Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 19, 1-18; http://www.math.u-szeged.hu/ejqtde/

Quasilinear degenerated equations with L^{1} datum and without coercivity in perturbation terms^{∗}

L. AHAROUCH, E. AZROUL and A. BENKIRANE

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

Abstract

In this paper we study the existence of solutions for the generated boundary value problem, with initial datum being an element of L^{1}(Ω) + W^{−1,p′}(Ω, w^{∗})

−diva(x, u, ∇u) + g(x, u, ∇u) = f − divF

where a(.) is a Carathéodory function satisfying the classical condition of type Leray-Lions hypothesis, while g(x, s, ξ) is a non-linear term which has a growth condition with respect to ξ and no growth with respect to s, but it satisfies a sign condition on s.

1 Introduction .

Let Ω be a bounded subset of IR^{N} (N ≥ 2), 1 < p < ∞, and w = {w_{i}(x); i = 0, ..., N} , be a collection of weight functions on Ω i.e., each w_{i} is a measurable and strictly positive function everywhere on Ω and satisfying some integrability conditions (see section 2). Let us consider the non-linear elliptic partial differential operator of order 2 given in divergence form

Au = −div(a(x, u, ∇u))  (1.1)

It is well known that equation Au = h is solvable by Drabek, Kufner and Mustonen in [7] in the case where h ∈ W^{−1,p′}(Ω, w^{∗}). In this paper we investigate the problem of existence solutions of the following Dirichlet problem

Au + g(x, u, ∇u) = μ in Ω.  (1.2)

∗AMS Subject Classification: 35J15, 35J20, 35J70.

Key words and phrases: Weighted Sobolev spaces, boundary value problems, truncations, unilateral problems

where μ ∈ L^{1}(Ω) + ∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{1−p′}).

In this context of nonlinear operators, if μ belongs to W^{−1,p′}(Ω, w^{∗}) existence results for problem (1.2) have been proved in [2], where the authors have used the approach based on the strong convergence of the positive part u_{ε}^{+} (resp. ngative part u_{ε}^{−} ). The case where μ ∈ L^{1}(Ω) is investigated in [3] under the following coercivity condition,

|g(x, s, ξ)| ≥ β∑_{i=1}^{N}w_{i}|ξ_{i}|^{p} for |s| ≥ γ,  (1.3)

Let us recall that the results given in [2, 3] have been proved under some additional conditions on the weight function σ and the parameter q introduced in Hardy inequality. The main point in our study to prove an existence result for some class of problem of the kind (1.2), without assuming the coercivity condition (1.3). Moreover, we didn’t supose any restriction for weight function σ and parameter q. It would be interesting at this stage to refer the reader to our previous work [1]. For different appproach used in the setting of Orlicz Sobolev space the reader can refer to [4], and for same results in the L^{p} case, to [10]. The plan of this is as follows : in the next section we will give some preliminaries and some technical lemmas, section 3 is concerned with main results and basic assumptions, in section 4 we prove main results and we study the stability and the positivity of solution.

2 Preliminaries .

Let Ω be a bounded open subset of IR^{N}(N ≥ 2). Let 1 < p < ∞, and let 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 0 ≤ i ≤ N

w_{i} ∈ L_{loc}^{1}(Ω) and w_{i}^{− \frac{1}{p−1}} ∈ L_{loc}^{1}(Ω).  (2.1)

We define the weighted space with weight γ in Ω as

L^{p}(Ω, γ) = \{u(x) : uγ^{\frac{1}{p}} ∈ L^{1}(Ω)\},

which is endowed with, we define the norm

∥u∥_{p,γ} = (∫_{Ω} |u(x)|^{p}γ(x) dx)^{\frac{1}{p}} .

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

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

This set of functions forms a Banach space under the norm

∥u∥_{1,p,w} = (∫_{Ω} |u(x)|^{p}w_{0} dx + ∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{\frac{1}{p}} .  (2.2)

To deal with the Dirichlet problem, we use the space

X = W_{0}^{1,p}(Ω, w)

defined as the closure of C_{0}^{∞}(Ω) with respect to the norm (2.2). Note that, C_{0}^{∞}(Ω) is dense in W_{0}^{1,p}(Ω, w) and (X, ∥.∥_{1,p,w}) is a reflexive Banach space. We recall that the dual of the 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 = 1, ..., N and p′ is the conjugate of p i.e. p′ = \frac{p}{p−1}. For more details we refer the reader to [8]. We introduce the functional spaces, we will need later. For p ∈ (1, ∞), T_{0}^{1,p}(Ω, w) is defined as the set of measurable functions u : Ω ⟶ IR such that for k > 0 the truncated functions T_{k}(u) ∈ W_{0}^{1,p}(Ω, w). We give the following lemma which is a generalization of Lemma 2.1 [5] in weighted Sobolev spaces.

Lemma 2.1. For every u ∈ T_{0}^{1,p}(Ω, w), there exists a unique measurable function v : Ω ⟶ IR^{N} such that

∇T_{k}(u) = vχ_{{|u|<k}}, almost everywhere in Ω, for every k > 0.

We will define the gradient of u as the function v, and we will denote it by v = ∇u.

Lemma 2.2. Let λ ∈ IR and let u and v be two functions which are finite almost everywhere, and which belongs to T_{0}^{1,p}(Ω, w). Then,

∇(u + λv) = ∇u + λ∇v a.e. in Ω,

where ∇u, ∇v and ∇(u + λv) are the gradients of u, v and u + λv introduced in Lemma 2.1.

The proof of this lemma is similar to the proof of Lemma 2.12 [6] for the non weighted case.

