2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 9–24. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Pseudo-monotonicity and degenerate elliptic operators of second order ^{∗}

Youssef Akdim & Elhoussine Azroul

Abstract

Extending the theory of pseudo-monotone mappings in weighted Sobolev spaces, we prove some existence results for degenerate or singular elliptic equations generated by the second-order differential operator

Au(x) = − div a(x, u, ∇u)) + a_{0}(x, u, ∇u),

(in particular, when only large monotonicity is satisfied)

1 Introduction

Let Ω be a open subset of ℝ^{N} (N ≥ 1) and p > 1 be a real number and ω = {ω_{0}, ω_{1}, …, ω_{N}} be a collection of weight functions on Ω, i.e, each ω_{i} is a measurable and positive almost everywhere in Ω, and satisfying some integrability condition (see section 2 below).

Let us consider the second-order differential operator

Au(x) = A_{1}u(x) + A_{0}u(x)  (1.1)

where

A_{1}u(x) = − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} a_{i}(x, u, ∇u)  (1.2)

is the top order part of A and where

A_{0}u(x) = a_{0}(x, u, ∇u)  (1.3)

is the lower order part of A and where {a_{i}(x, η, ζ), 0 ≤ i ≤ N} are functions defined on Ω×ℝ×ℝ^{N} and satisfy a suitable regularity and growth assumptions.

∗Mathematics Subject Classifications: 35J25, 35J70.

Key words: Weighted Sobolev spaces, pseudo-monotonicity,

nonlinear degenerate elliptic operators.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

Our objective in this paper, is to extend the theory of pseudo-monotone mappings in weighted Sobolev spaces. It’s well known that, the essential condition which allows to do this, is the so-called Leray-Lions condition,

∑_{i=1}^{N}(a_{i}(x, η, ζ) − a_{i}(x, η, ζ̄))(ζ_{i} − ζ̄_{i}) > 0,  (1.4)

for a.e. x ∈ Ω, all η ∈ ℝ and all ζ ≠ ζ̄ ∈ ℝ^{N} (resp. the so-called weak Leray-Lions condition,

∑_{i=1}^{N}(a_{i}(x, η, ζ) − a_{i}(x, η, ζ̄))(ζ_{i} − ζ̄_{i}) ≥ 0,  (1.5)

for a.e. x ∈ Ω, all (η, ζ, ζ̄) ∈ ℝ × ℝ^{N} × ℝ^{N}). Let us state the following assumptions:

(H1) The expression

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

is a norm on X = W_{0}^{1,p}(Ω, ω) equivalent to the usual norm (2.3)(see section 2). There exist a weight function ω̄ on Ω and a parameter q, 1 < q < ∞, such that the (Hardy) inequality

( ∫_{Ω} |u(x)|^{q}ω̄(x))^{1/q} ≤ c( ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}ω_{i}(x) dx)^{1/p}  (1.6)

holds for every u ∈ W_{0}^{1,p}(Ω, ω) with a constant c > 0 independent of u, and moreover, the imbedding expressed by (1.6) is compact, i.e.

W_{0}^{1,p}(Ω, ω) ↪↪ L^{q}(Ω, ω̄).  (1.7)

(H2) Each a_{i}(x, η, ζ) (1 ≤ i ≤ N) is a Carathéodory function and

|a_{i}(x, η, ζ)| ≤ C_{i}ω_{i}^{1/p}(x)[g_{i}(x) + ω̄^{\frac{1}{p′}} |η|^{q/p′} + ∑_{j=1}^{N} ω_{j}^{1/p′}(x)|ζ_{j}|^{p−1}], (1.8)

for a.e. x ∈ Ω, some constants C_{i} > 0, some functions g_{i}(x) ∈ L^{p′}(Ω), all (η, ζ) ∈ ℝ^{N+1} and all i = 1, …, N.

Recently, Drabek, Kufner and Mustonen [2] proved that the mapping T_{1} defined from X to its dual X^{∗} associated to the top order part A_{1} is pseudo-monotone in X, under the weak conditions (1.5), (H1), (H2). Hence, the authors obtained the existence result for the Dirichlet problem associated to the A_{1}u = f ∈ X^{∗} by assuming some degeneracy.

Our first purpose in this paper, is to extend the previous result [2] in the operator A from (1.1) where the lower order part A_{0} is affine with respect to the gradient, i.e., A_{0} is of the form

A_{0}u(x) = c_{0}(x, u(x)) + ∑_{i=1}^{N} c_{i}(x, u(x))\frac{∂u(x)}{∂x_{i}},  (1.9)

where c_{i}(x, η), 0 ≤ i ≤ N are some Carathéodory functions defined on Ω × ℝ and satisfy

\begin{matrix}|c_{0}(x, η)| ≤ C_{0}ω̄^{1/q}(x)[g_{0}(x) + ω̄^{\frac{1}{q′}} (x)|η|^{\frac{q}{q′}}] \\ |c_{i}(x, η)| ≤ C_{i}ω_{i}^{1/p}(x)ω̄^{1/q}(x)[γ_{i}(x) + ω̄^{\frac{1}{r}} (x)|η|^{\frac{q}{r}}] for all i = 1, …, N, \end{matrix} (1.10)

for a.e. x ∈ Ω, some constants C_{0} > 0, C_{i} > 0, some functions g_{0} ∈ L^{q′}(Ω) and γ_{i}(x) ∈ L^{r}(Ω) with

\frac{1}{r} + \frac{1}{p} + \frac{1}{q} < 1  (1.11)

and where ω̄(x) and q are from (1.6). More precisely, we prove the following theorem,

Theorem 1.1 Assume that (H1), (H2), (1.10), (1.5) hold. Then the mapping T associated to the operator A from (1.1) and (1.9) is pseudo-monotone in X.

