𝜉|^𝑝, where 𝐶₀ and 𝐶₁ are positive constants.(H10)𝑓/𝜔₁∈𝐿^𝑞(Ω,𝜔₁), with 𝑟₂/(𝑟₂−1)<𝑞<∞ (where 𝑟₂>1 as in Theorem 2.5) and 𝜔₂/𝜔₁∈𝐿^𝑞(Ω,𝜔₁).
Let 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) be a solution of problem (𝑃). Then there exists a constant 𝐶>0, which depends only on Ω, 𝑛,𝑝,𝛼₁,𝛼₀,𝐶₀,𝐶₁ and ‖𝑓/𝜔₁‖_{𝐿𝑞(Ω,𝜔1)}, such that ‖𝑢‖_𝐿^∞_(Ω)≤𝐶.

The main result of this article is given in the next theorem, which is proved in Section 4.

Theorem 1.2. Assume that (H1)–(H9) hold true and suppose that
(H11)𝑓/𝜔₁∈𝐿^𝑞(Ω,𝜔₁), with 𝑝′𝑟₂/(𝑟₂−1)<𝑞<∞;(H12)𝐻(𝑥,𝜂,𝜉)𝜂≥0, for all 𝜂∈ℝ.
Then there exists at least one solution 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) of the problem 𝑃.

Theorem 1.2 will be proved by approximating problem (𝑃) with the following problems:−div𝜔₂𝒜(𝑥,𝑢,∇𝑢)+𝑔(𝑥,𝑢)𝜔₁+𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜔₂(𝑥)𝑢(𝑥)=𝑓,onΩ,=0,on𝜕Ω, (𝑃_𝑚) where 𝐻_𝑚(𝑥,𝜂,𝜉)=𝐻(𝑥,𝜂,𝜉)/1+(1/𝑚)|𝐻(𝑥,𝜂,𝜀)|, for 𝑚∈ℕ. Note that |𝐻_𝑚|≤|𝐻| and that |𝐻_𝑚|≤𝑚.

2. Definitions and Basic Results

Let 𝜔 be a locally integrable nonnegative function in ℝ^𝑁 and assume that 0<𝜔(𝑥)<∞ almost everywhere. We say that 𝜔 belongs to the Muckenhoupt class 𝐴_𝑝, 1<𝑝<∞, or that 𝜔 is an 𝐴_𝑝-weight, if there is a constant 𝐶=𝐶_𝑝,𝜔 such that1|𝐵|_𝐵𝜔(𝑥)1𝑑𝑥|𝐵|_𝐵𝜔^1/1−𝑝(𝑥)𝑑𝑥^𝑝−1≤𝐶_𝑝,𝜔 (2.1) for all balls 𝐵⊂ℝ^𝑁, where |⋅| denotes the 𝑁-dimensional Lebesgue measure in ℝ^𝑁. If 1<𝑞≤𝑝, then 𝐴_𝑞⊂𝐴_𝑝 (see [4, 7, 12] or [13] for more information about 𝐴_𝑝-weights). The weight 𝜔 satisfies the doubling condition if 𝜇(2𝐵)≤𝐶𝜇(𝐵), for all balls 𝐵⊂ℝ^𝑛, where 𝜇(𝐵)=∫_𝐵𝜔(𝑥)𝑑𝑥 and 2𝐵 denotes the ball with the same center as 𝐵 which is twice as large. If 𝜔∈𝐴_𝑝, then 𝜔 is doubling (see Corollary  15.7 in [4]).

As an example of 𝐴_𝑝-weight, the function 𝜔(𝑥)=|𝑥|^𝛼, 𝑥∈ℝ^𝑁, is in 𝐴_𝑝 if and only if −𝑁<𝛼<𝑁(𝑝−1) (see Corollary  4.4, Chapter IX in [7]). If 𝜑∈BMO(ℝ^𝑁), then 𝜔(𝑥)=e^{𝛼𝜑(𝑥)}∈𝐴₂ for some 𝛼>0 (see [6]).

Definition 2.1. Let 𝜔 be a weight, and let Ω⊂ℝ^𝑁 be open. For 0<𝑝<∞, we define 𝐿^𝑝(Ω,𝜔) as the set of measurable functions 𝑓 on Ω such that 𝑓_{𝐿𝑝(Ω,𝜔)}=_Ω𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝<∞. (2.2)

Remark 2.2. If 𝜔∈𝐴_𝑝, 1<𝑝<∞, then since 𝜔^{−1/(𝑝−1)} is locally integrable, we have 𝐿^𝑝(Ω,𝜔)⊂𝐿¹_loc(Ω) for every open set Ω (see Remark  1.2.4 in [13]). It thus makes sense to talk about weak derivatives of functions in 𝐿^𝑝(Ω,𝜔).

Definition 2.3. Let Ω⊂ℝ^𝑁 be open, 1<𝑝<∞, and let 𝜔₁ and 𝜔₂ be 𝐴_𝑝-weights, 1<𝑝<∞. We define the weighted Sobolev space 𝑊^1,𝑝(Ω,𝜔₁,𝜔₂) as the set of functions 𝑢∈𝐿^𝑝(Ω,𝜔₁) with weak derivatives 𝐷_𝑗𝑢∈𝐿^𝑝(Ω,𝜔₂), for 𝑗=1,…,𝑁. The norm of 𝑢 in 𝑊^1,𝑝(Ω,𝜔₁,𝜔₂) is given by ‖𝑢‖_𝑊^1,𝑝_{(Ω,𝜔1,𝜔2)}=_Ω|𝑢(𝑥)|^𝑝𝜔₁(𝑥)𝑑𝑥+^𝑁_𝑗=1Ω𝐷_𝑗𝑢(𝑥)^𝑝𝜔₂(𝑥)𝑑𝑥^1/𝑝. (2.3) The space 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is the closure of 𝐶^∞₀(Ω) with respect to the norm ‖𝑢‖_𝑊^1,𝑝_{(Ω,𝜔1,𝜔2)}. The dual space of 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is the space 𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂^∗=𝑊^−1,𝑝′Ω,𝜔₁,𝜔₂=𝑇=𝑓−div𝑔∶𝑔=𝑔₁,…,𝑔_𝑁,𝑓𝜔₁∈𝐿^𝑝′Ω,𝜔₁,𝑔_𝑗𝜔₂∈𝐿^𝑝′Ω,𝜔₂. (2.4)

Remark 2.4. (a) If 𝜔∈𝐴_𝑝, 1<𝑝<∞, then 𝐶^∞(Ω) is dense in 𝑊^1,𝑝(Ω,𝜔)=𝑊^1,𝑝(Ω,𝜔,𝜔) (see Corollary  2.16 in [13]).
(b) If 𝜔₁≤𝜔₂, then 𝑊₀^1,𝑝Ω,𝜔₂⊂𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂⊂𝑊₀^1,𝑝Ω,𝜔₁. (2.5)

In this paper we use the following four results.

