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

Albo Carlos Cavalheiro

doi:10.2478/amsil-2024-0024

1. Introduction

In this paper we prove the existence of (weak) solutions in the weighted Sobolev space $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ (see Definition 2.2) for the Dirichlet problem

(P)

\begin{array}{l} Lu (x) = f_{0} (x) - \sum_{j = 1}^{n} D_{j} f_{j} (x) & in Ω, \\ u (x) = 0 on \partial Ω, \end{array}

where L is the partial differential operator

(1.1)

\begin{array}{l} Lu (x) = & - div 𝒜 (x, u, \nabla u) ω_{1} + ℬ (x, u, \nabla u) ν_{1}] + ℋ (x, u, \nabla u) ν_{2} \\ + {u|}^{p - 2} u ω_{2} - \sum_{i, j = 1}^{n} D_{j} (a_{ij} (x) D_{i} u (x)) \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
$𝒜 (x, η, ξ) - 𝒜 (x, η^{'}, ξ^{'}), (ξ - ξ^{'})〉 \geq θ_{1} {ξ - ξ^{'}|}^{p},$
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) $𝒜 (x, η, ξ)| \leq K_{1} (x) + h_{1} (x) {(\frac{ω_{2} (x)}{ω_{1} (x)})}^{1 / p^{'}} {η|}^{p / p^{'}} + h_{2} (x) {ξ|}^{p / p^{'}}$ , 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
$λ ν_{3} (x) {ξ|}^{2} \leq \sum_{i, j = 1}^{n} a_{ij} (x) ξ_{i} ξ_{j} \leq Λ {ξ|}^{2} ν_{3} (x)$
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.

Theorem 1.1.

Let 1 < q, s < p, 2 < p < ∞, and assume (H1)–(H13). If

(i) ω₁, ω₂∈A_p, ν₁, ν₂ and ν₃∈𝒲(Ω), $\frac{ν_{1}}{ω_{1}} \in L^{r_{1}} (Ω, ω_{1})$ , $\frac{ν_{1}}{ω_{2}} \in L^{r_{1}} (Ω, ω_{2})$ , $\frac{ν_{2}}{ω_{1}} \in L^{r_{2}} (Ω, ω_{1})$ , $\frac{ν_{2}}{ω_{2}} \in L^{r_{2}} (Ω, ω_{2})$ and $\frac{ν_{3}}{ω_{1}} \in L^{r_{3}} (Ω, ω_{1})$ , where r₁ = p/(p − q), r₂ = p/(p − s) and r₃ = 2/(p − 2);
(ii) f₀/ν₁∈L^{q^′} (Ω, ν₁) and f_j/ω₁∈L^{p^′} (Ω, ω₁) (j = 1, . . . , n);

then the problem (P) has a unique solution $u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ . Moreover, there is a constant C > 0 such that

{u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} \leq C {(C_{p, q} {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})})}^{1 / (p - 1)},

where C_p,q is the constant defined in Remark 2.5(i).

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).

2. Definitions and basic results

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

(\frac{1}{B|} \int_{B} ω (x) dx) {(\frac{1}{B|} \int_{B} ω^{1 / (1 - p)} (x) dx)}^{p - 1} \leq 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

{(\frac{E|}{B|})}^{p} \leq C \frac{μ (E)}{μ (B)},

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 $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ .

Definition 2.1.

Let ω be a weight, and let Ω ⊂ ℝⁿ be open. For 1 < p < ∞ we define L^p(Ω, ω) as the set of measurable functions f on Ω such that

{f‖}_{L^{p} (Ω, ω)} = {(\int_{Ω} {f|}^{p} ω dx)}^{1 / p} < \infty .

If ω ∈ A_p, 1 < p < ∞, then ω^−1/(p−1) is locally integrable and $L^{p} (Ω, ω) \subset L_{loc}^{1} (Ω)$ 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).

Definition 2.2.

Let Ω ⊂ ℝⁿ be a bounded open set 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, ω₂) with weak derivatives D_ju ∈ L^p(Ω, ω₁). The norm of u in W^1,p(Ω, ω₁, ω₂) is defined by

(2.1)

{u‖}_{W^{1, p} (Ω, ω_{1}, ω_{2})} = {(\int_{Ω} {u|}^{p} ω_{2} dx + \int_{Ω} {\nabla u|}^{p} ω_{1} dx)}^{1 / p} .

The space $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ is the closure of $C_{0}^{\infty} (Ω)$ with respect to the norm (2.1). Equipped with this norm, $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ 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 $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ is the space

\begin{array}{l} {W_{0}^{1, p} (Ω, ω_{1}, ω_{2})]}^{*} & = T = f_{0} - div (F), F = (f_{1}, \dots, f_{n}) : \\ \frac{f_{0}}{ω_{2}} \in L^{p^{'}} (Ω, ω_{2}), \frac{f_{j}}{ω_{1}} \in L^{p^{'}} (Ω, ω_{1}), j = 1, \dots, n} . \end{array}

If $T \in {W_{0}^{p} (Ω, ω_{1}, ω_{2})]}^{*}$ and $φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ , we denote

\begin{array}{l} (T | φ) = \int_{Ω} f_{0} φ dx + \sum_{j = 1}^{n} f_{j} D_{j} φ dx, \\ {T‖}_{*} = {f_{0} / ω_{2}‖}_{L^{p^{'}} (Ω, ω_{2})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}, \\ (T | φ)| \leq {T‖}_{*} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

If ω = ω₁ = ω₂, we denote $W_{0}^{1, p} (Ω, ω) = W_{0}^{1, p} (Ω, ω, ω)$ .

In this paper we use the following results.

Theorem 2.1.

Let ω ∈ A_p, 1 < p < ∞, and let Ω be a bounded open set in ℝⁿ. If u_m→u in L^p(Ω, ω) then there exist a subsequence {u_{m_k}} and a function Φ ∈ L^p(Ω, ω) such that

(i) u_{m_k} (x)→u(x), m_k → ∞ a.e. on Ω;
(ii) |u_{m_k} (x)| ≤ Φ(x) a.e. on Ω.

Proof

The proof of this theorem follows the lines of Theorem 2.8.1 in [13].

Theorem 2.2 (The weighted Sobolev inequality).

Let Ω be an open bounded set in ℝⁿ and ω ∈ A_p (1 < p < ∞). There exist positive constants C_Ω and δ such that for all $u \in W_{0}^{1, p} (Ω, ω)$ and all k satisfying 1 ≤ k ≤ n/(n−1)+δ,

(2.2)

{u‖}_{L^{kp} (Ω, ω)} \leq C_{Ω} {\nabla u|‖}_{L^{p} (Ω, ω)},

where C_Ω depends only on n, p, the A_p-constant C(p, ω) of ω and the diameter of Ω.

Proof

It suffices to prove the inequality for functions $u \in C_{0}^{\infty} (Ω)$ (see Theorem 1.3 in [9]). To extend the estimate (2.2) to arbitrary $u \in W_{0}^{1, p} (Ω, ω)$ , we let {u_m} be a sequence of $C_{0}^{\infty} (Ω)$ functions tending to u in $W_{0}^{1, p} (Ω, ω)$ . Applying the estimate (2.2) to differences u_m₁ − u_m₂ , we see that {u_m} will be a Cauchy sequence in L^kp(Ω, ω). Consequently the limit function u will lie in the desired spaces and satisfy (2.2).

