Existence and Uniqueness of Solutions for the Navier Problems with Degenerate

Existence and Uniqueness of Solutions for the Navier Problems with Degenerate

Nonlinear Elliptic Equations

Albo Carlos Cavalheiro

Department of Mathematics, Sate University of Londrina accava@gmail.com

Received: 31.10.2014; accepted: 23.3.2015.

Abstract. In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations

∆(v(x) |∆u|^p−2∆u)−

n

X

j=1

Djω(x)Aj(x, u, ∇u) +b(x, u, ∇u) ω(x) = f0(x)−

n

X

j=1

Djfj(x), in Ω

in the setting of the Weighted Sobolev Spaces

Keywords:Degenerate nonlinear elliptic equations, Weighted Sobolev Spaces.

MSC 2010 classification:primary 35J70, secondary 35J60

Introduction

In this work we prove the existence and uniqueness of (weak) solutions in the weighted Sobolev space X = W^2,p(Ω, v) ∩ W₀^1,p(Ω, ω) (see Definition 3 and Definition 4) for the Navier problem

(P )







Lu(x) = f0(x) −

n

X

j=1

Djfj(x), in Ω u(x) = ∆u(x) = 0, on ∂Ω

where L is the partial differential operator

Lu(x) = ∆(v(x)|∆u|^p−2∆u)−

n

X

j=1

D_jω(x)A_j(x, u(x), ∇u(x))+b(x, u, ∇u) ω(x)

where Dj = ∂/∂xj, Ω is a bounded open set in Rⁿ, ω and v are two weight functions, ∆ denotes the Laplacian operator, 2 ≤ p < ∞ and the functions

http://siba-ese.unisalento.it/ c 2015 Universit`a del Salento

A_j : Ω×R×Rⁿ→R (j = 1, ..., n) and b : Ω×R×Rⁿ→R satisfy the following assumptions:

(H1) The function x7→Aj(x, η, ξ) is measurable on Ω for all (η, ξ) ∈ R×Rⁿ. The function (η, ξ)7→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, η⁰, ξ⁰)].(ξ − ξ⁰)≥θ1|ξ − ξ⁰|^p,

whenever ξ, ξ⁰∈Rⁿ, ξ6=ξ⁰, where A(x, η, ξ) = (A1(x, η, ξ), ..., An(x, η, ξ)) (where a dot denote here the Euclidian scalar product in Rⁿ).

(H3) A(x, η, ξ).ξ ≥ λ1|ξ|^p+ Λ1|η|^p, where λ1 and Λ1 are nonnegative constants.

(H4) |A(x, η, ξ)| ≤ K₁(x) + h₁(x)|η|^p/p0+ h₂(x)|ξ|^p/p0, where K₁, h₁ and h₂ are nonegative functions, with h₁ and h₂∈L^∞(Ω), and K₁∈L^p0(Ω, ω) (with 1/p + 1/p⁰ = 1).

(H5) The function x7→b(x, η, ξ) is measurable on Ω for all (η, ξ) ∈ R×Rⁿ. The function (η, ξ)7→b(x, η, ξ) is continuous on R×Rⁿ for almost all x∈Ω.

(H6) there exists a constant θ2> 0 such that

[b(x, η, ξ) − b(x, η⁰, ξ⁰)](η − η⁰) ≥ θ₂|η − η⁰|^p whenever η, η⁰∈R, η6=η⁰.

(H7) b(x, η, ξ)η ≥ λ2|ξ|^p+ Λ2|η|^p, where λ2 and Λ2 are nonnegative constants.

(H8) |b(x, η, ξ)| ≤ K₂(x) + h₃(x)|η|^p/p0 + h₄(x)|ξ|^p/p0, where K₂, h₃ and h₄ are nonnegative functions, with K₂∈L^p0(Ω, ω), h₃ and h₄∈L^∞(Ω).

(H9) λ1+ λ2 > 0 and Λ1+ Λ2> 0.

By a weight, we shall mean a locally integrable function ω on Rⁿ such that ω(x) > 0 for a.e. x ∈ Rⁿ. Every weight ω gives rise to a measure on the mea- surable subsets on Rⁿthrough integration. This measure will be denoted by µ.

