2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2005, pp. 53–71. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

HIGHER ORDER NONLINEAR DEGENERATE ELLIPTIC PROBLEMS WITH WEAK MONOTONICITY

YOUSSEF AKDIM, ELHOUSSINE AZROUL, MOHAMED RHOUDAF

Abstract. We prove the existence of solutions for nonlinear degenerate elliptic boundary-value problems of higher order. Solutions are obtained using pseudo-monotonicity theory in a suitable weighted Sobolev space.

1. Introduction and statement of results

Let Ω be an open subset of ℝ^{N} with finite measure and let m ≥ 1 be an integer and p > 1 be a real number. We will consider the degenerated partial differential operators

Au(x) = A^{m}u(x) + A^{m−1}u(x),  (1.1)

on Ω where

A^{m}u(x) = ∑_{|α|=m}(−1)^{|α|}D^{α}A_{α}(x, u, …, ∇^{m}u)  (1.2)

is the top order part of the degenerated quasilinear operator A. and where

A^{m−1}u(x) = ∑_{|α|≤m−1}(−1)^{|α|}D^{α}A_{α}(x, u, …, ∇^{m}u)  (1.3)

is the lower order part of A. The coefficients {A_{α}(x, η, ζ), |α| ≤ m} are real valued functions defined on Ω × ℝ^{N_{m−1}} × ℝ^{N_{m}} (with N_{m−1} = card{α ∈ ℕ^{N}, |α| ≤ m − 1} and N_{m} = card{α ∈ ℕ^{N}, |α| = m}) which satisfy suitable regularity and growth assumptions (see section 2). Let V be a subspace such that

W_{0}^{m,p}(Ω, w) ⊆ V ⊆ W^{m,p}(Ω, w),  (1.4)

where W^{m,p}(Ω, w) and W_{0}^{m,p}(Ω, w) are weighted Sobolev spaces associated to a vector of weights w = {w_{α} ≡ w_{α}(x), |α| ≤ m} on Ω satisfying some integrability conditions (see sections 2). We deal with the case where A^{m−1} is affine with respect to the top order derivatives of u, i.e, A^{m−1}u is of the form,

2000 Mathematics Subject Classification. 35J40, 35J70.

Key words and phrases. Weighted Sobolev spaces; pseudo-monotonicity;

nonlinear degenerate elliptic operators; boundary value problems.

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

Published September 20, 2006.

\begin{aligned}A^{m−1}u(x) & = ∑_{|α|≤m−1}(−1)^{|α|}D^{α}L_{α}(x, u, …, ∇^{m−1}u) \\ & + ∑_{|α|≤m−1} ∑_{|β|=m}(−1)^{|α|}D^{α}C_{αβ}(x, u, …, ∇^{m−1}u)D^{β}u \end{aligned}  (1.5)

where L_{α}(x, η) and C_{αβ}(x, η) are some real valued functions defined on Ω×ℝ^{N_{m−1}}. We will assume the following hypotheses:

(H1) For every u ∈ V and any multi-index |β| ≤ m − 1, there exists a parameter q(β) ≥ 1 and a weight function σ_{β} = σ_{β}(x) such that,

D^{β}u ∈ L^{q(β)}(Ω, σ_{β}),
∥D^{β}u(x)∥_{q(β),σ_{β}} ≤ c̃_{β}∥u∥_{m,p,w}

with some constant c̃_{β} > 0 independent of u and moreover, the compact imbedding,

V ↪↪ H^{m−1,q}(Ω, σ)  (1.6)

holds, where H^{m−1,q}(Ω, σ) = {u, D^{β}u ∈ L^{q(β)}(Ω, σ_{β}) for all |β| ≤ m − 1}. (H2) The functions {A_{α}, |α| = m}, {L_{α}, |α| ≤ m − 1} and {C_{αβ}, |α| ≤ m −

1 and |β| = m} are Carathéodory functions and there exists functions g_{α} ∈ L^{p′}(Ω) for all |α| = m, g̃_{α} ∈ L^{q′(α)}(Ω) for all |α| ≤ m−1, and γ_{αβ} ∈ L^{r_{α}}(Ω) for all |α| ≤ m − 1 and all |β| = m such that

(i) for all |α| = m,

|A_{α}(x, η, ζ)|

≤ c_{α}w_{α}^{1/p}(x)(g_{α}(x) + c̃_{α} ∑_{|β|=m} wβ^{\frac{1}{p′}} |ζ_{β}|^{p−1} + c̃_{α} ∑_{|β|≤m−1} σβ^{\frac{1}{p′}} |η_{β}|^{\frac{q(β)}{p′}} )

(ii) for all |α| ≤ m − 1,

|L_{α}(x, η)| ≤ c_{α}σα^{\frac{1}{q(α)}} (g̃_{α}(x) + c̃_{α} ∑_{|β|≤m−1} σβ^{\frac{1}{q′(α)}} |η_{β}|^{\frac{q(β)}{q′(α)}} )

(iii) for all |α| ≤ m − 1 and all |β| = m,

|C_{αβ}(x, η)|

≤ c_{αβ}σα^{\frac{1}{q(α)}} (x)w_{β}^{1/p}(x)(γ_{αβ}(x) + c̃_{αβ} ∑_{|λ|≤m−1} σλ^{\frac{1}{rα}} (x)|η_{λ}|^{\frac{q(λ)}{rα}} )

for a.e. x ∈ Ω, some positive constants c_{α}, c̃_{α} and c̃_{αβ}, every (η, ζ) ∈ ℝ^{N_{m−1}} × ℝ^{N_{m}} = ℝ^{d} and some exponent r_{α} such that

\frac{1}{r_{α}} + \frac{1}{p} + \frac{1}{q(α)} < 1 for all |α| ≤ m − 1.  (1.7)

For the existence of r_{α} see Remark 2.1 below. Let us consider the degenerated boundary value problem (DBVP) associated to

the equation,

Au = f ∈ V^{∗},  (1.8)

where V^{∗} is the dual space of V from (1.4). Recently, Drabeck, Kufner and Mustonen proved in [4] the existence result for Dirichlet degenerated problem of second order associated to the operator A of the form,

Au(x) = − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} a_{i}(x, u, ∇u)  (1.9)

where the Carathéodory functions a_{i}(x, η, ζ) satisfy some simple growth conditions, that is,

|a_{i}(x, η, ζ)| ≤ c_{1}w_{i}^{1/p}(x)(g(x) + w̄^{\frac{1}{p′}} (x)|η|^{\frac{q}{p′}} + ∑_{j=1}^{N} wj^{\frac{1}{p′}} |ζ|^{p−1})  (1.10)

where the exponent q and the weight function w̄(x) verify the so called Hardy-type inequality; i.e,

∫_{Ω} |u(x)|^{q}w̄(x) dx ≤ c ∑_{i=1}^{N} ∫_{Ω} |D_{i}u|^{p}w_{i}(x) dx  (1.11)

and the compact imbedding

W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, w̄).  (1.12)

The authors have proved that the mapping T associated to A from (1.9) is pseudomonotone in W_{0}^{1,p}(Ω, w), by assuming only 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.  (1.13)

Our first objective of this paper is to extend the previous result of [4] in the general class of operators A from (1.1), where the lower order part A^{m−1} is of the form (1.5) and where the growth conditions are of the most general form (H2). More precisely, we prove the following result.