Remark 2.3.

If $u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ then by Theorem 2.2 (with k = 1)

{u‖}_{L^{p} (Ω, ω_{1})} \leq C_{Ω} {\nabla u|‖}_{L^{p} (Ω, ω_{1})} \leq C_{Ω} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} .

Hence,

W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \subset W_{0}^{1, p} (Ω, ω_{1})

Proposition 2.4.

Let 1 < p < ∞.

(a) There exists a constant C_p such that
$ξ |^{p - 2} ξ -| η |^{p - 2} η| \leq C_{p} ξ - η| {(ξ| + η|)}^{p - 2}$
for all ξ, η ∈ ℝⁿ.
(b) There exist two positive constants β_p, γ_p such that for every x, y ∈ℝⁿ
$β_{p} {(x| + y|)}^{p - 2} {x - y|}^{2} \leq {x|}^{p - 2} x - {y|}^{p - 2} y), (x - y)〉 \leq γ_{p} {(x| + y|)}^{p - 2} {x - y|}^{2} .$

Proof

See Proposition 17.2 and Proposition 17.3 in [6].

Definition 2.3.

We say that an element $u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ is a (weak) solution of problem (P) if

\begin{array}{l} \int_{Ω} 𝒜 (x, u, \nabla u), \nabla φ〉 ω_{1} dx + \int_{Ω} ℬ (x, u, \nabla u), \nabla φ〉 ν_{1} dx \\ + \int_{Ω} ℋ (x, u, \nabla u) φ ν_{2} dx + \sum_{i, j = 1}^{n} \int_{Ω} a_{ij} (x) D_{i} u (x) D_{j} φ (x) dx \\ + \int_{Ω} {u|}^{p - 2} u φ ω_{2} dx = \int_{Ω} f_{0} φ dx + \sum_{j = 1}^{n} \int_{Ω} f_{j} D_{j} φ dx, \end{array}

for all

φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

Remark 2.5.

(i) If $\frac{ν_{1}}{ω_{1}} \in L^{r_{1}} (Ω, ω_{1})$ and $\frac{ν_{1}}{ω_{2}} \in L^{r_{1}} (Ω, ω_{2})$ (where r₁ = p/(p − q), 1 < q < p < ∞) then
${u‖}_{L^{q} (Ω, ν_{1})} \leq C_{p, q} {u‖}_{L^{p} (Ω, ω_{1})} and {u‖}_{L^{q} (Ω, ν_{1})} \leq {\tilde{C}}_{p, q} {u‖}_{L^{p} (Ω, ω_{2})},$
where $C_{p, q} = {ν_{1} / ω_{1}‖}_{L^{r_{1}} (Ω, ω_{1})}^{1 / q}$ and ${\tilde{C}}_{p, q} = {ν_{1} / ω_{2}‖}_{L^{r_{1}} (Ω, ω_{2})}^{1 / q}$ . In fact, by Hölder’s inequality we obtain
$\begin{array}{l} {u‖}_{L^{q} (Ω, ν_{1})}^{q} & = \int_{Ω} {u|}^{q} ν_{1} dx = \int_{Ω} {u|}^{q} \frac{ν_{1}}{ω_{1}} ω_{1} dx \\ \leq {(\int_{Ω} {u|}^{q p / q} ω_{1} dx)}^{q / p} {(\int_{Ω} {(ν_{1} / ω_{1})}^{p / (p - q)} ω_{1} dx)}^{(p - q) / p} \\ = {u‖}_{L^{p} (Ω, ω_{1})}^{q} {ν_{1} / ω_{1}‖}_{L^{r_{1}} (Ω, ω_{1})} . \end{array}$
Hence,
${u‖}_{L^{q} (Ω, ν_{1})} \leq C_{p, q} {u‖}_{L^{p} (Ω, ω_{1})} .$
(ii) Analogously, if $\frac{ν_{2}}{ω_{1}} \in L^{r_{2}} (Ω, ω_{1})$ and $\frac{ν_{2}}{ω_{2}} \in L^{r_{2}} (Ω, ω_{2})$ (where r₂ = p/(p−s), 1 < s < p < ∞) then
${u‖}_{L^{s} (Ω, ν_{2})} \leq C_{p, s} {u‖}_{L^{p} (Ω, ω_{1})} and {u‖}_{L^{s} (Ω, ν_{2})} \leq {\tilde{C}}_{p, s} {u‖}_{L^{p} (Ω, ω_{2})},$
where $C_{p, s} = {ν_{2} / ω_{1}‖}_{L^{r_{2}} (Ω, ω_{1})}^{1 / s}$ and ${\tilde{C}}_{p, s} = {ν_{2} / ω_{2}‖}_{L^{r_{2}} (Ω, ω_{2})}^{1 / s}$ .
(iii) If $\frac{ν_{3}}{ω_{1}} \in L^{r_{3}} (Ω, ω_{1})$ (where r₃ = 2/(p − 2), 2 < p < ∞) then
${u‖}_{L^{2} (Ω, ν_{3})} \leq C_{p, 2} {u‖}_{L^{p} (Ω, ω_{1})},$
where $C_{p, 2} = {ν_{3} / ω_{1}‖}_{L^{r_{3}} (Ω, ω_{1})}^{1 / 2}$ .

3. Proof of Theorem 1.1

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

Theorem 3.1.

Let A: X→X^∗ be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach space X. Then the following assertions hold:

(a) for each T ∈ X^∗ the equation Au = T has a solution u∈X;
(b) if the operator A is strictly monotone, then equation Au = T is uniquely solvable in X.

Proof

See Theorem 26.A in [20].

To prove Theorem 1.1, we define

B, B_{1}, B_{2}, B_{3}, B_{4}, B_{5} : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \times W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to ℝ

\begin{array}{l} B (u, φ) = B_{1} (u, φ) + B_{2} (u, φ) + B_{3} (u, φ) + B_{4} (u, φ) + B_{5} (u, φ), \\ B_{1} (u, φ) = \int_{Ω} 𝒜 (x, u, \nabla u), \nabla φ〉 ω_{1} dx, \\ B_{2} (u, φ) = \int_{Ω} ℬ (x, u, \nabla u), \nabla φ〉 ν_{1} dx, \\ B_{3} (u, φ) = \int_{Ω} ℋ (x, u, \nabla u) φ ν_{2} dx, \\ B_{4} (u, φ) = \int_{Ω} {u|}^{p - 2} u φ ω_{2} dx, \\ B_{5} (u, φ) = \sum_{i, j = 1}^{n} \int_{Ω} a_{ij} (x) D_{i} u (x) D_{j} φ (x) dx = \int_{Ω} ℳ (x) \nabla u (x), \nabla φ (x)〉 dx, \end{array}

and

T : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to ℝ

T (φ) = \int_{Ω} f_{0} φ dx + \sum_{j = 1}^{n} \int_{Ω} f_{j} D_{j} φ dx .

Then

u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

is a (weak) solution to problem (P) if

B (u, φ) = B_{1} (u, φ) + B_{2} (u, φ) + B_{3} (u, φ) + B_{4} (u, φ) + B_{5} (u, φ) = T (φ),

for all

φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

Step 1. For j = 1, . . . , n we define the operator $F_{j} : W_{0}^{1, p} (Ω, ω_{1}, ω_{2})) \to L^{p^{'}} (Ω, ω_{1})$ as

(F_{j} u) (x) = 𝒜_{j} (x, u (x), \nabla u (x)) .

