2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73–81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

LERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION

YOUSSEF AKDIM, ABDELMOUJIB BENKIRANE, MOHAMED RHOUDAF

Abstract. In this paper, we study the existence of solutions for the nonlinear degenerated elliptic problem

−div(a(x, u, ∇u)) = F in Ω, where Ω is a bounded domain of ℝ^{N}, N ≥ 2, a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function satisfying the coercivity condition, but they verify the general growth condition and only the large monotonicity. The second term F belongs to W^{−1,p′}(Ω, w^{∗}).

1. Introduction

Let Ω be a bounded open set of ℝ^{N}, p be a real number such that 1 < p < ∞ and w = {w_{i}(x), 0 ≤ i ≤ N} be a vector of weight functions (i.e., every component w_{i}(x) is a measurable function which is positive a.e. in Ω) satisfying some integrability conditions. The Objective of this paper is to study the following problem, in the weighted Sobolev space,

\begin{matrix}Au = F in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (1.1)

where A is a Leray-Lions operator from W_{0}^{1,p}(Ω, w) to its dual W^{−1,p′}(Ω, w^{∗}). The principal part A is a differential operator of second order in divergence form defined as,

Au = − div(a(x, u, ∇u))

where a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function (that is, measurable with respect to x in Ω for every (x, ξ) in ℝ × ℝ^{N} and continuous with respect to (s, ξ) in ℝ × ℝ^{N} for almost every x in Ω) satisfying the coercivity condition. But, on the one hand, they verify the general growth condition in this form

|a_{i}(x, s, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + |s|^{p−1} + ∑_{j=1}^{N}w_{j}^{1/p′}(x)[γ(s)|ξ_{j}|]^{p−1}]

2000 Mathematics Subject Classification. 35J15, 35J70, 35J85.

Key words and phrases. Weighted Sobolev spaces; truncations; L^{1}-version of Minty’s lemma;

Hardy inequality.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

instead the classical growth condition, where we introduce some continuous function γ(s). This type of the growth condition can not guaranteed the existence of the weak solution (See Remark 4.6), for that we overcame this difficulty by introduce an other type of solution so-called T-solution. On the other hand, they verify only the large monotonicity, that is

[a(x, s, ξ) − a(x, s, η)](ξ − η) ≥ 0 for all (ξ, η) ∈ ℝ^{N} × ℝ^{N}.

We overcome this difficulty of the not strict monotonicity thanks to a technique (the L^{1}-version of Minty’s lemma) similar to the one used in [5]. Recently in [6] Boccardo has studied the problem (1.1) in the classical Sobolev space W_{0}^{1,p}(Ω). For that the author has proved the existence of the T-solution. Other works in this direction can be found in [5] (where the right hand side f ∈ L^{1} and F ∈ L^{p′}(Ω)) and in [1] (where the existence and nonexistence results for some quasilinear elliptic equations involving the P-Laplaces have proved).

2. Preliminaries

Let Ω be a bounded open set of ℝ^{N}, p be a real number such that 1 < p < ∞ and w = {w_{i}(x), 0 ≤ i ≤ N} be a vector of weight functions, i.e., every component w_{i}(x) is a measurable function which is strictly positive a.e. in Ω. Further, we suppose in all our considerations that (for each w_{i} ≠ 0.)

w_{i} ∈ L_{loc}^{1}(Ω),  (2.1)
wi^{\frac{−1}{p−1}} ∈ L_{loc}^{1}(Ω),  (2.2)

for any 0 ≤ i ≤ N.

We denote by W^{1,p}(Ω, w) the space of all real-valued functions u ∈ L^{p}(Ω, w_{0}) such that the derivatives in the sense of distributions fulfill

\frac{∂u}{∂x_{i}} ∈ L^{p}(Ω, w_{i}) for i = 1, …, N.

Which is a Banach space under the norm

∥u∥_{1,p,w} = [ ∫_{Ω} |u(x)|^{p}w_{0}(x) dx + ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx]^{1/p}.  (2.3)

The condition (2.1) implies that C_{0}^{∞}(Ω) is a space of W^{1,p}(Ω, w) and consequently, we can introduce the subspace W_{0}^{1,p}(Ω, w) of W^{1,p}(Ω, w) as the closure of C_{0}^{∞}(Ω) with respect to the norm (2.3). Moreover, condition (2.2) implies that W^{1,p}(Ω, w) as well as W_{0}^{1,p}(Ω, w) are reflexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces W_{0}^{1,p}(Ω, w) is equivalent to W^{−1,p′}(Ω, w^{∗}), where w^{∗} = {w_{i}^{∗} = w_{i}^{1−p′}, i = 0, …, N} and where p′ is the conjugate of p i.e. p′ = \frac{p}{p−1}.