Theorem 1.1. Assume that (H1), (H2) and that

∑_{|α|=m}(A_{α}(x, η, ζ) − A_{α}(x, η, ζ̄))(ζ_{α} − ζ̄_{α}) ≥ 0  (1.14)

for a.e. x ∈ Ω, all η ∈ ℝ^{N_{m−1}} and all (ζ, ζ̄) ∈ ℝ^{N_{m}} × ℝ^{N_{m}} hold. Then the mapping T associated to the operator A from (1.1) and (1.5) is pseudo-monotone in V.

If in addition the degeneracy satisfies

∑_{|α|≤m} A_{α}(x, ξ)ξ_{α} ≥ c ∑_{|α|≤m} w_{α}(x)|ξ_{α}|^{p},  (1.15)

for a.e. x ∈ Ω, some c > 0 and all ξ ∈ ℝ^{N_{m−1}} × ℝ^{N_{m}}, then the DBVP associated to the equation (1.8) has at least one solution u ∈ V.

Remark 1.2. The statement of Theorem 1.1, is obviously contained in Theorem 3.1 below (it suffices to take J = ∅) where some general situation is considered.

On the other hand, Drabeck, Kufner and Nikolosi in [6] have studied the existence result for the DBVP from the equation (1.8) with A of the form (1.1) and with more general hypotheses (H1’), (H2’), (H3) (in section 2) and with the so-called Leray-Lions condition

∑_{|α|=m}(A_{α}(x, η, ζ) − A_{α}(x, η, ζ̄))(ζ_{α} − ζ̄_{α}) > 0.  (1.16)

The authors have assumed in addition to the previous hypotheses the compact imbedding,

V ↪↪ W^{m−1,p}(Ω, w)  (1.17)

and then, have proved that the mapping T satisfies the condition α(V) (see definition 2.5) and hence used the degree theory of general mappings of monotone type. The hypotheses (1.17) play an important role in the work [6], because it is related to some strong converges appearing in the α(V) condition.

Our second objective of this paper, is to prove the same result as in [6] without assuming the compact imbedding (1.17). This is possible by proving the pseudomonotonicity of the mapping T induced by the operator A from (1.1). More precisely, we have the following result.

Theorem 1.3. Assume that (H1’), (H2’), (H3) and (1.16). Then the mapping T associated to operator A from (1.1) is pseudo-monotone in V. If in addition the degeneracy (1.15) is satisfied, then, the DBVP from the equation (1.8) has at least one solution u ∈ V.

Remark 1.4. Theorem 1.3 is obviously a consequence of the more general Theorem 3.1 it suffices to take J^{c} = ∅).

Hence, this paper can be seen as an extension of the preceding papers [4, 5, 6] (where the second order case without lower order part is considered in the first paper. The degree theory is used in the two last papers) and as a continuation of the papers [2] and [3] (where the second order case with lower order part not equal to zero, is studied in the first paper and where the higher order case with A^{m−1} ≡ 0 or with A^{m−1} ≢ 0 but under restrictions w_{α} ≡ 1 for all |α| ≤ m − 1, is considered in the last paper). Finally, note that our approach (based on the theory of pseudo-monotone mappings) can be applied in the case of non reflexive Banach spaces. For example in the general settings of weighted Orlicz-Sobolev spaces (see [1] for this direction). This work is divided into five sections. We start with the introduction of a basic assumptions in section 2. Next, we give our main general result in section 3, which is proved in section 4. Finally, we study in section 5, some particular case (where our basic assumption are satisfied). In our work, we shall adopt many ideas introduced in [7], but the results are generalized and improved.

2. Preliminaries and basic assumptions

2.1. Weighted Sobolev spaces. Let Ω be an open subset of ℝ^{N} with finite measure. In the sequel we suppose that the vector of weights, on Ω, w = {w_{α}(x) : |α| ≤ m} satisfies the integrability conditions:

w_{α} ∈ L_{loc}^{1}(Ω),
w_{α}^{− \frac{1}{p−1}} ∈ L_{loc}^{1}(Ω)

for any |α| ≤ m. We denote by W^{m,p}(Ω, w) (1 < p < ∞) the space of all real-valued functions u such that the derivatives in the sense of distributions fulfil

D^{α}u ∈ L^{p}(Ω, w_{α}) for all |α| ≤ m.

The weighted Sobolev space W^{m,p}(Ω, w) is normed when equipped by the norm

∥u∥_{m,p,w} = ( ∑_{|α|≤m} ∫_{Ω} |D^{α}u|^{p}w_{α} dx)^{1/p}.  (2.1)

The space W_{0}^{m,p}(Ω, w) is defined as the closure of the set C_{0}^{∞}(Ω) with respect to the norm (2.1). Note that the conditions (2.1) and (2.1) imply that the spaces W^{m,p}(Ω, w) and W_{0}^{m,p}(Ω, w) are reasonably defined and are reflexive Banach spaces (for more details see [6]). We recall that the dual space of W_{0}^{m,p}(Ω, w) is equivalent to W^{−m,p′}(Ω, w^{∗}) where w^{∗} = \{w_{α}^{∗} = w_{α}^{1−p′} : |α| ≤ m\}, with p′ = \frac{p}{p−1} is the Hölder’s conjugate of p.

2.2. Basic assumptions. Let J be a subset of {α ∈ ℕ^{N}, |α| = m} and J^{c} its complement. We will suppose that the coefficients A_{α} of the operator A from (1.1) are such that

\begin{matrix}A_{α}(x, η, ζ) = B_{α}(x, η, ζ_{J}) ∀α ∈ J, \\ A_{α}(x, η, ζ) = B_{α}(x, η, ζ_{J^{c}}) ∀α ∈ J^{c}, \\ A_{α}(x, η, ζ) = L_{α}(x, η, ζ_{J}) + ∑_{β∈J^{c}}C_{αβ}(x, η, ζ_{J})ζ_{β} ∀|α| ≤ m − 1, \end{matrix}  (2.2)

for a.e. x ∈ Ω and where {B_{α}, |α| = m}, {L_{α}, |α| ≤ m − 1} and {C_{αβ}, |α| ≤ m − 1 and β ∈ J^{c}} are some Carathéodory functions and where ζ_{I} denoted ζ_{I} = {ζ_{α}, α ∈ I}. We denote by N_{I} = card{α ∈ ℕ^{N}, α ∈ I}. Let us introduce the following modified versions of (1.16) and (1.14),

∑_{α∈J}(B_{α}(x, η, ζ_{J}) − B_{α}(x, η, ζ̄_{J}))(ζ_{α} − ζ̄_{α}) > 0,  (2.3)

for a.e x ∈ Ω, all η ∈ ℝ^{N_{m−1}} and all ζ_{J} ≠ ζ̄_{J} ∈ ℝ^{N_{J}} and

∑_{α∈J^{c}}(B_{α}(x, η, ζ_{J^{c}}) − B_{α}(x, η, ζ̄_{J^{c}}))(ζ_{α} − ζ̄_{α}) ≥ 0,  (2.4)

for a.e x ∈ Ω and all (η, ζ_{J^{c}}, ζ̄_{J^{c}}) ∈ ℝ^{N_{m−1}} × ℝ^{N_{J^{c}}} × ℝ^{N_{J^{c}}}.