Remark 1.2 Theorem 1.1 is obviously a consequence of the more general result (Theorem 3.1, it suffices to take I = ∅).

Remark 1.3 About the existence of such r satisfying (1.11) see Remarks 2.1 and 4.2 below.

The second aim of this paper, is to prove the same result of the preceding without restriction on A_{0} and where (1.4) is applied. This is done in Theorem 3.1, if we take I^{c} = ∅.

This paper is divided into four sections. In section 2, we start our basic assumptions and we prove some preliminaries lemmas concerning some convergence and generalized Hölder’s inequality in weighted Sobolev space. In section 3, we give our general main result and its proof and we study an example which illustrate our abstract hypotheses. The section 4, is devoted to the study of some particular case where ω_{0} ≡ 1 on Ω and where some of our hypotheses (imbedding) are satisfied.

In our work, we shall adopt many ideas from [5] (where the authors have studied the non-degenerated elliptic case). But the results are generalized and improved. concerning the existence results for higher order nonlinear degenerated (or singular) elliptic equations, we refer the reader to [3, 4, 1] (where the degree theory is used in the two first papers and where the pseudo-monotonicity is used in the last but under some restrictions on the weighted). Finally, not that our approach based on the theory of pseudo-monotone mappings can be applied in the case of non reflexive Banach space, for example in weighted Orlicz-Sobolev spaces (see [1] for related topics).

2 Preliminaries and basic assumptions

1) Weighted Sobolev spaces. Let Ω be a open subset of ℝ^{N} (N ≥ 1), with finite measure, let 1 < p < ∞, and let ω = {ω_{i}(x) 0 ≤ i ≤ N} be a vector of weight functions, i.e. every component ω_{i}(x) is a measurable function which is positive a.e. in Ω. Further, we suppose that

ω_{i} ∈ L_{loc}^{1}(Ω)  (2.1)

and

ω_{i}^{− \frac{1}{p−1}} ∈ L_{loc}^{1}(Ω)  (2.2)

for any 0 ≤ i ≤ N hold in all our considerations.

Now, we denote by W^{1,p}(Ω, ω) the space of all real-valued functions u ∈ L^{p}(Ω, ω_{0}) such that the derivatives in the sense of distributions fulfil

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

which is a Banach space under the norm,

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

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

We recall that the dual space of weighted Sobolev spaces W_{0}^{1,p}(Ω, ω) is equivalent to W^{−1,p′}(Ω, ω^{∗}), where ω^{∗} = \{ω_{i}^{∗} = ω_{i}^{1−p′} ∀i = 0, …, N\}, with p′ = \frac{p}{p−1}. We shall suppose that the expression

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

is a norm defined on W_{0}^{1,p}(Ω, ω) and it’s equivalent to the norm (2.3). The reader can find conditions on the weight ω which guarantee this fact in [3]. Notice that (X, ∥|.|∥_{X}) is a uniformly convex (and thus reflexive) Banach space.

2) Basic assumptions. Let I be a subset of {1, 2, …, N} and I^{c} its complement, and let introduce the following modified versions of (1.4) and (1.5),

∑_{i∈I}(b_{i}(x, η, ζ_{I}) − b_{i}(x, η, ζ̄_{I}))(ζ_{i} − ζ̄_{i}) > 0,  (2.4)

for a.e. x ∈ Ω, all η ∈ ℝ and all ζ ≠ ζ̄ ∈ ℝ^{N} and

∑_{i∈I^{c}}(b_{i}(x, η, ζ_{I^{c}}) − b_{i}(x, η, ζ̄_{I^{c}}))(ζ_{i} − ζ̄_{i}) ≥ 0,  (2.5)

for a.e. x ∈ Ω, all η ∈ ℝ and all ζ, ζ̄ ∈ ℝ^{N} where ζ_{J} denoted ζ_{J} = {ζ_{i}, i ∈ J} and where a_{i}(x, η, ζ) are Carathéodory functions such that,

a_{i}(x, η, ζ) = b_{i}(x, η, ζ_{I}) for all i ∈ I, a_{i}(x, η, ζ) = b_{i}(x, η, ζ_{I^{c}}) for all i ∈ I^{c},

a_{0}(x, η, ζ) = c_{0}(x, η, ζ_{I}) + ∑_{i∈I^{c}}c_{i}(x, η, ζ_{I})ζ_{i},

(2.6)

for a.e. x ∈ Ω, all (η, ζ) ∈ ℝ^{N+1} and where b_{i} (i = 1, …, N), c_{0} and c_{i} (i ∈ I^{c}) are functions satisfying the Carathéodory conditions (i.e. measurable in x for any fixed ξ = (η, ζ) ∈ ℝ^{N+1} and continuous in ξ for almost all fixed x ∈ Ω).

We assume the following growth conditions:

(H2’) Each a_{i}(x, η, ζ) is a Carathéodory function and, that there exists some positives constants C_{i}, and some functions g_{i}(x) ∈ L^{p′}(Ω) i = 1, …, N, and g_{0} ∈ L^{q′}(Ω) and some γ_{i}(x) ∈ L^{r}(Ω) for all i ∈ I^{c}) such that

|b_{i}(x, η, ζ_{I})| ≤ C_{i}ω_{i}^{1/p}(x)[g_{i}(x) + ω̄^{\frac{1}{p′}} |η|^{\frac{q}{p′}} + ∑_{j∈I} ωj^{\frac{1}{p′}} (x)|ζ_{j}|^{p−1}] for i ∈ I

|b_{i}(x, η, ζ_{I^{c}})| ≤ C_{i}ω_{i}^{1/p}(x)[g_{i}(x) + ω̄^{\frac{1}{p′}} (x)|η|^{\frac{q}{p′}} + ∑_{j∈I^{c}} ωj^{\frac{1}{p′}} (x)|ζ_{j}|^{p−1}],

