On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

In this paper we prove the existence of (weak) solutions in the weighted Sobolev space W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) (see Definition 2.2) for the Dirichlet problem (P) Lux=f0x−∑j=1nDjfjxin Ω,ux=0 on ∂Ω, \left\{ {\begin{array}{*{20}{l}}{Lu\left( x \right) = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)} }&{{\rm{in}}\;\;\Omega ,}\\{u\left( x \right) = 0\;\;\;\;{\rm{on}}\;\partial \Omega ,}&{}\end{array}} \right. where L is the partial differential operator (1.1) Lux=− div𝒜x,u,∇uω1+ℬx,u,∇uν1+ℋx,u,∇uν2+ up−2 u ω2−∑i,j=1nDjaijxDiux \begin{array}{*{20}{l}}{Lu\left( x \right) = }&{ - \;{\rm{div}}\left[ {\mathcal{A}\left( {x,u,\nabla u} \right){\omega _1} + {\mathcal{B}}\left( {x,u,\nabla u} \right){\nu _1}} \right] + \mathcal{H}\left( {x,u,\nabla u} \right){\nu _2}}\\{}&{ + \;{{\left| u \right|}^{p - 2}}u{\omega _2} - \sum\limits_{i,j = 1}^n {{D_j}\left( {{a_{ij}}\left( x \right){D_i}u\left( x \right)} \right)} }\end{array} where D_j = ∂/∂x_j, Ω is a bounded open set in ℝⁿ, ω₁, ω₂, ν₁ and ν₂ are four weight functions (which represent the degeneration or singularity in the equation (1.1)), 1 < q, s < p < ∞ and the functions 𝒜_j : Ω×ℝ×ℝⁿ→ℝ, ℬ_j : Ω×ℝ×ℝⁿ→ℝ (j = 1, . . . , n) and ℋ: Ω×ℝ×ℝⁿ→ℝ satisfy the following conditions:

• (H1)

  x↦𝒜_j(x, η, ξ) is measurable on Ω for all (η, ξ)∈ℝ×ℝⁿ, (η, ξ)↦𝒜_j(x, η, ξ) is continuous on ℝ×ℝⁿ for almost all x∈Ω.

• (H2)

  There exists a constant θ₁ > 0 such that 𝒜x,η,ξ−𝒜x,η′,ξ′,ξ−ξ′ ≥θ1ξ−ξ′p, \left\langle {\mathcal{A}\left( {x,\eta ,\xi } \right) - \mathcal{A}\left( {x,\eta ',\xi '} \right),\left( {\xi - \xi '} \right)} \right\rangle \; \ge {\theta _1}{\left| {\xi - \xi '} \right|^p}, whenever ξ, ξ^′∈ℝⁿ, ξ≠ξ^′, and 𝒜(x, η, ξ) = (𝒜₁(x, η, ξ), . . . ,𝒜_n(x, η, ξ)) (where ⟨·, ·⟩ denotes here the Euclidian scalar product in ℝⁿ).

• (H3)

  ⟨𝒜(x, η, ξ), ξ⟩ ≥ λ₁|ξ|^p, where λ₁ is a positive constant.

• (H4)

  𝒜x,η,ξ≤K1x+h1xω2xω1x1/p′ηp/p′+h2xξp/p′ \left| {\mathcal{A}\left( {x,\eta ,\xi } \right)} \right| \le {K_1}\left( x \right) + {h_1}\left( x \right){\left( {\frac{{{\omega _2}\left( x \right)}}{{{\omega _1}\left( x \right)}}} \right)^{1/p'}}{\left| \eta \right|^{p/p'}} + {h_2}\left( x \right){\left| \xi \right|^{p/p'}} , where K₁, h₁ and h₂ are nonnegative functions, with h₁, h₂∈L^∞(Ω) and K₁∈L^p′(Ω, ω₁) (with 1/p + 1/p^′ = 1).

• (H5)

  x↦ℬ_j(x, η, ξ) is measurable on Ω for all (η, ξ)∈ℝ×ℝⁿ, (η, ξ)↦ℬ_j(x, η, ξ) is continuous on ℝ×ℝⁿ for almost all x∈Ω.

• (H6)

  ⟨ℬ(x, η, ξ) − ℬ(x, η^′, ξ^′), (ξ − ξ^′)⟩ > 0, whenever ξ, ξ^′∈ℝⁿ, ξ≠ξ^′, where ℬ(x, η, ξ) = (ℬ₁(x, η, ξ), . . . , ℬ_n(x, η, ξ)).

• (H7)

  ⟨ℬ(x, η, ξ), ξ⟩ ≥ λ₂|ξ|^q + Λ₂|η|^q, where λ₂ > 0 and Λ₂≥0 are constants.

• (H8)

  |ℬ(x, η, ξ)| ≤ K₂(x) + g₁(x)|η|^q/q′ + g₂(x)|ξ|^q/q′, where K₂, g₁ and g₂ are nonnegative functions, with g₁ and g₂∈L^∞(Ω), and K₂∈L^q′ (Ω, ν₁) (with 1/q + 1/q^′ = 1).

• (H9)

  x ↦ℋ(x, η, ξ) is measurable on Ω for all (η, ξ)∈ℝ×ℝⁿ, (η, ξ)↦ℋ(x, η, ξ) is continuous on ℝ×ℝⁿ for almost all x∈Ω.