We now show that the operator F_j is bounded and continuous.

(i) Using (H4), we obtain

(3.1)

\begin{matrix} {F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})}^{p^{'}} = \int_{Ω} {F_{j} u (x)|}^{p^{'}} ω_{1} dx = \int_{Ω} {𝒜_{j} (x, u, \nabla u)|}^{p^{'}} ω_{1} dx \\ \leq \int_{Ω} {(K_{1} + h_{1} {(ω_{2} / ω_{1})}^{1 / p^{'}} {u|}^{p / p^{'}} + h_{2} {\nabla u|}^{p / p^{'}})}^{p^{'}} ω_{1} dx \\ \leq C_{p} \int_{Ω} K_{1}^{p^{'}} ω_{1} dx + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} \int_{Ω} {u|}^{p} ω_{2} dx + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} \int_{Ω} {\nabla u|}^{p} ω_{1} dx] \\ \leq C_{p} {K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}^{p^{'}} + ({h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p}], \end{matrix}

where the constant C_p depends only on p. Therefore, in (3.1) we obtain

{F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})} \leq C_{p}^{1 / p^{'}} ({K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})} + ({h_{1}‖}_{L^{\infty} (Ω)} + {h_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1}) .

(ii) Let u_m→u in $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ 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 $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ , then u_m→u in L^p(Ω, ω₂) and |∇_{u_m}|→ |∇u| in L^p(Ω, ω₁). Using Theorem 2.1, there exist a subsequence {u_{m_k}} and functions Φ₂ ∈ L^p(Ω, ω₂), Φ₁ ∈ L^p(Ω, ω₁) such that

\begin{matrix} u_{m_{k}} (x) \to u (x) a.e. in Ω, \\ u_{m_{k}} (x)| \leq Φ_{2} (x) a.e. in Ω, \\ D_{j} u_{m_{k}} (x) \to D_{j} u (x) a.e. in Ω, \\ \nabla u_{m_{k}} (x)| \leq Φ_{1} (x) a.e. in Ω . \end{matrix}

Next, applying (H4) we obtain

\begin{array}{l} {F_{j} u_{m_{k}} (x) - F_{j} u (x)|}^{p^{'}} ω_{1} & = {𝒜_{j} (x, u_{m_{k}}, \nabla u_{m_{k}}) - 𝒜_{j} (x, u, \nabla u)|}^{p^{'}} ω_{1} \\ \leq C_{p} ({𝒜_{j} (x, u_{m_{k}}, \nabla u_{m_{k}})|}^{p^{'}} + {𝒜_{j} (x, u, \nabla u)|}^{p^{'}}) ω_{1} \\ \leq C_{p} {(K_{1} + h_{1} {(ω_{2} / ω_{1})}^{1 / p^{'}} {u_{m_{k}}|}^{p / p^{'}} + h_{2} {\nabla u_{m_{k}}|}^{p / p^{'}})}^{p^{'}} \\ + {(K_{1} + h_{1} {(ω_{2} / ω_{1})}^{1 / p^{'}} {u|}^{p / p^{'}} + h_{2} {\nabla u|}^{p / p^{'}})}^{p^{'}}] ω_{1} \\ \leq C_{p} (K_{1}^{p^{'}} + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} {u_{m_{k}}|}^{p} \frac{ω_{2}}{ω_{1}} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} {\nabla u_{m_{k}}|}^{p}) \\ + {K_{1}^{p^{'}} + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} {u|}^{p} \frac{ω_{2}}{ω_{1}} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} \nabla u|}^{p}] ω_{1} \\ \leq C_{p} (K_{1}^{p^{'}} + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{2}^{p} \frac{ω_{2}}{ω_{1}} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{1}^{p}) \\ + (K_{1}^{p^{'}} + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{2}^{p} \frac{ω_{2}}{ω_{1}} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{1}^{p})] ω_{1} \\ = 2 C_{p} K_{1}^{p^{'}} ω_{1} + {h_{1}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{2}^{p} ω_{2} + {h_{2}‖}_{L^{\infty} (Ω)}^{p^{'}} Φ_{1}^{p} ω_{1}] \in L^{1} (Ω) . \end{array}

By condition (H1), we have

F_{j} u_{m_{k}} (x) = 𝒜_{j} (x, u_{m_{k}} (x), \nabla u_{m_{k}} (x)) \to 𝒜_{j} (x, u (x), \nabla u (x)) = F_{j} u (x),

as m_k → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain ‖F_ju_{m_k} − F_ju‖_{L^{p^′}(Ω,ω₁)} → 0, that is, F_ju_{m_k} → 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)

F_{j} u_{m} \to F_{j} u in L^{p^{'}} (Ω, ω_{1}) .

Step 2. We define the operator $G_{j} : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to L^{q^{'}} (Ω, ν_{1})$ by

(G_{j} u) (x) = ℬ_{j} (x, u (x), \nabla u (x)) .

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

\begin{array}{l} {G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}^{q^{'}} & = \int_{Ω} {G_{j} u (x)|}^{q^{'}} ν_{1} dx = \int_{Ω} {ℬ_{j} (x, u, \nabla u)|}^{q^{'}} ν_{1} dx \\ \leq \int_{Ω} {(K_{2} + g_{1} {u|}^{q / q^{'}} + g_{2} {\nabla u|}^{q / q^{'}})}^{q^{'}} ν_{1} dx \\ \leq C_{q} \int_{Ω} (K_{2}^{q^{'}} + g_{1}^{q^{'}} {u|}^{q} + g_{2}^{q^{'}} {\nabla u|}^{q}) ν_{1}] dx \\ = C_{q} \int_{Ω} K_{2}^{q^{'}} ν_{1} dx + \int_{Ω} g_{1}^{q^{'}} {u|}^{q} ν_{1} dx + \int_{Ω} g_{2}^{q^{'}} {\nabla u|}^{q} ν_{1} dx] \\ \leq C_{q} ({K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})}^{q^{'}} + {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} {u‖}_{L^{q} (Ω, ν_{1})}^{q} + {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}} {\nabla u|‖}_{L^{q} (Ω, ν_{1})}^{q}) \\ \leq C_{q} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})}^{q^{'}} + {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} C_{p, q}^{q} {u‖}_{L^{p} (Ω, ω_{1})}^{q} \\ + C_{p, q}^{q} {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{q}) \\ \leq C_{q} ({K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})}^{q^{'}} + C_{p, q}^{q} (C_{Ω}^{q} {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} + {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{q}), \end{array}

where the constant C_q depends only on q. Therefore, we obtain

\begin{array}{l} {G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})} & \leq C_{q}^{1 / q^{'}} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} \\ + C_{p, q}^{q - 1} (C_{Ω}^{q - 1} {g_{1}‖}_{L^{\infty} (Ω)} + {g_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{q - 1}) . \end{array}

(ii) Let u_m→u in $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ 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 $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ , 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_{m_k}} and functions Φ₂∈L^p(Ω, ω₂) and Φ₁∈L^p(Ω, ω₁) such that

\begin{matrix} u_{m_{k}} (x) \to u (x) a.e. in Ω, \\ u_{m_{k}} (x)| \leq Φ_{2} (x) a.e. in Ω, \\ D_{j} u_{m_{k}} (x) \to D_{j} u (x) a.e. in Ω, \\ \nabla u_{m_{k}} (x)| \leq Φ_{1} (x) a.e. in Ω . \end{matrix}