Thus, µ(E) =R

Eω(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 [1], [2], [4], [8] and [13]).

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 [10]). These classes have found many useful applications in harmonic analysis (see [12]). Another reason for studying A_p-weights is the fact that powers of the distance to submanifolds of Rⁿ often belong to Ap (see [9]). There are, in fact, many interesting examples of weights (see [8] for p-admissible weights).

In the non-degenerate case (i.e. with ω(x) ≡ 1), for all f ∈ L^p(Ω) the Poisson equation associated with the Dirichlet problem

 − ∆u = f (x), in Ω u(x) = 0, on ∂Ω

is uniquely solvable in W^2,p(Ω) ∩ W₀^1,p(Ω) (see [7]), and the nonlinear Dirichlet problem

 − ∆_pu = f (x), in Ω u(x) = 0, on ∂Ω

is uniquely solvable in W₀^1,p(Ω) (see [3]), where ∆pu = div(|∇u|^p−2∇u) is the p- Laplacian operator. In the degenerate case, the weighted p-Biharmonic operator has been studied by many authors (see [11] and the references therein), and the degenerated p-Laplacian has been studied in [4]. The problem with degenerated p-Laplacian and p-Biharmonic operators

 ∆(ω(x)|∆u|^p−2∆u) − div[ω(x)|∇u|^p−2∇u] = f (x) − div(G(x)), in Ω u(x) = ∆u(x) = 0, on ∂Ω

has been studied by the author in [2].

The following theorem will be proved in section 3.

Theorem 1. Assume (H1)-(H9). If ω, v ∈ A_p (with 2 ≤ p < ∞) and fj/ω ∈ L^p0(Ω, ω) (j = 0, 1, ..., n) then the problem (P) has a unique solution u ∈ X = W^2,p(Ω, v) ∩ W₀^1,p(Ω, ω). Moreover, we have

kuk_X≤ 1 γ^p0/p



kf₀/ωk_Lp 0(Ω,ω)+

n

X

j=1

kf_j/ωk_Lp 0(Ω,ω)

p⁰/p

,

where γ = min {λ₁+ λ₂, Λ₁+ Λ₂, 1}.

1 DEFINITIONS AND BASIC RESULTS

Let ω be a locally integrable nonnegative function in Rⁿ 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 that

 1

|B|

Z

B

ω(x)dx

 1

|B|

Z

B

ω^1/(1−p)(x)dx

p−1

≤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 [6],[8] or [12] 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; 2 r)) ≤ C µ(B(x; r)) for every ball B = B(x; r) ⊂ Rⁿ, where µ(B) =R

Bω(x) dx. If ω∈A_p, then µ is doubling (see Corollary 15.7 in [8]).

As an example of 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 [12]).

If ω∈A_p, then  |E|

|B|

p

≤ Cµ(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 [8]). Therefore, if µ(E) = 0 then |E| = 0.

Definition 1. Let ω be a weight, and let Ω ⊂ Rⁿ be open. For 0 < p < ∞ we define L^p(Ω, ω) as the set of measurable functions f on Ω such that

kf k_Lp(Ω,ω)=

 Z

Ω

|f (x)|^pω(x)dx

1/p

< ∞.

If ω ∈ A_p, 1 < p < ∞, then ω^−1/(p−1) is locally integrable and 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. Let Ω ⊂ Rⁿ be open, k be a nonnegative integer and ω ∈ Ap

(1 < p < ∞). We define the weighted Sobolev space W^k,p(Ω, ω) as the set of functions u ∈ L^p(Ω, ω) with weak derivatives D^αu ∈ L^p(Ω, ω) for 1 ≤ |α| ≤ k. The norm of u in W^k,p(Ω, ω) is defined by

kuk_Wk,p(Ω,ω)=

 Z

Ω

|u(x)|^pω(x) dx + X

1 ≤ |α| ≤ k

Z

Ω

|D^αu(x)|^pω(x) dx

1/p

. (1.1)

We also define W₀^k,p(Ω, ω) as the closure of C₀^∞(Ω) with respect to the norm k.k_Wk,p(Ω,ω).