• (H10)

  [ℋ(x, η, ξ) − ℋ(x, η^′, ξ^′)](η − η^′) > 0, whenever η, η^′∈ℝ, η≠η^′.

• (H11)

  ℋ(x, η, ξ)η ≥ λ₃|ξ|^s + Λ₃|η|^s, where λ₃ and Λ₃ are nonnegative constants.

• (H12)

  |ℋ(x, η, ξ)| ≤ K₃(x) + h₃(x)|η|^s/s′ + h₄(x)|ξ|^s/s′, where K₃, h₃ and h₄ are nonnegative functions, with K₃∈L^s′ (Ω, ν₂) (with 1/s + 1/s^′ = 1), h₃ and h₄∈L^∞(Ω).

• (H13)

  a_ij : Ω→ℝ are measurable functions, the coefficient matrix ℳ(x) = (a_ij(x)) is symmetric and satisfies the degenerate elliptic condition λν3xξ2≤∑i,j=1naijxξiξj≤Λξ2ν3x \lambda {\nu _3}\left( x \right){\left| \xi \right|^2} \le \sum\limits_{i,j = 1}^n {{a_{ij}}\left( x \right){\xi _i}{\xi _j} \le \Lambda {{\left| \xi \right|}^2}{\nu _3}\left( x \right)} for all ξ ∈ℝⁿ and almost every x ∈ Ω, λ > 0 and Λ > 0 are constants, ν₃ is a weight function.

Let Ω be a bounded open set in ℝⁿ. By the symbol 𝒲(Ω) we denote the set of all measurable a.e. in Ω positive and finite functions ω = ω(x), x ∈ Ω. Elements of 𝒲(Ω) will be called weight functions. Every weight ω gives rise to a measure on the measurable subsets of ℝⁿ through integration. This measure will be denoted by μ. Thus, μ(E) = ∫_E ω(x) dx for measurable sets E⊂ℝⁿ.

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, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [3], [4], [5] and [8]). In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is disturbed in the sense that some degeneration or singularity appears. There are several very concrete problems from practice which lead to such differential equations, e.g. from glaceology, non-Newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, petroleum extraction and reaction-diffusion problems (see some examples of applications of degenerate elliptic equations in [2] and [7]).

A class of weights, which is particularly well understood, is the class of A_p-weights (or Muckenhoupt class) that was introduced by B. Muckenhoupt (see [15]). These classes have found many useful applications in harmonic analysis (see [17]). Another reason for studying A_p-weights is the fact that powers of distance to submanifolds of ℝⁿ often belong to A_p (see [12]). There are, in fact, many interesting examples of weights (see [11] for p-admissible weights).

The following theorem will be proved in Section 3.

The paper is organized as follows. In Section 2 we present the definitions and basic results. In Section 3 we prove our main result about existence and uniqueness of solutions for problem (P).

We recall here some standard notations, properties and results which will be used throughout the paper.

Let ω be a locally integrable nonnegative function in ℝⁿ and assume that 0 < ω < ∞ 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 that 1B∫Bωxdx1B∫Bω1/1−pxdxp−1≤C, \left( {\frac{1}{{\left| B \right|}}\int_B {\omega \left( x \right)dx} } \right){\left( {\frac{1}{{\left| B \right|}}\int_B {{\omega ^{1/\left( {1 - p} \right)}}\left( x \right)dx} } \right)^{p - 1}} \le C, for all balls B ⊂ ℝⁿ, where | · | denotes the n-dimensional Lebesgue measure in ℝⁿ. If 1 < q ≤ p, then A_q ⊂ A_p (see [10], [11] or [17] for more information about A_p-weights). The weight ω satisfies the doubling condition if there exists a positive constant C such that μ(B(x; 2r)) ≤ C μ(B(x; r)), for every ball B = B(x; r) ⊂ ℝⁿ, where μ(B) = ∫_B ω(x) dx. If ω∈A_p, then μ is doubling (see Corollary 15.7 in [11]).

As an example of a A_p-weight, the function ω(x) = |x|^α, x∈ℝⁿ, is in A_p if and only if −n < α < n(p − 1) (see Corollary 4.4, Chapter IX in [17]). Other example, we have ω(x) = |x|^α(max{1, −ln(|x|)})^β is an A₁-weight if and only if −n < α < 0 or α = 0 ≤ β (see Proposition 7.2 in [1]).

If ω∈A_p, then EBp≤CμEμB, {\left( {\frac{{\left| E \right|}}{{\left| B \right|}}} \right)^p} \le C\frac{{\mu \left( E \right)}}{{\mu \left( B \right)}}, whenever B is a ball in ℝⁿ and E is a measurable subset of B (see 15.5 strong doubling property in [11]). Therefore, if μ(E) = 0 then |E| = 0. The measure μ and the Lebesgue measure | · | are mutually absolutely continuous, i.e., they have the same zero sets (μ(E) = 0 if and only if |E| = 0); so there is no need to specify the measure when using the ubiquitous expression almost everywhere and almost every, both abbreviated a.e..

In order to discuss the problem (P), we need some elementary results for weighted Lebesgue spaces L^p(Ω, ω) and the weighted Sobolev spaces W^1,p(Ω, ω₁, ω₂) and W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) .

If ω ∈ A_p, 1 < p < ∞, then ω^−1/(p−1) is locally integrable and LpΩ,ω⊂Lloc1Ω {L^p}\left( {\Omega ,\omega } \right) \subset L_{{\rm{loc}}}^1\left( \Omega \right) for every open set Ω (see Remark 1.2.4 in [18]). It thus makes sense to talk about weak derivatives of functions in L^p(Ω, ω).

