2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 11–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ON THE SOLVABILITY OF DEGENERATED QUASILINEAR ELLIPTIC PROBLEMS

YOUSSEF AKDIM, ELHOUSSINE AZROUL, MOHAMED RHOUDAF

Abstract. In this article, we study the quasilinear elliptic problem

Au = − div(a(x, u, ∇u)) = f(x, u, ∇u) in D′(Ω)

u = 0 on ∂Ω ,

where A is a Leray-Lions operator from W_{0}^{1,p}(Ω, w) to its dual W^{−1,p′}(Ω, w^{∗}). We show that there exists a solution in W_{0}^{1,p}(Ω, w) provided that

|f(x, r, ξ)| ≤ σ^{1/q}[g(x) + |r|^{η}σ^{η/q} + ∑_{i=1}^{N} w_{i}^{δ/p}(x)|ξ_{i}|^{δ}],

where g(x) is a positive function in L^{q′}(Ω) and σ(x) is weight function and 0 ≤ η < min(p − 1, q − 1), 0 ≤ δ < (p − 1)/q′.

1. Introduction

Let Ω be a bounded open set in ℝ^{N}, N ≥ 2, and p be a real number such that 1 < p < ∞. Let w = {w_{i}(x), 0 ≤ i ≤ N} be a vector weight functions on Ω; i.e., each w_{i}(x) is a measurable a.e. strictly positive function on Ω, satisfying some integrability conditions (see section2). Let us consider the problem

\begin{matrix}Au = f(x, u, ∇u) in D′(Ω) \\ u = 0 on ∂Ω, \end{matrix}  (1.1)

where A is a Leray-Lions operator Au = − div(a(x, u, ∇u)) and f(x, r, ξ) : Ω × ℝ × ℝ^{N} → ℝ is a Carathéodory function. Boccardo, Murat and Puel in [3] studied the problem (1.1) in the non weighted case, with f satisfying the condition

|f(x, r, ξ)| ≤ h(|r|)(1 + |ξ|^{p}),

where h is increasing function from ℝ^{+} into ℝ^{+}. The existence result is proved assuming the existence of the subsolution and supersolution in W^{1,∞}(Ω), which play an important roll in their work. Further in [2] the author’s studied the problem (1.1) with f satisfies the hypotheses

2000 Mathematics Subject Classification. 35J20, 35J25, 35J70.

Key words and phrases. Weighted Sobolev spaces; variational calculus, Hardy inequality.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

|f(x, r, ξ)| ≤ β[g(x) + |r|^{p−1} + |ξ|^{p−1}],  (1.2)
f(x, r, ξ)r ≥ α|r|^{p}.  (1.3)

Recently, Tsang-Hai Kuo and Chiung-Chion Tsai [8] proved an existence result under the assumption

|f(x, r, ξ)| ≤ c(1 + |r|^{δ} + |ξ|^{η}).

Our objective in this paper, is to study the problem (1.1) in weighted Sobolev spaces where f satisfying only the growth condition

|f(x, r, ξ)| ≤ σ^{1/q}[g(x) + |r|^{η}σ^{\frac{η}{q}} + ∑_{i=1}^{N} w_{i}^{δ/p}(x)|ξ_{i}|^{δ}],

where g(x) is a positive function in L^{q′}(Ω), σ is a weight function, and

0 ≤ η < min(p − 1, q − 1), 0 ≤ δ < \frac{p − 1}{q′}.

Note that we obtain the existence result without assuming the condition (1.3) and without knowing a priori the existence of subsolutions and supersolutions. Let us point out that this work can be see as a generalization of the work in [2] and [8].

2. Preliminaries and Basic Assumptions

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

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

for any 0 ≤ i ≤ N. We denote by W^{1,p}(Ω, w) the space of real-valued functions u ∈ L^{p}(Ω, w_{0}) such that their derivatives in the sense of distributions satisfies

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

Which is a Banach space under the norm

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

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

We recall that the dual space of weighted Sobolev spaces W_{0}^{1,p}(Ω, w) is equivalent to W^{−1,p′}(Ω, w^{∗}), where w^{∗} = {w_{i}^{∗} = w_{i}^{1−p′}, i = 1, …, N} and p′ is the conjugate of p, i.e. p′ = \frac{p}{p−1}. For more details we refer the reader to [5]. We start by stating the following assumptions:

(H1) The expression

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

is a norm defined on W_{0}^{1,p}(Ω, w) and its equivalent to the norm (2.3). And there exist a weight function σ on Ω and a parameter 0 < q < ∞, such that the Hardy inequality