Next, applying (H8) and Remark 2.5(i) we obtain

\begin{array}{l} {G_{j} u_{m_{k}} (x) - G_{j} u (x)|}^{q^{'}} ν_{1} & = {ℬ_{j} (x, u_{m_{k}}, \nabla u_{m_{k}}) - ℬ_{j} (x, u, \nabla u)|}^{q^{'}} ν_{1} \\ \leq C_{q} {ℬ_{j} (x, u_{m_{k}}, \nabla u_{m_{k}})|}^{q^{'}} + {ℬ_{j} (x, u, \nabla u)|}^{q^{'}}] ν_{1} \\ \leq C_{q} {(K_{2} + g_{1} {u_{m_{k}}|}^{q / q^{'}} + g_{2} {\nabla u_{m_{k}}|}^{q / q^{'}})}^{q^{'}} \\ + {(K_{2} + g_{1} {u|}^{q / q^{'}} + g_{2} {\nabla u|}^{q / q^{'}})}^{q^{'}}] ν_{1} \\ \leq C_{q} (K_{2}^{q^{'}} + {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} {u_{m_{k}}|}^{q} + {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}} {\nabla u_{m_{k}}|}^{q}) \\ + (K_{2}^{q^{'}} + {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} {u|}^{q} + {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}} {\nabla u|}^{q})] ν_{1} \\ \leq 2 C_{q} K_{2}^{q^{'}} ν_{1} + {g_{1}‖}_{L^{\infty} (Ω)}^{q^{'}} Φ_{2}^{q} ν_{1} + {g_{2}‖}_{L^{\infty} (Ω)}^{q^{'}} Φ_{1}^{q} ν_{1}] \in L^{1} (Ω), \end{array}

since

\int_{Ω} Φ_{1}^{q} ν_{1} dx \leq C_{p, q}^{q} \int_{Ω} Φ_{1}^{p} ω_{1} dx

and

\int_{Ω} Φ_{2}^{q} ν_{1} dx \leq {\tilde{C}}_{p, q}^{q} \int_{Ω} Φ_{2}^{p} ω_{2} dx

. By condition (H5), we have

G_{j} u_{m_{k}} (x) = ℬ_{j} (x, u_{m_{k}} (x), \nabla u_{m_{k}} (x)) \to ℬ_{j} (x, u (x), \nabla u (x)) = G_{j} u (x),

as m_k → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain

{G_{j} u_{m_{k}} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})} \to 0,

that is,

G_{j} u_{m_{k}} \to G_{j} u in L^{q^{'}} (Ω, ν_{1}) .

We conclude from the Convergence Principle in Banach spaces (see Proposition 10.13 in [19]) that

(3.3)

G_{j} u_{m} \to G_{j} u in L^{q^{'}} (Ω, ν_{1}) .

Step 3. We define the operator $H : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to L^{s^{'}} (Ω, ν_{2})$ by

(Hu) (x) = ℋ (x, u (x), \nabla u (x)) .

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

\begin{array}{l} {Hu‖}_{L^{s^{'}} (Ω, ν_{2})}^{s^{'}} & = \int_{Ω} {Hu|}^{s^{'}} ν_{2} dx = \int_{Ω} {ℋ (x, u, \nabla u)|}^{s^{'}} ν_{2} dx \\ \leq \int_{Ω} {(K_{3} + h_{3} {u|}^{s / s^{'}} + h_{4} {\nabla u|}^{s / s^{'}})}^{s^{'}} ν_{2} dx \\ \leq C_{s} \int_{Ω} (K_{3}^{s^{'}} + h_{3}^{s^{'}} {u|}^{s} + h_{4}^{s^{'}} {\nabla u|}^{s}) ν_{2} dx \\ \leq C_{s} \int_{Ω} K_{3}^{s^{'}} ν_{2} dx + {h_{3}‖}_{L^{\infty} (Ω)}^{s^{'}} \int_{Ω} {u|}^{s} ν_{2} dx + {h_{4}‖}_{L^{\infty} (Ω)}^{s^{'}} \int_{Ω} {\nabla u|}^{s} ν_{2} dx] \\ \leq C_{s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})}^{s^{'}} + {h_{3}‖}_{L^{\infty} (Ω)}^{s^{'}} C_{p, s}^{s} {u‖}_{L^{p} (Ω, ω_{1})}^{s} \\ + {h_{4}‖}_{L^{\infty} (Ω)}^{s^{'}} C_{p, s}^{s} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{s}) \\ \leq C_{s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})}^{s^{'}} + {h_{3}‖}_{L^{\infty} (Ω)}^{s^{'}} C_{p, s}^{s} C_{Ω}^{s} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{s} \\ + {h_{4}‖}_{L^{\infty} (Ω)}^{s^{'}} C_{p, s}^{s} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{s}) \\ \leq C_{s} ({K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})}^{s^{'}} + C_{p, s}^{s} (C_{Ω}^{s} {h_{3}‖}_{L^{\infty} (Ω)}^{s^{'}} + {h_{4}‖}_{L^{\infty} (Ω)}^{s^{'}}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{s}), \end{array}

where the constant C_s depends only on s. Hence, we obtain

\begin{array}{l} {Hu‖}_{L^{s^{'}} (Ω, ν_{2})} & \leq C_{s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})} \\ + C_{p, s}^{s - 1} (C_{Ω}^{s - 1} {h_{3}‖}_{L^{\infty} (Ω)} + {h_{4}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{s - 1}] . \end{array}

(ii) Applying (H12) and Remark 2.5(ii), by the same argument used in Step 2(ii), we obtain analogously, if u_m→u in $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ then

(3.4)

H u_{m} \to Hu in L^{s^{'}} (Ω, ν_{2}) .

Step 4. We define the operator $J : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to L^{p^{'}} (Ω, ω_{2})$ by

(Ju) (x) = {u (x)|}^{p - 2} u (x) .

We also have that the operator J is continuous and bounded. In fact:

(i) For all $u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ ,

\begin{array}{l} {Ju‖}_{L^{p^{'}} (Ω, ω_{2})}^{p^{'}} & = \int_{Ω} {Ju|}^{p^{'}} ω_{2} dx = \int_{Ω} {u|}^{(p - 1) p^{'}} ω_{2} dx \\ = \int_{Ω} {u|}^{p} ω_{2} dx \leq {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p} . \end{array}

Hence,

{Ju‖}_{L^{p^{'}} (Ω, ω_{2})} \leq {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1}

(ii) Let u_m→u in $W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ . Then u_m→u in L^p(Ω, ω₂). Using Theorem 2.1, there exist a subsequence {u_{m_k}} and a function Φ₂ ∈ L^p(Ω, ω₂) such that

\begin{matrix} u_{m_{k}} (x) \to u (x) a.e. in Ω, \\ u_{m_{k}} (x)| \leq Φ_{2} (x) a.e. in Ω . \end{matrix}