Theorem 2.5 (The Weighted Sobolev Inequality). Let Ω be an open bounded set in ℝ^𝑁(𝑁≥2) and 𝜔₂∈𝐴_𝑝(1<𝑝<∞). There exist constants 𝐶_Ω and 𝛿 positive such that for all 𝑢∈𝐶^∞₀(Ω) and all 𝑟₂ satisfying 1≤𝑟₂≤𝑁/(𝑁−1)+𝛿, ‖𝑢‖_𝐿𝑝^∗_(Ω,𝜔2)≤𝐶_Ω‖∇𝑢‖_𝐿^𝑝_(Ω,𝜔2), (2.6) where 𝑝^∗=𝑝𝑟₂.

Proof. See Theorem  1.3 in [3].

The following lemma is due to Stampacchia (see [14], Lemma  4.1).

Lemma 2.6. Let 𝛼, 𝛽, 𝐶, and 𝑘₀ be real positive numbers, where 𝛽>1.
Let 𝜑∶ℝ₊→ℝ₊ be a decreasing function such that 𝜑(𝑙)≤𝐶(𝑙−𝑘)^𝛼𝜑(𝑘)^𝛽 (2.7) for all 𝑙>𝑘≥𝑘₀. Then 𝜑(𝑘₀+𝑑)=0, where 𝑑^𝛼=𝐶[𝜑(𝑘₀)]^𝛽−12^{𝛼𝛽/(𝛽−1)}.

Lemma 2.7. If 𝜔∈𝐴_𝑝, then (|𝐸|/|𝐵|)^𝑝≤𝐶_𝑝,𝜔(𝜇(𝐸)/𝜇(𝐵)), whenever 𝐵 is a ball in ℝ^𝑁 and 𝐸 is a measurable subset of 𝐵.

Proof. See Theorem  15.5 Strong doubling of 𝐴_𝑝-weights in [4].

By Lemma 2.7, if 𝜇(𝐸)=0, then |𝐸|=0.

Lemma 2.8. Let 𝜔₁ and 𝜔₂ be 𝐴_𝑝-weights, 1<𝑝<∞, 𝜔₁≤𝜔₂, and a sequence {𝑢_𝑛}, 𝑢_𝑛∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) satisfies the following: (i)𝑢_𝑛⇀𝑢 in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) and 𝜇₂-a.e. in Ω;(ii)∫_Ω⟨𝒜(𝑥,𝑢_𝑛,∇𝑢_𝑛)−𝒜(𝑥,𝑢_𝑛,∇𝑢),∇(𝑢_𝑛−𝑢)⟩𝜔₂𝑑𝑥→0 with 𝑛→∞.
Then 𝑢_𝑛→𝑢 in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂).

Proof. The proof of this lemma follows the line of Lemma  5 in [10].

Definition 2.9. We say that 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) is a (weak) solution of problem (𝑃) if _Ω𝜔₂𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜑𝑑𝑥+_Ω𝑔(𝑥,𝑢)𝜑𝜔₁𝑑𝑥+_Ω𝐻(𝑥,𝑢,∇𝑢)𝜑𝜔₂𝑑𝑥=_Ω𝑓𝜑𝑑𝑥, (2.8) for all 𝜑∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω).

3. Proof of Theorem 1.1

Set 𝜆=(𝐶₁/𝛼₁)+1 and define for 𝑘>0 the functions 𝜙∈𝐶¹(ℝ) and 𝐺_𝑘∈𝑊^1,∞(ℝ) by𝜙(𝑠)=e^𝜆𝑠−1,if−𝑠≥0,e^−𝜆𝑠+1,if𝐺𝑠≤0,_𝑘(𝑠)=𝑠−𝑘,if𝑠≥𝑘,0,if−𝑘≤𝑠≤𝑘,𝑠+𝑘,if𝑠≤−𝑘. (3.1)

If 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) is a solution of problem (𝑃), define the set 𝐴(𝑘)={𝑥∈Ω∶|𝑢(𝑥)|>𝑘}. We will use the test functions 𝑣(𝑥)=𝜙(𝐺_𝑘(𝑢(𝑥))). Since 𝑢∈𝑊₀^1,2(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) and Ω is a bounded set in ℝ^𝑁, then 𝑣(𝑥)=𝜙(𝐺_𝑘(𝑢(𝑥)))∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) and 𝑣(𝑥)=𝜙(|𝑢|−𝑘)⁺𝜒_𝐴(𝑘)(𝑢),𝑢|)sign∇𝑣=𝜙′(|−𝑘⁶𝜒_𝐴(𝑘)∇𝑢, (3.2) where 𝜒_𝐴(𝑘) is the characteristic function of the set 𝐴(𝑘) (by sign(𝑢(𝑥)) we mean the function equal to +1 for 𝑢(𝑥)>0 and to −1 for 𝑢(𝑥)<0).

Since 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω), we have that 𝑣∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω).