Let us denote by m_{1} = m − \frac{N}{p} and suppose that m_{1} > 0 i.e, mp > N. We denote by C(Ω, w_{α}) the weighted spaces of continuous functions, more precisely C(Ω, w_{α}) = {u = u(x) continuous on Ω, ∥u∥_{C(Ω,w_{α})} = sup_{x∈Ω} |u(x)w_{α}(x)| < ∞}. (H1’) Let u ∈ V.

(i) For |β| < m_{1}, there is a weight function σ_{β} = σ_{β}(x) such that, D^{β}u ∈ C(Ω, σ_{β}) and moreover,

sup_{x∈Ω} |D^{β}u(x)σ_{β}(x)| ≤ c̃_{β}∥u∥_{m,p,w}  (2.5)

with some constant c̃_{β} > 0 independent of u. When we denote by k(x, u(x)) the expression ∑_{|β|<m_{1}} |σ_{β}(x)D^{β}u(x)|, then, in view of (2.5),

|k(x, u(x))| ≤ c∥u∥_{m,p,w} for all u ∈ V.  (2.6)

(ii) For m_{1} ≤ |β| ≤ m − 1, there is a parameter q(β) ≥ 1 and a weight function σ_{β} = σ_{β}(x) such that D^{β}u ∈ L^{q(β)}(Ω, σ_{β}) and moreover,

∥D^{β}u(x)∥_{q(β),σ_{β}} ≤ c̃_{β}∥u∥_{m,p,w}  (2.7)

for some constant c̃_{β} > 0 independent of u.

(iii) The imbedding V ↪↪ H^{m−1,q}(Ω, σ) compact, where H^{m−1,q}(Ω, σ) = {u, D^{β}u ∈ X_{β}, for all |β| ≤ m − 1} with X_{β} = L^{q(β)}(Ω, σ_{β}) for m_{1} ≤ |β| ≤ m − 1 and X_{β} = C(Ω, σ_{β}) for |β| < m_{1}.

(H2’) There exists functions g_{α} ∈ L^{p′}(Ω) for |α| = m, g̃_{α} ∈ L^{q′(α)}(Ω) for m_{1} ≤ |α| ≤ m − 1, ĝ_{α} ∈ L^{1}(Ω) for |α| < m_{1}, γ_{αβ} ∈ L^{r_{α}}(Ω) for all |α| ≤ m − 1 and β ∈ J^{c} and some positive constants c̃_{α} and c̃_{αβ}, moreover there exists a positive continuous, non decreasing function G(t), t ≥ 0, such that the following estimates hold:

(i) For α ∈ J,

|B_{α}(x, η, ζ_{J})|

≤ G(k(x, κ))w_{α}^{1/p}(g_{α}(x) + c̃_{α} ∑_{β∈J} wβ^{\frac{1}{p′}} |ζ_{β}|^{p−1} + c̃_{α} ∑_{m_{1}≤|β|≤m−1} σβ^{\frac{1}{p′}} |η_{β}|^{\frac{q(β)}{p′}} )

(ii) for α ∈ J^{c},

|B_{α}(x, η, ζ_{J^{c}})|

≤ G(k(x, κ))w_{α}^{1/p}(g_{α}(x) + c̃_{α} ∑_{β∈J^{c}} wβ^{\frac{1}{p′}} |ζ_{β}|^{p−1} + c̃_{α} ∑_{m_{1}≤|β|≤m−1} σβ^{\frac{1}{p′}} |η_{β}|^{\frac{q(β)}{p′}} )

(iii) for m_{1} ≤ |α| ≤ m − 1,

|L_{α}(x, η, ζ_{J})|

≤ G(k(x, κ))σα^{\frac{1}{q(α)}} (g̃_{α}(x) + c̃_{α} ∑_{β∈J} wβ^{\frac{1}{q′(α)}} |ζ_{β}|^{\frac{p}{q′(α)}} + c̃_{α} ∑_{m_{1}≤|β|≤m−1} σβ^{\frac{1}{q′(α)}} |η_{β}|^{\frac{q(β)}{q′(α)}} )

(iv) for |α| < m_{1},

|L_{α}(x, η, ζ_{J})|

≤ G(k(x, κ))σ_{α}(ĝ_{α}(x) + c̃_{α} ∑_{β∈J} w_{β}|ζ_{β}|^{p} + c̃_{α} ∑_{m_{1}≤|β|≤m−1} σ_{β}|η_{β}|^{q(β)})

(v) for m_{1} ≤ |α| ≤ m − 1 and β ∈ J^{c},

|C_{αβ}(x, η, ζ_{J})|

≤ G(k(x, κ))σα^{\frac{1}{q(α)}} w_{β}^{1/p}(γ_{αβ}(x) + c̃_{αβ} ∑_{λ∈J} wλ^{\frac{1}{rα}} |ζ_{λ}|^{\frac{p}{rα}} + c̃_{αβ} ∑_{m_{1}≤|λ|≤m−1}σλ^{\frac{1}{rα}} |η_{λ}|^{\frac{q(λ)}{rα}} )

(vi) for |α| < m_{1} and β ∈ J^{c},

|C_{αβ}(x, η, ζ_{J})|

≤ G(k(x, κ))σ_{α}w_{β}^{1/p}(γ_{αβ}(x) + c̃_{αβ} ∑_{λ∈J} wλ^{\frac{1}{rα}} |ζ_{λ}|^{\frac{p}{rα}} + c̃_{αβ} ∑_{m_{1}≤|λ|≤m−1} σλ^{\frac{1}{rα}} |η_{λ}|^{\frac{q(λ)}{rα}} )

for a.e. x ∈ Ω, every η ∈ ℝ^{N_{m−1}} and every ζ_{I} ∈ ℝ^{N_{I}} where κ = {η_{β}, |β| < m_{1}} and

\frac{1}{r_{α}} + \frac{1}{p} + \frac{1}{q(α)} < 1

for any m_{1} ≤ |α| ≤ m − 1 and any β ∈ J^{c} and with

\frac{1}{r_{α}} + \frac{1}{p} < 1

for any |α| < m_{1} and any β ∈ J^{c}. Note that the exponent q′(α) denotes the Hölder’s conjugate of q(α).

Remark 2.1. For all m_{1} ≤ |α| ≤ m − 1, the such r_{α} satisfying \frac{1}{r_{α}} + \frac{1}{p} + \frac{1}{q(α)} < 1 exists when q(α) > p′. And we can choose r_{α} > p′ when |α| < m_{1}.

Remark 2.2. If m_{1} ≤ 0, then the set of multi-indices ξ_{β} with |β| < m_{1} is empty. Then we set G(t) ≡ 1 and since the cases iv)and vi) in (H′_{2}) are irrelevant, we obtain the growth condition of type C [6]. Further if we do not differ between |α| = m and |α| ≤ m − 1 i.e, if we take g̃_{α} = g_{α} ∈ L^{p′}(Ω) we immediately obtain the growth conditions of type (B) [6]. Finally if we choose q(β) = p and σ_{β} = w_{β}, we obtain the growth condition of type A [6].