Definition 2.1. Let Y be a reflexive Banach space, a bounded operator B from Y to its dual Y^{∗} is called pseudo-monotone if for any sequence u_{n} ∈ Y with u_{n} ⇀ u weakly in Y. Bu_{n} ⇀ χ weakly in Y^{∗} and lim sup_{n→∞}⟨Bu_{n}, u_{n}⟩ ≤ ⟨χ, u⟩, we have

Bu_{n} = Bu and ⟨Bu_{n}, u_{n}⟩ → ⟨χ, u⟩ as n → ∞.

Now, we state the following assumptions. (H_{1})-The expression

∥u∥_{X} = (∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{\frac{1}{p}} ,  (2.3)

is a norm defined on X and is equivalent to the norm (2.2). (Note that (X, ∥u∥_{X}) is a uniformly convex (and reflexive) Banach space. -There exist a weight function σ on Ω and a parameter q, 1 < q < ∞, such that the Hardy inequality

(∫_{Ω} |u|^{q}σ(x) dx)^{\frac{1}{q}} ≤ C (∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{\frac{1}{p}} ,  (2.4)

holds for every u ∈ X with a constant C > 0 independent of u. Moreover, the imbeding

X ↪ L^{q}(Ω, σ)  (2.5)

determined by the inequality (2.4) is compact. We state the following technical lemmas which are needed later.

Lemma 2.3 [2]. Let g ∈ L^{r}(Ω, γ) and let g_{n} ∈ L^{r}(Ω, γ), with ∥g_{n}∥_{Ω,γ} ≤ c, 1 < r < ∞. If g_{n}(x) → g(x) a.e. in Ω, then g_{n} ⇀ g weakly in L^{r}(Ω, γ).

Lemma 2.4 [2]. Assume that (H_{1}) holds. Let F : IR → IR be unifomly Lipschitzian, with F (0) = 0. Let u ∈ W_{0}^{1,p}(Ω, w). Then F (u) ∈ W_{0}^{1,p}(Ω, w). Moreover, if the set D of discontinuity points of F′ is finite, then

\frac{∂F (u)}{∂x_{i}} = \{ \begin{matrix}F′(u)\frac{∂u}{∂x_{i}} & a.e. & in & \{x ∈ Ω : u(x) ∉ D\} \\ 0 & a.e. & in & \{x ∈ Ω : u(x) ∈ D\}. \end{matrix}

From the previous lemma, we deduce the following.

Lemma 2.5 [2]. Assume that (H_{1}) holds. Let u ∈ W_{0}^{1,p}(Ω, w), and let T_{k}(u), k ∈ IR^{+}, be the usual truncation, then T_{k}(u) ∈ W_{0}^{1,p}(Ω, w). Moreover, we have

T_{k}(u) → u strongly in W_{0}^{1,p}(Ω, w).

3 Main results .

Let Ω be a bounded open subset of IR^{N} (N ≥ 2). Consider the second order operator A : W_{0}^{1,p}(Ω, w) ⟶ W^{−1,p′}(Ω, w^{∗}) in divergence form

Au = −div(a(x, u, ∇u)),

where a : Ω×IR×IR^{N} → IR^{N} is a Carathéodory function Satisfying the following assumptions: (H_{2}) For i = 1, ..., N

|a_{i}(x, s, ξ)| ≤ βwi^{\frac{1}{p}} (x)[k(x) + σ^{\frac{1}{p′}} |s|^{\frac{q}{p′}} + ∑_{j=1}^{N}wj^{\frac{1}{p′}} (x)|ξ_{j}|^{p−1}],  (3.1)

[a(x, s, ξ) − a(x, s, η)](ξ − η) > 0 for all ξ ≠ η ∈ IR^{N},  (3.2)
a(x, s, ξ)ξ ≥ α∑_{i=1}^{N}w_{i}(x)|ξ_{i}|^{p}.  (3.3)

where k(x) is a positive function in L^{p′}(Ω) and α, β are positive constants. Assume that g : Ω × IR × IR^{N} ⟶ IR is a Carathéodory function satisfying : (H_{3}) g(x, s, ξ) is a Carathéodory function satisfying

g(x, s, ξ).s ≥ 0,  (3.4)

|g(x, s, ξ)| ≤ b(|s|)(∑_{i=1}^{N}w_{i}(x)|ξ_{i}|^{p} + c(x)),  (3.5)

where b : IR^{+} → IR^{+} is a positive increasing function and c(x) is a positive function which belong to L^{1}(Ω). Furthermore we suppose that

μ = f − divF, f ∈ L^{1}(Ω), F ∈ ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{1−p′}),  (3.6)

Consider the nonlinear problem with Dirichlet boundary condition

(P) \{ \begin{matrix}u ∈ T_{0}^{1,p}(Ω, w), g(x, u, ∇u) ∈ L^{1}(Ω) \\ ∫_{Ω} a(x, u, ∇u)∇T_{k}(u − v) dx + ∫_{Ω} g(x, u, ∇u)T_{k}(u − v) dx \\ ≤ ∫_{Ω} fT_{k}(u − v) dx + ∫_{Ω} F ∇T_{k}(u − v) dx \\ ∀v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) ∀k > 0. \end{matrix}

We shall prove the following existence theorem

Theorem 3.1. Assume that (H_{1}) − (H_{3}) hold true. Then there exists at least one solution of the problèm (P).

Remark 3.1. If w_{i} = σ = q = 1, the result of the preceding theorem coincides with those of Porretta (see [10]).

4 Proof of main results .

In order to prove the existence theorem we need the following

Lemma 4.1 [2]. Assume that (H_{1}) and (H_{2}) are satisfied, and let (u_{n})_{n} be a sequence in W_{0}^{1,p}(Ω, w) such that

1) u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w),

2) ∫_{Ω}[a(x, u_{n}, ∇u_{n}) − a(x, u_{n}, ∇u)]∇(u_{n} − u) dx → 0

then,

u_{n} → u in W_{0}^{1,p}(Ω, w).