Next, applying Proposition 2.4(a), we have

\begin{array}{l} {J u_{m_{k}} - Ju‖}_{L^{p^{'}} (Ω, ω_{2})}^{p^{'}} & = \int_{Ω} {J u_{m_{k}} - Ju|}^{p^{'}} ω_{2} dx \\ = \int_{Ω} {{u_{m_{k}}|}^{p - 2} u_{m_{k}} - {u|}^{p - 2} u|}^{p^{'}} ω_{2} dx \\ \leq \int_{Ω} {C_{p} u_{m_{k}} - u| {(u_{m_{k}}| + u|)}^{p - 2}]}^{p^{'}} ω_{2} dx \\ = C_{p}^{p^{'}} \int_{Ω} {u_{m_{k}} - u|}^{p^{'}} {(u_{m_{k}}| + u|)}^{(p - 2) p^{'}} ω_{2} dx \\ \leq 2^{(p - 2) p^{'}} C_{p}^{p^{'}} \int_{Ω} {u_{m_{k}} - u|}^{p^{'}} Φ_{2}^{(p - 2) p^{'}} ω_{2} dx \\ \leq 2^{(p - 2) p^{'}} C_{p}^{p^{'}} {(\int_{Ω} {u_{m_{k}} - u|}^{p^{'} (p / p^{'})} ω_{2} dx)}^{p^{'} / p} \\ \times {(\int_{Ω} Φ_{2}^{(p - 2) p^{'} p / (p - p^{'})} ω_{2} dx)}^{(p - p^{'}) / p} \\ = 2^{(p - 2) p^{'}} C_{p}^{p^{'}} {(\int_{Ω} {u_{m_{k}} - u|}^{p} ω_{2} dx)}^{p^{'} / p} {(\int_{Ω} Φ_{2}^{p} ω_{2} dx)}^{(p - p^{'}) / p} \\ = 2^{(p - 2) p^{'}} C_{p}^{p^{'}} {u_{m_{k}} - u‖}_{L^{p} (Ω, ω_{2})}^{p^{'}} {Φ_{2}‖}_{L^{p} (Ω, ω_{2})}^{p - p^{'}} . \end{array}

Hence ‖Ju_{m_k} − Ju‖_{L^{p^′}(Ω,ω₂)}→0 as m_k→∞. We conclude from the Convergence Principle in Banach spaces that

(3.5)

J u_{m} \to Ju in L^{p^{'}} (Ω, ω_{2}) .

Step 5. By (H13) and Remark 2.5(iii) we obtain

\begin{array}{l} B_{5} (u, φ)| & \leq \int_{Ω} ℳ (x) \nabla u (x), \nabla φ (x)〉| dx \\ \leq \int_{Ω} {ℳ (x) \nabla u (x), \nabla u (x)〉}^{1 / 2} {ℳ (x) \nabla φ (x), \nabla φ (x)〉}^{1 / 2} dx \\ \leq {(\int_{Ω} ℳ (x) \nabla u (x), \nabla u (x)〉 dx)}^{1 / 2} {(\int_{Ω} ℳ (x) \nabla φ (x), \nabla φ (x)〉 dx)}^{1 / 2} \\ \leq {(\int_{Ω} Λ {\nabla u (x)|}^{2} ν_{3} dx)}^{1 / 2} {(\int_{Ω} Λ {\nabla φ (x)|}^{2} ν_{3} dx)}^{1 / 2} \\ = Λ {\nabla u|‖}_{L^{2} (Ω, ν_{3})} {\nabla φ (x)|‖}_{L^{2} (Ω, ν_{3})} \\ \leq Λ C_{p, 2}^{2} {\nabla u|‖}_{L^{p} (Ω, ω_{1})} {\nabla φ (x)|‖}_{L^{p} (Ω, ω_{1})} \\ \leq Λ C_{p, 2}^{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

Step 6. Since $\frac{f_{0}}{ν_{1}} \in L^{q^{'}} (Ω, ν_{1})$ and $\frac{f_{j}}{ω_{1}} \in L^{p^{'}} (Ω, ω_{1}) (j = 1, \dots, n)$ then $T \in {W_{0}^{1, p} (Ω, ω_{1}, ω_{2})]}^{*}$ . Moreover, by Remark 2.5(i), we have

\begin{array}{l} T (φ)| & \leq \int_{Ω} f_{0}| φ| dx + \sum_{j = 1}^{n} \int_{Ω} f_{j}| D_{j} φ| dx \\ = \int_{Ω} \frac{f_{0}|}{ν_{1}} φ| ν_{1} dx + \sum_{j = 1}^{n} \int_{Ω} \frac{f_{j}|}{ω_{1}} D_{j} φ| ω_{1} dx \\ \leq {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} {φ‖}_{L^{q} (Ω, ν_{1})} + (\sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}) {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ \leq (C_{p, q} {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}) {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

Moreover, we also have

(3.6)

\begin{array}{l} B (u, φ)| & \leq B_{1} (u, φ) + B_{2} (u, φ)| + B_{3} (u, φ)| + B_{4} (u, φ)| + B_{5} (u, φ)| \\ \leq \int_{Ω} 𝒜 (x, u, \nabla u)| \nabla φ| ω_{1} dx + \int_{Ω} ℬ (x, u, \nabla u)| \nabla φ| ν_{1} dx \\ + \int_{Ω} ℋ (x, u, \nabla u)| φ| ν_{2} + \int_{Ω} {u|}^{p - 1} φ| ω_{2} dx \\ + \int_{Ω} ℳ (x) \nabla u, \nabla φ〉| dx . \end{array}

In (3.6) we have, by (H4),

\begin{array}{l} \int_{Ω} 𝒜 (x, u, \nabla u)| \nabla φ| ω_{1} dx & \leq \int_{Ω} (K_{1} + h_{1} {(\frac{ω_{2}}{ω_{1}})}^{1 / p^{'}} {u|}^{p / p^{'}} + h_{2} {\nabla u|}^{p / p^{'}}) \nabla φ| ω_{1} dx \\ \leq \int_{Ω} K_{1} \nabla φ| ω_{1} dx + {h_{1}‖}_{L^{\infty} (Ω)} \int_{Ω} {(\frac{ω_{2}}{ω_{1}})}^{1 / p^{'}} {u|}^{p / p^{'}} \nabla φ| ω_{1} dx \\ + {h_{2}‖}_{L^{\infty} (Ω)} \int_{Ω} {\nabla u|}^{p / p^{'}} \nabla φ| ω_{1} dx \\ \leq {K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} + {h_{1}‖}_{L^{\infty} (Ω)} {u‖}_{L^{p} (Ω, ω_{2})}^{p - 1} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ + {h_{2}‖}_{L^{\infty} (Ω)} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{p - 1} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ \leq {K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})} + {h_{1}‖}_{L^{\infty} (Ω)} \\ + {h_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1}) {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

and by (H8) and Remark 2.5(i),

\begin{array}{l} \int_{Ω} ℬ (x, u, \nabla u)| \nabla φ| ν_{1} dx & \leq \int_{Ω} (K_{2} + g_{1} {u|}^{q / q^{'}} + g_{2} {\nabla u|}^{q / q^{'}}) \nabla φ| ν_{1} dx \\ \leq {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} {\nabla φ|‖}_{L^{q} (Ω, ν_{1})} + {g_{1}‖}_{L^{\infty} (Ω)} {u‖}_{L^{q} (Ω, ν_{1})}^{q / q^{'}} {\nabla φ|‖}_{L^{q} (Ω, ν_{1})} \\ + {g_{2}‖}_{L^{\infty} (Ω)} {\nabla u|‖}_{L^{q} (Ω, ν_{1})}^{q / q^{'}} {\nabla φ|‖}_{L^{q} (Ω, ν_{1})} \\ \leq C_{p, q} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ + C_{p, q}^{q - 1} {g_{1}‖}_{L^{\infty} (Ω)} {u‖}_{L^{p} (Ω, ω_{1})}^{q - 1} C_{p, q} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ + {g_{2}‖}_{L^{\infty} (Ω)} C_{p, q}^{q - 1} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{q - 1} C_{p, q} {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ \leq C_{p, q} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} + C_{p, q}^{q} {g_{1}‖}_{L^{\infty} (Ω)} \\ + C_{p, q}^{q} {g_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{q - 1}] {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