(H3) Let G_{1} be a continuous positive, nonincreasing function on [0, ∞), and let G_{2} be a continuous positive, nondecreasing function on [0, ∞), we will suppose that for every ξ = (κ, η, ζ) ∈ ℝ^{d} and for a.e. x ∈ Ω the ellipticity condition holds

∑_{|α|=m} A_{α}(x, κ, η, ζ)ζ_{α}

≥ G_{1}(h(x, κ)) ∑_{|β|=m} w_{β}|ζ_{β}|^{p} − G_{2}(h(x, κ)) ∑_{m_{1}≤|β|≤m−1} σ_{β}|η_{β}|^{q(β)},

where κ = {ξ_{β}, |β| < m_{1}} ∈ ℝ^{d_{1}}, η = {ξ_{β}, m_{1} ≤ |β| ≤ m − 1} ∈ ℝ^{d_{2}}, ζ = {ξ_{β}, |β| = m} ∈ ℝ^{N_{m}} and d_{1} + d_{2} = N_{m−1}.

Under these assumptions, the differential operator (1.1) generates a mapping T from V to its dual V^{∗} through the formula

\begin{aligned}⟨T u, v⟩ & = ∑_{α∈J} ∫_{Ω} B_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx \\ & + ∑_{α∈J^{c}} ∫_{Ω} B_{α}(x, η(u), ζ_{J^{c}}(∇^{m}u))D^{α}v dx \\ & + ∑_{|α|≤m−1} ∫_{Ω} L_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx \\ & + ∑_{|α|≤m−1} ∑_{β∈J^{c}} ∫_{Ω} C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))D^{β}uD^{α}v dx, \end{aligned}  (2.8)

for all u, v ∈ V and where ⟨., .⟩ denotes the duality pairing between V^{∗} and V. The mapping T is well defined and bounded, this can be easily seen by Hölder’s inequality and the following lemma.

Lemma 2.3. Let Ω be a subset of ℝ^{N} with finite measure and let f ∈ L^{p}(Ω, σ_{1}), g ∈ L^{q}(Ω, σ_{2}) 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}}) 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’s inequality we have,

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

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

Let us recall the following definitions.

Definition 2.4. A mapping T from X to its dual X^{∗}, is called pseudo-monotone, if for every 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.

Definition 2.5. Let X be a reflexive Banach space. The mapping T from X to X^{∗} is said to satisfy condition α(X) if the assumptions

u_{n} ⇀ u in X and lim sup_{n→∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0,

imply u_{n} → u in X.

Obviously, the class α(X) of operators is contained in the class of pseudomonotone operators.

3. Main general result

The aim of this section, is to prove the following result.

Theorem 3.1. Assume that (H1’), (H2’), (H3), (2.3) and (2.4) hold. Then, the mapping T defined by (2.8) is pseudo-monotone in V.

Remark 3.2. (1) When J = ∅, the previous theorem applies in particular to operators like (1.1) with A_{α}, |α| ≤ m − 1 affine with respect to ∇^{m}u. This gives from (1.14) a sufficient condition (see Theorem 1.1).

(2) When J = ∅, m = 1 and A_{0} ≡ 0, we immediately obtain [4, proposition 1].

(3) When A_{α} ≡ 0 for all |α| ≤ m − 1 and J = ∅ (resp. J^{c} = ∅) we obtain Theorem 8.1 (resp. Theorem 8.3) of [1] with some simple the growth conditions.

Remark 3.3. Since the hypothesis (H3) concerns only the terms L_{α} with |α| < m_{1} (see Remark 4.5 below), then the statement of Theorem 3.1 remains true without assuming (H3), when m_{1} ≤ 0.

Remark 3.4. If we take m_{1} ≤ 0, q(β) = p and σ_{β} = w_{β}, then X_{β} = L^{p}(Ω, w_{β}) for all |β| ≤ m − 1, hence the growth condition (H2’) is of the type A (see [6]) and the statement of Theorem 3.1 remains true without assuming (H3).

Applying the previous theorem, we obtain the following existence results, which generalize the corresponding (cf. [1, 4]) and extend the corresponding in [5, 6].

corollary 3.5. Assume the hyptheses in Theorem 3.1 and the condtion on the degeneracy (1.15). Then the DBVP from the equation (1.8) has at least one solution u ∈ V.

Remark 3.6. If the expression,

∥|u|∥_{V} = ( ∑_{|α|=m} ∫_{Ω} w_{α}(x)|D^{α}u|^{p} dx)^{1/p}

is a norm in V equivalent to the usual norm (2.1) (see section 5 where this fact is verified for V = W_{0}^{m,p}(Ω, w)), then we can replace in Corollary 3.5, the degeneracy (1.15) by the weaker condition

∑_{|α|≤m} A_{α}(x, ξ)ξ_{α} ≥ c ∑_{|α|=m} w_{α}|ξ_{α}|^{p}.  (3.1)

4. Proof of Theorem 3.1

For this goal, we need the following lemmas.

Lemma 4.1. Let (g_{n})_{n} be a sequence of L^{p}(Ω, σ̃) and let g ∈ L^{p}(Ω, σ̃) (1 < p < ∞), where σ̃ is a weight function in Ω. If g_{n} → g in measure (in particular a.e in Ω) and it is bounded in L^{p}(Ω, σ̃), then g_{n} → g in L^{q}(Ω, σ̃^{\frac{q}{p}}) for all q < p.

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

∫_{Ω} |g_{n} − g|^{q}σ̃^{\frac{q}{p}} dx = ∫_{A_{n}} |g_{n} − g|^{q}σ̃^{\frac{q}{p}} dx + ∫_{A_{n}^{c}} |g_{n} − g|^{q}σ̃^{\frac{q}{p}} dx

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

By Hölder inequality, one can see that

∫_{A_{n}^{c}} |g_{n} − g|^{q}σ̃^{\frac{q}{p}} dx ≤ ( ∫_{Ω} |g_{n} − g|^{p}σ̃ dx)^{\frac{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, 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}σ̃^{\frac{q}{p}} dx ≤ \frac{ε}{2}.

□

The following lemma is a generalization of [9, Lemma 3.2] in weighted spaces.

Lemma 4.2. Let g ∈ L^{q}(Ω, σ̃) and let g_{n} ∈ L^{q}(Ω, σ̃), with ∥g_{n}∥_{q,σ̃} ≤ c (1 < q < ∞). If g_{n}(x) → g(x) a.e. in Ω, then g_{n} ⇀ g in L^{q}(Ω, σ̃), where ⇀ denotes weak convergence.

Proof. Since g_{n}σ̃^{\frac{1}{q}} is bounded in L^{q}(Ω) and g_{n}(x)σ̃^{\frac{1}{q}} (x) → g(x)σ̃^{\frac{1}{q}} (x), a.e. in Ω, then by [9, lemma 3.2],

g_{n}σ̃^{\frac{1}{q}} ⇀ gσ̃^{\frac{1}{q}} in L^{q}(Ω).