for i ∈ I^{c}

|c_{0}(x, η, ζ_{I})| ≤ C_{0}ω̄^{1/q}[g_{0}(x) + ω̄^{\frac{1}{q′}} (x)|η|^{\frac{q}{q′}} + ∑_{j∈I} ωj^{\frac{1}{q′}} (x)|ζ_{j}|^{\frac{p}{q′}}]

|c_{i}(x, η, ζ_{I})| ≤ C_{i}ω_{i}^{1/p}(x)ω̄^{1/q}(x)[γ_{i}(x) + ω̄^{\frac{1}{r}} (x)|η|^{\frac{q}{r}} + ∑_{j∈I} ωj^{\frac{1}{r}} (x)|ζ_{j}|^{\frac{p}{r}}],

for i ∈ I^{c},

for a.e. x ∈ Ω, all η ∈ ℝ, ζ ∈ ℝ^{N}, with

\frac{1}{r} + \frac{1}{p} + \frac{1}{q} < 1.  (2.7)

Remark 2.1 1) The such r satisfying (2.7), exists when q > p′ (it suffices to choose r > \frac{pq}{pq−p−q} > 1).

2) If q ≤ p′, we can not found any r satisfying (2.7) (since \frac{1}{p} + \frac{1}{p′} = 1 ≤ \frac{1}{p} + \frac{1}{q}).

Before to give main general result, let us give and prove the following lemmas which are needed below.

Lemma 2.2 Let Ω be a subset of ℝ^{N} with finite measure and let f ∈ L^{p}(Ω, σ_{1}) (1 < p < ∞), g ∈ L^{q}(Ω, σ_{2}) (1 < q < ∞) where σ_{1} and σ_{2} are weight functions in Ω and let h ∈ L^{r}(Ω, σ_{1}^{− \frac{r}{p}} σ_{2}^{− \frac{r}{q}}) (1 < r < ∞) with \frac{1}{p} + \frac{1}{q} + \frac{1}{r} ≤ 1, then fgh ∈ L^{1}(Ω).

Indeed: Let \frac{1}{s} = \frac{1}{p} + \frac{1}{q} + \frac{1}{r} ≤ 1. By Hölder inequality we have,

∫_{Ω} |fgh|^{s} ≤ ( ∫_{Ω} f^{p}σ_{1})^{s/p}( ∫_{Ω} g^{q}σ_{2})^{s/q}( ∫_{Ω} h^{r}σ_{1}^{−r/p}σ_{2}^{−r/q})^{s/r} < ∞,

then fgh ∈ L^{s}(Ω) which implies that fgh ∈ L^{1}(Ω).

Lemma 2.3 Let (g_{n})_{n} be a sequence of L^{p}(Ω, σ) and let g ∈ L^{p}(Ω, σ) (1 < p < ∞), where σ is a weight function in Ω. If g_{n} → g in measure (in particular a.e. in Ω) and is bounded in L^{p}(Ω, σ), then g_{n} → g in L^{q}(Ω, σ^{q/p}) for all q < p.

Proof. Let ε > 0 and set A_{n} = \{x ∈ Ω : |g_{n}(x)−g(x)|σ^{1/p}(x) ≤ (\frac{ε}{2 meas(Ω)})^{1/q}\}, we have

∫_{Ω} |g_{n} − g|^{q}σ^{q/p} dx = ∫_{A_{n}} |g_{n} − g|^{q}σ^{q/p} dx + ∫_{A_{n}^{c}} |g_{n} − g|^{q}σ^{q/p} dx
≤ \frac{ε}{2} + ∫_{A_{n}^{c}} |g_{n} − g|^{q}σ^{q/p} dx.

By Hölder inequality,

∫_{A_{n}^{c}} |g_{n} − g|^{q}σ^{q/p} dx ≤ ( ∫_{Ω} |g_{n} − g|^{p}σ dx)^{q/p}( meas(A_{n}^{c}))^{1− \frac{q}{p}}
≤ M (meas(A_{n}^{c}))^{1− \frac{q}{p}} ,

where M is a constant does not depend on n. On the other hand since g_{n} → g in measure we have

meas(A_{n}^{c}) → 0 as n → ∞,

then there exists some n_{0} ∈ ℕ such that for all n ≥ n_{0},

∫_{A_{n}^{c}} |g_{n} − g|^{q}σ^{q/p} dx ≤ \frac{ε}{2}.

Remark 2.4 We can also give an other proof of the last lemma, by using the non-weighted case, i.e., g_{n}σ^{1/p} is bounded in L^{p}(Ω) and g_{n}(x)σ^{1/p}(x) → g(x)σ^{1/p}(x), in measure, hence g_{n}σ^{1/p} → gσ^{1/p} in L^{q}(Ω) for all q < p.

The following lemma is a generalization of [7, Lemma 3.2] in weighted spaces.

Lemma 2.5 Let g ∈ L^{q}(Ω, σ) and let g_{n} ∈ L^{q}(Ω, σ), with ∥g_{n}∥_{q,σ} ≤ c (1 < q < ∞). If g_{n}(x) → g(x) a.e. in Ω, then g_{n} ⇀ g in L^{q}(Ω, σ), where ⇀ denotes weak convergence.

Proof. Since g_{n}σ^{1/q} is bounded in L^{q}(Ω) and g_{n}(x)σ^{1/q}(x) → g(x)σ^{1/q}(x), a.e. in Ω, by the [7, Lemma 3.2], we have

g_{n}σ^{1/q} ⇀ gσ^{1/q} in L^{q}(Ω).

Moreover for all φ ∈ L^{q′}(Ω, σ^{1−q′}), we have φσ^{−1/q} ∈ L^{q′}(Ω), then

∫_{Ω} g_{n}φ dx → ∫_{Ω} gφ dx, i.e. g_{n} ⇀ g in L^{q}(Ω, σ).