We give now the proof of theorem 3.1. STEP 1. The approximate problem. Let f_{n} be a sequence of smooth functions which strongly converges to f in L^{1}(Ω). We Consider the sequence of approximate problems:

\{ \begin{matrix}u_{n} ∈ W_{0}^{1,p}(Ω, w), \\ ∫_{Ω} a(x, u_{n}, ∇u_{n})∇v dx + ∫_{Ω} g_{n}(x, u_{n}, ∇u_{n})v dx \\ = ∫_{Ω} f_{n}v dx + ∫_{Ω} F ∇v dx \\ ∀v ∈ W_{0}^{1,p}(Ω, w). \end{matrix}  (4.1)

where g_{n}(x, s, ξ) = \frac{g(x,s,ξ)}{1+ \frac{1}{n} |g(x,s,ξ)|}θ_{n}(x) with θ_{n}(x) = nT_{1/n}(σ^{1/q}(x)). Note that g_{n}(x, s, ξ) satisfises the following conditions

g_{n}(x, s, ξ)s ≥ 0, |g_{n}(x, s, ξ)| ≤ |g(x, s, ξ)| and |g_{n}(x, s, ξ)| ≤ n.

We define the operator G_{n} : X ⟶ X^{∗} by,

⟨G_{n}u, v⟩ = ∫_{Ω} g_{n}(x, u, ∇u)v dx

and

⟨Au, v⟩ = ∫_{Ω} a(x, u, ∇u)∇v dx

Thanks to Hölder’s inequality, we have for all u ∈ X and v ∈ X,

\begin{aligned}|∫_{Ω} g_{n}(x, u, ∇u)v dx| & ≤ & (∫_{Ω} |g_{n}(x, u, ∇u)|^{q′}σ^{− \frac{q′}{q}} dx)^{\frac{1}{q′}} (∫_{Ω} |v|^{q}σ dx)^{\frac{1}{q}} \\ & ≤ & n (∫_{Ω} σ^{q′/q}σ^{−q′/q} dx)^{\frac{1}{q′}} ∥v∥_{q,σ} \\ & ≤ & C_{n}∥v∥_{X}, \end{aligned}  (4.2)

the last inequality is due to (2.3) and (2.4).

Lemma 4.2. The operator B_{n} = A + G_{n} from X into its dual X^{∗} is pseudomonotone. Moreover, B_{n} is coercive, in the following sense:

\frac{< B_{n}v, v >}{∥v∥_{X}} ⟶ +∞ if ∥v∥_{X} ⟶ +∞, v ∈ W_{0}^{1,p}(Ω, w).

This Lemma will be proved below. In view of Lemma 4.2, there exists at least one solution u_{n} of (4.1) (cf. Theorem 2.1 and Remark 2.1 in Chapter 2 of [11] ). STEP 2. A priori estimates. Taking v = T_{k}(u_{n}) as test function in (4.1), gives

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n}) dx + ∫_{Ω} g_{n}(x, u_{n}, ∇u_{n})T_{k}(u_{n}) dx \\ = ∫_{Ω} f_{n}T_{k}(u_{n}) dx + ∫_{Ω} F_{n}∇T_{k}(u_{n}) dx \end{matrix}

and by using in fact that g_{n}(x, u_{n}, ∇u_{n})T_{k}(u_{n}) ≥ 0, we obtain

∫_{{|u_{n}|≤k}} a(x, u_{n}, ∇u_{n})∇u_{n} dx ≤ ck + ∫_{Ω} F_{n}∇T_{k}(u_{n}) dx.

Thank’s to Young’s inequality and (3.3), one easily has

\frac{α}{2}_{∫Ω}∑_{i=1}^{N}|\frac{∂T_{k}(u_{n})}{∂x_{i}}|^{p}w_{i}(x) dx ≤ c_{1}k.  (4.3)

STEP 3. Almost everywhere convergence of u_{n}. We prove that u_{n} converges to some function u locally in measure (and therefore, we can aloways assume that the convergence is a.e. after passing to a suitable subsequence). To prove this, we show that u_{n} is a Cauchy sequence in measure in any ball B_{R}. Let k > 0 large enough, we have

k meas({|u_{n}| > k} ∩ B_{R}) = ∫_{{|u_{n}|>k}∩B_{R}} |T_{k}(u_{n})| dx ≤ ∫_{B_{R}} |T_{k}(u_{n})| dx
≤ (∫_{Ω} |T_{k}(u_{n})|^{p}w_{0} dx)^{\frac{1}{p}} (∫_{B_{R}} w_{0}^{1−p′} dx)^{\frac{1}{q′}}
≤ c_{0} (∫_{Ω} ∑_{i=1}^{N}|\frac{∂T_{k}(u_{n})}{∂x_{i}}|^{p}w_{i}(x) dx)^{\frac{1}{p}}
≤ c_{1}k^{\frac{1}{p}} .

Which implies that

meas(\{|u_{n}| > k\} ∩ B_{R}) ≤ \frac{c_{1}}{k^{1− \frac{1}{p}}} ∀k > 1.  (4.4)

Moreover, we have, for every δ > 0,

meas({|u_{n} − u_{m}| > δ} ∩ B_{R}) ≤ meas({|u_{n}| > k} ∩ B_{R}) + meas({|u_{m}| > k} ∩ B_{R})
+meas{|T_{k}(u_{n}) − T_{k}(u_{m})| > δ}.