If ω ∈ Ap, then W^k,p(Ω, ω) is the closure of C^∞(Ω) with respect to the norm (1.1) (see Theorem 2.1.4 in [13]). The spaces W^k,p(Ω, ω) and W₀^k,p(Ω, ω) are Banach spaces.

It is evident that the weight function ω which satisfies 0 < c1≤ ω(x) ≤ c₂for x ∈ Ω (c1 and c2 positive constants), gives nothing new (the space W^k,p₀ (Ω, ω) is then identical with the classical Sobolev space W₀^k,p(Ω)). Consequently, we study all such weight functions ω that either vanish in Ω ∪ ∂Ω or increase to infinity (or both).

In this article we use the following results.

Theorem 2. 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_mk} and a function Φ ∈ L^p(Ω, ω) such that

(i) u_mk(x)→ u(x), m_k→ ∞, µ-a.e. on Ω;

(ii) |u_mk(x)| ≤ Φ(x), µ-a.e. on Ω;

(where µ(E) =R

Eω(x) dx).

Proof. The proof of this theorem follows the lines of Theorem 2.8.1 in [5].

QED

Lemma 1. Let 1 < p < ∞. (a) There exists a constant αp such that 

|x|^p−2x − |y|^p−2y    

≤ α_p|x − y|(|x| + |y|)^p−2, for all x, y ∈ 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≤ (|x|^p−2x − |y|^p−2y).(x − y) ≤ γ_p(|x| + |y|)^p−2|x − y|². Proof. See [3], Proposition 17.2 and Proposition 17.3. ^QED Definition 3. We denote by X = W^2,p(Ω, v) ∩ W₀^1,p(Ω, ω) with the norm

kuk_X =

 Z

Ω

|u|^pω dx + Z

Ω

|∇u|^pω dx + Z

Ω

|∆u|^pv dx

1/p

.

Definition 4. We say that an element u ∈ X = W^2,p(Ω, v) ∩ W₀^1,p(Ω, , ω) is a (weak) solution of problem (P) if, for all ϕ ∈ X,

Z

Ω

|∆u|^p−2∆u ∆ϕ v dx +

n

X

j=1

Z

Ω

ω Aj(x, u(x), ∇u(x))Djϕ(x)dx

+ Z

Ω

b(x, u, ∇u)ϕ ω dx = Z

Ω

f0(x)ϕ(x)dx +

n

X

j=1

Z

Ω

fj(x)Djϕ(x)dx.

2 PROOF OF THEOREM 1

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

Theorem 3. 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 A u = T is uniquely solvable in X.

Proof. See Theorem 26.A in [15]. ^QED

We define B, B₁, B₂, B₃ : X × X → R and T : X → R by B(u, ϕ) = B1(u, ϕ) + B2(u, ϕ) + B3(u, ϕ)

B₁(u, ϕ) =

n

X

j=1

Z

Ω

ω A_j(x, u, ∇u)D_jϕdx = Z

Ω

ω A(x, u, ∇u).∇ϕ dx

B₂(u, ϕ) = Z

Ω

|∆u|^p−2∆u ∆ϕ v dx B₃(u, ϕ) =

Z

Ω

b(x, u, ∇u) ϕ ω dx T (ϕ) =

Z

Ω

f₀(x) ϕ(x) dx +

n

X

j=1

Z

Ω

f_j(x) D_jϕ(x) dx.

Then u ∈ X is a (weak) solution to problem (P) if, for all ϕ ∈ X, we have B(u, ϕ) = B₁(u, ϕ) + B₂(u, ϕ) + B₃(u, ϕ) = T (ϕ).

Step 1. For j = 1, ..., n we define the operator Fj : X →L^p0(Ω, ω) by (F_ju)(x) = A_j(x, u(x), ∇u(x)).