Lemma 2.6 Let g_{n} ∈ L^{p}(Ω, σ_{1}) and let g ∈ L^{p}(Ω, σ_{1}) (1 < p < ∞). If g_{n} ⇀ g in L^{p}(Ω, σ_{1}), then

g_{n}v ⇀ gv in L^{s}(Ω, σ_{1}^{s/p}σ_{2}^{s/q}) for any v ∈ L^{q}(Ω, σ_{2}),

with q > 1 and \frac{1}{s} = \frac{1}{p} + \frac{1}{q}.

Proof. Letφ ∈ L^{s′}(Ω, σ_{1}^{\frac{s}{p} (1−s′)}σ_{2}^{\frac{s}{q} (1−s′)}). For any v ∈ L^{q}(Ω, σ_{2}) we have, vφ ∈ L^{p′}(Ω, σ_{1}^{1−p′}). Indeed, since \frac{1}{p′} = \frac{1}{s′} + \frac{1}{q}, we have by Hölder’s inequality,

∫_{Ω} |vφ|^{p′}σ_{1}^{1−p′}(x) dx
= ∫_{Ω} |vσ_{2}^{1/q}(x)|^{p′}|φ|^{p′}σ_{1}^{1−p′}(x)σ_{2}^{−p′/q}(x) dx
≤ ( ∫_{Ω} |v|^{q}σ_{2}(x) dx)^{p′/q}( ∫_{Ω} |φ|^{s′}σ_{1}^{\frac{s′}{p′} (1−p′)}(x)σ_{2}^{−s′/q}(x) dx)^{p′/s′}
= ( ∫_{Ω} |v|^{q}σ_{2} dx)^{p′/q}( ∫_{Ω} |φ|^{s′}σ_{1}^{\frac{s}{p} (1−s′)}(x)σ_{2}^{\frac{s}{q} (1−s′)}(x) dx)^{p′/s′} < ∞.

Finally, since g_{n} ⇀ g in L^{p}(Ω, σ_{1}), then

∫_{Ω} g_{n}vφ dx → ∫_{Ω} gvφ dx i.e. g_{n}v ⇀ gv in L^{s}(Ω, σ_{1}^{s/p}σ_{2}^{s/q}) ∀v ∈ L^{q}(Ω, σ_{2}).

Lemma 2.7 Let Ω be a subset of ℝ^{N} with finite measure and let 1 ≤ p ≤ q then, we have the continuous imbedding L^{q}(Ω, σ) ↪ L^{p}(Ω, σ^{p/q}) where σ is a weight function in Ω.

The proof of this lemma can be deduced easily from Hölder’s inequality.

3 Main general result

Under the previous assumptions, the differential operator (1.1) (with coefficients satisfying (2.6), generates a mapping T from X = W_{0}^{1,p}(Ω, ω) to its dual X^{∗} through the formula,

\begin{aligned}⟨T u, v⟩ = & ∫_{Ω} ∑_{i∈I} b_{i}(x, u, ζ_{I}(∇u))\frac{∂v}{∂x_{i}} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u, ζ_{I^{c}}(∇u))\frac{∂v}{∂x_{i}} dx \\ & + ∫_{Ω} c_{0}(x, u, ζ_{I}(∇u))v dx + ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u, ζ_{I}(∇u))\frac{∂u}{∂x_{i}} v dx, \end{aligned} (3.1)

for all u, v ∈ X and where ⟨, ⟩ denotes the duality pairing between X^{∗} and X. When we have adopted the notation ζ_{J}(∇u) = \{\frac{∂u}{∂x_{i}} , i ∈ J\}.

We recall that the mapping T is well defined and bounded, this can be easily seen by Lemma 2.2 and Hölder’s inequality.

Definition A bounded mapping T from X to X^{∗} is called pseudo-monotone if for any sequence u_{n} ∈ X with u_{n} ⇀ u in X and lim sup_{n→∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0, one has

lim inf_{n→∞}⟨T u_{n}, u_{n} − v⟩ ≥ ⟨T u, u − v⟩ for all v ∈ X.

Theorem 3.1 Assume that (H1), (H2’), (2.4) and (2.5) hold. Then the corresponding mapping T defined by (3.1) is pseudo-monotone in X = W_{0}^{1,p}(Ω, ω).

Remark 3.2 1) When I = ∅, the previous theorem applies in particular to operators like (1.1) with A_{0} affine with respect to the gradient variable, this gives from (1.5) a sufficient condition (theorem 1.1 in the introduction).

2) When I = ∅ and A_{0} ≡ 0, we immediately obtain [2, Proposition 1].

3) When I^{c} = ∅, we obtain [1, Theorem 7.4] and when A_{0} ≡ 0, I = ∅, we give in [1, Theorem 7.2].

4) Theorem 3.1 generalizes [5, Theorem 3.1] in the weighted case.

Applying the previous theorem, we obtain the following existence results, which generalizes the corresponding (cf. [1, 2]).

Corollary 3.3 Let Ω be a bounded open subset of ℝ^{N} and assume the hypotheses in Theorem 3.1. Also assume the degenerate ellipticity condition

∑_{i=0}^{N} a_{i}(x, ξ)ξ_{i} ≥ C_{0} ∑_{i=1}^{N} ω_{i}(x)|ξ_{i}|^{p}

for a.e. x ∈ Ω, some C_{0} > 0 and all ξ ∈ ℝ^{N+1}. Then for any f ∈ X^{∗} the Dirichlet associated problem

∫_{Ω} ∑_{i∈I} b_{i}(x, u, ζ_{I}(∇u))\frac{∂v}{∂x_{i}} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u, ζ_{I^{c}}(∇u))\frac{∂v}{∂x_{i}} dx

+ ∫_{Ω} c_{0}(x, u, ζ_{I}(∇u))v dx + ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u, ζ_{I}(∇u))\frac{∂u}{∂x_{i}} v dx = ∫_{Ω} fv dx

for all v ∈ X has at least one solution u ∈ X.

