Introduction

Let Ω be a regular bounded domain of R^N , N > 1 and let us consider the problem:

where −div(a(x,u,∇u)) is a Leray-Lions operator acting from W) into its dual W ^−1,p0 (Ω,w^1−p0 ) with p > 1 and = 1, g is a nonlinearity which satisfies the growth condition, but it does not satisfy any sign condition. And f satisfies suitable summability assumption.

In [1], the authors proved the existence results in the setting of weighted Sobolev spaces for quasilinear degenerated elliptic problems associated with the

following equation − div af , where g satisfies the sign condition.

In [2], Benkirane and Bennouna studied L^∞ esti-

1,p mates of the solutions in W₀ (Ω,w) of the problem − div a(x,u,∇u) − div φ(u) + g(x,u) = f with a nonuniform elliptic condition, and g satisfies the sign condition.

In [3], the authors proved the existence of bounded solutions of the problem − div a(x,u,∇u) = g − div f , whose principal part has a degenerate coercivity, in

1,p the setting of weighted Sobolev spaces W₀                    (Ω,w).

The equations like (1) have been studied by many authors in the non-degenerate case (i.e. with w(x) ≡ 1) (see, e.g., [4] and the references therein).

The aim of this article is to establish a bounded solution for the problem (1) based on rearrangement properties. The results of this work can be considered as an extension of the results in [4] to the weighted case.

In order to perform L^∞-Estimates, the paper is organized in the following way. In section 1, we presented the introduction of the current work. In Section 2 we will state some basic knowledge of Sobolev spaces with weight and properties of the relative rearrangement. Finally in Section 3, we will introduce the essential assumptions, and we will prove our main result.

2 Preliminary results

2.1        Sobolev spaces with weight

In order to discuss the problem (1), we need some theories on W^1,p(Ω,w) which is called Sobolev spaces with weight.Firstly we state some basic properties of spaces W1;p(;w) which will be used later (for details, see [5]). Let be an open subset of RN with N  2, and 1  p <1 a real number. Let w = w(x) be a weighted function which is measurable and positive function a.e. in . Define Lp(;w) = fu measurable : uw 1 p 2 Lp g. We shall denote by W1;p(;w) the function space which consists of all real functions u 2 Lp such that their weak derivatives @u @xi , for all i = 1; : : : ;N (in the sense of distributions) satisfy @u @xi 2 Lp(;w), for all i = 1; : : : ;N. Endowed with the norm

W1;p(;w) is a Banach space. Further more, we suppose that

Due to condition (3), C1 is a subset of W1;p(;w).
Since we are dealing with compactness methods to get solutions of nonlinear elliptic equations, a compact imbedding is necessary. This leads us to suppose that the weight function w also satisfies

Condition (5) ensures that the imbedding

We remark that condition (4) implies that W1;p(;w) as well as W 1;p 0 (;w) are reflexive Banca spaces if 1 < p <1. Let us give the following lemmas which will be needed later.
Lemma 2.1 (See [1]) Assume that (6) holds. Let F : R! R be a uniformly Lipschitz function such that F(0) = 0. Then, F maps W 1;p 0 (;w) into itself. Moreover, if the set D of discontinuity points of F0 is finite.

Lemma 2.2 (See [1]) Let u 2 Lr (;w) and let un 2 Lr (;w), with jjunjjLr (;w)  c, 1 < r < 1. If un ! ua.e. in , then un * u in Lr (;w), where * denotes weak convergence.

2.2. Properties of the relative rearrangement

In this paragraph, we recall some standard notations and properties about decreasing rearrangements which will be used throughout this paper.
Let RN be a bounded domain, and let v : R be a measurable function. If one denotes by jEj the Lebesgue measure of a set E, one can define the distribution function.

The theory of rearrangements is well known, and its exhaustive treatments can be found for example in [6, 7,8]. Now we recall two notions which allow us to define a ”generalized” concept of rearrangement of a function f with respect to a given function v.

The two notions are equivalent in some precise sense (see [6]). For this reason we will denote both f v and f  v by Fv. We only recall a few results which hold for both the pseudo- and the relative rearrangements. If f and v are non-negative and it is possible to prove the following properties:

The proofs of (8) and (9) can be found in [9] (for pseudo-rearrangements) and in [11, 12] (for relative rearrangements). We finally recall the following chain of inequalities which holds for any non-negative v 2where CN denotes the measure of the unit ball in RN. It is a consequence of the Fleming-Rishel formula [13], the isoperimetric inequality [14] and the H¨older’s inequality.

3. Assumptions and main results

Let us now give the precise hypotheses on the problem (1), we assume that the following assumptions: is a bounded open set of RN (N > 1 ), 1 < p < +1, Let w be a non-negative real valued measurable function defined on which satisfies (3), (4) and (5). Let a : RRN !RN be a Carath´eodory function, such that

Furthermore, let g(x; s;e) :  R  RN ! R is a Carath´eodory function which satisfies, for almost every x 2 and for all s 2 R;  2 RN, the following condition

Finally, the right hand side we assume that

Now, we give the definition of weak solutions of problem (1).

Lemma 3.1 Let u be a solution of (1) and let us assume that (3)–(5) and (11)–(15) hold true. Define

Then the decreasing rearrangement of ‘ satisfies the following differential inequality:

Proof. Let us define two real functions 1(z), 2(z), z 2 R, as follows:

Furthermore, for t > 0, h > 0, let us put

We use in (1) the test function v 2W 1;p 0 (;w)\L1 defined by

Taking into account (22) and Young’s inequality, it follows that

using (14), and the ellipticity condition (11), we obtain

By(23), it follows that

Using (23) and the definition of 1, 2 in (21), the above inequality gives:

with F(x) = jf (x)jp0 w p (x).
Using Hardy-Littlewood’s inequality and the inequality (8). It follows that

where F’ is a pseudo-rearrangement (or the relative rearrangement) of jf jp0 with respect to ‘. On the other hand, thanks to H¨older’s inequality, we can easily check that

Using the Fleming–Rishel formula (see [8]), we can write

for almost every t > 0. Combining (28), (29), (30) and (31), we obtain

Proof of Theorem 3.2. By using Young’s inequality and (20) of Lemma 3.1 implies:

Since ‘ attains its maximum at 0, we can write

In order to estimate I, we use the H¨older’s inequality and (7), obtaining

The assumptions on exponents (5) and (14) allow us to get

We now turn to estimate J. Since p > N > 1, we can consider the H¨older conjugate exponent. The conjugate exponent satisfies the identity

Then (32) implies (18).
Proof of Theorem 3.1. Let us define for ” > 0 the approximation

Since the operator A g” : W 1;p (;w) !W?1;p0 (;w1?p0 ) is bounded, coercive, and pseudomonotone
operator, where A(u) = div a(x;u;ru) there exists at least one solution u” 2 W 1;p 0 (;w) ofthe problems (36)(see [15], [16] and in the weighted case [1]). Using Stampacchia’s method [17], one can prove that any solution u” of (36) belongs to L1 for fixed “.
Finally using the L1 estimate obtained in Theorem 3.2 and working as in [15] and [1], but with obvious modifications, we obtain Theorem 3.1.

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