3. Basic assumptions and statement of results

Assumption (H1). The expression

∥|u|∥_{X} = ( ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx)^{1/p}  (3.1)

is a norm defined on X and is equivalent to the norm (2.3). There exist a weight function σ on Ω and a parameter q, 1 < q < ∞, such that the Hardy inequality,

( ∫_{Ω} |u(x)|^{q}σ dx)^{1/q} ≤ c( ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx)^{1/p},  (3.2)

holds for every u ∈ X with a constant c > 0 independent of u, and moreover, the imbedding

X ↪↪ L^{q}(Ω, σ),  (3.3)

expressed by the inequality (3.2) is compact.

Note that (X, ∥|.|∥_{X}) is a uniformly convex (and thus reflexive) Banach space.

Remark 3.1. If we assume that w_{0}(x) ≡ 1 and in addition the integrability condition: There exists ν ∈]\frac{N}{P} , +∞ [∩[\frac{1}{P −1}, +∞[ such that

w_{i}^{−ν} ∈ L^{1}(Ω) for all i = 1, …, N.  (3.4)

Note that the assumptions (2.1) and (3.4) imply that,

∥|u∥| = (∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{1/p},  (3.5)

is a norm defined on W_{0}^{1,p}(Ω, w) and its equivalent to (2.3) and that, the imbedding

W_{0}^{1,p}(Ω, w) ↪ L^{p}(Ω),  (3.6)

is compact for all 1 ≤ q ≤ p_{1}^{∗} if p.ν < N(ν + 1) and for all q ≥ 1 if p.ν ≥ N(ν + 1) where p_{1} = \frac{pν}{ν+1} and p_{1}^{∗} is the Sobolev conjugate of p_{1} [see [9], pp 30-31].

Assumption (H2).

|a_{i}(x, s, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + |s|^{p−1} + ∑_{j=1}^{N}w_{j}^{1/p′}(x)[γ(s)|ξ_{j}|]^{p−1}],  (3.7)
[a(x, s, ξ) − a(x, s, η)](ξ − η) ≥ 0 for all (ξ, η) ∈ ℝ^{N} × ℝ^{N},  (3.8)
a(x, s, ξ).ξ ≥ α∑_{i=1}^{N}w_{i}|ξ_{i}|^{p},  (3.9)

where k(x) is a positive function in L^{p′}(Ω), γ(s) is a continuous function and α, β are strictly positive constants.

We recall that, for k > 1 and s in ℝ, the truncation is defined as,

T_{k}(s) = \begin{cases}s & if s ≤ k \\ k \frac{s}{|s|} & if |s| > k. \end{cases}

4. Existence results

Consider the problem

u ∈ W_{0}^{1,p}(Ω, w), F ∈ W^{−1,p′}(Ω, w^{∗})
− div(a(x, u, ∇u)) = F in Ω
u = 0 on ∂Ω.

(4.1)

Definition 4.1. A function u in W_{0}^{1,p}(Ω, w) is a T -solution of (4.1) if

∫_{Ω} a(x, u, ∇u)∇T_{k}[u − φ] dx = ⟨F, T_{k}[u − φ]⟩ ∀φ ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω).

Theorem 4.2. Assume that (H1) and(H2). Then the problem (4.1) has at least one T -solution u.

Remark 4.3. Recall that an existence result for the problem (4.1) can be found in [8] by using the approach of pseudo monotonicity with some particular growths condition, that is γ(s) = 1.

Remark 4.4. In [9] the authors study the problem (4.1) under the strong hypotheses

[a(x, s, ξ) − a(x, s, η)](ξ − η) > 0, for all ξ ≠ η ∈ ℝ^{N},

|a_{i}(x, s, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + |s|^{p−1} + ∑_{j=1}^{N}w_{j}^{1/p′}(x)|ξ_{j}|^{p−1}],

instead of (3.8) and (3.7) (respectively ). Then the operator A associated to the problem (4.1) verifies the (S^{+}) condition and is coercive. Hence A is surjective from W_{0}^{1,p}(Ω, w) into its dual W^{−1,p′}(Ω, w^{∗}).

Proof of Theorem 4.2. Consider the approximate problem

\begin{matrix}u_{n} ∈ W_{0}^{1,p}(Ω, w) \\ − div(a(x, T_{n}(u_{n}), ∇u_{n})) = F. \end{matrix}  (4.2)

under the following assumptions: Assertion (a): A priori estimates The problem (4.2) has a solution by a classical result in [8]. Moreover, by using u_{n} as test function in (4.2) we have,

∫_{Ω} a(x, T_{n}(u_{n}), ∇u_{n}).∇u_{n} dx = ∫_{Ω} F u_{n} dx.