(4.5)

Since T_{k}(u_{n}) is bounded in W_{0}^{1,p}(Ω, w), there exists some v_{k} ∈ W_{0}^{1,p}(Ω, w), such that

T_{k}(u_{n}) ⇀ v_{k} weakly in W_{0}^{1,p}(Ω, w)
T_{k}(u_{n}) → v_{k} strongly in L^{q}(Ω, σ) and a.e. in Ω.

Consequently, we can assume that T_{k}(u_{n}) is a Cauchy sequence in measure in Ω. Let ε > 0, then, by (4.4) and (4.5), there exists some k(ε) > 0 such that meas({|u_{n} − u_{m}| > δ} ∩ B_{R}) < ε for all n, m ≥ n_{0}(k(ε), δ, R). This proves that (u_{n})_{n} is a Cauchy sequence in measure in B_{R}, thus converges almost everywhere to some measurable function u. Then

T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w),
T_{k}(u_{n}) → T_{k}(u) strongly in L^{q}(Ω, σ) and a.e. in Ω.

STEP 4. Strong convergence of truncations. We fix k > 0, and let h > k > 0. We shall use in (4.1) the test function

\{ \begin{matrix}v_{n} & = & ϕ(w_{n}) \\ w_{n} & = & T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)), \end{matrix}  (4.6)

with ϕ(s) = se^{γs^{2}}, γ = (\frac{b(k)}{α})^{2}. It is well known that

ϕ′(s) − \frac{b(k)}{α}|ϕ(s)| ≥ \frac{1}{2} ∀s ∈ IR,  (4.7)

It follows that

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx + ∫_{Ω} g_{n}(x, u_{n}, ∇u_{n})ϕ(w_{n}) dx \\ = ∫_{Ω} f_{n}ϕ(w_{n}) dx + ∫_{Ω} F ∇ϕ(w_{n}) dx. \end{matrix}  (4.8)

Since ϕ(w_{n})g_{n}(x, u_{n}, ∇u_{n}) > 0 on the subset {x ∈ Ω, |u_{n}(x)| > k}, we deduce from (4.8) that

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx + ∫_{\{|u_{n}|≤k\}} g_{n}(x, u_{n}, ∇u_{n})ϕ(w_{n}) dx \\ ≤ ∫_{Ω} f_{n}ϕ(w_{n}) dx + ∫_{Ω} F ∇ϕ(w_{n}) dx. \end{matrix}  (4.9)

Denote by ε_{h}^{1}(n), ε_{h}^{2}(n), ... various sequences of real numbers which converge to zero as n tends to infinity for any fixed value of h. We will deal with each term of (4.9). First of all, observe that

∫_{Ω} f_{n}ϕ(w_{n}) dx = ∫_{Ω} fϕ(T_{2k}(u − T_{h}(u))) dx + ε_{h}^{1}(n)  (4.10)

and

∫_{Ω} F ∇ϕ(w_{n}) dx = ∫_{Ω} F ∇T_{2k}(u − T_{h}(u))ϕ′(T_{2k}(u − T_{h}(u))) dx + ε_{h}^{2}(n).  (4.11)

Splitting the first integral on the left hand side of (4.9) where |u_{n}| ≤ k and |u_{n}| > k, we can write,

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx \\ = ∫_{\{|u_{n}|≤k\}} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx \\ +∫_{\{|u_{n}|>k\}} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx. \end{matrix} (4.12)

Setting m = 4k + h, using a(x, s, ξ)ξ ≥ 0 and the fact that ∇w_{n} = 0 on the set where |u_{n}| > m, we have

\begin{matrix}∫_{\{|u_{n}|>k\}} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx \\ ≥ −ϕ′(2k)∫_{\{|u_{n}|>k\}} |a(x, T_{m}(u_{n}), ∇T_{m}(u_{n}))||∇T_{k}(u)| dx, \end{matrix}

(4.13)

and since a(x, s, 0) = 0 ∀s ∈ IR, we have

\begin{matrix}∫_{\{|u_{n}|≤k\}} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx \\ = ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx. \end{matrix}  (4.14)

Combining (4.13) and (4.14), we get

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx \\ ≥ ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx \\ −ϕ′(2k)∫_{\{|u_{n}|>k\}} |a(x, T_{m}(u_{n}), ∇T_{m}(u_{n}))||∇T_{k}(u)| dx. \end{matrix}  (4.15)

The second term of the right hand side of the last inequality tends to 0 as n tends to infinity. Indeed. Since the sequence (a(x, T_{m}(u_{n}), ∇T_{m}(u_{n})))_{n} is bounded in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{1−p′}) while

∇T_{k}(u)χ_{|u_{n}|>k} tends to 0 strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}), which yields

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx \\ ≥ ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx + ε_{h}^{3}. \end{matrix} (4.16)

On the other hand, the term of the right hand side of (4.16) reads as

\begin{matrix}∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx \\ = ∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))] \\ ×[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx \\ +∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u))∇T_{k}(u_{n})ϕ′(T_{k}(u_{n}) − T_{k}(u)) dx \\ −∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u))∇T_{k}(u)ϕ′(w_{n}) dx. \end{matrix} (4.17)

Since a_{i}(x, T_{k}(u_{n}), ∇T_{k}(u))ϕ′(T_{k}(u_{n})−T_{k}(u)) → a_{i}(x, T_{k}(u), ∇T_{k}(u))ϕ′(0) strongly in L^{p′}(Ω, w_{i}^{1−p′}) by using the continuity of the Nymetskii operator, while \frac{∂(T_{k}(u_{n}))}{∂x_{i}} ⇀ \frac{∂(T_{k}(u))}{∂x_{i}} weakly in L^{p}(Ω, w_{i}), we have

\begin{matrix}∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u))∇T_{k}(u_{n})ϕ′(T_{k}(u_{n}) − T_{k}(u)) dx \\ = ∫_{Ω} a(x, T_{k}(u), ∇T_{k}(u))∇T_{k}(u)ϕ′(0) dx + ε_{h}^{4}(n). \end{matrix}  (4.18)