The space W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) is the closure of C0∞Ω C_0^\infty \left( \Omega \right) with respect to the norm (2.1). Equipped with this norm, W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) is a reflexive Banach space (see [14] or [16] for more information about the spaces W^1,p(Ω, ω₁, ω₂)). The dual of space W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) is the space W01,pΩ,ω1,ω2*=T=f0−divF, F=f1,…,fn: f0ω2∈Lp′Ω,ω2,fjω1∈Lp′Ω,ω1,j=1,…,n}. \begin{array}{*{20}{l}}{{{\left[ {W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)} \right]}^*}}&{ = \left\{ {T = {f_0} - {\rm{div}}\left( F \right),\;F = \left( {{f_1}, \ldots ,{f_n}} \right):} \right.}\\{}&{\;\;\;\;\;\;\;\frac{{{f_0}}}{{{\omega _2}}} \in {L^{p'}}\left( {\Omega ,{\omega _2}} \right),\frac{{{f_j}}}{{{\omega _1}}} \in {L^{p'}}\left( {\Omega ,{\omega _1}} \right),j = 1, \ldots ,n\} .}\end{array}

If T∈W0pΩ,ω1,ω2* T \in {\left[ {W_0^p\left( {\Omega ,{\omega _1},{\omega _2}} \right)} \right]^*} and φ∈W01,pΩ,ω1,ω2 \varphi \in W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) , we denote T|φ=∫Ωf0 φ dx+∑j=1nfj Djφ dx, T*=f0/ω2Lp′Ω,ω2+∑j=1nfj/ω1Lp′Ω,ω1,T|φ≤T*φW01,pΩ,ω1,ω2. \begin{array}{*{20}{l}}{\;\;\left( {T|\varphi } \right) = \int_\Omega {{f_0}\varphi \;dx} + \sum\limits_{j = 1}^n {{f_j}\;{D_j}\varphi \;dx} ,}\\{\;\;\;{{\left\| T \right\|}_*} = {{\left\| {{f_0}/{\omega _2}} \right\|}_{{L^{p'}}\left( {\Omega ,{\omega _2}} \right)}} + \sum\limits_{j = 1}^n {{{\left\| {{f_j}/{\omega _1}} \right\|}_{{L^{p'}}\left( {\Omega ,{\omega _1}} \right)}}} ,}\\{\left| {\left( {T|\varphi } \right)} \right| \le {{\left\| T \right\|}_*}{{\left\| \varphi \right\|}_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}}.}\end{array}

If ω = ω₁ = ω₂, we denote W01,pΩ,ω=W01,pΩ,ω,ω W_0^{1,p}\left( {\Omega ,\omega } \right) = W_0^{1,p}\left( {\Omega ,\omega ,\omega } \right) .

In this paper we use the following results.

The basic idea is to reduce the problem (P) to an operator equation Au = T and apply the theorem below.

To prove Theorem 1.1, we define B,B1,B2,B3,B4,B5:W01,pΩ,ω1,ω2×W01,pΩ,ω1,ω2→ℝ {\bf{B}},{{\bf{B}}_{\bf{1}}},{{\bf{B}}_{\bf{2}}},{{\bf{B}}_{\bf{3}}},{{\bf{B}}_{\bf{4}}},{{\bf{B}}_{\bf{5}}}:W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) \times W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) \to \mathbb{R} by Bu,φ=B1u,φ+B2u,φ+B3u,φ+B4u,φ+B5u,φ,B1u,φ=∫Ω𝒜x,u,∇u,∇φω1 dx,B2u,φ=∫Ωℬx,u,∇u,∇φν1 dx,B3u,φ=∫Ωℋx,u,∇uφ ν2 dx,B4u,φ=∫Ωup−2u φ ω2 dx,B5u,φ=∑i,j=1n∫ΩaijxDiuxDj φxdx=∫Ωℳx∇ux,∇φx dx, \begin{array}{*{20}{l}}{{\bf{B}}\left( {u,\varphi } \right) = {{\bf{B}}_{\bf{1}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{2}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{3}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{4}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{5}}}\left( {u,\varphi } \right),}\\{{{\bf{B}}_{\bf{1}}}\left( {u,\varphi } \right) = \int_\Omega {\left\langle {\mathcal{A}\left( {x,u,\nabla u} \right),\nabla \varphi } \right\rangle {\omega _1}\;dx} ,}\\{{{\bf{B}}_{\bf{2}}}\left( {u,\varphi } \right) = \int_\Omega {\left\langle {{\mathcal{B}}\left( {x,u,\nabla u} \right),\nabla \varphi } \right\rangle {\nu _1}\;dx} ,}\\{{{\bf{B}}_{\bf{3}}}\left( {u,\varphi } \right) = \int_\Omega {\mathcal{H}\left( {x,u,\nabla u} \right)\varphi \;{\nu _2}\;dx} ,}\\{{{\bf{B}}_{\bf{4}}}\left( {u,\varphi } \right) = \int_\Omega {{{\left| u \right|}^{p - 2}}u\;\varphi \;{\omega _2}\;dx} ,}\\{{{\bf{B}}_{\bf{5}}}\left( {u,\varphi } \right) = \sum\limits_{i,j = 1}^n {\int_\Omega {{a_{ij}}\left( x \right){D_i}u\left( x \right){D_j}\varphi \left( x \right)dx} } = \int_\Omega {\left\langle {{\mathcal{M}}\left( x \right)\nabla u\left( x \right),\nabla \varphi \left( x \right)} \right\rangle \;dx} ,}\end{array} and T:W01,pΩ,ω1,ω2→ℝ {\bf{T}}:W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) \to \mathbb{R} by Tφ=∫Ωf0 φ dx+∑j=1n∫Ωfj Dj φ dx. {\bf{T}}\left( \varphi \right) = \int_\Omega {{f_0}\;\varphi \;dx} + \sum\limits_{j = 1}^n {\int_\Omega {{f_j}\;{D_j}\varphi \;dx} } . Then u∈W01,pΩ,ω1,ω2 u \in W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) is a (weak) solution to problem (P) if Bu,φ=B1u,φ+B2u,φ+B3u,φ+B4u,φ+B5u,φ=Tφ, {\bf{B}}\left( {u,\varphi } \right) = {{\bf{B}}_{\bf{1}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{2}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{3}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{4}}}\left( {u,\varphi } \right) + {{\bf{B}}_{\bf{5}}}\left( {u,\varphi } \right) = {\bf{T}}\left( \varphi \right), for all φ∈W01,pΩ,ω1,ω2 \varphi \in W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) .