Proof of Theorem 3.1. Let (u_{n})_{n} be a sequence in X such that:

u_{n} ⇀ u in X  (3.2)

and

lim sup_{n→∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0,  (3.3)

i.e.

lim sup_{n→∞} \{ ∫_{Ω} ∑_{i∈I} b_{i}(x, u_{n}, ζ_{I}(∇u_{n}))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx

+ ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx

+ ∫_{Ω} c_{0}(x, u_{n}, ζ_{I}(∇u_{n}))(u_{n} − u) dx

+ ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} (u_{n} − u) dx\} ≤ 0.

a) We shall prove that

⟨T u_{n}, v⟩ → ⟨T u, v⟩ as n → ∞ for all v ∈ X.  (3.4)

First step. We show that

lim_{n→∞} ∫_{Ω} ∑_{i∈I}(b_{i}(x, u_{n}, ζ_{I}(∇u_{n})) − b_{i}(x, u_{n}, ζ_{I}(∇u)))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx = 0 (3.5)

and

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}}(b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))−b_{i}(x, u_{n}, ζ_{I^{c}}(∇u)))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx = 0. (3.6)

Indeed: First, we can choose q_{1} such that 1 < q_{1} < r, and \frac{1}{q_{1}} + \frac{1}{p} + \frac{1}{q} < 1 (due to \frac{1}{r} + \frac{1}{p} + \frac{1}{q} < 1). It follows from the compact imbedding (1.7) that, for a subsequence,

\begin{matrix}u_{n} → u in L^{q}(Ω, ω̄) \\ u_{n}(x) → u(x) a.e. in Ω. \end{matrix}  (3.7)

By (H2’), the sequences \{c_{0}(x, u_{n}, ζ_{I}(∇u_{n}))\} (resp. \{c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} (i ∈

I^{c})\}) remains bounded in L^{q′}(Ω, ω̄^{1−q′}) (resp. Ls̃(Ω, ω̄^{\frac{−s̃}{q}}) with \frac{1}{s̃} = \frac{1}{p} + \frac{1}{r}). Indeed,

∫_{Ω} |ω̄^{−1/q}c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} |s̃

≤ ( ∫_{Ω} ω_{i}^{−r/p}ω̄^{−r/q}|c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))|^{r})^{s̃/r}( ∫_{Ω} |\frac{∂u_{n}}{∂x_{i}} |^{p}ω_{i})^{s̃/p} < c.

Thanks to Lemma 2.7 and since q′ ≤ s̃, we have

Ls̃(Ω, ω̄^{−s̃/q}) ↪ L^{q′}(Ω, ω̄^{−q′/q}),

then \{c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} (i ∈ I^{c})\}) is bounded in L^{q′}(Ω, ω̄^{1−q′})). Hence, using (3.7) we conclude that

lim_{n→∞} ∫_{Ω} c_{0}(x, u_{n}, ζ_{I}(∇u_{n}))(u_{n} − u) dx = 0  (3.8)

and

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} (u_{n} − u) dx = 0.  (3.9)

On the other hand, in virtue of (3.7) and continuity of the Nemytskii operators (see [3]), we have

b_{i}(x, u_{n}, ζ_{I}(∇u)) → b_{i}(x, u, ζ_{I}(∇u)) in L^{p′}(Ω, ω_{i}^{∗}), i ∈ I
b_{i}(x, u_{n}, ζ_{I^{c}}(∇u)) → b_{i}(x, u, ζ_{I^{c}}(∇u)) in L^{p′}(Ω, ω_{i}^{∗}), i ∈ I^{c},

which implies

lim_{n→∞} ∫_{Ω} ∑_{i∈I} b_{i}(x, u_{n}, ζ_{I}(∇u))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx = 0  (3.10)

and

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx = 0.  (3.11)

Combining (2.4), (2.5), (3.3), (3.8), (3.9), (3.10) and (3.11) we conclude the assertions (3.5) and (3.6).

Second step. For to prove of the relation (3.4) it suffices to show the following assertions:

(i) For every v ∈ X,

lim_{n→∞} ∫_{Ω} c_{0}(x, u_{n}, ζ_{I}(∇u_{n}))v dx = ∫_{Ω} c_{0}(x, u, ζ_{I}(∇u))v dx.  (3.12)

(ii) For every v ∈ X,

lim_{n→∞} ∫_{Ω} ∑_{i∈I} b_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂v}{∂x_{i}} dx = ∫_{Ω} ∑_{i∈I} b_{i}(x, u, ζ_{I}(∇u))\frac{∂v}{∂x_{i}} dx. (3.13)

(iii) For every v ∈ X,

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} v dx = ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u, ζ_{I}(∇u))\frac{∂u}{∂x_{i}} v dx.

(3.14)

(iv) For every v ∈ X,

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂v}{∂x_{i}} dx = ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u, ζ_{I^{c}}(∇u))\frac{∂v}{∂x_{i}} dx.

(3.15)

Proof of (i)and (ii). Invoking Landes [6, Lemma 6], we obtain from (3.5) and the strict monotonicity (2.4) that,

\frac{∂u_{n}}{∂x_{i}} → \frac{∂u}{∂x_{i}} a.e. in Ω for each i ∈ I,  (3.16)

which gives

c_{0}(x, u_{n}, ζ_{I}(∇u_{n})) → c_{0}(x, u, ζ_{I}(∇u)) a.e. in Ω,
b_{i}(x, u_{n}, ζ_{I}(∇u_{n})) → b_{i}(x, u, ζ_{I}(∇u)) a.e. in Ω ∀i ∈ I.