Using the function 𝑣 in (2.8) we obtain_Ω𝜔₂𝒜(𝑥,𝑢,∇𝑢)⋅∇𝑣𝑑𝑥+_Ω𝑔(𝑥,𝑢)𝑣𝜔₁𝑑𝑥+_Ω𝐻(𝑥,𝑢,∇𝑢)𝑣𝜔₂𝑑𝑥=_Ω𝑓𝑣𝑑𝑥. (3.3) We have the following estimates.(i) By (H3) we obtain_Ω𝜔₂𝒜(𝑥,𝑢,∇𝑢)⋅∇𝑣𝑑𝑥=_𝐴(𝑘)𝜙′(|𝑢|−𝑘)⁺𝒜(𝑥,𝑢,∇𝑢)⋅∇𝑢𝜔₂𝑑𝑥≥𝛼_1𝐴(𝑘)|∇𝑢|^𝑝𝜙′(|𝑢|−𝑘)⁺𝜔₂𝑑𝑥. (3.4)(ii)By (H7) we obtain_Ω𝑔(𝑥,𝑢)𝑣𝜔₁𝑑𝑥=_𝐴(𝑘)𝑔(𝑥,𝑢)𝜙(|𝑢|−𝑘)⁺𝜔₁𝑑𝑥≥𝛼_0𝐴(𝑘)|𝑢|^𝑝−1𝜙(|𝑢|−𝑘)⁺𝜔₁𝑑𝑥. (3.5)(iii) Using (H9) we obtain_Ω𝐻(𝑥,𝑢,∇𝑢)𝑣𝜔₂𝑑𝑥≤_Ω|𝐻(𝑥,𝑢,∇𝑢)||𝑣|𝜔₂≤𝑑𝑥_𝐴(𝑘)𝐶₀+𝐶₁|∇𝑢|^𝑝𝜙(|𝑢|−𝑘)⁺𝜔₂𝑑𝑥. (3.6) And we also have𝑓𝑣𝑑𝑥≤_Ω𝐴(𝑘)𝑓𝜙(|𝑢|−𝑘)⁺𝑑𝑥. (3.7) Hence in (3.3) we obtain𝛼_1𝐴(𝑘)|∇𝑢|^𝑝𝜙^′(|𝑢|−𝑘)⁺𝜔₂𝑑𝑥+𝛼_0𝐴(𝑘)|𝑢|^𝑝−1𝜙(|𝑢|−𝑘)⁺𝜔₁≤𝑑𝑥_𝐴(𝑘)𝐶₀+𝐶₁|∇𝑢|^𝑝𝜙(|𝑢|−𝑘)⁺𝜔₂𝑑𝑥+_𝐴(𝑘)𝑓𝜙(|𝑢|−𝑘)⁺𝑑𝑥. (3.8)

Since 𝜆=(𝐶₁/𝛼₁)+1, we have for 𝑠≥0𝛼₁𝜙^′(𝑠)−𝐶₁𝜙(𝑠)=𝛼₁𝜆e^𝜆𝑠−𝐶₁e^𝜆𝑠=𝛼−1₁𝜆−𝐶₁e^𝜆𝑠+𝐶₁=𝛼₁e^𝜆𝑠+𝐶₁≥𝛼₁e^𝜆𝑠=𝛼₁𝜆^𝑝𝜆e^{𝜆𝑠/𝑝𝑝}=𝛼₁𝜆^𝑝𝜙^′𝑠𝑝^𝑝. (3.9) Hence in (3.8) we obtain_𝐴(𝑘)𝛼₁|∇𝑢|^𝑝𝜙′(|𝑢|−𝑘)⁺−𝐶₁|∇𝑢|^𝑝𝜙(|𝑢|−𝑘)⁺𝜔₂𝑑𝑥+𝛼_0𝐴(𝑘)|𝑢|^𝑝−1𝜙(|𝑢|−𝑘)⁺𝜔₁𝑑𝑥≤_𝐴(𝑘)𝑓+𝐶₀𝜔₂𝜙(|𝑢|−𝑘)⁺𝑑𝑥. (3.10) Using (3.9) and 𝑘<|𝑢(𝑥)| if 𝑥∈𝐴(𝑘), we obtain𝛼₁𝜆^𝑝_𝐴(𝑘)𝜙^′(|𝑢|−𝑘)⁺𝑝∇𝑢^𝑝𝜔₂𝑑𝑥+𝛼₀𝑘^𝑝−1_𝐴(𝑘)𝜙(|𝑢|−𝑘)⁺𝜔₁≤𝑑𝑥_𝐴(𝑘)𝑓+𝐶₀𝜔₂𝜙(|𝑢|−𝑘)⁺𝑑𝑥. (3.11) Let us define the function 𝜓_𝑘 by 𝜓_𝑘(𝑥)=𝜙((|𝑢(𝑥)|−𝑘)⁺/𝑝). We have that 𝜓_𝑘∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) and∇𝜓_𝑘=1𝑝𝜙^′(|𝑢|−𝑘)⁺𝑝𝜒_𝐴(𝑘)sign(𝑢)∇𝑢. (3.12) We have the following.(a)For all 𝑠≥0, e^𝜆𝑠−1≥(e^{𝜆𝑠/𝑝}−1)^𝑝.(b)There exists, a constant 𝐶₂>0 (𝐶₂=𝐶₂(𝜆,𝑝)) such that for all 𝑠≥1e^𝜆𝑠−1≤𝐶₂e^{𝜆𝑠/𝑝}−1^𝑝,𝜆e^𝜆𝑠≤𝐶₂𝜆e^{𝜆𝑠/𝑝}−1^𝑝. (3.13)