Step 1. For j = 1, . . . , n we define the operator Fj:W01,pΩ,ω1,ω2→Lp'Ω,ω1 {F_j}:W_0^{1,p}\left. {\left( {\Omega ,{\omega _1},{\omega _2}} \right)} \right) \to {L^{{p^\prime}}}\left( {\Omega ,{\omega _1}} \right) as Fjux=𝒜jx,ux,∇ux. \left( {{F_j}u} \right)\left( x \right) = {\mathcal{A}_j}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right). We now show that the operator F_j is bounded and continuous.

(i) Using (H4), we obtain (3.1) FjuLp′Ω,ω1p′=∫ΩFjuxp′ω1 dx=∫Ω𝒜jx,u,∇up′ω1 dx≤∫ΩK1+h1(ω2/ω1)1/p′up/p′+h2∇up/p′p′ω1 dx≤Cp∫ΩK1p′ω1 dx+h1L∞Ωp′∫Ωupω2 dx+h2L∞Ωp′∫Ω∇upω1 dx≤CpK1Lp′Ω,ω1p′+h1L∞Ωp′+h2L∞Ωp′uW01,pΩ,ω1,ω2p, \begin{array}{*{20}{c}}{\left\| {{F_j}u} \right\|_{{L^{{p^\prime}}}\left( {\Omega ,{\omega _1}} \right)}^{{p^\prime}} = \int_\Omega {{{\left| {{F_j}u\left( x \right)} \right|}^{p'}}{\omega _1}\;dx} = \int_\Omega {{{\left| {{\mathcal{A}_j}\left( {x,u,\nabla u} \right)} \right|}^{p'}}{\omega _1}\;dx} }\\{ \le \int_\Omega {{{\left( {{K_1} + {h_1}{{({\omega _2}/{\omega _1})}^{1/p'}}{{\left| u \right|}^{p/p'}} + {h_2}{{\left| {\nabla u} \right|}^{p/p'}}} \right)}^{p'}}{\omega _1}\;dx} }\\{ \le {C_p}\left[ {\int_\Omega {K_1^{p'}{\omega _1}\;dx + \left\| {{h_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{p'}\int_\Omega {{{\left| u \right|}^p}{\omega _2}\;dx + \left\| {{h_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{p'}} \int_\Omega {{{\left| {\nabla u} \right|}^p}{\omega _1}\;dx} } } \right]}\\{ \le {C_p}\left[ {\left\| {{K_1}} \right\|_{{L^{{p^\prime}}}\left( {\Omega ,{\omega _1}} \right)}^{{p^\prime}} + \left( {\left\| {{h_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{p'} + \left\| {{h_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{p'}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^p} \right],}\end{array} where the constant C_p depends only on p. Therefore, in (3.1) we obtain FjuLp′Ω,ω1≤Cp1/p′K1Lp′Ω,ω1+h1L∞Ω+h2L∞ΩuW01,pΩ,ω1,ω2p−1. {\left\| {{F_j}u} \right\|_{{L^{p'}}\left( {\Omega ,{\omega _1}} \right)}} \le C_p^{1/p'}\left( {{{\left\| {{K_1}} \right\|}_{{L^{p'}}\left( {\Omega ,{\omega _1}} \right)}} + \left( {{{\left\| {{h_1}} \right\|}_{{L^\infty }\left( \Omega \right)}} + {{\left\| {{h_2}} \right\|}_{{L^\infty }\left( \Omega \right)}}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^{p - 1}} \right).