The growth conditions (H2’) imply that, the sequences {c_{0}(x, u_{n}, ζ_{I}(∇u_{n}))} (resp. {b_{i}(x, u_{n}, ζ_{I}(∇u_{n})) i ∈ I}) remains bounded in L^{q′}(Ω, ω̄^{1−q′}) (resp. L^{p′}(Ω, ω_{i}^{∗})). Hence by Lemma 2.5 we conclude (i) and (ii). Proof of (iii). Similarly, by (3.7) and (3.16) we can write,

c_{i}(x, u_{n}, ζ_{I}(∇u_{n})) → c_{i}(x, u, ζ_{I}(∇u)) a.e. in Ω for all i ∈ I^{c}.

And by the growth conditions (H2’) also {c_{i}(x, u_{n}, ζ_{I}(∇u_{n})), i ∈ I^{c}} is bounded in L^{r}(Ω, ω_{i}^{− \frac{r}{p}} ω̄^{− \frac{r}{q}}), then in virtue of Lemma 2.3, we have

c_{i}(x, u_{n}, ζ_{I}(∇u_{n})) → c_{i}(x, u, ζ_{I}(∇u)) in L^{q_{1}}(Ω, ωi^{\frac{−q_{1}}{p}} ω̄^{\frac{−q_{1}}{q}}) ∀i ∈ I^{c}.

Let s > 1 such that \frac{1}{s} = \frac{1}{p} + \frac{1}{q}. Since \frac{1}{s′} + \frac{1}{s} = 1 > \frac{1}{s} + \frac{1}{q_{1}} i.e. s′ < q_{1}, we have (as in the proof of Lemma 2.7),

∫_{Ω} |v|^{s′}ω_{i}^{−s′/p}ω̄^{−s′/q} dx ≤ ( ∫_{Ω} |v|^{q_{1}}ω_{i}^{−q_{1}/p}ω̄^{\frac{−q_{1}}{q}} dx)^{s′/q_{1}}(meas(Ω))^{1− \frac{s′}{q_{1}}}

for all v ∈ L^{q_{1}}(Ω, ω_{i}^{−q_{1}/p}ω̄^{−q_{1}/q}). Then

L^{q_{1}}(Ω, ωi^{\frac{−q_{1}}{p}} ω̄^{\frac{−q_{1}}{q}}) ↪ L^{s′}(Ω, ωi^{\frac{−s′}{p}} ω̄^{\frac{−s′}{q}}),

which implies

c_{i}(x, u_{n}, ζ_{I}(∇u_{n})) → c_{i}(x, u, ζ_{I}(∇u)) in L^{s′}(Ω, ω_{i}^{−s′/p}ω̄^{−s′/q}) ∀i ∈ I^{c}.

On the other hand, from Lemma 2.6 we obtain,

\frac{∂u_{n}}{∂x_{i}} v ⇀ \frac{∂u}{∂x_{i}} v in L^{s}(Ω, ω_{i}^{s/p}ω̄^{s/q}),

for any v ∈ L^{q}(Ω, ω̄) and so for any v ∈ X,

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} v dx = ∫_{Ω} ∑_{i∈I^{c}} c_{i}(x, u, ζ_{I}(∇u))\frac{∂u}{∂x_{i}} v dx

for any v ∈ X. Proof of (iv). As before, the growth conditions (H2’) implies that, the sequence {b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n})) i ∈ I^{c}} is bounded in L^{p′}(Ω, ω_{i}^{∗}). Next, we show that,

∫_{Ω} ∑_{i∈I^{c}}\{b_{i}(x, u, ζ_{I^{c}}(v)) − h_{i}\}(v_{i} − \frac{∂u}{∂x_{i}}) dx ≥ 0 for all v = (v_{i}) ∈ ∏_{i=1}^{N} L^{p}(Ω, ω_{i}),

(3.17)

here h_{i} stands for the weak limit of {b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n})), i ∈ I^{c}} in L^{p′}(Ω, ω_{i}^{1−p′}). Indeed by (3.6) we have

lim sup_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} dx ≤ ∫_{Ω} ∑_{i∈I^{c}} h_{i} \frac{∂u}{∂x_{i}} dx  (3.18)

and from (2.5), we obtain for any v = (v_{i}) ∈ ∏_{i=1}^{N} L^{p}(Ω, ω_{i}),

∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} dx

≥ ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))v_{i} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(v))(\frac{∂u_{n}}{∂x_{i}} − v_{i}) dx.

Letting n → ∞ we conclude by (3.18) that,

∫_{Ω} ∑_{i∈I^{c}} h_{i} \frac{∂u}{∂x_{i}} dx ≥ ∫_{Ω} ∑_{i∈I^{c}} h_{i}v_{i} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u, ζ_{I^{c}}(v))(\frac{∂u}{∂x_{i}} − v_{i}) dx

and hence (3.17) follows. Choosing v = ∇u + tw̃ with t > 0, w̃ = (w̃_{i}) ∈ ∏_{i=1}^{N} L^{p}(Ω, ω_{i}) and letting t → 0 we obtain,

h_{i} = b_{i}(x, u, ζ_{I^{c}}(∇u)) a.e. in Ω,

which gives,

lim_{n→∞} ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂v}{∂x_{i}} dx = ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u, ζ_{I^{c}}(∇u))\frac{∂v}{∂x_{i}} dx

for all v ∈ X. □

b) We shall prove that

lim inf_{n→∞}⟨T u_{n}, u_{n}⟩ ≥ ⟨T u, u⟩  (3.19)

by (2.4) and (2.5) we have

∫_{Ω} ∑_{i∈I}b_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂u_{n}}{∂x_{i}} dx

≥ ∫_{Ω} ∑_{i∈I} b_{i}(x, u_{n}, ζ_{I}(∇u_{n}))\frac{∂u}{∂x_{i}} dx + ∫_{Ω} ∑_{i∈I} b_{i}(x, u_{n}, ζ_{I}(∇u))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx

+ ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u_{n}))\frac{∂u}{∂x_{i}} dx + ∫_{Ω} ∑_{i∈I^{c}} b_{i}(x, u_{n}, ζ_{I^{c}}(∇u))(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx,

then letting n → ∞, and using (3.8) and (3.9), we conclude (3.19).