According to (H12) and Remark 2.5(ii),

\begin{array}{l} \int_{Ω} ℋ (x, u, \nabla u)| φ| ν_{2} dx & \leq \int_{Ω} (K_{3} + h_{3} {u|}^{s / s^{'}} + h_{4} {\nabla u|}^{s / s^{'}}) φ| ν_{2} dx \\ \leq \int_{Ω} K_{3} φ| ν_{2} dx + {h_{3}‖}_{L^{\infty} (Ω)} \int_{Ω} {u|}^{s / s^{'}} φ| ν_{2} dx \\ + {h_{4}‖}_{L^{\infty} (Ω)} \int_{Ω} {\nabla u|}^{s / s^{'}} φ| ν_{2} dx \\ \leq {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})} {φ‖}_{L^{s} (Ω, ν_{2})} + {h_{3}‖}_{L^{\infty} (Ω)} {u‖}_{L^{s} (Ω, ν_{2})}^{s / s^{'}} {φ‖}_{L^{s} (Ω, ν_{2})} \\ + {h_{4}‖}_{L^{\infty} (Ω)} {\nabla u‖}_{L^{s} (Ω, ν_{1})}^{s - 1} {φ‖}_{L^{s} (Ω, ν_{2})} \\ \leq C_{p, s} {K_{3}‖}_{L^{s^{'}} (Ω)} {φ‖}_{L^{p} (Ω, ω_{1})} + {h_{3}‖}_{L^{\infty} (Ω)} C_{p, s}^{s - 1} {u‖}_{L^{p} (Ω, ω_{1})}^{s - 1} C_{p, s} {φ‖}_{L^{p} (Ω, ω_{1})} \\ + {h_{4}‖}_{L^{\infty} (Ω)} C_{p, s}^{s - 1} {\nabla u|‖}_{L^{p} (Ω, ω_{1})}^{s - 1} C_{p, s} {φ‖}_{L^{p} (Ω, ω_{1})} \\ \leq C_{p, s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})} + C_{p, s}^{s} {h_{3}‖}_{L^{\infty} (Ω)} \\ + {h_{4}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{s - 1}] {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

and

\begin{array}{l} \int_{Ω} {u|}^{p - 1} φ| ω_{2} dx & \leq {(\int_{Ω} {u|}^{p} ω_{2} dx)}^{1 / p^{'}} {(\int_{Ω} {φ|}^{p} ω_{2} dx)}^{1 / p} \\ \leq C_{Ω} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

and by Step 5,

B_{5} (u, φ)| \leq Λ C_{p, 2}^{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} .

Hence, in (3.6) we obtain, for all

u, φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

\begin{array}{l} B (u, φ)| \leq {K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})} + ({h_{1}‖}_{L^{\infty} (Ω)} + {h_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1} \\ + C_{p, q} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} + C_{p, q}^{q} ({g_{1}‖}_{L^{\infty} (Ω)} + {g_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{q - 1} \\ + C_{p, s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})} + C_{p, s}^{s} ({h_{3}‖}_{L^{\infty} (Ω)} + {h_{4}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{s - 1} \\ + C_{Ω} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1} + Λ C_{p, 2}^{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}] {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

Since B(u, .) is linear, for each

u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

, there exists a linear and continuous functional on

W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

denoted by Au such that (Au|φ) = B(u, φ) for all

u, φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

(here (f|x) denotes the value of the linear functional f at the point x). Moreover

\begin{array}{l} {A u‖}_{*} \leq {K_{1}‖}_{L^{p^{'}} (Ω, ω_{1})} + ({h_{1}‖}_{L^{\infty} (Ω)} + {h_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1} \\ + C_{p, q} {K_{2}‖}_{L^{q^{'}} (Ω, ν_{1})} + C_{p, q}^{q} ({g_{1}‖}_{L^{\infty} (Ω)} + {g_{2}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{q - 1} \\ + C_{p, s} {K_{3}‖}_{L^{s^{'}} (Ω, ν_{2})} + C_{p, s}^{s} ({h_{3}‖}_{L^{\infty} (Ω)} + {h_{4}‖}_{L^{\infty} (Ω)}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{s - 1} \\ + C_{Ω} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p - 1} + Λ C_{p, 2}^{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

where

{Au‖}_{*} = sup {(Au | φ)| = B (u, φ)| : φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2}), {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} = 1}

is the norm of the operator Au. Hence, we obtain the operator

A : W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) \to {W_{0}^{1, p} (Ω, ω_{1}, ω_{2})]}^{*}, u \mapsto Au .

Consequently, problem (P) is equivalent to the operator equation

Au = T, u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2}) .

Step 7. Using (H2), (H6), (H10), (H13) and Proposition 2.4(b), we obtain, for u₁, $u_{2} \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ , u₁ ≠ u₂,

\begin{array}{l} (A u_{1} - A u_{2} | u_{1} - u_{2}) & = B (u_{1}, u_{1} - u_{2}) - B (u_{2}, u_{1} - u_{2}) \\ = \int_{Ω} 𝒜 (x, u_{1} \nabla u_{1}), \nabla (u_{1} - u_{2})〉 ω_{1} dx + \int_{Ω} ℬ (x, u_{1}, \nabla u_{1}), \nabla (u_{1} - u_{2})〉 ν_{1} dx \\ + \int_{Ω} ℋ (x, u_{1}, \nabla u_{1}) (u_{1} - u_{2}) ν_{2} dx + \int_{Ω} {u_{1}|}^{p - 2} u_{1} (u_{1} - u_{2}) ω_{2} dx \\ + \int_{Ω} ℳ (x) \nabla u_{1} (x), \nabla (u_{1} - u_{2})〉 dx \\ - \int_{Ω} 𝒜 (x, u_{2}, \nabla u_{2}), \nabla (u_{1} - u_{2})〉 ω_{1} dx - \int_{Ω} ℬ (x, u_{2}, \nabla u_{2}, \nabla (u_{1} - u_{2})〉 ν_{1} dx \\ - \int_{Ω} ℋ (x, u_{2}, \nabla u_{2}) (u_{1} - u_{2}) ν_{2} dx - \int_{Ω} {u_{2}|}^{p - 2} u_{2} (u_{1} - u_{2}) ω_{2} dx \\ - \int_{Ω} ℳ (x) \nabla u_{2} (x), \nabla (u_{1} - u_{2})〉 dx \\ = \int_{Ω} 𝒜 (x, u_{1} \nabla u_{1}) - 𝒜 (x, u_{2}, \nabla u_{2}), \nabla (u_{1} - u_{2})〉 ω_{1} dx \\ + \int_{Ω} ℬ (x, u_{1}, \nabla u_{1}) - ℬ (x, u_{2}, \nabla u_{2}), \nabla (u_{1} - u_{2})〉 ν_{1} dx \\ + \int_{Ω} (ℋ (x, u_{1}, \nabla u_{1}) - ℋ (x, u_{2}, \nabla u_{2})) (u_{1} - u_{2}) v_{2} dx \\ + \int_{Ω} ({u_{1}|}^{p - 2} u_{1} - {u_{2}|}^{p - 2} u_{2}) (u_{1} - u_{2}) ω_{2} dx \\ + \int_{Ω} ℳ (x) \nabla (u_{1} - u_{2}), \nabla (u_{1} - u_{2})〉 dx \\ \geq θ_{1} \int_{Ω} {\nabla (u_{1} - u_{2})|}^{p} ω_{1} dx + β_{p} \int_{Ω} {(u_{1} +| u_{2}|)}^{p - 2} {u_{1} - u_{2}|}^{2} ω_{2} dx \\ + Λ \int_{Ω} {\nabla (u_{1} - u_{2})|}^{2} ν_{3} dx \\ \geq θ_{1} \int_{Ω} {\nabla (u_{1} - u_{2})|}^{p} ω_{1} dx + β_{p} \int_{Ω} {u_{1} - u_{2}|}^{p - 2} {u_{1} - u_{2}|}^{2} ω_{2} dx \\ = θ_{1} \int_{Ω} {\nabla (u_{1} - u_{2})|}^{p} ω_{1} dx + β_{p} \int_{Ω} {u_{1} - u_{2}|}^{p} ω_{2} dx \geq γ_{1} {u_{1} - u_{2}‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p}, \end{array}