In the same way, we have

\begin{matrix}−∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u))∇T_{k}(u)ϕ′(w_{n}) dx \\ = −∫_{Ω} a(x, T_{k}(u), ∇T_{k}(u))∇T_{k}(u)ϕ′(0) dx + ε_{h}^{5}(n). \end{matrix}  (4.19)

Combining (4.16)-(4.19), we get

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇w_{n}ϕ′(w_{n}) dx \\ ≥ ∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))] \\ ×[∇T_{k}(u_{n}) − ∇T_{k}(u)]ϕ′(w_{n}) dx + ε_{h}^{6}(n). \end{matrix}  (4.20)

The second term of the left hand side of (4.9), can be estimated as

|∫_{\{|u_{n}|≤k\}} g_{n}(x, u_{n}, ∇u_{n})ϕ(w_{n}) dx| ≤ ∫_{\{|u_{n}|≤k\}} b(k) (c(x) + ∑_{i=1}^{N}w_{i}|\frac{∂T_{k}(u_{n})}{∂x_{i}}|^{p}) |ϕ(w_{n})| dx

≤ b(k)∫_{Ω} c(x)|ϕ(w_{n})| dx

+\frac{b(k)}{α}∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})|ϕ(w_{n})| dx.

(4.21)

EJQTDE, 2006 No. 19, p. 9

Since c(x) belongs to L^{1}(Ω) it is easy to see that

b(k)∫_{Ω} c(x)|ϕ(w_{n})| dx = b(k)∫_{Ω} c(x)|ϕ(T_{2k}(u − T_{h}(u)))| dx + ε_{h}^{7}(n).  (4.22)

On the other side, we have

\begin{matrix}∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u_{n})|ϕ(w_{n})| dx \\ = ∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))] \\ ×[∇T_{k}(u_{n}) − ∇T_{k}(u)]|ϕ(w_{n})| dx \\ +∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u)|ϕ(w_{n})| dx \\ +∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u))[∇T_{k}(u_{n}) − ∇T_{k}(u)]|ϕ(w_{n})| dx. \end{matrix} (4.23)

As above, by letting n go to infinity, we can easily see that each one of last two integrals of the right-hand side of the last equality is of the form ε_{h}^{8}(n) and then

\begin{matrix}| & ∫_{\{|u_{n}|≤k\}} g_{n}(x, u_{n}, ∇u_{n})ϕ(w_{n}) dx & | \\ ≤ ∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))] \\ ×[∇T_{k}(u_{n}) − ∇T_{k}(u)]|ϕ(w_{n})| dx \\ +b(k)∫_{Ω} c(x)|ϕ(T_{2k}(u − T_{h}(u)))| dx + ε_{h}^{9}(n). \end{matrix}  (4.24)

(4.9)-(4.11), (4.20) and (4.24), we get

∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))]
×[∇T_{k}(u_{n}) − ∇T_{k}(u)](ϕ′(w_{n}) − \frac{b(k)}{α}|ϕ(w_{n})|) dx
≤ b(k)∫_{Ω} c(x)|ϕ(T_{2k}(u − T_{h}(u)))| dx + ∫_{Ω} fϕ(T_{2k}(u − T_{h}(u))) dx,
+∫_{Ω} F ∇T_{2k}(u − T_{h}(u))ϕ′(T_{2k}(u − T_{h}(u))) dx + ε_{h}^{10}(n),

which and (4.7) implies that

∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))]
×[∇T_{k}(u_{n}) − ∇T_{k}(u)] dx
≤ 2b(k)∫_{Ω} c(x)|ϕ(T_{2k}(u − T_{h}(u)))| dx + 2∫_{Ω} fϕ(T_{2k}(u − T_{h}(u))) dx,
+2∫_{Ω} F ∇T_{2k}(u − T_{h}(u))ϕ′(T_{2k}(u − T_{h}(u))) dx + ε_{h}^{11}(n),

in which, we can pass to the limit as n → +∞ to obtain

lim sup_{n→∞}∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))][∇T_{k}(u_{n}) − ∇T_{k}(u)] dx
≤ 2b(k)∫_{Ω} c(x)|ϕ(T_{2k}(u − T_{h}(u)))| dx + 2∫_{Ω} fϕ(T_{2k}(u − T_{h}(u))) dx,
+2∫_{Ω} F ∇T_{2k}(u − T_{h}(u))ϕ′(T_{2k}(u − T_{h}(u))) dx.

(4.25)

It remains to show, for our purposes, that the all terms on the right hand side of (4.25) converge to zero as h goes to infinity. The only difficulty that exists is in the last term. For the other terms it suffices to apply Lebesgue’s theorem. We deal with this term. Let us observe that, if we take ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) as test function in (4.1) and use (3.3), we obtain

α∫_{\{h≤|u_{n}|≤2k+h\}} ∑_{i=1}^{N}|\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx

+∫_{Ω} g_{n}(x, u_{n}, ∇u_{n})ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) dx
≤ ∫_{{h≤|u_{n}|≤2k+h}} F ∇u_{n}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx
∫_{Ω} f_{n}ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) dx,

and thanks to the sign condition (3.4), we get

α∫_{\{h≤|u_{n}|≤2k+h\}} ∑_{i=1}^{N}|\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx

≤ ∫_{{h≤|u_{n}|≤2k+h}} F ∇u_{n}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx
+∫_{Ω} f_{n}ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) dx.