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

(i) Using (H4) we obtain kF_juk^p0

L^{p 0}(Ω,ω) = Z

Ω

|F_ju(x)|^p0ω dx = Z

Ω

|A_j(x, u, ∇u)|^p0ω dx

≤ Z

Ω



K1+ h1|u|^p/p0 + h2|∇u|^p/p0

p⁰

ω dx

≤ C_p Z

Ω



(K₁^p0+ h^p₁⁰|u|^p+ h^p₂⁰|∇u|^p)ω

 dx

= C_p

 Z

Ω

K₁^p0ω dx + Z

Ω

h^p₁⁰|u|^pω dx +

Z

Ω

h^p₂⁰|∇u|^pω dx



, (2.1)

where the constant C_p depends only on p. We have, Z

Ω

h^p₁⁰|u|^pω dx ≤ kh1k^p_L⁰∞(Ω)

Z

Ω

|u|^pω dx ≤ kh1k^p_L⁰∞(Ω)kuk^p_X,

and Z

Ω

h^p₂⁰|∇u|^pω dx ≤ kh₂k^p_L⁰∞(Ω)

Z

Ω

|∇u|^pω dx ≤ kh₂k^p_L⁰∞(Ω)kuk^p_X. Therefore, in (2.1) we obtain

kF_juk_Lp 0(Ω,ω) ≤ C_p



kK₁k_Lp 0(Ω,ω)+ (kh₁k_L∞(Ω)+ kh₂k_L∞(Ω)) kuk^p/p_X ⁰

 . (ii) Let u_m→ u in X as m → ∞. We need to show that F_ju_m→F_ju in L^p0(Ω, ω).

If um→ u in X, then u_m→ u in L^p(Ω, ω) and |∇um|→ |∇u| in L^p(Ω, ω). Using Theorem 2, there exist a subsequence {u_mk} and two functions Φ₁ and Φ₂ in L^p(Ω, ω) such that

umk(x)→u(x), µ − a.e. in Ω,

|u_mk(x)|≤Φ1(x), µ − a.e. in Ω,

|∇u_mk(x)|→|∇u(x)|, µ − a.e. in Ω,

|∇u_mk(x)|≤Φ₂(x), µ − a.e. in Ω.

Hence, using (H4), we obtain kF_ju_mk− F_juk^p0

L^{p 0}(Ω,ω) = Z

Ω

|F_ju_mk(x) − F_ju(x)|^p0ω dx

= Z

Ω

|A_j(x, u_mk, ∇u_mk) − A_j(x, u, ∇u)|^p0ω dx

≤ C_p Z

Ω



|A_j(x, u_mk, ∇u_mk)|^p0+ |A_j(x, u, ∇u)|^p0

 ω dx

≤ C_p

 Z

Ω



K1+ h1|u_mk|^p/p0 + h2|∇u_mk|^p/p0

p⁰

ω dx +

Z

Ω



K₁+ h₁|u|^p/p0 + h₂|∇u|^p/p0

p⁰

ω dx



≤ 2 C_p Z

Ω



K₁+ h₁Φ^p/p₁ ⁰ + h₂Φ^p/p₂ ⁰

p⁰

ω dx

≤ 2 C_p

 Z

Ω

K₁^p0ω dx + Z

Ω

h^p₁⁰Φ^p₁ω dx + Z

Ω

h^p₂⁰Φ^p₂ω dx



≤ 2 C_p

 kK₁k^p0

L^{p 0}(Ω,ω)+ kh1k^p_L⁰∞(Ω)

Z

Ω

Φ^p₁ω dx

+ kh₂k^p_L⁰∞(Ω)

Z

Ω

Φ^p₂ω dx



≤ 2 C_p

 kK₁k^p0

L^{p 0}(Ω,ω)+ kh1k^p_L⁰∞(Ω) kΦ₁k^p_Lp(Ω,ω)

+ kh2k^p_L⁰∞(Ω)kΦ₂k^p_Lp(Ω,ω)

 . By condition (H1), we have

F_ju_m(x) = A_j(x, u_m(x), ∇u_m(x))→ A_j(x, u(x), ∇u(x)) = F_ju(x), as m → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain

kF_ju_mk− F_juk_Lp 0(Ω,ω)→ 0, that is,

Fjumk→ F_ju in L^p0(Ω, ω).

By the Convergence Principle in Banach spaces (see Proposition 10.13 in [14]), we have

Fjum→ F_ju in L^p0(Ω, ω). (2.2) Step 2. We define the operator

G : X → L^p0(Ω, v)

(Gu)(x) = |∆u(x)|^p−2∆u(x)

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

(i) We have

kGuk^p0

L^{p 0}(Ω,v) = Z

Ω

|∆u|^p−2∆u 

p⁰

v dx

= Z

Ω

|∆u|^{(p−2) p0}|∆u|^p0v dx

= Z

Ω

|∆u|^pv dx ≤ kuk^p_X.