This implies the following.(I1)𝜙((|𝑢|−𝑘)⁺)=e^{𝜆(|𝑢|−𝑘)+}−1≥(e^{𝜆(|𝑢|−𝑘)+/𝑝}−1)^𝑝=|𝜓_𝑘|^𝑝 a.e. on Ω.(I2) If 𝑥∈A(𝑘+1), then𝜙(|𝑢|−𝑘)⁺=e^{𝜆(|𝑢|−𝑘)+}−1≤𝐶₂e^{𝜆(|𝑢|−𝑘)+/𝑝}−1=𝐶₂𝜓_𝑘^𝑝,𝜙^′(|𝑢|−𝑘)⁺=(|𝑢|−𝑘)⁺e^{𝜆(|𝑢|−𝑘)+}≤𝐶₂𝜆e^{𝜆(|𝑢|−𝑘)+/𝑝}−1^𝑝=𝐶₂𝜆𝜓_𝑘^𝑝. (3.14) Combining (I1) and (I2) with (3.11) and (3.12) we obtain𝛼₁𝑝^𝑝𝜆^𝑝_Ω∇𝜓_𝑘^𝑝𝜔₂𝑑𝑥+𝛼₀𝑘^𝑝−1_Ω𝜓_𝑘^𝑝𝜔₁≤𝑑𝑥_𝐴(𝑘)𝑓+𝐶₀𝜔₂𝜙(|𝑢|−𝑘)⁺≤𝑑𝑥_𝐴()𝑘+1𝑓+𝐶₀𝜔₂𝐶₂𝜓_𝑘^𝑝+𝑑𝑥_{𝐴(𝑘)()−𝐴𝑘+1}𝑓+𝐶₀𝜔₂𝜙(|𝑢|−𝑘)⁺𝑑𝑥. (3.15) Define the function ℎ=|𝑓|+𝐶₀𝜔₂. Since 𝑓/𝜔₁∈𝐿^𝑞(Ω,𝜔₁) and 𝜔₂/𝜔₁∈𝐿^𝑞(Ω,𝜔₁), we have that ℎ/𝜔₁∈𝐿^𝑞(Ω,𝜔₁). Hence_Ωℎ𝜓_𝑘^𝑝𝑑𝑥=_Ωℎ𝜔₁𝜓_𝑘^𝑝𝜔₁^1/𝑞𝜔₁^1/𝑞′≤ℎ𝑑𝑥𝜔_{1𝐿𝑞(Ω,𝜔1)}𝜓_𝑘^𝑝_{𝐿′𝑝𝑞Ω,𝜔1}. (3.16) If 𝑥∈𝐴(𝑘)−𝐴(𝑘+1), we have 𝑘<|𝑢|<𝑘+1. Hence𝜙(|𝑢|−𝑘)⁺=e^{𝜆(|𝑢|−𝑘)+}−1≤e^𝜆−1, (3.17) and we obtain_{𝐴(𝑘)()−𝐴𝑘+1}𝑓+𝐶₀𝜔₂𝜙(|𝑢|−𝑘)⁺𝑑𝑥≤_{𝐴(𝑘)()−𝐴𝑘+1}e^𝜆≤−1ℎ𝑑𝑥e^𝜆_𝐴(𝑘)ℎ𝑑𝑥. (3.18)

By Theorem 2.5, (3.15), and (3.18) we have𝛼₁𝑝^𝑝𝜆^𝑝1𝐶^𝑝_Ω^Ω𝜓_𝑘^𝑝∗𝜔₂𝑑𝑥^𝑝/𝑝∗+𝛼₀𝑘₀^𝑝−1_Ω𝜓_𝑘^𝑝𝜔₁𝑑𝑥≤𝐶_2Ωℎ𝜓_𝑘^𝑝𝑑𝑥+e^𝜆_𝐴(𝑘)ℎ𝑑𝑥. (3.19) Therefore, there exist positive constants 𝐶₃ and 𝐶₄ (depending only on Ω, 𝛼₁, 𝑝, 𝜆, 𝐶₂, and 𝐶_Ω) such that𝐶_3Ω𝜓_𝑘^𝑝∗𝜔₂𝑑𝑥^𝑝/𝑝∗+𝐶₄𝛼₀𝑘₀^𝑝−1_Ω𝜓_𝑘^𝑝𝜔₁≤𝑑𝑥_Ωℎ𝜓_𝑘^𝑝𝑑𝑥+_𝐴(𝑘)ℎ𝑑𝑥. (3.20) Since 𝑟₂/(𝑟₂−1)<𝑞, then 𝑞^′<𝑟₂ and 𝑝<𝑝𝑞^′<𝑝^∗. For 0<𝜃<1 such that 1/𝑝𝑞′=(𝜃/𝑝)+(1−𝜃)/𝑝^∗, using an interpolation inequality, Young’s inequality (with 0<𝛾<∞), and Hölder’s inequality with exponents 𝑞 and 𝑞′ we thus obtain_Ωℎ𝜓_𝑘^𝑝ℎ𝑑𝑥≤𝜔_{1𝐿𝑞(Ω,𝜔1)}𝜓_𝑘^𝑝_{𝐿′𝑝𝑞Ω,𝜔1}≤ℎ𝜔_{1𝐿𝑞(Ω,𝜔1)}𝜓_𝑘𝐿^𝜃𝑝_𝑝Ω,𝜔1𝜓_𝑘𝐿^{(1−𝜃)𝑝}_{𝑝∗Ω,𝜔1}≤(1−𝜃)𝛾^1/(1−𝜃)𝜓_𝑘^𝑝_{𝐿𝑝∗Ω,𝜔1}+𝜃𝛾^−1/𝜃ℎ𝜔_1𝐿^1/𝜃_{𝑞(Ω,𝜔1)}𝜓_𝑘^𝑝_{𝐿𝑝Ω,𝜔1}. (3.21) Hence in (3.20) we obtain𝐶_3Ω𝜓_𝑘^𝑝∗𝜔₂𝑑𝑥^𝑝/𝑝∗+𝐶₄𝛼₀𝑘₀^𝑝−1_Ω𝜓_𝑘^𝑝𝜔₁≤()𝛾𝑑𝑥1−𝜃^1/(1−𝜃)𝜓_𝑘^𝑝_{𝐿𝑝∗Ω,𝜔1}+𝜃𝛾^−1/𝜃ℎ𝜔_1𝐿^1/𝜃_{𝑞(Ω,𝜔1)}𝜓_𝑘^𝑝_{𝐿𝑝Ω,𝜔1}+_𝐴(𝑘)ℎ𝑑𝑥≤(1−𝜃)𝛾^1/(1−𝜃)𝜓_𝑘^𝑝_{𝐿𝑝∗Ω,𝜔2}+𝜃𝛾^−1/𝜃ℎ𝜔_1𝐿^1/𝜃_{𝑞(Ω,𝜔1)}𝜓_𝑘^𝑝_{𝐿𝑝Ω,𝜔1}+_𝐴(𝑘)ℎ𝑑𝑥. (3.22) Now, we can choose 𝛾 in order to have (1−𝜃)𝛾^1/(1−𝜃)=𝐶₃/2 and 𝑘₀ such that𝐶₄𝛼₀𝑘₀^𝑝−1=𝜃𝛾^−1/𝜃ℎ𝜔_1𝐿^1/𝜃_𝑞Ω,𝜔1. (3.23) We obtain, from (3.22), that for every 𝑘≥𝑘₀ it results𝐶₃2_Ω𝜓_𝑘^𝑝∗𝜔₂𝑑𝑥^𝑝/𝑝∗≤_𝐴(𝑘)ℎℎ𝑑𝑥≤𝜔_{1𝐿𝑞(Ω,𝜔1)}𝜇₁𝐴_𝑘^1/𝑞′, (3.24) where 𝜇₁(𝐴_𝑘)=∫_𝐴𝑘𝜔₁𝑑𝑥. Hence for all 𝑘≥𝑘₀ we have_Ω𝜓_𝑘^𝑝∗𝜔₂2𝑑𝑥≤𝐶₃ℎ𝜔_{1𝐿𝑞(Ω,𝜔1)}𝜇₁(𝐴(𝑘))^{1/𝑞′𝑝∗/𝑝}=2𝐶₃^𝑝∗/𝑝ℎ𝜔₁^𝑝∗_𝐿^/𝑝_𝑞Ω,𝜔1𝜇₁(𝐴(𝑘))^{𝑝∗/𝑝𝑞′}=𝐶₅𝜇₁(𝐴(𝑘))^{𝑝∗/𝑝𝑞′}. (3.25) Let us now take 𝑙>𝑘≥𝑘₀. Then we have𝜇₁(𝐴(𝑙))𝜆𝑙−𝑘𝑝^𝑝∗≤𝜇₁(𝐴(𝑘))𝜙𝑙−𝑘𝑝^𝑝∗≤_𝐴(𝑘)𝜓_𝑘^𝑝∗𝜔₁𝑑𝑥≤_Ω𝜓_𝑘^𝑝∗𝜔₂𝑑𝑥. (3.26) Therefore for all 𝑙>𝑘≥𝑘₀ we obtain (by (3.25) and (3.26))(𝑙−𝑘)^𝑝∗𝜇₁(𝐴(𝑙≤𝑝))^𝑝∗𝜆^𝑝∗𝐶₅𝜇₁(𝐴(𝑘))^{𝑝∗/𝑝𝑞′}=𝐶₆𝜇₁(𝐴(𝑘))^{𝑝∗/𝑝𝑞′}, (3.27) that is, 𝜇₁(𝐴(𝑙))≤(𝐶₆/(𝑙−𝑘)^𝑝∗)[𝜇₁(𝐴(𝑘))]^{𝑝∗/𝑝𝑞′}.