Thanks to assumption (3.9), we have

∫_{Ω} a(x, T_{n}(u_{n}), ∇u_{n}).∇u_{n} dx ≥ α∑_{i=1}^{N}∫_{Ω} |\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i}(x) dx = α∥|u_{n}∥|^{p}

i.e.,

α∥|u_{n}∥|^{p} ≤ ⟨F, u_{n}⟩ ≤ ∥F ∥_{−1,p′,w^{∗}}∥|u_{n}∥|,

which implies α∥|u_{n}∥|^{p} ≤ C_{1}∥|u_{n}∥| for p > 1, with C_{1} is a constant positive, then the sequence u_{n} is bounded in W_{0}^{1,p}(Ω, w), thus, there exists a function u ∈ W_{0}^{1,p}(Ω, w) and a subsequence u_{n_{j}} such that u_{n_{j}} converges weakly to u in W_{0}^{1,p}(Ω, w). Assertion (b) We shall prove that for φ in W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω), we have

∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx ≤ ⟨F, T_{k}[u_{n_{j}} − φ]⟩.  (4.3)

Let n_{j} large enough (n_{j} > k + ∥φ∥_{L^{∞}(Ω)}), we have by choosing T_{k}[u_{n_{j}} − φ] as test function in (4.2)

∫_{Ω} a(x, u_{n_{j}}, ∇u_{n_{j}})∇T_{k}[u_{n_{j}} − φ] dx = ⟨F, T_{k}[u_{n_{j}} − φ]⟩,

i.e.,

∫_{Ω} a(x, u_{n_{j}}, ∇u_{n_{j}})∇T_{k}[u_{n_{j}} − φ] dx + ∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx
− ∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx
= ⟨F, T_{k}[u_{n_{j}} − φ]⟩,

which implies

\begin{matrix}∫_{Ω}[a(x, u_{n_{j}}, ∇u_{n_{j}}) − a(x, u_{n_{j}}, ∇φ)]∇T_{k}[u_{n_{j}} − φ] dx \\ + ∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx & = ⟨F, T_{k}[u_{n_{j}} − φ]⟩. \end{matrix} (4.4)

Thanks to assumption (3.8) and the definition of truncating function, we have,

∫_{Ω}[a(x, u_{n_{j}}, ∇u_{n_{j}}) − a(x, u_{n_{j}}, ∇φ)]∇T_{k}[u_{n_{j}} − φ] dx ≥ 0.  (4.5)

Combining (4.4) and (4.5), we obtain (4.3). Assertion (c) We claim that,

∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx → ∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] dx

and that

⟨F, T_{k}[u_{n_{j}} − φ]⟩ → ⟨F, T_{k}[u − φ]⟩.

Indeed, first, by virtue of u_{n_{j}} ⇀ u weakly in W_{0}^{1,p}(Ω, w), and [3, Lemma 2.4], we have

T_{k}(u_{n_{j}} − φ) ⇀ T_{k}(u − φ) in W_{0}^{1.p}(Ω, w).  (4.6)

Which gives

\frac{∂T_{k}}{∂x_{i}} (u_{n_{j}} − φ) ⇀ \frac{∂T_{k}}{∂x_{i}} (u_{n_{j}} − φ) in L^{p}(Ω, w_{i}).  (4.7)

Note that ∇T_{k}(u_{n_{j}} −φ) is not zero on the subset {x ∈ Ω : |u_{n_{j}} −φ(x)| ≤ k} (subset of {x ∈ Ω : |u_{n_{j}}(x)| ≤ k + ∥φ∥_{L^{∞}(Ω)}},). Thus thanks to assumption (3.7), we have

\begin{aligned}|a_{i}(x, u_{n_{j}}, ∇φ)|^{p′}w_{i}^{−p′/p} & ≤ [k(x) + |u_{n_{j}}|^{p−1} + γ_{0}^{p−1} ∑_{k=1}^{N} |\frac{∂φ}{∂x_{k}} |^{p−1}w_{k}^{1/p′}]^{p′} \\ & ≤ β[k(x)^{p′} + |u_{n_{j}}|^{p} + γ_{0}^{p} ∑_{k=1}^{N} |\frac{∂φ}{∂x_{k}} |^{p}w_{k}]. \end{aligned} (4.8)

where {γ_{0} = sup|γ(s)|, |s| ≤ k + ∥φ∥_{∞}}. Since u_{n_{j}} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, σ), it follows that u_{n_{j}} → u strongly in L^{q}(Ω, σ) and u_{n_{j}} → u a.e. in Ω. Combining (4.7), (4.8) and By Vitali’s theorem we obtain,

∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] dx → ∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] dx.  (4.9)

Secondly, we show that

⟨F, T_{k}[u_{n_{j}} − φ]⟩ → ⟨F, T_{k}[u − φ]⟩.  (4.10)

In view of (4.6) and since F ∈ W^{−1,p′}(Ω, w^{∗}), we get

⟨F, T_{k}[u_{n_{j}} − φ]⟩ → ⟨F, T_{k}[u − φ]⟩.  (4.11)

The convergence (4.9) and (4.11) allow to pass to the limit in the inequality (4.3), and to obtain

∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] dx ≤ ⟨F, T_{k}[u − φ]⟩.  (4.12)

Now we introduce the following Lemma which will be proved later and which is considered as an L^{1} version of Minty’s lemma (in weighted Sbolev spaces).

Result (4.12) and the following lemma complete the proof of Theorem 4.2. □

Lemma 4.5. Let u be a measurable function such that T_{k}(u) belongs to W_{0}^{1,p}(Ω, w) for every k > 0. Then the following two statements are equivalent:

(i) For every φ in W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) and every k > 0,

∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] dx ≤ ∫_{Ω} F ∇T_{k}(u − φ) dx .
(ii) For every φ in W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) and every k > 0,
∫_{Ω} a(x, u, ∇u)∇T_{k}[u − φ] dx = ∫_{Ω} F ∇T_{k}(u − φ) dx .

Proof. Note that (ii) implies (i) is easily proved adding and subtracting

∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] dx,

and then using assumption (3.8). Thus, it only remains to prove that (i) implies (ii). Let h and k be positive real numbers, let λ ∈] − 1, 1[ and ψ ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω). Choosing, φ = T_{h}(u−λT_{k}(u−ψ)) ∈ W_{0}^{1,p}(Ω, w)∩L^{∞}(Ω) as test function in (4.12), we have,

I ≤ J,  (4.13)

with

I = ∫_{Ω} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx,
J = ⟨F, T_{k}(u − T_{h}(u − λT_{k}(u − ψ))⟩.

Put A_{hk} = {x ∈ Ω : |u − T_{h}(u − λT_{k}(u − ψ))| ≤ k} and B_{h} = {x ∈ Ω : |u − λT_{k}(u − ψ)| ≤ h}. Then, we have

I = ∫_{A_{kh}∩B_{h}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx

+ ∫_{A_{kh}∩B_{h}^{C}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx

+ ∫_{A_{kh}^{C}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ))) dx.

Since ∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) is zero in A_{kh}^{C}, we obtain

∫_{A_{kh}^{C}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx = 0. (4.14)

Moreover, if x ∈ B_{h}^{C}, we have ∇T_{h}(u − λT_{k}(u − ψ) = 0 which implies,

∫_{A_{kh}∩B_{h}^{C}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx
= ∫_{A_{kh}∩B_{h}^{C}} a(x, u, 0)∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx.

Now, thanks to assumption (3.9), we have a(x, u, 0) = 0. Then

∫_{A_{kh}∩B_{h}} a(x, u, 0)∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx = 0.  (4.15)

Combining (4.14) and (4.15), we obtain

I = ∫_{A_{kh}∩B_{h}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx,

letting h → +∞, we have

A_{kh} → {x, |T_{k}(u − ψ)| ≤ k} = Ω,  (4.16)

and B_{h} → Ω which implies

A_{kh} ∩ B_{h} → Ω.  (4.17)

Then

\begin{matrix}lim_{h→+∞} ∫_{A_{kh}∩B_{h}} a(x, u, ∇T_{h}(u − λT_{k}(u − ψ))∇T_{k}(u − T_{h}(u − λT_{k}(u − ψ)) dx \\ = λ ∫_{Ω} a(x, u, ∇(u − λT_{k}(u − ψ)∇T_{k}(u − ψ) dx. \end{matrix}

(4.18)

On the other hand, we have

J = ⟨F, T_{k}[u − T_{h}(u − λT_{k}(u − ψ)]⟩.