Moreover, for all φ ∈ L^{q′}(Ω, σ̃^{1−q′}), we have φσ̃^{− \frac{1}{q}} ∈ L^{q′}(Ω). Then

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

62  Y. AKDIM, E. AZROUL, M. RHOUDAF  EJDE/CONF/14

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

lim sup_{n→∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0,  (4.1)

i.e.,

lim sup_{n→∞} { ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx

+ ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx

+ ∫_{Ω} ∑_{|α|≤m−1} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx

+ ∫_{Ω} ∑_{|α|≤m−1} ∑_{β∈J^{c}} C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u) dx} ≤ 0.

(a) We shall prove that

⟨T u_{n}, v⟩ → ⟨T u, v⟩ asn → ∞ ∀v ∈ V.  (4.2)

By (H1’)(iii), the compact imbedding implies that for a subsequence

\begin{matrix}D^{α}u_{n} → D^{α}u in X_{α} \\ D^{α}u_{n} → D^{α}u a.e. in Ω∀ |α| ≤ m − 1. \end{matrix}  (4.3)

Step (1) We shall prove that

lim_{n→∞} ∑_{|α|≤m−1} ∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) = 0.  (4.4)

(i) We show that

lim_{n→∞} ∑_{m_{1}≤|α|≤m−1} ∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx = 0.  (4.5)

Let m_{1} ≤ |α| ≤ m − 1 be fixed. Thanks to (H2’), we have ∫_{Ω} |L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u)| dx

≤ ∫_{Ω} G(k(x, u_{n}(x)))σα^{\frac{1}{q(α)}} |(D^{α}u_{n} − D^{α}u)||g̃_{α}| dx

+ c̃_{α} ∑_{β∈J} ∫_{Ω} G(k(x, u_{n}(x)))σα^{\frac{1}{q(α)}} |(D^{α}u_{n} − D^{α}u)|wβ^{\frac{1}{q′(α)}} |D^{β}u_{n}(x)|^{\frac{p}{q′(α)}} dx

+ c̃_{α} ∑_{m_{1}≤|β|≤m−1} ∫_{Ω} G(k(x, u_{n}(x)))σα^{\frac{1}{q(α)}} |(D^{α}u_{n} − D^{α}u)|σβ^{\frac{1}{q′(α)}} |D^{β}u_{n}(x)|^{\frac{q(β)}{q′(α)}} dx.

By (2.6) we have,

G(k(x, u_{n}(x)) ≤ G(c∥u_{n}∥_{m,p,w}).

Applying the Hölder’s inequality with exponents q(α) and q′(α) we obtain

∫_{Ω} |L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u)| dx

≤ G(c∥u_{n}∥_{m,p,w})∥D^{α}u_{n} − D^{α}u∥_{q(α),σ_{α}}(∥g̃_{α}∥_{q′(α)}

+ c̃_{α} ∑_{β∈J} ∥D^{β}u_{n}∥p,w_{β}^{\frac{p}{q′(α)}} + c̃_{α} ∑_{m_{1}≤|β|≤m−1} ∥D^{β}u_{n}∥q(β),σ_{β}^{\frac{q(β)}{q′(α)}}).

Thanks to (2.7), we have ∥D^{β}u_{n}∥_{q(β),σ_{β}} ≤ c̃_{β}∥u_{n}∥_{m,p,w} for all m_{1} ≤ |β| ≤ m − 1. Since ∥D^{β}u_{n}∥_{p,w_{β}} ≤ ∥u_{n}∥_{m,p,w} for all β ∈ J, we conclude that

∫_{Ω} |L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u)| dx
≤ ∥D^{α}u_{n} − D^{α}u∥_{q(α),σ_{α}}R_{α}(∥u_{n}∥_{m,p,w})

with,

R_{α}(t) = G(c_{1}t)(∥g̃_{α}∥_{q′(α)} + c_{2}t^{\frac{p}{q′(α)}} + c_{3} ∑_{m_{1}≤|β|≤m−1} t^{\frac{q(β)}{q′(α)}} )

which is a positive continuous function, hence R_{α}(∥u_{n}∥_{m,p,w}) is bounded. Moreover, by (4.3) we have,

∥D^{α}u_{n} − D^{α}u∥_{q(α),σ_{α}} → 0 as n → ∞.

then

∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx → 0,

which yields (4.5). (ii) We show that

lim_{n→∞} ∑_{|α|<m_{1}} ∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx = 0.  (4.6)

Let |α| < m_{1} be fixed. Similarly by virtue of (H2’),

∫_{Ω} |L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u)| dx

≤ G(c∥u_{n}∥_{m,p,w}) sup_{x∈Ω}(|(D^{α}u_{n} − D^{α}u)σ_{α}|)(∥ĝ_{α}∥_{1} + c̃_{α} ∑_{β∈J} ∥D^{β}u_{n}∥_{p,w_{β}}^{p}

+ c̃_{α} ∑_{m_{1}≤|β|≤m−1} ∥D^{β}u_{n}∥_{q(β),σ_{β}}^{q(β)}).

It follows from (2.7) and ∥D^{β}u_{n}∥_{p,w_{β}} ≤ ∥u_{n}∥_{m,p,w} for all β ∈ J that

∫_{Ω} |L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u)| dx
≤ ∥D^{α}u_{n} − D^{α}u∥_{C(Ω,σ_{α})}R̃_{α}(∥u_{n}∥_{m,p,w}),

where

R̃_{α}(t) = G(c_{1}t)(∥ĝ_{α}∥_{1} + c_{2}t^{p} + c_{3} ∑_{m_{1}≤|β|≤m−1} t^{q(β)}).

This function is also positive and continuous, hence R̃_{α}(∥u_{n}∥_{m,p,w}) is bounded. Since, by (4.3) ∥D^{α}u_{n} − D^{α}u∥_{C(Ω,σ_{α})} → 0 as n → ∞, it follows that

∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx → 0,

which yields (4.6). Thus, due to (4.5) and (4.6) we conclude (4.4). Step (2) We shall prove that

lim_{n→∞} ∑_{|α|≤m−1} ∑_{β∈J^{c}} ∫_{Ω} C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u) dx = 0. (4.7)

(i) Let m_{1} ≤ |α| ≤ m − 1 and β ∈ J^{c} be fixed. And let s_{α} such that,

\frac{1}{s_{α}} = \frac{1}{q(α)} + \frac{1}{p} + \frac{1}{r_{α}} < 1.

By Hölder’s inequality, we have

∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u)|^{s_{α}} dx
≤ ( ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))|^{r_{α}}σ_{α}^{− \frac{rα}{q(α)}} w_{β}^{− \frac{rα}{p}} dx)^{\frac{sα}{rα}}
× ( ∫_{Ω} |D^{β}u_{n}|^{p}w_{β} dx)^{\frac{sα}{p}} ( ∫_{Ω} |D^{α}u_{n} − D^{α}u|^{q(α)}σ_{α} dx)^{\frac{sα}{q(α)}} .