Let 𝜑(𝑘)=𝜇₁(𝐴(𝑘)). Since 𝛽=𝑝^∗/𝑝𝑞′>1, by Lemma 2.6 there exists a constant 𝐶₇>0 such that𝜇₁(𝐴(𝑘))=0,∀𝑘≥𝐶₇. (3.28)

Using Lemma 2.7 we have |𝐴(𝑘)|=0 for all 𝑘≥𝐶₇. Therefore any solution 𝑢 of problem (𝑃) satisfies the estimate ‖𝑢‖_𝐿^∞_(Ω)≤𝐶₇.

4. Proof of Theorem 1.2

Step 1. Let us define for 𝑚∈ℕ the approximation 𝐻_𝑚=𝐻𝑥,𝜂,𝜉𝑥,𝜂,𝜉(𝐻.1+1/𝑚)𝑥,𝜂,𝜉 (4.1) We have that |𝐻_𝑚(𝑥,𝜂,𝜉)|≤|𝐻(𝑥,𝜂,𝜉)|, |𝐻_𝑚(𝑥,𝜂,𝜉)|<𝑚, and 𝐻_𝑚(𝑥,𝜂,𝜉) satisfies the conditions (H9) and (H12). We consider the approximate problem −div𝜔₂𝒜(𝑥,𝑢,∇𝑢)+𝑔(𝑥,𝑢)𝜔₁+𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜔₂𝑢(𝑥),onΩ,=0,on𝜕Ω. (𝑃_𝑚) We say that 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is (weak) solution of problem (𝑃_𝑚) if _Ω𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜑𝜔₂𝑑𝑥+_Ω𝑔(𝑥,𝑢)𝜑𝜔₁+𝑑𝑥_Ω𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜑𝜔₂𝑑𝑥=_Ω𝑓𝜑𝑑𝑥, (4.2) for all 𝜑∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂). We will prove that there exists at least one solution 𝑢_𝑚 of the problem (𝑃_𝑚). For 𝑢,𝑣,𝜑∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) we define 𝐵𝑢,𝑣,𝜑=_Ω𝜔₂𝒜()𝐵𝑥,𝑢,∇𝑣⋅∇𝜑𝑑𝑥,_𝑚𝑢,𝜑=_Ω𝑔(𝑥,𝑢)𝜑𝜔₁𝑑𝑥+_Ω𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜑𝜔₂𝑇𝜑𝑑𝑥,=_Ω𝑓𝜑𝑑𝑥. (4.3) Then 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is a (weak) solution of problem (𝑃_𝑚) if 𝐵𝑢,𝑢,𝜑+𝐵_𝑚𝑢,𝜑=𝑇𝜑,forall𝜑∈𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂. (4.4) Let 𝑎(𝑢,𝑣,𝜑)=𝐵(𝑢,𝑣,𝜑)+𝐵_𝑚(𝑢,𝜑).(i)Using (H4) we obtain𝐵𝑢,𝑣,𝜑≤𝐾_{2𝐿𝑝′(Ω,𝜔2)}+ℎ_1𝐿∞(Ω)‖𝑢‖^𝑝/𝑝′_{𝑊01,𝑝Ω,𝜔1,𝜔2}+ℎ_2𝐿∞(Ω)‖𝑣‖^𝑝/𝑝′_{𝑊01,𝑝Ω,𝜔1,𝜔2}×𝜑_𝑊0^1,𝑝_{(Ω,𝜔1,𝜔2)}. (4.5)(ii) Using (H6) and |𝐻_𝑚(𝑥,𝜂,𝜉)|≤𝑚, we obtain𝐵_𝑚𝑢,𝜑≤𝐾_{1𝐿𝑝′(Ω,𝜔1)}+ℎ_3𝐿∞(Ω)‖𝑢‖^𝑝′_𝑊^/𝑝_{01,𝑝Ω,𝜔1,𝜔2}+𝑚𝐶_Ω𝜇₂(Ω)^1′/𝑝×𝜑_𝑊0^1,𝑝_{(Ω,𝜔1,𝜔2)}. (4.6) Hence, 𝑎𝑢,𝑣,𝜑≤𝐾_{2𝐿𝑝′(Ω,𝜔2)}+ℎ_1𝐿∞(Ω)+ℎ_3𝐿∞(Ω)‖𝑢‖_𝑊^𝑝/𝑝′_{01,𝑝Ω,𝜔1,𝜔2}+𝐾_{1𝐿𝑝′(Ω,𝜔1)}+ℎ_2𝐿∞(Ω)‖𝑣‖^𝑝/𝑝′_{𝑊01,𝑝Ω,𝜔1,𝜔2}+𝑚𝐶_Ω𝜇₂(Ω)^1/𝑝′𝜑_𝑊0^1,𝑝_{(Ω,𝜔1,𝜔2)}. (4.7) For each (𝑢,𝑣)∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)×𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) we have that 𝑎(𝑢,𝑣,.) is linear and continuous. Hence, there exists a linear and continuous operator 𝐴(𝑢,𝑣)∶𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂⟶𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂^∗ (4.8) such that ⟨𝐴(𝑢,𝑣),𝜑⟩=𝑎(𝑢,𝑣,𝜑). We set 𝐴(𝑢)=𝐴(𝑢,𝑢),∀𝑢∈𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂. (4.9) The operator 𝐴∶𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)→[𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗ is semimonotone; that is, by similar arguments as in the proof of Theorem  2 in [11] we have the following.(i)⟨𝐴(𝑢,𝑢)−𝐴(𝑢,𝑣),𝑢−𝑣⟩≥0 for all 𝑢,𝑣∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂).(ii)For each 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂), the operator 𝑣↦𝐴(𝑢,𝑣) is hemicontinuous and bounded from 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) to [𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗ and,for each 𝑣∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) the operator 𝑢↦𝐴(𝑢,𝑣) is hemicontinuous and bounded from 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) to [𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗.(iii) If 𝑢_𝑛⇀𝑢 in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) and ⟨𝐴(𝑢_𝑛,𝑢_𝑛)−𝐴(𝑢_𝑛,𝑢),𝑢_𝑛−𝑢⟩→0, then 𝐴(𝑢_𝑛,𝑢)⇀𝐴(𝑢,𝑣) in [𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗ as 𝑛→∞ for all 𝑣∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂).(iv)If 𝑣∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂), 𝑢_𝑛⇀𝑢 in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂), and 𝐴(𝑢_𝑛,𝑣)⇀𝑣 in [𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗, then ⟨𝐴(𝑢_𝑛,𝑣),𝑢_𝑛⟩→⟨𝑣,𝑢⟩ as 𝑛→∞.(v)The operator 𝐴∶𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)→[𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗ is bounded.Hence the operator 𝐴∶𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)→[𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗ is pseudomonotone (see [15]).(vi)By (H3), (H7), and (H12) we have𝐴(𝑢),𝑢≥𝛼_1Ω|∇𝑢|^𝑝𝜔₂𝑑𝑥+𝛼_0Ω|𝑢|^𝑝𝜔₁𝑑𝑥≥𝛼‖𝑢‖^𝑝_{𝑊01,𝑝Ω,𝜔1,𝜔2}, (4.10) where 𝛼=min{𝛼₀,𝛼₁}. Since 𝑝>1, we have 𝐴(𝑢),𝑢‖𝑢‖_{𝑊01,𝑝Ω,𝜔1,𝜔2}⟶∞on‖𝑢‖_𝑊0^1,𝑝_{(Ω,𝜔1,𝜔2)}⟶∞; (4.11) that is, the operator 𝐴 is coercive. Then, by Theorem  27.B in [15], for each 𝑇∈[𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)]^∗, the equation 𝐴𝑢=𝑇,𝑢∈𝑊₀^1,𝑝Ω,𝜔₁,𝜔₂ (4.12) has a solution. Therefore, the problem (𝑃_𝑚), has a solution 𝑢_𝑚∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂).