where γ₁ = min{θ₁, β_p}. Therefore, the operator A is strictly monotone. Moreover, from (H3), (H7), (H11) and (H13) we obtain

\begin{array}{l} (Au | u) & = B (u, u) = B_{1} (u, u) + B_{2} (u, u) + B_{3} (u, u) + B_{4} (u, u) + B_{5} (u, u) \\ = \int_{Ω} 𝒜 (x, u, \nabla u), \nabla u〉 ω_{1} dx + \int_{Ω} ℬ (x, u, \nabla u), \nabla u〉 ν_{1} dx \\ + \int_{Ω} ℋ (x, u, \nabla u) u ν_{2} dx + \int_{Ω} {u|}^{p} ω_{2} dx + \int_{Ω} ℳ (x) \nabla u, \nabla u〉 dx \\ \geq λ_{1} \int_{Ω} {\nabla u|}^{p} ω_{1} dx + λ_{2} \int_{Ω} {\nabla u|}^{q} ν_{1} dx + Λ_{2} \int_{Ω} {u|}^{q} ν_{1} dx \\ + λ_{3} \int_{Ω} {\nabla u|}^{s} ν_{2} dx + Λ_{3} \int_{Ω} {u|}^{s} ν_{2} dx \\ + \int_{Ω} {u|}^{p} ω_{2} dx + Λ \int_{Ω} {\nabla u|}^{2} ν_{3} dx \\ \geq λ_{1} \int_{Ω} {\nabla u|}^{p} ω_{1} dx + \int_{Ω} {u|}^{p} ω_{2} dx \geq γ_{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}), ω_{2}}^{p}, \end{array}

where γ₂ = min{λ₁, 1}. Hence, since 1 < q, s < p < ∞, we have

\frac{(Au | u)}{{u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}} \to + \infty, as {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} \to + \infty,

that is, A is coercive.

Step 8. We need to show that the operator A is continuous. Let u_m→u in X as m → ∞. We have,

\begin{array}{l} B_{1} (u_{m}, φ) - B_{1} (u, φ)| & \leq \sum_{j = 1}^{n} \int_{Ω} 𝒜_{j} (x, u_{m}, \nabla u_{m}) - 𝒜_{j} (x, u, \nabla u)| D_{j} φ| ω_{1} dx \\ = \sum_{j = 1}^{n} \int_{Ω} F_{j} u_{m} - F_{j} u| D_{j} φ| ω_{1} dx \\ \leq (\sum_{j = 1}^{n} {F_{j} u_{m} - F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})}) {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ \leq (\sum_{j = 1}^{n} {F_{j} u_{m} - F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})}) {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

By Remark 2.5(i), we obtain

\begin{array}{l} B_{2} (u_{m}, φ) - B_{2} (u, φ)| & \leq \sum_{j = 1}^{n} \int_{Ω} ℬ_{j} (x, u_{m}, \nabla u_{m}) - ℬ_{j} (x, u, \nabla u)| D_{j} φ| ν_{1} dx \\ = \sum_{j = 1}^{n} \int_{Ω} G_{j} u_{m} - G_{j} u| D_{j} φ| ν_{1} dx \\ \leq (\sum_{j = 1}^{n} {G_{j} u_{m} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}) {\nabla φ|‖}_{L^{q} (Ω, ν_{1})} \\ \leq C_{p, q} (\sum_{j = 1}^{n} {G_{j} u_{m} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}) {\nabla φ|‖}_{L^{p} (Ω, ω_{1})} \\ \leq C_{p, q} (\sum_{j = 1}^{n} {G_{j} u_{m} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}) {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

and, by Remark 2.5(ii),

\begin{array}{l} B_{3} (u_{m}, φ) - B_{3} (u, φ)| & \leq \int_{Ω} ℋ (x, u_{m}, \nabla u_{m}) - ℋ (x, u, \nabla u)| φ| ν_{2} dx \\ = \int_{Ω} H u_{m} - Hu| φ| ν_{2} dx \\ \leq {H u_{m} - Hu‖}_{L^{s^{'}} (Ω, ν_{2})} {φ‖}_{L^{s} (Ω, ν_{2})} \\ \leq C_{p, s} {H u_{m} - Hu‖}_{L^{s^{'}} (Ω, ν_{2})} {φ‖}_{L^{p} (Ω, ω_{1})} \\ \leq C_{p, s} {H u_{m} - Hu‖}_{L^{s^{'}} (Ω, ν_{2})} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

On account of Step 4,

\begin{array}{l} B_{4} (u_{m}, φ) - B_{4} (u, φ)| & \leq \int_{Ω} {u_{m}|}^{p - 2} u_{m} - {u|}^{p - 2} u| φ| ω_{2} dx \\ = \int_{Ω} J u_{m} - Ju| φ| ω_{2} dx \\ \leq {J u_{m} - Ju‖}_{L^{p^{'}} (Ω, ω_{2})} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

and by Step 5,

\begin{array}{l} B_{5} (u_{m}, φ) - B_{5} (u, φ)| & = \int_{Ω} ℳ (x) \nabla (u_{m} - u) \nabla φ〉 dx| \\ \leq Λ C_{p, 2}^{2} {u_{m} - u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

for all

φ \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})

. Hence,

\begin{array}{l} B (u_{m}, φ) - B (u, φ)| \leq B_{1} (u_{m}, φ) - B_{1} (u, φ)| + B_{2} (u_{m}, φ) - B_{2} (u, φ)| \\ + B_{3} (u_{m}, φ) - B_{3} (u, φ)| + B_{4} (u_{m}, φ) - B_{4} (u, φ)| + B_{5} (u_{m}, φ) - B_{5} (u, φ)| \\ \leq \sum_{j = 1}^{n} ({F_{j} u_{m} - F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})} + C_{p, q} {G_{j} u_{m} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}) \\ + C_{p, s} {H u_{m} - H u‖}_{L^{s^{'}} (Ω, ν_{2})} + {J u_{m} - J u‖}_{L^{p^{'}} (Ω, ω_{2})} \\ + Λ C_{p, 2}^{2} ‖ u_{m} - u ‖_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}] {φ‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