(ii) Let u_m→u in W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) as m → ∞. We need to show that F_ju_m→F_ju in L^p′ (Ω, ω₁). We will apply the Lebesgue Dominated Convergence Theorem. If u_m→u in W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) , then u_m→u in L^p(Ω, ω₂) and |∇_um|→ |∇u| in L^p(Ω, ω₁). Using Theorem 2.1, there exist a subsequence {u_mk} and functions Φ₂ ∈ L^p(Ω, ω₂), Φ₁ ∈ L^p(Ω, ω₁) such that umkx→ux a.e. in Ω,umkx≤Φ2x a.e. in Ω,Djumkx→Djux a.e. in Ω,∇umkx≤Φ1x a.e. in Ω. \begin{array}{*{20}{c}}{{u_{{m_k}}}\left( x \right) \to u\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{\left| {{u_{{m_k}}}\left( x \right)} \right| \le {\Phi _2}\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{{D_j}{u_{{m_k}}}\left( x \right) \to {D_j}u\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{\left| {\nabla {u_{{m_k}}}\left( x \right)} \right| \le {\Phi _1}\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega .}\end{array}

Next, applying (H4) we obtain Fjumkx−Fjuxp′ω1= 𝒜jx,umk,∇umk−𝒜jx,u,∇up′ω1≤Cp𝒜jx,umk,∇umkp′+𝒜jx,u,∇up′ω1≤CpK1+h1ω2/ω11/p′umkp/p′+h2∇umkp/p′p′ + K1+h1ω2/ω11/p′up/p′+h2∇up/p′p′ω1≤CpK1p′+h1L∞Ωp′umkpω2ω1+h2L∞Ωp′∇umkp + K1p′+h1L∞Ωp′upω2ω1+h2L∞Ωp′∇upω1≤CpK1p′+h1L∞Ωp′Φ2pω2ω1+h2L∞Ωp′Φ1p + K1p′+h1L∞Ωp′Φ2pω2ω1+h2L∞Ωp′Φ1pω1=2CpK1p′ω1+h1L∞Ωp′Φ2pω2+h2L∞Ωp′Φ1pω1∈L1Ω. \begin{array}{*{35}{l}} {{\left| {{F}_{j}}{{u}_{{{m}_{k}}}}\left( x \right)-{{F}_{j}}u\left( x \right) \right|}^{{{p}'}}}{{\omega }_{1}} & =\ {{\left| {{\mathcal{A}}_{j}}\left( x,{{u}_{{{m}_{k}}}},\nabla {{u}_{{{m}_{k}}}} \right)-{{\mathcal{A}}_{j}}\left( x,u,\nabla u \right) \right|}^{{{p}'}}}{{\omega }_{1}} \\ {} & \le {{C}_{p}}\left( {{\left| {{\mathcal{A}}_{j}}\left( x,{{u}_{{{m}_{k}}}},\nabla {{u}_{{{m}_{k}}}} \right) \right|}^{{{p}'}}}+{{\left| {{\mathcal{A}}_{j}}\left( x,u,\nabla u \right) \right|}^{{{p}'}}} \right){{\omega }_{1}} \\ {} & \le {{C}_{p}}\left[ {{\left( {{K}_{1}}+{{h}_{1}}{{\left( {{\omega }_{2}}/{{\omega }_{1}} \right)}^{1/{p}'}}{{\left| {{u}_{{{m}_{k}}}} \right|}^{p/{p}'}}+{{h}_{2}}{{\left| \nabla {{u}_{{{m}_{k}}}} \right|}^{p/{p}'}} \right)}^{{{p}'}}} \right. \\ {} & \left. \ \ \ +\ {{\left( {{K}_{1}}+{{h}_{1}}{{\left( {{\omega }_{2}}/{{\omega }_{1}} \right)}^{1/{p}'}}{{\left| u \right|}^{p/{p}'}}+{{h}_{2}}{{\left| \nabla u \right|}^{p/{p}'}} \right)}^{{{p}'}}} \right]{{\omega }_{1}} \\ {} & \le {{C}_{p}}\left[ \left( K_{1}^{{{p}'}}+\left\| {{h}_{1}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}{{\left| {{u}_{{{m}_{k}}}} \right|}^{p}}\frac{{{\omega }_{2}}}{{{\omega }_{1}}}+\left\| {{h}_{2}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}{{\left| \nabla {{u}_{{{m}_{k}}}} \right|}^{p}} \right) \right. \\ {} & \left. \ \ \ +\ \left( K_{1}^{{{p}'}}+\left\| {{h}_{1}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}{{\left| u \right|}^{p}}\frac{{{\omega }_{2}}}{{{\omega }_{1}}}+\left\| {{h}_{2}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}{{\left| \nabla u \right|}^{p}} \right. \right]{{\omega }_{1}} \\ {} & \le {{C}_{p}}\left[ \left( K_{1}^{{{p}'}}+\left\| {{h}_{1}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{2}^{p}\frac{{{\omega }_{2}}}{{{\omega }_{1}}}+\left\| {{h}_{2}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{1}^{p} \right) \right. \\ {} & \ \ \ \left. +\ \left( K_{1}^{{{p}'}}+\left\| {{h}_{1}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{2}^{p}\frac{{{\omega }_{2}}}{{{\omega }_{1}}}+\left\| {{h}_{2}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{1}^{p} \right) \right]{{\omega }_{1}} \\ {} & =2{{C}_{p}}\left[ K_{1}^{{{p}'}}{{\omega }_{1}}+\left\| {{h}_{1}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{2}^{p}{{\omega }_{2}}+\left\| {{h}_{2}} \right\|_{{{L}^{\infty }}\left( \Omega \right)}^{{{p}'}}\Phi _{1}^{p}{{\omega }_{1}} \right]\in {{L}^{1}}\left( \Omega \right). \\ \end{array} By condition (H1), we have Fjumkx=𝒜jx,umkx,∇umkx→𝒜jx,ux,∇ux=Fjux, {F_j}{u_{{m_k}}}\left( x \right) = {\mathcal{A}_j}\left( {x,{u_{{m_k}}}\left( x \right),\nabla {u_{{m_k}}}\left( x \right)} \right) \to {\mathcal{A}_j}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right) = {F_j}u\left( x \right), as m_k → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain ‖F_ju_mk − F_ju‖_{L^p′(Ω,ω1)} → 0, that is, F_ju_mk → F_ju in L^p′ (Ω, ω₁). We conclude from the Convergence Principle in Banach spaces (see Proposition 10.13 in [19]) that (3.2) Fjum→Fju in Lp′Ω,ω1. {F_j}{u_m} \to {F_j}u\;\;\;{\rm{in}}\;\;\;{L^{p'}}\left( {\Omega ,{\omega _1}} \right).