Hence, kGuk_Lp 0(Ω,v)≤ kuk^p/p_X ⁰.

(ii) If um→ u in X then ∆u_m→ ∆u in L^p(Ω, v).By Theorem 2, there exist a subsequence {um_k} and a function Φ₃∈ L^p(Ω, v) such that

∆u_mk(x) → ∆u(x), µ₁− a.e. in Ω

|∆u_mk(x)| ≤ Φ₃(x), µ₁− a.e. in Ω,

where µ₁(E) =R

Ev(x) dx. Hence, using Lemma 1 (a), we obtain, if p 6= 2 kGu_mk− Guk^p0

L^{p 0}(Ω,v)= Z

Ω

| Gu_mk− Gu|^p0v dx

= Z

Ω

|∆u_mk|^p−2∆u_mk − | ∆u|^p−2∆u    

p⁰

v dx

≤ Z

Ω



α_p|∆u_mk− ∆u| ( |∆u_mk| + |∆u|)^(p−2)

p⁰

v dx

≤ α^p_p⁰ Z

Ω

| ∆u_mk− ∆u|^p0(2 Φ3)^{(p−2) p0}v dx

≤ α^p_p⁰2^(p−2)p0

 Z

Ω

|∆u_mk− ∆u|^pv dx

p⁰/p

×

×

 Z

Ω

Φ^{(p−2) p p}

0/(p−p⁰)

3 v dx

(p−p⁰)/p

≤ α^p_p⁰2^{(p−2) p0}ku_mk− uk^p_X⁰kΦk^p−p_Lp(Ω,v)⁰ ,

since (p − 2) p p⁰/(p − p⁰) = p if p 6= 2. If p = 2, we have

kGu_mk − Guk²_L2(Ω,v) = Z

Ω

|∆u_mk− ∆u|²v dx ≤ ku_mk− uk²_X.

Therefore (for 2≤p < ∞), by the Lebesgue Dominated Convergence Theorem, we obtain

kGu_mk− Guk_X→ 0,

that is, Gu_mk→ Gu in L^p0(Ω, v). By the Convergence Principle in Banach spaces (see Proposition 10.13 in [14]), we have

Gu_m→ Gu in L^p0(Ω, v). (2.3)

Step 3. We define the operator H : X → L^p0(Ω, ω) by (Hu)(x) = b(x, u(x), ∇u(x)).

We also have that the operator H is continuous and bounded. In fact,

(i) Using (H8) we obtain

kHuk^p0

L^{p 0}(Ω,ω) = Z

Ω

|Hu|^p0ω dx

= Z

Ω

|b(x, u, ∇u)|^p0ω dx

≤ Z

Ω



K₂+ h₃|u|^p/p0+ h₄|∇u|^p/p0

p⁰

ω dx

≤ C_p Z

Ω



(K₂^p0+ h^p₃⁰|u|^p+ h^p₄⁰|∇u|^p)ω

 dx

= Cp

 Z

Ω

K₂^p0ω dx + Z

Ω

h^p₃⁰|u|^pω dx + Z

Ω

h^p₄⁰|∇u|^pω dx



≤ C_p

 kK₂k^p0

L^{p 0}(Ω,ω)+ (kh₃k^p_L⁰∞(Ω)+ kh₄k^p_L⁰∞(Ω)) kuk_X

 .

Hence,

kHuk_Lp 0(Ω,ω)≤ C_p



kK₂k_Lp 0(Ω,ω)+ (kh₃k_L∞(Ω)+ kh₄k_L∞(Ω))kuk^p/p_X ⁰

 .

(ii) By the same argument used in Step 1(ii), we obtain analogously, if um→ u in X then

Hum→ Hu, in L^p0(Ω, ω). (2.4) Step 4. We also have

|T (ϕ)| ≤ Z

Ω

|f₀||ϕ| dx +

n

X

j=1

Z

Ω

|f_j||D_jϕ| dx

= Z

Ω

|f₀|

ω |ϕ|ω dx +

n

X

j=1

Z

Ω

|f_j|

ω |D_jϕ| ω dx

≤ kf₀/ωk_Lp 0(Ω,ω)kϕk_Lp(Ω,ω)+

n

X

j=1

kf_j/ωk_Lp 0(Ω,ω)kD_jϕk_Lp(Ω,ω)

≤



kf₀/ωk_Lp 0(Ω,ω)+

n

X

j=1

kf_j/ωk_Lp 0(Ω,ω)

 kϕk_X.