Step 2. We will show that 𝑢_𝑚∈𝐿^∞(Ω) and ‖𝑢_𝑚‖_𝐿∞(Ω)≤𝐶, where 𝐶 is independent of 𝑚. If 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is a solution of problem (𝑃_𝑚), we define 𝑢_𝑛(𝑥)=𝑇_𝑛(𝑢(𝑥=𝑢(𝑥),))if|𝑢(𝑥)|≤𝑛,𝑛,if𝑢(𝑥)>𝑛,−𝑛,if𝑢(𝑥)<−𝑛. (4.13) We have 𝐷_𝑖𝑢_𝑛=𝐷_𝑖𝑢 if |𝑢(𝑥)|≤𝑛. For 𝑘>0, let us define the function 𝜓_𝑛(𝑥)=sign(𝑢_𝑛(𝑥))max{|𝑢_𝑛(𝑥)|−𝑘,0}. We have 𝜓_𝑛∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω).
Now consider the function Φ(𝑡)=𝑡+𝑘,if𝑡≤−𝑘,0,if|𝑡|≤𝑘,𝑡−𝑘,if𝑡≥𝑘. (4.14) Since Φ is a Lipschitz function and Φ(0)=0, then Φ(𝜓_𝑛)∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂). Moreover, 𝐷_𝑖Φ(𝜓_𝑛)=Φ′(𝜓_𝑛)𝐷_𝑖𝜓 and Φ^′𝑢_𝑛∇𝑢_𝑛⟶Φ^′(𝑢)∇𝑢,𝜇₂−𝑎.𝑒.inΩ. (4.15) We also have, for all measurable subset 𝐸⊂Ω, _𝐸Φ^′𝑢_𝑛∇𝑢_𝑛^𝑝𝜔₂𝑑𝑥≤_𝐸∇𝑢_𝑛^𝑝𝜔₂𝑑𝑥. (4.16) By applying Vitali’s Convergence Theorem, with 𝜓=Φ(𝑢), we obtain ∇𝜓_𝑛⟶∇𝜓in𝐿^𝑝Ω,𝜔₂. (4.17) Since 𝜔₁≤𝜔₂, we obtain 𝜓_𝑛−𝜓_{𝐿𝑝(Ω,𝜔1)}≤𝜓_𝑛−𝜓_{𝐿𝑝(Ω,𝜔2)}≤𝐶_Ω∇𝜓_𝑛−∇𝜓_{𝐿𝑝(Ω,𝜔2)}. (4.18) Hence 𝜓_𝑛→𝜓in 𝐿^𝑝(Ω,𝜔₁). Since 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) is a solution of problem (𝑃_𝑚) and 𝜓_𝑛∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω), we have _Ω𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜓_𝑛𝜔₂𝑑𝑥+_Ω𝑔(𝑥,𝑢)𝜓_𝑛𝜔₁𝑑𝑥+_Ω𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜓_𝑛𝜔₂𝑑𝑥=_Ω𝑓𝜓_𝑛𝑑𝑥. (4.19) Using (H4), (H6), |𝐻_𝑚(𝑥,𝜂,𝜉)|≤𝑚, (4.17), and (4.18), we obtain in (4.19) as 𝑛→∞_Ω𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜓𝜔₂𝑑𝑥+_Ω𝑔(𝑥,𝑢)𝜓𝜔₁𝑑𝑥+_Ω𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜓𝜔₂𝑑𝑥=_Ω𝑓𝜓𝑑𝑥. (4.20) Using 𝜑=𝜓𝜒_𝐴(𝑘) in (4.2) (where 𝐴(𝑘)={𝑥∈Ω∶|𝑢(𝑥)|>𝑘}) we obtain _𝐴(𝑘)𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜓𝜔₂𝑑𝑥+_𝐴(𝑘)𝑔(𝑥,𝑢)𝜓𝜔₁𝑑𝑥+_𝐴(𝑘)𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜓𝜔₂=𝑑𝑥_𝐴(𝑘)𝑓𝜓𝑑𝑥. (4.21) Since 𝜓=Φ(𝑢)=𝑢+𝑘,if𝑢≤−𝑘,0,if|𝑢|≤𝑘,𝑢−𝑘,if𝑢≥𝑘, (4.22) we obtain the following.