Using the Young inequality we have

\begin{aligned}\frac{α}{2} & ∫_{\{h≤|u_{n}|≤2k+h\}} ∑_{i=1}^{N}|\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx \\ & ≤ ∫_{Ω} f_{n}ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) dx + c_{k}∫_{\{h≤|u_{n}|\}} |w^{\frac{−1}{p}} .F |^{p′} dx, \end{aligned}  (4.26)

so that, since ϕ′ ≥ 1, we have

\begin{matrix}∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u − T_{h}(u))}{∂x_{i}} |^{p} w_{i} dx \\ ≤ ∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u − T_{h}(u))}{∂x_{i}} |^{p} w_{i}ϕ′(T_{2k}(u − T_{h}(u))) dx, \end{matrix}

again because the norm is lower semi-continuity, we get

∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u − T_{h}(u))}{∂x_{i}} |^{p} w_{i}ϕ′(T_{2k}(u − T_{h}(u))) dx

≤ c_{k}∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u − T_{h}(u))}{∂x_{i}} |^{p} w_{i} dx

≤ lim inf_{n→∞} ∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u_{n} − T_{h}(u_{n}))}{∂x_{i}} |^{p} w_{i} dx

≤ c_{k} lim inf_{n→∞} ∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u_{n} − T_{h}(u_{n}))}{∂x_{i}} |^{p} w_{i}ϕ′(T_{2k}(u_{n} − T_{h}(u_{n}))) dx

(4.27)

Consequently, in view of (4.26) and (4.27), we obtain

∫_{Ω} ∑_{i=1}^{N} |\frac{∂T_{2k}(u − T_{h}(u))}{∂x_{i}} |^{p} w_{i}ϕ′(T_{2k}(u − T_{h}(u))) dx

≤ lim inf_{n→∞} c_{k}∫_{\{h≤|u_{n}|\}} |w^{\frac{−1}{p}} F |^{p′} dx

+ lim inf_{n→∞} ∫_{Ω} f_{n}ϕ(T_{2k}(u_{n} − T_{h}(u_{n}))) dx.

Finally, the strong convergence in L^{1}(Ω) of f_{n}, we have, as first n and then h tend to infinity,

lim sup_{h→∞}_{∫\{h≤|u|≤2k+h\}}∑_{i=1}^{N}|\frac{∂u}{∂x_{i}} |^{p}w_{i}ϕ′(T_{2k}(u − T_{h}(u))) dx = 0,

hence

lim_{h→∞}∫_{Ω} F ∇T_{2k}(u − T_{h}(u))ϕ′(T_{2k}(u − T_{h}(u))) dx = 0.

Therefore by (4.25), letting h go to infinity, we conclude,

lim_{n→∞}∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))][∇T_{k}(u_{n}) − ∇T_{k}(u)] dx = 0,

which and using Lemma 4.1 implies that

T_{k}(u_{n}) → T_{k}(u) strongly in W_{0}^{1,p}(Ω, w) ∀k > 0.  (4.28)

STEP 5. Passing to the limit. By using T_{k}(u_{n} − v) as test function in (4.1), with v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω), we get

\begin{matrix}∫_{Ω} a(x, T_{k+∥v∥_{∞}}(u_{n}), ∇T_{k+∥v∥_{∞}}(u_{n}))∇T_{k}(u_{n} − v) dx + ∫_{Ω} g_{n}(x, u_{n}, ∇u_{n})T_{k}(u_{n} − v) dx \\ = ∫_{Ω} f_{n}T_{k}(u_{n} − v) dx + ∫_{Ω} F ∇T_{k}(u_{n} − v) dx. \end{matrix}

(4.29)

By Fatou’s lemma and the fact that

a(x, T_{k+∥v∥_{∞}}(u_{n}), ∇T_{k+∥v∥_{∞}}(u_{n})) ⇀ a(x, T_{k+∥v∥_{∞}}(u), ∇T_{k+∥v∥_{∞}}(u))

weakly in ∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{1−p′}) one easily sees that

∫_{Ω} a(x, T_{k+∥v∥_{∞}}(u), ∇T_{k+∥v∥_{∞}}(u))∇T_{k}(u − v) dx

≤ lim inf_{n→∞}∫_{Ω} a(x, T_{k+∥v∥_{∞}}(u_{n}), ∇T_{k+∥v∥_{∞}}(u_{n}))∇T_{k}(u_{n} − v) dx.

(4.30)

For the second term of the right hand side of (4.29), we have

∫_{Ω} F ∇T_{k}(u_{n} − v) dx ⟶ ∫_{Ω} F ∇T_{k}(u − v) dx as n → ∞.  (4.31)

since ∇T_{k}(u_{n} − v) ⇀ ∇T_{k}(u − v) weakly in ∏_{i=1}^{N}L^{p}(Ω, w_{i}), while F ∈ ∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{1−p′}).

On the other hand, we have

∫_{Ω} f_{n}T_{k}(u_{n} − v) dx ⟶ ∫_{Ω} fT_{k}(u − v) dx as n → ∞.  (4.32)

To conclude the proof of theorem, it only remains to prove

g_{n}(x, u_{n}, ∇u_{n}) → g(x, u, ∇u) strongly in L^{1}(Ω),  (4.33)

in particular it is enough to prove the equiintegrable of g_{n}(x, u_{n}, ∇u_{n}). To this purpose, we take T_{l+1}(u_{n}) − T_{l}(u_{n}) as test function in (4.1), we obtain

∫_{{|u_{n}|>l+1}} |g_{n}(x, u_{n}, ∇u_{n})| dx ≤ ∫_{{|u_{n}|>l}} |f_{n}| dx.