Step 2. We define the operator Gj:W01,pΩ,ω1,ω2→Lq′Ω,ν1 {G_j}:W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) \to {L^{q'}}\left( {\Omega ,{\nu _1}} \right) by Gjux=ℬjx,ux,∇ux. \left( {{G_j}u} \right)\left( x \right) = {\mathcal{B}_j}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right). This operator is continuous and bounded. In fact:

(i) Using (H8), Remark 2.5(i) and Theorem 2.2 (since ω₁ ∈ A_p) we obtain GjuLq′Ω,ν1q′=∫ΩGjuxq′ν1 dx=∫Ωℬjx,u,∇uq′ν1 dx≤∫ΩK2+g1uq/q′+g2∇uq/q′q′ν1 dx≤Cq∫ΩK2q′+g1q′uq+g2q′∇uqν1dx=Cq∫ΩK2q′ν1 dx+∫Ωg1q′uqν1 dx+∫Ωg2q′∇uqν1 dx≤CqK2Lq′Ω,ν1q′+g1L∞Ωq′uLqΩ,ν1q+g2L∞Ωq′∇uLqΩ,ν1q≤CqK2Lq′Ω,ν1q′+g1L∞Ωq′Cp,qquLpΩ,ω1q + Cp,qqg2L∞Ωq′∇uLpΩ,ω1q≤CqK2Lq′Ω,ν1q′+Cp,qqCΩqg1L∞Ωq′+g2L∞Ωq′uW01,pΩ,ω1,ω2q, \begin{array}{*{20}{l}}{\left\| {{G_j}u} \right\|_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}^{q'}}&{ = \int_\Omega {{{\left| {{G_j}u\left( x \right)} \right|}^{q'}}{\nu _1}\;dx} = \int_\Omega {{{\left| {{\mathcal{B}_j}\left( {x,u,\nabla u} \right)} \right|}^{q'}}{\nu _1}\;dx} }\\{}&{ \le \int_\Omega {{{\left( {{K_2} + {g_1}{{\left| u \right|}^{q/q'}} + {g_2}{{\left| {\nabla u} \right|}^{q/q'}}} \right)}^{q'}}{\nu _1}\;dx} }\\{}&{ \le {C_q}\int_\Omega {\left[ {\left( {K_2^{q'} + g_1^{q'}{{\left| u \right|}^q} + g_2^{q'}{{\left| {\nabla u} \right|}^q}} \right){\nu _1}} \right]dx} }\\{}&{ = {C_q}\left[ {\int_\Omega {K_2^{q'}{\nu _1}\;dx} + \int_\Omega {g_1^{q'}{{\left| u \right|}^q}{\nu _1}\;dx} + \int_\Omega {g_2^{q'}{{\left| {\nabla u} \right|}^q}{\nu _1}\;dx} } \right]}\\{}&{ \le {C_q}\left( {\left\| {{K_2}} \right\|_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}^{q'} + \left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}\left\| u \right\|_{{L^q}\left( {\Omega ,{\nu _1}} \right)}^q + \left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}\left\| {\left| {\nabla u} \right|} \right\|_{{L^q}\left( {\Omega ,{\nu _1}} \right)}^q} \right)}\\{}&{ \le {C_q}\left( {\left\| {{K_2}} \right\|_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}^{q'} + \left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}C_{p,q}^q\left\| u \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^q} \right.}\\{}&{\left. {\;\;\; + \;C_{p,q}^q\left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}\left\| {\left| {\nabla u} \right|} \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^q} \right)}\\{}&{ \le {C_q}\left( {\left\| {{K_2}} \right\|_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}^{q'} + C_{p,q}^q\left( {C_\Omega ^q\left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'} + \left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^q} \right),}\end{array} where the constant C_q depends only on q. Therefore, we obtain GjuLq′Ω,ν1≤Cq1/q′K2Lq′Ω,ν1 + Cp,qq−1CΩq−1g1L∞Ω+g2L∞ΩuW01,pΩ,ω1,ω2q−1. \begin{array}{*{20}{l}}{{{\left\| {{G_j}u} \right\|}_{{L^{q'}}\left( {\Omega ,{\nu_1}} \right)}}}&{ \le C_q^{1/q'}\left( {{{\left\| {{K_2}} \right\|}_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}}} \right.}\\{}&{\left. {\;\;\; + \;C_{p,q}^{q - 1}\left( {C_\Omega ^{q - 1}{{\left\| {{g_1}} \right\|}_{{L^\infty }\left( \Omega \right)}} + {{\left\| {{g_2}} \right\|}_{{L^\infty }\left( \Omega \right)}}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^{q - 1}} \right).}\end{array}