Then

lim_{h→+∞}⟨F, T_{k}(u − T_{h}(u − λT_{k}(u − ψ))⟩ = λ⟨F, T_{k}[u − ψ]⟩.  (4.19)

Together (4.18), (4.19) and passing to the limit in (4.13), we obtain

λ ∫_{Ω} a(x, u, ∇(u − λT_{k}(u − ψ)∇T_{k}(u − ψ) dx ≤ λ⟨F, T_{k}[u − ψ]⟩

for every ψ ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω), and for k > 0. Choosing λ > 0 dividing by λ, and then letting λ tend to zero , we obtain

∫_{Ω} a(x, u, ∇u)∇T_{k}(u − ψ) dx ≤ ⟨F, T_{k}[u − ψ]⟩.  (4.20)

For λ < 0 , dividing by λ, and then letting λ tend to zero , we obtain

∫_{Ω} a(x, u, ∇u)∇T_{k}(u − ψ) dx ≥ ⟨F, T_{k}[u − ψ]⟩,  (4.21)

Combining (4.20) and (4.21), we conclude that

∫_{Ω} a(x, u, ∇u)∇T_{k}(u − ψ) dx = ⟨F, T_{k}[u − ψ]⟩.

This completes the proof of Lemma. □

Remark 4.6. (1) The fact that the terms T_{n}(u_{n}) is introduced in (4.2) and also γ(s) is a continuous function, allow to have a weak solution for the a approximate problem.

(2) Since in the formulation of the problem (4.1), we have a(x, u, ∇u) instead of a(x, T_{n}(u_{n}), ∇u_{n}), then the term a(x, u, ∇u) may not belongs in L^{p′}(Ω, w^{∗}) and not in L^{1}(Ω), thus the problem (4.1) can have a T-solutions but, not a weak solution.

For example if w_{i} ≡ 1, i = 1, …, N and a(x, u, ∇u) = e^{|u|}|∇u|^{p−2}∇u, with γ(s) = e^{|s|} then

u ∈ W_{0}^{1,p}(Ω, w), F ∈ W^{−1,p′}(Ω, w^{∗})
− div(e^{|u|}|∇u|^{p−2}∇u) = F in Ω
u = 0 on ∂Ω.

our simple problem has a T -solutions, but not a weak solution

Example 4.7. Let us consider the special case:

a_{i}(x, η, ξ) = e^{|s|}w_{i}(x)|ξ_{i}|^{p−1} sgn(ξ_{i}) i = 1, …, N,

with w_{i}(x) is a weight function (i = 1, …, N). For simplicity, we shall suppose that w_{i}(x) = w(x), for i = 1, …, N − 1, and w_{N}(x) ≡ 0 it is easy to show that the a_{i}(x, s, ξ) are Caracthéodory function satisfying the growth condition (3.7) and the coercivity (3.8). On the other hand the monotonicity condition is verified. In fact,

∑_{i=1}^{N}(a_{i}(x, s, ξ) − a_{i}(x, s, ξ̂))(ξ_{i} − ξ̂_{i})

= e^{|s|}w(x) ∑_{i=1}^{N−1}(|ξ_{i}|^{p−1} sgn(ξ_{i}) − |ξ̂_{i}|^{p−1} sgn(ξ̂_{i}))(ξ_{i} − ξ̂_{i}) ≥ 0

for almost all x ∈ Ω and for all ξ, ξ̂ ∈ ℝ^{N}. This last inequality can not be strict, since for ξ ≠ ξ̂ with ξ_{N} ≠ ξ̂_{N} and ξ_{i} = ξ̂_{i}, i = 1, …, N − 1. The corresponding expression is zero. In particular, let us use special weight functions w expressed in terms of the distance to the bounded ∂Ω. Denote d(x) = dist(x, ∂Ω) and set w(x) = d^{λ}(x), such that,

λ < min(\frac{p}{N} , p − 1)  (4.22)

Remark 4.8. Condition (4.22) is sufficient for (3.4) to hold [see [10],pp 40-41].

Finally, the hypotheses of Theorem 4.2 are satisfied. Therefore, for all F ∈ ∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{∗}) the following problem has at last one solution:

T_{k}(u) ∈ W_{0}^{1,p}(Ω, w),

∫_{Ω} ∑_{i=1}^{N}w_{i}(x)e^{|u|}|\frac{∂u}{∂x_{i}} |^{p−1} sgn (\frac{∂u}{∂x_{i}} )\frac{∂T_{k}(u − φ)}{∂x_{i}} dx = ∫_{Ω} F T_{k}(u − φ) dx

∀φ ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) .

References

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[3] Y. Akdim, E. Azroul, and A. Benkirane, Existence of Solution for quasilinear degenerated Elliptic Unilateral Problems, Ann. Math. Blaise pascal vol 10 (2003) pp 1-20.

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Youssef Akdim Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: akdimyoussef@yahoo.fr

Abdelmoujib Benkirane Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: abenkirane@fsdmfes.ac.ma

Mohamed Rhoudaf Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: rhoudaf mohamed@yahoo.fr