Let ε > 0. Then there exists l(ε) ≥ 1 such that

∫_{{|u_{n}|>l(ε)}} |g_{n}(x, u_{n}, ∇u_{n})| dx < ε/2.  (4.34)

For any measurable subset E ⊂ Ω, we have

\begin{aligned}∫_{E} |g_{n}(x, u_{n}, ∇u_{n})| dx & ≤ & ∫_{E} b(l(ε)) (c(x) + ∑_{i=1}^{N}w_{i}|\frac{∂(T_{l(ε)}(u_{n}))}{∂x_{i}}|^{p}) dx \\ & & +∫_{\{|u_{n}|>l(ε)\}} |g_{n}(x, u_{n}, ∇u_{n})| dx. \end{aligned}

In view of (4.28) there exists η(ε) > 0 such that

∫Eb(l(ε)) (c(x) + ∑_{i=1}^{N}w_{i}|\frac{∂(T_{l(ε)}(u_{n}))}{∂x_{i}}|^{p}) dx < ε/2

for all E such that meas E < η(ε).

(4.35)

Finally, by combining (4.34) and (4.35) one easily has

∫_{E} |g_{n}(x, u_{n}, ∇u_{n})| dx < ε for all E such that meas E < η(ε),

which shows that g_{n}(x, u_{n}, ∇u_{n}) are uniformly equintegrable in Ω as required. Thanks to (4.30)-(4.33) we can pass to the limit in (4.29) and we obtain that u is a solution of the problem (P). This completes the proof of Theorem 3.1.

Remark 4.1. Note that, we obtain the existence result withowt assuming the coercivity condition. However one can overcome this difficulty by introduced the function w_{n} = T_{2k}(u_{n}− T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) in the test function (4.6).

Proof of Lemma 4.2. From Hölder’s inequality, the growth condition (3.1) we can show that A is bounded, and by using (4.2), we have B_{n} bounded. The coercivity folows from (3.3) and (3.4). it remain to show that B_{n} is pseudo-monotone. Let a sequence (u_{k})_{k} ∈ W_{0}^{1,p}(Ω, w) such that

u_{k} ⇀ u weakly in W_{0}^{1,p}(Ω, w),
B_{n}u_{k} ⇀ χ weakly in W^{−1,p′}(Ω, w^{∗}),

and lim sup_{k→∞}⟨B_{n}u_{k}, u_{k}⟩ ≤ ⟨χ, u⟩.

We will prove that

χ = B_{n}u and ⟨B_{n}u_{k}, u_{k}⟩ → ⟨χ, u⟩ as k → +∞.

Since (u_{k})_{k} is a bounded sequence in W_{0}^{1,p}(Ω, w), we deduce that (a(x, u_{k}, ∇u_{k}))_{k} is bounded

in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{1−p′}), then there exists a function h ∈ ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{1−p′}) such that

a(x, u_{k}, ∇u_{k}) ⇀ h weakly in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{1−p′}) as k → ∞,

similarly, it is easy to see that (g_{n}(x, u_{k}, ∇u_{k}))_{k} is bounded in L^{q′}(Ω, σ^{1−q′}), then there exists a function k_{n} ∈ L^{q′}(Ω, σ^{1−q′}) such that

g_{n}(x, u_{k}, ∇u_{k}) ⇀ k_{n} weakly in Ł^{q′}(Ω, σ^{1−q′}) as k → ∞.

It is clear that, for all w ∈ W_{0}^{1,p}(Ω, w), we have

⟨χ, w⟩ = lim_{k→+∞}⟨B_{n}u_{k}, w⟩
= lim_{k→+∞} ∫_{Ω} a(x, u_{k}, ∇u_{k})∇w dx

+ lim_{k→+∞} ∫_{Ω} g_{n}(x, u_{k}, ∇u_{k}).w dx.

Consequently, we get

⟨χ, w⟩ = ∫_{Ω} h∇w dx + ∫_{Ω} k_{n}.w dx ∀w ∈ W_{0}^{1,p}(Ω, w).  (4.36)

On the one hand, we have

∫_{Ω} g_{n}(x, u_{k}, ∇u_{k}).u_{k} dx ⟶ ∫_{Ω} k_{n}.u dx as k → ∞,  (4.37)

and, by hypotheses, we have

lim sup_{k→∞} {∫_{Ω} a(x, u_{k}, ∇u_{k})∇u_{k} dx + ∫_{Ω} g_{n}(x, u_{k}, ∇u_{k}).u_{k} dx}
≤ ∫_{Ω} h∇u dx + ∫_{Ω} k_{n}.u dx,

therefore

lim sup_{k→∞} ∫_{Ω} a(x, u_{k}, ∇u_{k})∇u_{k} dx ≤ ∫_{Ω} h∇u dx.  (4.38)

By virtue of (3.2), we have

∫_{Ω}(a(x, u_{k}, ∇u_{k}) − a(x, u_{k}, ∇u))(∇u_{k} − ∇u) dx > 0.  (4.39)

Consequently

\begin{aligned}∫_{Ω} a(x, u_{k}, ∇u_{k})∇u_{k} dx & ≥ & − ∫_{Ω} a(x, u_{k}, ∇u)∇u dx + ∫_{Ω} a(x, u_{k}, ∇u_{k})∇u dx \\ & & + ∫_{Ω} a(x, u_{k}, ∇u)∇u_{k} dx, \end{aligned}

hence

lim inf_{k→∞} ∫_{Ω} a(x, u_{k}, ∇u_{k})∇u_{k} dx ≥ ∫_{Ω} h∇u dx.

This implies by using (4.38)

lim_{k→∞} ∫_{Ω} a(x, u_{k}, ∇u_{k})∇u_{k} dx = ∫_{Ω} h∇u dx.  (4.40)

By means of (4.36), (4.37) and (4.40), we obtain

⟨B_{n}u_{k}, u_{k}⟩ → ⟨χ, u⟩ as k ⟶ +∞.