By (H2’) the sequences {C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})), m_{1} ≤ |α| ≤ m − 1, β ∈ J^{c}}

(resp \{D^{β}u_{n}, β ∈ J^{c}\}) remain bounded in L^{r_{α}}(Ω, σ_{α}^{− \frac{rα}{q(α)}} w_{β}^{− \frac{rα}{p}}) (resp L^{p}(Ω, w_{β})). Moreover, ∥D^{α}u_{n} − D^{α}u∥_{q(α),σ_{α}}^{s_{α}} → 0 as n → ∞. Then

lim_{n→∞} ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u)|^{s_{α}} dx = 0.

Consequently,

lim_{n→∞} ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u)| dx = 0,

i.e,

lim_{n→∞} ∑_{m_{1}≤|α|≤m−1} ∑_{β∈J^{c}} ∫_{Ω} C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u) = 0. (4.8)

(ii) Let |α| < m_{1} and β ∈ J^{c} be fixed. By (H2’) the sequences

{σ_{α}^{−1}C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}, |α| < m_{1}, β ∈ J^{c}}

remain bounded in L^{s_{α}}(Ω) with \frac{1}{s_{α}} = \frac{1}{p} + \frac{1}{r_{α}} < 1. Indeed,

∫_{Ω} |σ_{α}^{−1}C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}|^{s_{α}} dx

≤ ( ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))|^{r_{α}}σ_{α}^{−r_{α}}w_{β}^{− \frac{rα}{p}} dx)^{\frac{sα}{rα}} ( ∫_{Ω} |D^{β}u_{n}|^{p}w_{β} dx)^{\frac{sα}{p}} .

The right hand side is bounded because {C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))} is bounded in

L^{r}(Ω, σ_{α}^{−r_{α}}w_{β}^{− \frac{rα}{p}}) and \{D^{β}u_{n}\} is bounded in L^{p}(Ω, w_{β}).

Thanks to s_{α} ≥ 1, the sequences {σ_{α}^{−1}C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}, |α| < m_{1}, β ∈ J^{c}} remain bounded in L^{1}(Ω). Since

∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u)| dx

≤ sup_{x∈Ω}(|(D^{α}u_{n} − D^{α}u)σ_{α}|) ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}σ_{α}^{−1}| dx

it follows that

lim_{n→∞} ∫_{Ω} |C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u)| dx = 0

(because sup_{x∈Ω}(|(D^{α}u_{n} − D^{α}u)σ_{α}|) → 0). Which gives

lim_{n→∞} ∑_{|α|<m_{1}} ∑_{β∈J^{c}} ∫_{Ω} C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}(D^{α}u_{n} − D^{α}u) dx = 0. (4.9)

Combining (4.8) and (4.9) we obtain (4.7). Step (3) We shall prove that

\begin{matrix}lim_{n→∞} ∑_{α∈J} ∫_{Ω} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) \\ − B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u)))(D^{α}u_{n} − D^{α}u) dx = 0 \end{matrix}  (4.10)

and that

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{α∈J^{c}}(B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n})) \\ − B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u)))(D^{α}u_{n} − D^{α}u) dx = 0. \end{matrix}  (4.11)

Combining (4.1), (4.4) and (4.7) one obtain

lim sup_{n→∞} ∑_{α∈J} ∫_{Ω} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx

+ ∑_{α∈J^{c}} ∫_{Ω} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx ≤ 0.

(4.12)

Thanks to (4.3) and (H2’) one deduce that

B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u)) → B_{α}(x, η(u), ζ_{J}(∇^{m}u)) in L^{p′}(Ω, w_{α}^{∗}), α ∈ J
B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u)) → B_{α}(x, η(u), ζ_{J^{c}}(∇^{m}u)) in L^{p′}(Ω, w_{α}^{∗}), α ∈ J^{c}.

Since D^{α}u_{n} ⇀ D^{α}u in L^{p}(Ω, w_{α}) for all |α| = m, one can write

lim_{n→∞} ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u))(D^{α}u_{n} − D^{α}u) dx = 0 ,

lim_{n→∞} ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u))(D^{α}u_{n} − D^{α}u) dx = 0 .

(4.13)

Combining (4.12), (4.13), (2.3) and (2.4) we conclude the assertions (4.10) and (4.11).

Step (4) To prove the relation (4.2), it suffices to show the following assertions:

(i) For every v ∈ V,

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v dx \\ = ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx. \end{matrix}  (4.14)

(ii) For every v ∈ V,

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{m_{1}≤|α|≤m−1} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v dx \\ = ∫_{Ω} ∑_{m_{1}≤|α|≤m−1} L_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx. \end{matrix}  (4.15)

(iii) For every v ∈ V,

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{|α|<m_{1}} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v dx \\ = ∫_{Ω} ∑_{|α|<m_{1}} L_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx. \end{matrix}  (4.16)

(iv) For every v ∈ V,

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{|α|≤m−1} ∑_{β∈J^{c}} C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{β}u_{n}D^{α}v \\ = ∫_{Ω} ∑_{|α|≤m−1} ∑_{β∈J^{c}} C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))D^{β}uD^{α}v. \end{matrix}  (4.17)

(v) For every v ∈ V,

\begin{matrix}lim_{n→∞} ∫_{Ω} ∑_{α∈J^{c}}(B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))D^{α}v dx \\ = ∫_{Ω} ∑_{α∈J^{c}}(B_{α}(x, η(u), ζ_{J^{c}}(∇^{m}u))D^{α}v dx. \end{matrix}  (4.18)

Proof of assertions (i)and (ii). Invoking Landes [8, lemma 6], we obtain from (4.10) and the strict monotonicity (2.3) that

D^{α}u_{n} → D^{α}u a.e in Ω for each α ∈ J,  (4.19)

which gives

B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → B_{α}(x, η(u), ζ_{J}(∇^{m}u)) a.e. in Ω ∀α ∈ J ,
L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → L_{α}(x, η(u), ζ_{J}(∇^{m}u))
a.e in Ω ∀m_{1} ≤ |α| ≤ m − 1.

¿From the growth condition (H2’), the sequence {B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})), α ∈ J} (resp {L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) m_{1} ≤ |α| ≤ m − 1}) are bounded in L^{p′}(Ω, w_{α}^{∗}) (resp. L^{q′(α)}(Ω, σ_{α}^{∗})), hence by Lemma 4.2 we have

B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) ⇀ B_{α}(x, η(u), ζ_{J}(∇^{m}u))

in L^{p′}(Ω, w_{α}^{∗}) for all α ∈ J and

L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) ⇀ L_{α}(x, η(u), ζ_{J}(∇^{m}u))

in L^{q′(α)}(Ω, σ_{α}^{∗}) for all m_{1} ≤ |α| ≤ m − 1, which implies (i) and (ii). □

Proof of assertion (iii). In virtue of the growth condition (H2’) we have for all v ∈ V and all |α| < m_{1}

|L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v| ≤ |D^{α}v|G(k(x, η(u_{n})))σ_{α}(ĝ_{α}(x) + c̃_{α} ∑_{β∈J} w_{β}|D^{β}u_{n}|^{p}

+ c̃_{α} ∑_{m_{1}≤|β|≤m−1} σ_{β}|D^{β}u_{n}|^{q(β)}).