Moreover, using (H4), (H8) and the H¨older inequality, we also have

|B(u, ϕ)| ≤ |B₁(u, ϕ)| + |B₂(u, ϕ)| + |B₃(u, ϕ)|

≤

n

X

j=1

Z

Ω

|A_j(x, u, ∇u)||Djϕ| ω dx + Z

Ω

| ∆u|^p−2| ∆u| | ∆ϕ| v dx

+ Z

Ω

|b(x, u, ∇u)| |ϕ| ω dx. (2.5)

In (2.5) we have Z

Ω

|A(x, u, ∇u)| |∇ϕ| ω dx ≤ Z

Ω



K1+ h1|u|^p/p0+ h2|∇u|^p/p0



|∇ϕ| ω dx

≤ kK₁k_Lp 0(Ω,ω)k∇ϕk_Lp(Ω,ω)+ kh₁k_L∞(Ω)kuk^p/p_Lp(Ω,ω)⁰ k∇ϕk_Lp(Ω,ω)

+ kh₂k_L∞(Ω)k∇uk^p/p_Lp(Ω,ω)⁰ k∇ϕk_Lp(Ω,ω)

≤



kK₁k_Lp 0(Ω,ω)+ (kh1k_L∞(Ω)+ kh2k_L∞(Ω))kuk^p/p

0

X

 kϕk_X, and

Z

Ω

| ∆u|^p−2| ∆u| | ∆ ϕ| v dx = Z

Ω

| ∆u|^p−1| ∆ ϕ| v dx

≤

 Z

Ω

| ∆u|^pv dx

1/p⁰ Z

Ω

| ∆ϕ|^pv dx

1/p

≤ kuk^p/p_X ⁰kϕk_X, and

Z

Ω

|b(x, u, ∇u)| |ϕ| ω dx ≤ Z

Ω



K2+ h3|u|^p/p0 + h4|∇u|^p/p0



|ϕ| ω dx

≤ Z

Ω

K2|ϕ| ω dx + kh₃k_L∞(Ω)

Z

Ω

|u|^p/p0|ϕ| ω dx +kh4k_L∞(Ω)

Z

Ω

|∇u|^p/p0|ϕ| ω dx

≤



kK₂k_Lp 0(Ω,ω)+ kh₃k_L∞(Ω)kuk^p/p_X ⁰ + kh₄k_L∞(Ω)kuk^p/p_X ⁰

 kϕk_X Therefore, in (2.5) we obtain, for all u, ϕ ∈ X

|B(u, ϕ)| ≤



kK₁k_Lp 0(Ω,ω)+ kK₂k_Lp 0(Ω,ω)

+ (kh₁k_L∞(Ω)+ kh₂k_L∞(Ω,ω)+ kh₃k_L∞(Ω)

+ kh₄k_L∞(Ω,ω)+ 1)kuk^p/p

0

X

 kϕk_X.

Since B(u, .) is linear, for each u ∈ X, there exists a linear and continuous operator A : X → X^∗ such that hAu, ϕi = B(u, ϕ), for all u, ϕ ∈ X (where hf, xi denotes the value of the linear functional f at the point x) and

kAuk_∗ ≤ kK₁k_Lp 0(Ω,ω)+ kK₂k_Lp 0(Ω,ω)

+ (kh₁k_L∞(Ω)+ kh₂k_L∞(Ω,ω)+ kh₃k_L∞(Ω)+ kh₄k_L∞(Ω,ω)+ 1)kuk^p/p_X ⁰. Consequently, problem (P) is equivalent to the operator equation

Au = T, u ∈ X.