On the other hand, by (4.40) and the fact that a(x, u_{k}, ∇u) → a(x, u, ∇u) strongly in

∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{1−p′}) it can be easily seen that

lim_{k→+∞} ∫_{Ω}(a(x, u_{k}, ∇u_{k}) − a(x, u_{k}, ∇u))(∇u_{k} − ∇u) dx = 0,

and so, thanks to Lemma 4.1

∇u_{n} → ∇u a.e. in Ω.

We deduce then that

a(x, u_{k}, ∇u_{k}) → a(x, u, ∇u) weakly in ∏_{i=1}^{N}L^{p′}(Ω, w_{i}^{1−p′}),

and g_{n}(x, u_{k}, ∇u_{k}) → g(x, u, ∇u) weakly in L^{q′}(Ω, w_{i}^{1−q′}).

Thus implies that χ = B_{n}u.

Corollary 4.1. Let 1 < p < ∞. Assume that the hypothesis (H_{1}) − (H_{3}) holds, let f_{n} be any sequence of functions in L^{1}(Ω) which converge to f weakly in L^{1}(Ω) and let u_{n} the solution of the following problem

(P′_{n}) \{ \begin{matrix}u_{n} ∈ T_{0}^{1,p}(Ω, w), g(x, u_{n}, ∇u_{n}) ∈ L^{1}(Ω) \\ ∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − v) dx + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(u_{n} − v) dx \\ ≤ ∫_{Ω} f_{n}T_{k}(u_{n} − v) dx, \\ ∀ v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω), ∀k > 0. \end{matrix}

Then there exists a subsequence of u_{n} still denoted u_{n} such that u_{n} converges to u almost everywhere and T_{k}(u_{n}) ⇀ T_{k}(u) strongly in W_{0}^{1,p}(Ω, w), further u is a solution of the problem (P) (with F = 0).

Proof. We give a brief proof. Step 1. A priori estimates. As before we take v = 0 as test function in (P′_{n}), we get

∫_{Ω} ∑_{i=1}^{N}w_{i}|\frac{∂T_{k}(u_{n})}{∂x_{i}}|^{p} dx ≤ C_{1}k.  (4.41)

Hence, by the same method used in the first step in the proof of Theorem 3.1 there exists a function u ∈ T_{0}^{1,p}(Ω, w) and a subsequence still denoted by u_{n} such that

u_{n} → u a.e. in Ω, T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w), ∀k > 0.

Step 2. Strong convergence of truncation. The choice of v = T_{h}(u_{n} − ϕ(w_{n})) as test function in (P′_{n}), we get, for all l > 0

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{l}(u_{n} − T_{h}(u_{n} − ϕ(w_{n}))) dx + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{l}(u_{n} − T_{h}(u_{n} − ϕ(w_{n})) dx \\ ≤ ∫_{Ω} f_{n}T_{l}(u_{n} − T_{h}(u_{n} − ϕ(w_{n}))) dx. \end{matrix}

Which implies that

∫_{{|u_{n}−ϕ(w_{n})|≤h}} a(x, u_{n}, ∇u_{n})∇T_{l}(ϕ(w_{n}))
+∫_{Ω} g(x, u_{n}, ∇u_{n})T_{l}(u_{n} − T_{h}(u_{n} − ϕ(w_{n}))) dx
≤ ∫_{Ω} f_{n}T_{l}(u_{n} − T_{h}(u_{n} − ϕ(w_{n}))) dx.

Letting h tend to infinity and choosing l large enough, we deduce

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇ϕ(w_{n}) dx + ∫_{Ω} g(x, u_{n}, ∇u_{n})ϕ(w_{n}) dx \\ ≤ ∫_{Ω} f_{n}ϕ(w_{n}) dx, \end{matrix}

the rest of the proof of this step is the same as in step 4 of the proof of Theorem 3.1. Step 3. Passing to the limit. This step is similar to the step 5 of the proof of Theorem 3.1, by using the Egorov’s theorem in the last term of (P′_{n}).

Remark 4.2. In the case where F = 0, if we suppose that the second member is nonnegative, then we obtain a nonnegative solution.

Indeed. If we take v = T_{h}(u^{+}) in (P), we have

∫_{Ω} a(x, u, ∇u)∇T_{k}(u − T_{h}(u^{+})) dx
+∫_{Ω} g(x, u, ∇u)T_{k}(u − T_{h}(u^{+})) dx
≤ ∫_{Ω} fT_{k}(u − T_{h}(u^{+})) dx.

Since g(x, u, ∇u)T_{k}(u − T_{h}(u^{+})) ≥ 0, we deduce

∫_{Ω} a(x, u, ∇u)∇T_{k}(u − T_{h}(u^{+})) dx ≤ ∫_{Ω} fT_{k}(u − T_{h}(u^{+})) dx,

and remark also that by using f ≥ 0 we have

∫_{Ω} fT_{k}(u − T_{h}(u^{+})) dx ≤ ∫_{{u≥h}} fT_{k}(u − T_{h}(u)) dx.

On the other hand, thanks to (3.3), we conclude

α∫_{Ω} ∑_{i=1}^{N}w_{i}|\frac{∂T_{k}(u^{−})}{∂x_{i}}|^{p} dx ≤ ∫_{\{u≥h\}} fT_{k}(u − T_{h}(u)) dx.

Letting h tend to infinity, we can easily deduce

T_{k}(u^{−}) = 0, ∀k > 0,

which implies that

u ≥ 0.

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L. Ahrouch (E-mail: l_{−}aharouch@yahoo.fr) E. Aazroul (E-mail: azroul_{−}elhoussine@yahoo.fr )

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

(Received October 15, 2005)