Since G(k(x, η(u_{n}))) ≤ c_{1} and sup_{x∈Ω}(|D^{α}vσ_{α}|) ≤ c_{2} for all |α| < m_{1}, where c_{i}(i = 1, 2) are some positive constants, it follows that

|L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v|

≤ c(ĝ_{α}(x) + c_{α} ∑_{β∈J} w_{β}|D^{β}u_{n}|^{p} + c_{α} ∑_{m_{1}≤|β|≤m−1} σ_{β}|D^{β}u_{n}|^{q(β)}) = g_{n}.

It follows from (4.3) and (4.19) that

L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → L_{α}(x, η(u), ζ_{J}(∇^{m}u)) a.e. in Ω ∀|α| < m_{1}

and

g_{n} → g = c(ĝ_{α}(x) + c_{α} ∑_{β∈J} w_{β}|D^{β}u|^{p} + c_{α} ∑_{m_{1}≤|β|≤m−1} σ_{β}|D^{β}u|^{q(β)})a.ea.e. in Ω.

Lemma 4.3. D^{β}u_{n} → D^{β}u as n → ∞ in L^{p}(Ω, w_{β}) for all β ∈ J.

By (4.3) and Lemma 4.3 we obtain

∫_{Ω} g_{n} dx → ∫_{Ω} g dx.

By the generalized Lebesgue theorem we have,

∫_{Ω} L_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}v dx → ∫_{Ω} L_{α}(x, η(u), ζ_{J}(∇^{m}u))D^{α}v dx

for all |α| < m_{1} which implies (4.16). □

Proof of assertion (iv). By (4.3) and (4.19) we have for each |α| ≤ m − 1 and each β ∈ J^{c},

C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → C_{αβ}(x, η(u), ζ_{J}(∇^{m}u)) a.e. in Ω.

So, from (H2’) the sequences {C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})), m_{1} ≤ |α| ≤ m−1 and β ∈ J^{c}} (resp. {C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})), |α| < m_{1} and β ∈ J^{c}}) remain bounded in

L^{r_{α}}(Ω, σ_{α}^{− \frac{rα}{q(α)}} w_{β}^{− \frac{rα}{p}}) (resp. L^{r_{α}}(Ω, σ_{α}^{−r_{α}}w_{β}^{− \frac{rα}{p}})). Then Lemma 4.1 yields

C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))

in L^{q}(Ω, σ_{α}^{− \frac{q}{q(α)}} w_{β}^{− \frac{q}{p}}) for all q < r_{α}, all m_{1} ≤ |α| ≤ m − 1 and all β ∈ J^{c}. Lemma 4.1 also yields

C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))

in L^{q}(Ω, σ_{α}^{−q}w_{β}^{− \frac{q}{p}}) for all q < r_{α}, all |α| < m_{1} and all β ∈ J^{c}.

Let s_{α} such that \frac{1}{s_{α}} = \frac{1}{p} + \frac{1}{q(α)}. Remark that r_{α} > s′_{α} = \frac{s_{α}}{s_{α}−1} for m_{1} ≤ |α| ≤ m − 1 and since p′ < r_{α} for |α| < m_{1} one has

C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n})) → C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))

in L^{s′_{α}}(Ω, σ_{α}^{− \frac{s′_{α}}{q(α)}} w_{β}^{− \frac{s′_{α}}{p}}) for all m_{1} ≤ |α| ≤ m − 1. Also one has

C_{αβ}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))σ_{α}^{−1} → C_{αβ}(x, η(u), ζ_{J}(∇^{m}u))σ_{α}^{−1}  (4.20)

in L^{p′}(Ω, w_{β}^{− \frac{p′}{p}}) for all |α| < m_{1}.

Lemma 4.4. For all v ∈ V, one has

(1) D^{β}u_{n}D^{α}v ⇀ D^{β}uD^{α}v in L^{s_{α}}(Ω, σα^{\frac{sα}{q(α)}} wβ^{\frac{sα}{p}}) for each m_{1} ≤ |α| ≤ m − 1

and each β ∈ J^{c}.

(2) D^{β}u_{n}D^{α}vσ_{α} ⇀ D^{β}uD^{α}vσ_{α} in L^{p}(Ω, w_{β}) for each |α| < m_{1} and each β ∈ J^{c}.

In view of (4.20) and Lemma 4.4 we conclude (4.17). □

Proof of assertion (v). First we show that

∫_{Ω} ∑_{α∈J^{c}}(B_{α}(x, η(u), ζ_{J^{c}}(v)) − h_{α})(v_{α} − D^{α}u) dx ≥ 0  (4.21)

for all v = (v_{α}) ∈ ∏_{|α|=m} L^{p}(Ω, w_{α}), where h_{α} stands for the weak limit of {B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n})), α ∈ J^{c}} in L^{p′}(Ω, w_{α}^{∗}). Indeed by (4.11) we have,

lim sup_{n→∞} ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))(D^{α}u_{n} − D^{α}u) dx ≤ 0,

implies

lim sup_{n→∞} ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))D^{α}u_{n} dx ≤ ∫_{Ω} ∑_{α∈J^{c}} h_{α}D^{α}u dx (4.22)

and from weak Leray-Lions condition (2.4), for any v = (v_{α}) ∈ ∏_{|α|=m} L^{p}(Ω, w_{α}), we obtain

∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))D^{α}u_{n} dx

≥ ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))v_{α} dx

+ ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(v))(D^{α}u_{n} − v_{α}) dx.

Letting n → ∞ we conclude by (4.22) that

∫_{Ω} ∑_{α∈J^{c}} h_{α}D^{α}u dx ≥ ∫_{Ω} ∑_{α∈J^{c}} h_{α}v_{α} dx + ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u), ζ_{J^{c}}(v))(D^{α}u − v_{α}) dx

and hence (4.21) follows. Choosing v = D^{α}u + tŵ with t > 0, ŵ = (ŵ_{α}) ∈ ∏_{|α|=m} L^{p}(Ω, w_{α}) and letting t → 0 we obtain h_{α} = B_{α}(x, η(u), ζ_{J^{c}}(∇^{m}u)) a.e. in Ω which implies (4.18). □

(b) We shall prove that

lim inf_{n→∞}⟨T u_{n}, u_{n}⟩ ≥ ⟨T u, u⟩.  (4.23)

In view of monotonicity condition (2.3) and (2.4) we have

∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}u_{n} + ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))D^{α}u_{n}

≥ ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u_{n}))D^{α}u

+ ∫_{Ω} ∑_{α∈J} B_{α}(x, η(u_{n}), ζ_{J}(∇^{m}u))(D^{α}u_{n} − D^{α}u)

+ ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u_{n}))D^{α}u

+ ∫_{Ω} ∑_{α∈J^{c}} B_{α}(x, η(u_{n}), ζ_{J^{c}}(∇^{m}u))(D^{α}u_{n} − D^{α}u).

Letting n → ∞ and using (4.4) and (4.7)we obtain (4.23).