(ii) Let u_m→u in W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) as m → ∞. We need to show that G_ju_m→G_ju in L^q′ (Ω, ν₁). We will apply the Lebesgue Dominated Theorem. If u_m→u in W01,pΩ,ω1,ω2 W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) , then u_m→u in L^p(Ω, ω₂ and |∇u_m|→ |∇u| in L^p(Ω, ω₁). Analogously to Step 1(ii), there exist a subsequence {u_mk} and functions Φ₂∈L^p(Ω, ω₂) and Φ₁∈L^p(Ω, ω₁) such that umkx→ux a.e. in Ω,umkx≤Φ2x a.e. in Ω,Djumkx→Djux a.e. in Ω,∇umkx≤Φ1x a.e. in Ω. \begin{array}{*{20}{c}}{{u_{{m_k}}}\left( x \right) \to u\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{\left| {{u_{{m_k}}}\left( x \right)} \right| \le {\Phi _2}\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{{D_j}{u_{{m_k}}}\left( x \right) \to {D_j}u\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega ,}\\{\left| {\nabla {u_{{m_k}}}\left( x \right)} \right| \le {\Phi _1}\left( x \right)\;\;\;{\rm{a.e.}}\;{\rm{in}}\;\Omega .}\end{array} Next, applying (H8) and Remark 2.5(i) we obtain Gjumkx−Gjuxq′ν1=ℬjx,umk,∇umk−ℬjx,u,∇uq′ν1≤Cqℬjx,umk,∇umkq′+ℬjx,u,∇uq′ν1≤CqK2+g1umkq/q′+g2∇umkq/q′q′ + K2+g1uq/q′+g2∇uq/q′q′ν1≤CqK2q′+g1L∞Ωq′umkq+g2L∞Ωq′∇umkq + K2q′+g1L∞Ωq′uq+g2L∞Ωq′∇uqν1≤2CqK2q′ν1+g1L∞Ωq′Φ2qν1+g2L∞Ωq′Φ1qν1∈L1Ω, \begin{array}{*{20}{l}}{{{\left| {{G_j}{u_{{m_k}}}\left( x \right) - {G_j}u\left( x \right)} \right|}^{q'}}{\nu _1}}&{ = {{\left| {{\mathcal{B}_j}\left( {x,{u_{{m_k}}},\nabla {u_{{m_k}}}} \right) - {\mathcal{B}_j}\left( {x,u,\nabla u} \right)} \right|}^{q'}}{\nu _1}}\\{}&{ \le {C_q}\left[ {{{\left| {{\mathcal{B}_j}\left( {x,{u_{{m_k}}},\nabla {u_{{m_k}}}} \right)} \right|}^{q'}} + {{\left| {{\mathcal{B}_j}\left( {x,u,\nabla u} \right)} \right|}^{q'}}} \right]{\nu _1}}\\{}&{ \le {C_q}\left[ {{{\left( {{K_2} + {g_1}{{\left| {{u_{{m_k}}}} \right|}^{q/q'}} + {g_2}{{\left| {\nabla {u_{{m_k}}}} \right|}^{q/q'}}} \right)}^{q'}}} \right.}\\{}&{\left. {\;\;\; + \;{{\left( {{K_2} + {g_1}{{\left| u \right|}^{q/q'}} + {g_2}{{\left| {\nabla u} \right|}^{q/q'}}} \right)}^{q'}}} \right]{\nu _1}}\\{}&{ \le {C_q}\left[ {\left( {K_2^{q'} + \left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}{{\left| {{u_{{m_k}}}} \right|}^q} + \left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}{{\left| {\nabla {u_{{m_k}}}} \right|}^q}} \right)} \right.}\\{}&{\;\;\;\left. { + \;\left( {K_2^{q'} + \left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}{{\left| u \right|}^q} + \left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}{{\left| {\nabla u} \right|}^q}} \right)} \right]{\nu _1}}\\{}&{ \le 2{C_q}\left[ {K_2^{q'}{\nu _1} + \left\| {{g_1}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}\Phi _2^q{\nu _1} + \left\| {{g_2}} \right\|_{{L^\infty }\left( \Omega \right)}^{q'}\Phi _1^q{\nu _1}} \right] \in {L^1}\left( \Omega \right),}\end{array} since ∫ΩΦ1qν1 dx≤Cp,qq∫ΩΦ1pω1 dx \int_\Omega {\Phi _1^q{\nu _1}\;dx} \le C_{p,q}^q\int_\Omega {\Phi _1^p{\omega _1}\;dx} and ∫ΩΦ2qν1 dx≤C˜p,qq∫ΩΦ2pω2 dx \int_\Omega {\Phi _2^q{\nu _1}\;dx} \le \tilde C_{p,q}^q\int_\Omega {\Phi _2^p{\omega _2}\;dx} . By condition (H5), we have Gjumkx=ℬjx,umkx,∇umkx→ℬjx,ux,∇ux=Gjux, {G_j}{u_{{m_k}}}\left( x \right) = {\mathcal{B}_j}\left( {x,{u_{{m_k}}}\left( x \right),\nabla {u_{{m_k}}}\left( x \right)} \right) \to {\mathcal{B}_j}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right) = {G_j}u\left( x \right), as m_k → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain Gjumk−GjuLq′Ω,ν1→0, {\left\| {{G_j}{u_{{m_k}}} - {G_j}u} \right\|_{{L^{q'}}\left( {\Omega ,{\nu _1}} \right)}} \to 0, that is, Gjumk→Gju in Lq′Ω,ν1. {G_j}{u_{{m_k}}} \to {G_j}u\;\;\;{\rm{in}}\;{L^{q'}}\left( {\Omega ,{\nu _1}} \right). We conclude from the Convergence Principle in Banach spaces (see Proposition 10.13 in [19]) that (3.3) Gjum→Gju in Lq′Ω,ν1. {G_j}{u_m} \to {G_j}u\;\;{\rm{in}}\;{L^{q'}}\left( {\Omega ,{\nu _1}} \right).