Example Many ideas in this example have adapted from the corresponding examples 1-2 in [2]. We shall suppose that the weight functions satisfy: ω_{i_{0}}(x) ≡ 0 for some i_{0} ∈ I^{c}, and ω_{i}(x) = ω(x), x ∈ Ω, for all i ∈ I ∪ I^{c} and i ≠ i_{0} with ω(x) > 0 a.e. in Ω. Then, we can consider the Hardy inequality in the form

( ∫_{Ω} |u(x)|^{q}ω̄(x) dx)^{1/q} ≤ c( ∑_{i≠i_{0}} ∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}ω)^{1/p}  (3.20)

for every u ∈ X with a constant c > 0 independent of u and for some q ≥ p′. Let us consider the Carathéodory functions:

b_{i}(x, η, ζ_{I}) = ω|ζ_{i}|^{p−1} sgn ζ_{i} + ω_{0}A_{0}(η) for i ∈ I

b_{i}(x, η, ζ_{I^{c}}) = ω|ζ_{i}|^{p−1} sgn ζ_{i} + ω_{0}A_{0}(η) for i ∈ I^{c} and i ≠ i_{0}

b_{i_{0}}(x, η, ζ_{I^{c}}) = ω_{0}A_{0}(η)

c_{0}(x, η, ζ_{I}) = ∑_{j∈I} ω^{1/q′}ω̄^{1/q}|ζ_{j}|^{\frac{p}{q′}} + ω_{0}B_{0}(η)

c_{i}(x, η, ζ_{I}) = ∑_{j∈I} ω^{1/p+1/r}ω̄^{1/q}|ζ_{j}|^{p/r} + ω_{0}B_{1}(η) for i ∈ I^{c},

(3.21)

with 1/p + 1/r + 1/q < 1. The above functions define by (3.21) satisfies the growth conditions (H2’) if we suppose that

\begin{matrix}|ω_{0}A_{0}(η)| ≤ β_{1}ω^{1/p}ω̄^{1/p′}|η|^{q/p′} \\ |ω_{0}B_{0}(η)| ≤ β_{2}ω̄|η|^{q/q′} \\ |ω_{0}B_{1}(η)| ≤ β_{3}ω^{1/p}ω̄^{1/q+1/r}|η|^{q/r}, \end{matrix}  (3.22)

with β_{j} j = 1, 2, 3 are some positive constants. In particular, let us use the special weight functions ω_{0}, ω, ω̄ expressed in terms of the distance to the boundary ∂Ω: denote d(x) = dist(x, ∂Ω) and set

ω(x) = d^{λ}(x), ω_{0}(x) = d^{λ_{0}}(x), ω̄(x) = d^{μ}(x).

In this case the condition (3.22) writes as

\begin{matrix}|A_{0}(η)| ≤ β_{1}d^{λ/p+μ/p′−λ_{0}}|η|^{q/p} \\ |B_{0}(η)| ≤ β_{2}d^{μ−λ_{0}}|η|^{q/q′} \\ |B_{1}(η)| ≤ β_{3}d^{λ/p+μ/q+μ/r−λ_{0}}|η|^{q/r}, \end{matrix}  (3.23)

and the Hardy inequality reads

( ∫_{Ω} |u(x)|^{q} d^{μ}(x) dx)^{1/q} ≤ c( ∑_{i≠i_{0}} ∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p} d^{λ}(x) dx)^{1/p},  (3.24)

and the corresponding imbedding (1.7) is compact for 1 ≤ p ≤ q < ∞ (resp. 1 ≤ q < p < ∞,), if and only if λ ≠ p−1, \frac{N}{q} − \frac{N}{p} +1 ≥ 0, \frac{μ}{q} − \frac{λ}{p} + \frac{N}{q} − \frac{N}{p} +1 > 0, (resp. λ ∈ ℝ, \frac{μ}{q} − \frac{λ}{p} + \frac{1}{q} − \frac{1}{p} > 0) (see [8]). Moreover, the two monotonicity conditions (2.4) and (2.5) are satisfied:

∑_{i∈I} (b_{i}(x, η, ζ_{I}) − b_{i}(x, η, ζ̄_{I})) (ζ_{i} − ζ̄_{i})

= ω(x) ∑_{i∈I} (|ζ_{i}|^{p−1} sgn ζ_{i} − |ζ̄_{i}|^{p−1} sgn ζ̄_{i}) (ζ_{i} − ζ̄_{i}) > 0

for almost all x ∈ Ω and for all ζ, ζ̄ ∈ ℝ^{N} with ζ_{I} ≠ ζ̄_{I}, since ω > 0 a.e. in Ω; and

∑_{i∈I^{c}} (b_{i}(x, η, ζ_{I^{c}}) − b_{i}(x, η, ζ̄_{I^{c}})) (ζ_{i} − ζ̄_{i})

= ω(x) ∑_{i≠i_{0}^{i∈I^{c}}} (|ζ_{i}|^{p−1}sgnζ_{i} − |ζ̄_{i}|^{p−1}sgnζ̄_{i}) (ζ_{i} − ζ̄_{i}) ≥ 0

for almost all x ∈ Ω and for all ζ, ζ̄ ∈ ℝ^{N}. This last inequality can not be strict, since for ζ_{I^{c}} ≠ ζ̄_{I^{c}} with ζ_{i_{0}} ≠ ζ̄_{i_{0}} but ζ_{i} = ζ̄_{i} for all i ∈ I^{c} and i ≠ i_{0}, the corresponding expression is zero. Finally, the hypotheses of theorem 3.1 are verify, then the mapping T defined as (3.1) corresponding to (3.21) is pseudomonotone.