Step 5. Using condition (H2), (H6) and Lemma 1 (b), we have hAu₁− Au₂, u1− u₂i = B(u₁, u1− u₂) − B(u2, u1− u₂)

= Z

Ω

ω A(x, u1, ∇u1).∇(u1− u₂) dx + Z

Ω

| ∆u₁|^p−2∆u1∆(u1− u₂) v dx +

Z

Ω

b(x, u₁, ∇u₁)(u₁− u₂) ω dx

− Z

Ω

ω A(x, u₂, ∇u₂).∇(u₁− u₂) dx − Z

Ω

| ∆u₂|^p−2∆u₂∆(u₁− u₂) v dx

− Z

Ω

b(x, u₂, ∇u₂)(u₁− u₂) ω dx

= Z

Ω

ω



A(x, u₁, ∇u1) − A(x, u2, ∇u2)



.∇(u1− u₂) dx +

Z

Ω

(| ∆u1|^p−2∆u1− | ∆u₂|^p−2∆u2) ∆(u1− u₂) v dx +

Z

Ω

(b(x, u1, ∇u1) − b(x, u2, ∇u2))(u1− u₂) ω dx

≥ θ₁ Z

Ω

ω |∇(u1− u₂)|^pdx + βp

Z

Ω

(| ∆u1| + | ∆u₂|)^p−2|∆u₁− ∆u₂|²v dx + θ2

Z

Ω

|u₁− u₂|^pω dx

≥ θ₁ Z

Ω

ω |∇(u1− u₂)|^pdx + βp

Z

Ω

(| ∆u1− ∆u₂|)^p−2|∆u₁− ∆u₂|²v dx + θ₂

Z

Ω

|u₁− u₂|^pω dx

= θ₁ Z

Ω

ω |∇(u₁− u₂)|^pdx + β_p Z

Ω

|∆u₁− ∆u₂|^pv dx + θ₂ Z

Ω

|u₁− u₂|^pω dx

≥ θ ku₁− u₂k^p_X

where θ = min {θ₁, θ₂, β_p}.

Therefore, the operator A is strongly monotone, and this implies that the operator A is strictly monotone. Moreover, using (H3) and (H9), we obtain

hAu, ui = B(u, u) = B₁(u, u) + B2(u, u) + B3(u, u)

= Z

Ω

ω A(x, u, ∇u).∇u dx + Z

Ω

| ∆u|^p−2∆u ∆u v dx + Z

Ω

b(x, u, ∇u) u ω dx

≥ Z

Ω

(λ₁|∇u|^p+ Λ₁|u|^p) ω dx + Z

Ω

| ∆u|^pv dx + Z

Ω

(λ₂|∇u|^p+ Λ₂|u|^p) ω dx

= (Λ₁+ Λ₂) Z

Ω

|u|^pω dx + (λ₁+ λ₂) Z

Ω

|∇u|^pω dx + Z

Ω

|∆u|^pv dx

≥ γ kuk^p_X

where γ = min {λ₁+ λ₂, Λ₁+ Λ₂, 1}. Hence, since p ≥ 2, we have hAu, ui

kuk_X → + ∞, as kuk_X→ + ∞, that is, A is coercive.

Step 6. We need to show that the operator A is continuous.

Let um→ u in X as m → ∞. We have,

|B₁(um, ϕ) − B1(u, ϕ)| ≤

n

X

j=1

Z

Ω

|A_j(x, um, ∇um) − Aj(x, u, ∇u)||Djϕ| ω dx

=

n

X

j=1

Z

Ω

|F_ju_m− F_ju||D_jϕ| ω dx

≤

n

X

j=1

kF_ju_m− F_juk_Lp 0(Ω,ω)kD_jϕk_Lp(Ω,ω)

≤

n

X

j=1

kF_jum− F_juk_Lp 0(Ω,ω)kϕk_X, and

|B₂(u_m, ϕ) − B₂(u, ϕ)|

=     Z

Ω

| ∆u_m|^p−2∆um∆ϕ v dx − Z

Ω

| ∆u|^p−2∆u ∆ϕ v dx    

≤ Z

Ω

| ∆u_m|^p−2∆u_m− | ∆u|^p−2∆u    

| ∆ϕ| v dx

= Z

Ω

|Gu_m− Gu| |∆ϕ| v dx

≤ kGu_m− Guk_Lp 0(Ω,v)kϕk_X,