Step 3. We define the operator H:W01,pΩ,ω1,ω2→Ls′Ω,ν2 H:W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right) \to {L^{s'}}\left( {\Omega ,{\nu _2}} \right) by Hux=ℋx,ux,∇ux. \left( {Hu} \right)\left( x \right) = \mathcal{H}\left( {x,u\left( x \right),\nabla u\left( x \right)} \right). We also have that the operator H is continuous and bounded. In fact:

(i) Using (H12), Remark 2.5(ii) and Theorem 2.2 we obtain HuLs′Ω,ν2s′=∫ΩHus′ν2 dx=∫Ωℋx,u,∇us′ν2 dx≤∫Ω(K3+h3us/s′+h4∇us/s′)s′ν2 dx≤Cs∫ΩK3s′+h3s′us+h4s′∇usν2 dx≤Cs∫ΩK3s′ ν2 dx+h3L∞Ωs′∫Ωusν2 dx+h4L∞Ωs′∫Ω∇usν2 dx≤CsK3Ls′Ω,ν2s′+h3L∞Ωs′Cp,ssuLpΩ,ω1s + h4L∞Ωs′Cp,ss∇uLpΩ,ω1s≤CsK3Ls′Ω,ν2s′+h3L∞Ωs′Cp,ssCΩs∇uLpΩ,ω1s + h4L∞Ωs′Cp,ss∇uLpΩ,ω1s≤CsK3Ls′Ω,ν2s′+Cp,ssCΩsh3L∞Ωs′+h4L∞Ωs′uW01,pΩ,ω1,ω2s, \begin{array}{*{20}{l}}{\left\| {Hu} \right\|_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}^{s'}}&{ = \int_\Omega {{{\left| {Hu} \right|}^{s'}}{\nu _2}\;dx} = \int_\Omega {{{\left| {\mathcal{H}\left( {x,u,\nabla u} \right)} \right|}^{s'}}{\nu _2}\;dx} }\\{}&{ \le \int_\Omega {{{({K_3} + {h_3}{{\left| u \right|}^{s/s'}} + {h_4}{{\left| {\nabla u} \right|}^{s/s'}})}^{s'}}{\nu _2}\;dx} }\\{}&{ \le {C_s}\int_\Omega {\left( {K_3^{s'} + h_3^{s'}{{\left| u \right|}^s} + h_4^{s'}{{\left| {\nabla u} \right|}^s}} \right){\nu _2}\;dx} }\\{}&{ \le {C_s}\left[ {\int_\Omega {K_3^{s'}\;{\nu _2}\;dx + \left\| {{h_3}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}\int_\Omega {{{\left| u \right|}^s}{\nu _2}\;dx + \left\| {{h_4}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}\int_\Omega {{{\left| {\nabla u} \right|}^s}{\nu _2}\;dx} } } } \right]}\\{}&{ \le {C_s}\left( {\left\| {{K_3}} \right\|_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}^{s'} + \left\| {{h_3}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}C_{p,s}^s\left\| u \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^s} \right.}\\{}&{\left. {\;\;\; + \;\left\| {{h_4}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}C_{p,s}^s\left\| {\left| {\nabla u} \right|} \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^s} \right)}\\{}&{ \le {C_s}\left( {\left\| {{K_3}} \right\|_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}^{s'} + \left\| {{h_3}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}C_{p,s}^sC_\Omega ^s\left\| {\left| {\nabla u} \right|} \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^s} \right.}\\{}&{\left. {\;\;\; + \;\left\| {{h_4}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}C_{p,s}^s\left\| {\left| {\nabla u} \right|} \right\|_{{L^p}\left( {\Omega ,{\omega _1}} \right)}^s} \right)}\\{}&{ \le {C_s}\left( {\left\| {{K_3}} \right\|_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}^{s'} + C_{p,s}^s\left( {C_\Omega ^s\left\| {{h_3}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'} + \left\| {{h_4}} \right\|_{{L^\infty }\left( \Omega \right)}^{s'}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^s} \right),}\end{array} where the constant C_s depends only on s. Hence, we obtain HuLs′Ω,ν2≤CsK3Ls′Ω,ν2 + Cp,ss−1CΩs−1h3L∞Ω+h4L∞ΩuW01,pΩ,ω1,ω2s−1. \begin{array}{*{20}{l}}{{{\left\| {Hu} \right\|}_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}}}&{ \le {C_s}\left[ {{{\left\| {{K_3}} \right\|}_{{L^{s'}}\left( {\Omega ,{\nu _2}} \right)}}} \right.}\\{}&{\;\;\;\left. { + \;C_{p,s}^{s - 1}\left( {C_\Omega ^{s - 1}{{\left\| {{h_3}} \right\|}_{{L^\infty }\left( \Omega \right)}} + {{\left\| {{h_4}} \right\|}_{{L^\infty }\left( \Omega \right)}}} \right)\left\| u \right\|_{W_0^{1,p}\left( {\Omega ,{\omega _1},{\omega _2}} \right)}^{s - 1}} \right].}\end{array}