(i) By (H7) we have 𝑔(𝑥,𝜂)𝜂≥0 for all 𝜂∈ℝ, and_𝐴(𝑘)𝑔(𝑥,𝑢)𝜓𝜔₁𝑑𝑥=_{{}𝑢≤−𝑘}𝑔(𝑥,𝑢)(𝑢+𝑘)𝜔₁+𝑑𝑥_{{}𝑢≥𝑘}𝑔(𝑥,𝑢)(𝑢−𝑘)𝜔₁𝑑𝑥≥0. (4.23)(ii) Using (H12) we have 𝐻_𝑚(𝑥,𝜂,𝜉)𝜂≥0 for all 𝜂∈ℝ, and_𝐴(𝑘)𝐻_𝑚(𝑥,𝑢,∇𝑢)𝜓𝜔₂𝑑𝑥=_{{𝑢≤−𝑘}}𝐻_𝑚(𝑥,𝑢,∇𝑢)(𝑢+𝑘)𝜔₂+𝑑𝑥_{{𝑢≥𝑘}}𝐻_𝑚(𝑥,𝑢,∇𝑢)(𝑢−𝑘)𝜔₂𝑑𝑥≥0. (4.24) We have ∇𝜓=∇𝑢 in 𝐴(𝑘). Using (H3), (i), and (ii) we obtain in (4.21) 𝛼_1𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥≤_𝐴(𝑘)𝑓𝜓𝑑𝑥. (4.25) By Theorem  2.1.14 in [13] there is a positive constant 𝐶 such that _𝜔𝜓^𝑝𝜔₁𝑑𝑥≤_Ω𝜓^𝑝𝜔₂𝑑𝑥≤𝐶_Ω∇𝜓^𝑝𝜔₂𝑑𝑥. (4.26) Then we obtain _𝐴(𝑘)𝑓𝜓𝑑𝑥≤_𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁^1/𝑝′_𝐴(𝑘)𝜓^𝑝𝜔₁𝑑𝑥^1/𝑝≤𝐶_𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁^1/𝑝′_𝐴(𝑘)∇𝜓^𝑝𝜔₂𝑑𝑥^1/𝑝=𝐶_𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁^1/𝑝′_𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥^1/𝑝. (4.27) Using (4.27) and Young’s inequality we obtain in (4.25) (for all 𝜀>0) 𝛼_1𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥≤𝐶_𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁^1/𝑝′_𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥^1/𝑝≤𝐶𝜀_𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥+𝐶(𝜀)_𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁𝑑𝑥, (4.28) where 𝐶(𝜀)=(𝜀𝑝)^{−𝑝′/𝑝}/𝑝′. We can choose 𝜀>0 so that 𝐶𝜀=𝛼₁/2, and there exists a constant 𝐶₈ such that _𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥≤𝐶_8𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₁𝑑𝑥. (4.29) Using Sobolev’s inequality (Theorem 2.5) and Hölder’s inequality with exponents 𝑞 and 𝑞′ we obtain (since 𝑞>𝑝^′(𝑟/(𝑟−1))>𝑝^′) (|𝑢_𝐴(𝑘)|−𝑘)^𝑝∗𝜔₁𝑑𝑥^𝑝/𝑝∗=_𝐴(𝑘)𝜓^𝑝∗𝜔₁𝑑𝑥^𝑝/𝑝∗≤_𝐴(𝑘)𝜓^𝑝∗𝜔₂𝑑𝑥^𝑝/𝑝∗≤𝐶_𝐴(𝑘)∇𝜓^𝑝𝜔₂𝑑𝑥=𝐶_𝐴(𝑘)|∇𝑢|^𝑝𝜔₂𝑑𝑥≤𝐶𝐶_8𝐴(𝑘)𝑓𝜔₁^𝑝′𝜔₂𝑑𝑥≤𝐶_9Ω𝑓𝜔₁^𝑞𝜔₁𝑑𝑥^𝑝′/𝑞𝜇₁(Ω)^{1−(𝑝′/𝑞)}. (4.30) Let us now take 𝑙>𝑘>0 and observe that 𝐴(𝑙)⊂𝐴(𝑘). Then, from the previous inequality, it follows that 𝜇₁(𝐴(𝑙))(𝑙−𝑘)^𝑝∗=_𝐴(𝑙)(𝑙−𝑘)^𝑝∗𝜔₁≤𝑑𝑥_𝐴(𝑙)(|𝑢|−𝑘)^𝑝∗𝜔₁𝑑𝑥≤_𝐴(𝑘)(|𝑢|−𝑘)^𝑝∗𝜔₁𝑑𝑥≤𝐶_9Ω𝑓𝜔₁^𝑞𝜔₁𝑑𝑥^{𝑝′𝑝∗/𝑞𝑝}𝜇₁(𝐴(𝑘))^{(1−(𝑝′/𝑞))(𝑝∗/𝑝)}. (4.31) Hence we obtain 𝜇₁(𝐴(𝑙≤𝐶))⁹∫_Ω𝑓/𝜔₁^𝑞𝜔₁𝑑𝑥^{(𝑝′𝑝∗)/𝑞𝑝}(𝑙−𝑘)^𝑝∗𝜇₁(𝐴(𝑘))^{(1−(𝑝′/𝑞))(𝑝∗/𝑝)}. (4.32) Since (1−(𝑝^′/𝑞))(𝑝^∗/𝑝)>1, by Lemma 2.6 there exists a constant 𝐶₁₀>0 such that 𝜇₁(𝐴(𝑘))=0 for all 𝑘≥𝐶₁₀, and using Lemma 2.7 we obtain |𝐴(𝑘)|=0. Therefore if 𝑢_𝑚 is a solution of problem (𝑃_𝑚), we have ‖𝑢_𝑚‖_𝐿^∞_(Ω)≤𝐶₁₀ and 𝐶₁₀ is independent of 𝑚.