4 Specific case

Let Ω be a bounded open subset of ℝ^{N} satisfying the cone condition. In this section we assume in addition that the collection of weight functions ω = {ω_{i}(x) i = 0, …, N} satisfy ω_{0}(x) = 1 and the integrability condition: There exists ν ∈]\frac{N}{p} , ∞[∩[\frac{1}{p−1}, ∞[ such that

ω_{i}^{−ν} ∈ L^{1}(Ω) ∀i = 1, …, N.  (4.1)

Note that (4.1) is stronger than (2.2).

Remark 4.1 ([3]) 1. Assumptions (2.1) and (4.1) imply that,

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

is a norm defined on W_{0}^{1,p}(Ω, ω) and it’s equivalent to (2.3), and that

W_{0}^{1,p}(Ω, ω) ↪↪ L^{q}(Ω)  (4.2)

for all 1 ≤ q < p_{1}^{∗} if pν < N(ν + 1) and q ≥ 1 is arbitrary if pν ≥ N(ν + 1) where p_{1} = \frac{pν}{ν+1} and p_{1}^{∗} = \frac{Np_{1}}{N−p_{1}} = \frac{Npν}{N(ν+1)−pν} is the Sobolev conjugate of p_{1}.

2. Hypotheses (H1) holds for all q such that 1 < q < p_{1}^{∗} and ω̄ ≡ 1.

In the sequel, we replace (4.1) by the hypothesis

(H̃1) If \frac{2N}{N+1} < p < N there exists ν ∈]\frac{N}{p} , ∞[∩]\frac{1}{(p−1)− \frac{p}{p^{∗}}} , ∞[ such that ω_{i}^{−ν} ∈

L^{1}(Ω), for all i = 1, …, N. If p = N there exists ν ∈]1, ∞[ such that ω_{i}^{−ν} ∈ L^{1}(Ω) for all i = 1, …, N. If p > N there exist ν ∈]\frac{N}{p−N} , ∞[∩[\frac{1}{(p−1)}, ∞[

such that ω_{i}^{−ν} ∈ L^{1}(Ω) for all i = 1, …, N.

Remark 4.2 1. Hypothesis (H̃1) guarantees the existence of r satisfying \frac{1}{r} + \frac{1}{p} + \frac{1}{p_{1}^{∗}} < 1, where p_{1}^{∗} is the Sobolev conjugate of p_{1} in the case \frac{2N}{N+1} < p ≤ N and where p_{1}^{∗} = ∞ in the case p > N (since p_{1} > N due to ν > \frac{N}{p−N}).

2. If 1 < p ≤ \frac{2N}{N+1} we can’t find a real r > 1 such that \frac{1}{r} + \frac{1}{p} + \frac{1}{p_{1}^{∗}} < 1, since \frac{1}{p} + \frac{1}{p_{1}^{∗}} > \frac{1}{p} + \frac{1}{p^{∗}} ≥ 1.

3. Note that (H̃1) is stronger than (4.1), then the compact imbedding (4.2) is satisfied whenever (H̃1) is assumed.

Theorem 4.3 Let Ω be a bounded open subset of ℝ^{N}. And assume that (2.1), (H̃1), (H2’), (2.4) and (2.5) are satisfied. Then the operator T defined in (3.1) is pseudo-monotone in X = W_{0}^{1,p}(Ω, ω). Moreover, assume the degenerate ellipticity condition

∑_{i=0}^{N} a_{i}(x, ξ)ξ_{i} ≥ c_{0} ∑_{i=1}^{N} ω_{i}(x)|ξ_{i}|^{p}

for a.e. x ∈ Ω, some c_{0} > 0 and all ξ ∈ ℝ^{N+1}. Then for any f ∈ X^{∗} the Dirichlet associated problem

⟨T u, v⟩ = ⟨f, v⟩ for all v ∈ X,

has at least one solution u ∈ X.

References

[1] E. Azroul, Sur certains problèmes elliptiques (non linéaires dégénérés ou singuliers) et calcul des variations, Thèse de Doctorat d’Etat, Faculté des Sciences Dhar-Mahraz, Fès, Maroc, janvier 2000.

[2] P. Drabek, A. Kufner and V. Mustonen, Pseudo-monotonicity and degenerated or singular elliptic operators, Bull. Austral. Math. Soc. Vol. 58 (1998), 213-221.

[3] P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations, singular and degenerate cases, University of West Bohemia, (1996).

[4] P. Drabek, A. Kufner and F. Nicolosi, On the solvability of degenerated quasilinear elliptic equations of higher order, Journal of differential equations, 109 (1994), 325-347.

[5] J.P. Gossez and V. Mustonen, Pseudo-monotoncity and the Leray-Lions condition, Differential Integral Equations 6 (1993), 37-45.

[6] R. Landes, On Galerkin’s method in the existence theory of quasilinear elliptic equations, J. Functional Analysis 39 (1980), 123-148.

[7] J. Leray and J.L. Lions, Quelques résultats de Vis̃ik sur des problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bul. Soc. Math. France 93 (1965), 97-107.

[8] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series 219(Longman Scientific and Technical, Harlow, 1990).

Youssef Akdim (e-mail y.akdim1@caramail.com) Elhoussine Azroul (e-mail elazroul@caramail.com) Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz B.P 1796 Atlas Fès, Maroc