Then we obtain

\begin{array}{l} {A u_{m} - Au‖}_{*} \leq \sum_{j = 1}^{n} ({F_{j} u_{m} - F_{j} u‖}_{L^{p^{'}} (Ω, ω_{1})} + C_{p, q} {G_{j} u_{m} - G_{j} u‖}_{L^{q^{'}} (Ω, ν_{1})}) \\ + C_{p, s} {H u_{m} - Hu‖}_{L^{s^{'}} (Ω, ν_{2})} + {J u_{m} - Ju‖}_{L^{p^{'}} (Ω, ω_{2})} \\ + Λ C_{p, 2}^{2} {u_{m} - u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} . \end{array}

Hence, using (3.2), (3.3), (3.4) and (3.5) we have ‖Au_m − Au‖_∗→0 as m → +∞, that is, A is continuous and this implies that A is hemicontinuous.

Therefore, by Theorem 3.1, the operator equation Au = T has a unique solution $u \in W_{0}^{1, p} (Ω, ω_{1}, ω_{2})$ and it is the unique solution for problem (P).

Step 8. Estimates for ${u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}$ . In particular, by setting φ = u in Definition 2.3, we have

(3.7)

B (u, u) = B_{1} (u, u) + B_{2} (u, u) + B_{3} (u, u) + B_{4} (u, u) + B_{5} (u, u) = T (u) .

Hence, using (H3), (H7), (H11) and (H13) we obtain

(3.8)

\begin{array}{l} B_{1} (u, u) + B_{2} (u, u) + B_{3} (u, u) + B_{4} (u, u) + B_{5} (u, u) \\ = \int_{Ω} 𝒜 (x, u, \nabla u), \nabla u〉 ω_{1} dx + \int_{Ω} B (x, u, \nabla u), \nabla u〉 ν_{1} dx \\ + \int_{Ω} ℋ (x, u, \nabla u) u ν_{2} dx + \int_{Ω} {u|}^{p - 2} u^{2} ω_{2} dx + \int_{Ω} ℳ (x) \nabla u, \nabla u〉 dx \\ \geq λ_{1} \int_{Ω} {\nabla u|}^{p} ω_{1} dx + λ_{2} \int_{Ω} {\nabla u|}^{q} ν_{1} dx + Λ_{2} \int_{Ω} {u|}^{q} ν_{1} dx \\ + λ_{3} \int_{Ω} {\nabla u|}^{s} ν_{2} dx + Λ_{3} \int_{Ω} {u|}^{s} ν_{2} dx + \int_{Ω} {u|}^{p} ω_{2} dx + Λ \int_{Ω} {\nabla u|}^{2} ν_{3} dx \\ \geq λ_{1} \int_{Ω} {\nabla u|}^{p} ω_{1} dx + \int_{Ω} {u|}^{p} ω_{2} dx \geq γ_{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p}, \end{array}

where γ₂ = min{λ₁, 1}, and by Remark 2.5(i)

(3.9)

\begin{array}{l} T (u) & = \int_{Ω} f_{0} u dx + \sum_{j = 1}^{n} \int_{Ω} f_{j} D_{j} u dx \\ \leq {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} {u‖}_{L^{q} (Ω, ν_{1})} + (\sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}) {\nabla u|‖}_{L^{p} (Ω, ω_{1})} \\ \leq (C_{p, q} {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}) {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} \\ = M {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}, \end{array}

where

M = C_{p, q} {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})}

. Hence in (3.7), using (3.8) and (3.9), we obtain

γ_{2} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}^{p} \leq M {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})}

. Therefore,

\begin{array}{l} {u‖}_{W_{0}^{1, p} (Ω, ω_{1}, ω_{2})} & \leq {(\frac{M}{γ_{2}})}^{1 / (p - 1)} \\ = C {(C_{p, q} {f_{0} / ν_{1}‖}_{L^{q^{'}} (Ω, ν_{1})} + \sum_{j = 1}^{n} {f_{j} / ω_{1}‖}_{L^{p^{'}} (Ω, ω_{1})})}^{1 / (p - 1)}, \end{array}

where C = (1/γ₂)^1/(p−1).

Example

Let Ω = {(x, y) ∈ ℝ² : x² + y² < 1}, the weight functions ω₁(x, y) = (x² + y²)^−1/2, ω₂(x, y) = (x² + y²)^−3/2, ν₁(x, y) = (x² + y²)^−1/3, ν₂(x, y) = (x² + y²)⁻¹ and ν₃(x, y) = (x² + y²)^−1/2 (ω₁, ω₂ ∈ A₄, p = 4, q = 3 and s = 2), the function

𝒜 : Ω \times ℝ \times ℝ^{2} \to ℝ^{2}, 𝒜 ((x, y), η, ξ) = h_{1} (x, y) | ξ |^{2} ξ,

where h₁(x, y) = 2 e^(x²+y²), and

ℬ : Ω \times ℝ \times ℝ^{2} \to ℝ^{2}, ℬ ((x, y), η, ξ) = g_{2} (x, y) ξ| ξ,

where g₂(x, y) = 2 + cos(x² + y²), and

ℋ : Ω \times ℝ \times ℝ^{2} \to ℝ, ℋ ((x, y), η, ξ) = η h_{2} (x, y),

where h₂(x, y) = 1 + cos²(xy) and the coefficient matrix

ℳ (x, y) = (a_{i, j} (x, y)) = (\begin{matrix} λ {(x^{2} + y^{2})}^{- 1 / 2} & 0 \\ 0 & Λ {(x^{2} + y^{2})}^{- 1 / 2} \end{matrix}),

where 0 < λ < Λ.

Let us consider the partial differential operator

\begin{array}{l} Lu (x, y) & = - div (𝒜 ((x, y), \nabla u) ω_{1} (x, y) + ℬ ((x, y), u, \nabla u) ν_{1} (x, y)) \\ + ℋ ((x, y), u, \nabla u) ν_{2} (x, y) + {u|}^{2} u ω_{2} (x, y) - \sum_{i, j = 1}^{2} D_{j} (a_{ij} (x) D_{i} u (u)) . \end{array}

Therefore, by Theorem 1.1, the problem

\begin{array}{l} Lu (x) = \frac{cos (xy)}{(x^{2} + y^{2})} - \frac{\partial}{\partial x} (\frac{sin (xy)}{(x^{2} + y^{2})}) - \frac{\partial}{\partial y} (\frac{sin (xy)}{(x^{2} + y^{2})}) & in Ω, \\ u (x) = 0 on \partial Ω, \end{array}

has a unique solution

u \in W_{0}^{1, 4} (Ω, ω_{1}, ω_{2})

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

Full Article

1. Introduction

(P)

(1.1)

Theorem 1.1.

2. Definitions and basic results

Definition 2.1.

Definition 2.2.

(2.1)

Theorem 2.1.

Proof

Theorem 2.2 (The weighted Sobolev inequality).

(2.2)

Proof

Remark 2.3.

Proposition 2.4.

Proof

Definition 2.3.

Remark 2.5.

3. Proof of Theorem 1.1

Theorem 3.1.

Proof

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

Example

Paradigm

My account