Step 3. Since 𝑢_𝑚∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) and ‖𝑢_𝑚‖_𝐿^∞_(Ω)≤𝐶₁₀, then the sequence {𝑢_𝑚} is relative compact in the strong topology of 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) (by apply the analogous results of [10] and Lemma 2.8). Then, by extracting a subsequence {𝑢_𝑚𝑘} which strongly converges in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) (i.e., there exists 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂) such that 𝑢_𝑚𝑘→𝑢 in 𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)), we have for all 𝜑∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω)_Ω𝒜𝑥,𝑢_𝑚𝑘,∇𝑢_𝑚𝑘⋅∇𝜑𝜔₂+_Ω𝑔𝑥,𝑢_𝑚𝑘𝜑𝜔₁+𝑑𝑥_Ω𝐻_𝑚𝑘𝑥,𝑢_𝑚𝑘,∇𝑢_𝑚𝑘𝜑𝜔₂𝑑𝑥⟶_Ω𝒜(𝑥,𝑢,∇𝑢)⋅∇𝜑𝜔₂+_Ω𝑔(𝑥,𝑢)𝜑𝜔₁𝑑𝑥+_Ω𝐻(𝑥,𝑢,∇𝑢)𝜑𝜔₂𝑑𝑥. (4.33) Therefore 𝑢∈𝑊₀^1,𝑝(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω) is the solution of problem (𝑃).

Example 4.1. Let Ω={(𝑥,𝑦)∈ℝ²∶𝑥²+𝑦²<1}, and consider the weights 𝜔₁(𝑥,𝑦)=(𝑥²+𝑦²)^1/2 and 𝜔₂=(𝑥²+𝑦²)^1/4 (𝜔₁ and 𝜔₂∈𝐴₂), and the functions 𝒜∶Ω×ℝ×ℝ²→ℝ², 𝑔∶Ω×ℝ→ℝ, and 𝐻∶Ω×ℝ×ℝ²→ℝ defined by 𝒜𝑥,𝑦,𝜂,𝜉=ℎ₂𝑔𝑥,𝑦𝜉,𝑥,𝑦,𝜂=𝜂cos²,𝐻=𝜉𝑥𝑦+1𝑥,𝑦,𝜂,𝜉²sin𝑥𝑦+arctan𝜂, (4.34) where ℎ₂(𝑥,𝑦)=2e^𝑥2+𝑦2. Let us consider the partial differential operator 𝐿𝑢𝑥,𝑦=−div𝜔₂𝑥,𝑦𝒜𝑥,𝑦,𝑢,∇𝑢+𝜔₁𝑥,𝑦𝑔𝑥,𝑦,𝑢+𝜔₂𝑥,𝑦𝐻𝑥,𝑦,𝜂,𝜉, (4.35) and 𝑓(𝑥,𝑦)=(𝑥²+𝑦²)^3/2𝑞cos(1/(𝑥²+𝑦²)), with 2𝑟₂/(𝑟₂−1)<𝑞<6. Therefore, by Theorem 1.2, the problem 𝐿𝑢𝑥,𝑦=𝑓𝑥,𝑦,onΩ,𝑢𝑥,𝑦=0,on𝜕Ω (𝑃) has a solution 𝑢∈𝑊₀^1,2(Ω,𝜔₁,𝜔₂)∩𝐿^∞(Ω).