Proof of Lemma 4.3. Let E be a measurable subset of Ω, in view of steps 1, 2, 3 and 4 in [6, Lemma 2.7], we obtain

lim_{meas E→0} ∫_{E} ∑_{β∈J} w_{β}|D^{β}u_{n}(x)|^{p} dx = 0  (4.24)

uniformly with respect to n ∈ ℕ, i.e the sequence (w_{β}|D^{β}u_{n}|^{p}) is equi-integrable. And due to (4.19) we have

w_{β}|D^{β}u_{n}|^{p} → w_{β}|D^{β}u|^{p} a.e. ∀β ∈ J.

Since meas(Ω) < ∞, by Vitali’s theorem we obtain D^{β}u_{n} → D^{β}u in L^{p}(Ω, w_{β}) for all β ∈ J. □

Proof of Lemma 4.4. (1) Let m_{1} ≤ |α| ≤ m − 1 and β ∈ J^{c} fixed. Let φ ∈

L^{s′_{α}}(Ω, σ_{α}^{− \frac{s′_{α}}{q(α)}} w_{β}^{− \frac{s′_{α}}{p}}). Since, \frac{1}{p′} = \frac{1}{s′_{α}} + \frac{1}{q(α)}, by Hölder’s inequality we obtain

∫_{Ω} |D^{α}vφ|^{p′}w_{β}^{1−p′} ≤ ( ∫_{Ω} |D^{α}v|^{q(α)}σ_{α})^{\frac{p′}{q(α)}} ( ∫_{Ω} |φ|^{s′_{α}}wβ^{\frac{s′_{α}(1−p′)}{p′}} σα^{\frac{−s′_{α}}{q(α)}} )^{\frac{p′}{s′_{α}}} < ∞

(because \frac{s′_{α}(1−p′)}{p′} = \frac{−s′_{α}}{p}). Then D^{α}vφ ∈ L^{p′}(Ω, w_{β}^{1−p′}) and since D^{β}u_{n} ⇀ D^{β}u in L^{p}(Ω, w_{α}), we have

∫_{Ω} D^{β}u_{n}D^{α}vφ → ∫_{Ω} D^{β}u_{n}D^{α}vφ for all φ ∈ L^{s′_{α}}(Ω, σ_{α}^{− \frac{s′_{α}}{q(α)}} w_{α}^{− \frac{s′_{α}}{p}});

i.e.,

D^{β}u_{n}D^{α}v ⇀ D^{β}uD^{α}v ∈ L^{s_{α}}(Ω, σα^{\frac{sα}{q(α)}} wβ^{\frac{sα}{p}})

for all m_{1} ≤ |α| ≤ m − 1 and all β ∈ J^{c}.

(2) Let |α| < m_{1} and β ∈ J^{c} and let φ ∈ L^{p′}(Ω, w_{β}^{∗}). Thanks to D^{α}v ∈ C(Ω, σ_{α}) ∀v ∈ V, we have D^{α}vσ_{α}φ ∈ L^{p′}(Ω, w_{β}^{∗}). Since D^{β}u_{n} ⇀ D^{β}u in L^{p}(Ω, w_{α}), we have

∫_{Ω} D^{β}u_{n}D^{α}vσ_{α}φ dx → ∫_{Ω} D^{β}uD^{α}vσ_{α}φ dx for all φ ∈ L^{p′}(Ω, w_{β}^{∗}).
□

Remark 4.5. Note that, the ellepticity condition (H3) is only used to prove (4.24) (see step 3 in [6, lemma 2.7], which concerns only the equality (4.16) corresponding to a terms L_{α} with |α| < m_{1}).

5. Specific case

Let Ω be a bounded open subset of ℝ^{N} satisfying the cone condition. In the sequel we assume in addition that the collection of weight functions w = {w_{α}(x), |α| ≤ m} satisfies w_{α}(x) = 1 for all |α| ≤ m − 1, and the integrability condition: There exists ν ∈]\frac{N}{P} , ∞[∩[\frac{1}{P −1}, ∞[ such that

w_{α}^{−ν} ∈ L^{1}(Ω) ∀|α| = m.  (5.1)

Note that (5.1) is stronger than (2.1). Assumptions (2.1) and (5.1) imply

∥|u|∥_{V} = ( ∑_{|α|=m} ∫_{Ω} |D^{α}u|^{p}w_{α}(x) dx)^{1/p}

is a norm defined on V = W_{0}^{m,p}(Ω, w) and it’s equivalent to (2.1). Let

m_{1} = \frac{mpν − N(ν + 1)}{pν} = m − \frac{N}{p_{1}} with p_{1} = \frac{pν}{ν + 1}.  (5.2)

Remark 5.1 ([5]). Under the above assumption the following continuous imbeddings hold: (i) For k < m_{1},

W^{m,p}(Ω, w) ↪ C^{k}(Ω̄).

(ii) For k = m_{1}, with arbitrary r, 1 < r < ∞,

W^{m,p}(Ω, w) ↪ W^{k,r}(Ω) .

(iii) For k > m_{1},

W^{m,p}(Ω, w) ↪ W^{k,r_{k}}(Ω)

where r_{k} satisfies 1 < r_{k} ≤ q_{k} = \frac{pνN}{N(ν+1)−pν(m−k)}.

Moreover the imbedding (i) and (ii) are compact and (iii) is compact if r_{k} < q_{k}.

Now, we define

H^{m−1,q}(Ω, σ) = ∏_{|β|≤m−1} X_{β},

where X_{β} = L^{q(β)}(Ω, σ_{β}), q(β) > 1 for m_{1} ≤ |β| ≤ m − 1 and X_{β} = C^{|β|}(Ω, σ_{β}) for |β| < m_{1}.

Also we define the assumption (H4) Let

1 < q(β) < q_{|β|} = \frac{pνN}{N(ν + 1) − pν(m − |β|)}

for m_{1} < |β| ≤ m − 1 and q(β) arbitrary if |β| = m_{1} and σ_{β} ≡ 1 for all

β ≤ m − 1.

Remark 5.2. If (H4) is satisfied, then by Remark 5.1,

W^{m,p}(Ω, w) ↪↪ H^{m−1,q}(Ω)

which implies immediately that (H2’)(iii) with σ ≡ 1.

Theorem 5.3. Let Ω be a bounded open subset of ℝ^{d}. And assume that (2.1), (5.1), (H1’), (H2’)(,i,ii), (H3), (H4), (2.3) and (2.4) are satisfied. Then the operator T defined in (2.8) is pseudo-monotone in V = W_{0}^{m,p}(Ω, w).

If in addition the degeneracy (3.1) is satisfied, then the degenerate boundaryvalue problem from (1.8) has at least one solution u ∈ V.

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] Y. Akdim, E. Azroul, A. Benkirane; Pseudo-monotonicity and degenerated elliptic operators of second order, (submitted).

[3] Y. Akdim, E. Azroul, A. Benkirane; Pseudo-Monotone and Nonlinear Degenerated Elliptic Boundary-value problems of higher order I, (submitted).

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

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

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

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

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

[9] J. Leray, 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.

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

   E-mail address: akdimyoussef@yahoo.fr

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

   E-mail address: azroul elhoussine@yahoo.fr

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