(∫_{Ω} |u(x)|^{q}σ(x) dx)^{1/q} ≤ c(∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{1/p},  (2.5)

holds for every u ∈ W_{0}^{1,p}(Ω, w) with a constant c > 0. Moreover, the imbedding

W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, σ),  (2.6)

is compact. Let A be a nonlinear operator from W_{0}^{1,p}(Ω, w) into its dual W^{−1,p′}(Ω, w^{∗}) defined by

A(u) = −div(a(x, u, ∇u)),

where a(x, r, ξ) : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory vector-valued function that satisfies the following assumption:

(H2) For i = 1, …, N,

|a_{i}(x, r, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + σ^{\frac{1}{p′}} |r|^{q/p′} + ∑_{j=1}^{N}wj^{\frac{1}{p′}} |ξ_{j}|^{p−1}]  (2.7)
[a(x, r, ξ) − a(x, r, η)](ξ − η) > 0 for all ξ ≠ η ∈ ℝ^{N};  (2.8)
a(x, r, ξ)ξ ≥ α∑_{i=1}^{N}w_{i}|ξ_{i}|^{p},  (2.9)

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

Let f(x, r, ξ) is a Carathéodory function satisfying the following assumptions: (H3)

|f(x, r, ξ)| ≤ σ^{1/q}[g(x) + |r|^{η}σ^{\frac{η}{q}} + ∑_{i=1}^{N}w_{i}^{δ/p}(x)|ξ_{i}|^{δ}],  (2.10)

where g(x) is a positive function in L^{q′}(Ω), and

0 ≤ η < min(p − 1, q − 1), 0 ≤ δ < \frac{p − 1}{q′}.  (2.11)

3. Main result

Consider the problem

− div a(x, u, ∇u) = f(x, u, ∇u) in D′(Ω)

u = 0 on ∂Ω . (3.1)

Theorem 3.1. Under hypotheses (H1)-(H3), there exist at least one solution to (3.1).

We first give some definition and some lemmas that will be used in the proof of this theorem. Definition Let Y be a separable reflexive Banach space, the operator B from Y to its dual Y^{∗} is called of the calculus of variations type, if B is bounded and is of the from,

B(u) = B(u, u),  (3.2)

where (u, v) → B(u, v) is an operator Y × Y into Y^{∗} satisfying the following properties:

For u ∈ Y, the mapping v ↦ B(u, v) is bounded and hemicontinuous

from Y to Y^{∗} and (B(u, u) − B(u, v), u − v) ≥ 0;

(3.3)

for v ∈ Y, the mapping u ↦ B(u, v) is bounded and hemicontinuous from Y to Y^{∗}; If u_{n} ⇀ u weakly in Y and if (B(u_{n}, u_{n}) − B(u_{n}, u), u_{n} − u) → 0,

then B(u_{n}, v) ⇀ B(u, v) weakly in Y^{∗}, for all v ∈ Y;

(3.4)

If u_{n} ⇀ u weakly in Y and if B(u_{n}, v) ⇀ ψ weakly in Y^{∗},

then (B(u_{n}, v), u_{n}) → (ψ, u). (3.5)

Lemma 3.2 ([1]). Let g ∈ L^{q}(Ω, γ), g_{n} ∈ L^{q}(Ω, γ), and ∥g_{n}∥_{q,γ} ≤ c (1 < q < ∞). If g_{n}(x) → g(x) a.e. in Ω, then g_{n} ⇀ g weakly in L^{q}(Ω, γ), where γ is a weight function on Ω.

Lemma 3.3. If u_{n} ⇀ u in W_{0}^{1,p}(Ω, w) and v ∈ W_{0}^{1,p}(Ω, w), then a_{i}(x, u_{n}, ∇v) → a_{i}(x, u, ∇v) in L^{p′}(Ω, w_{i}^{∗}).

Proof. From (H2), it follows that

\begin{aligned}|a_{i}(x, u_{n}, ∇v)|^{p′}wi^{\frac{−p′}{p}} & ≤ β[k(x) + |u_{n}|^{\frac{q}{p′}} σ^{\frac{1}{p′}} + ∑_{j=1}^{N} |\frac{∂v}{∂x_{j}} |^{p−1}wj^{\frac{1}{p′}}]^{p′} \\ & ≤ γ[k(x)^{p′} + |u_{n}|^{q}σ + ∑_{j=1}^{N} |\frac{∂v}{∂x_{j}} |^{p}w_{j}], \end{aligned}  (3.6)

where β and γ are positive constants. Since u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, σ), it follows that u_{n} → u strongly in L^{q}(Ω, σ) and u_{n} → u a.e. in Ω; hence

|a_{i}(x, u_{n}, ∇v)|^{p′}w_{i}^{∗} → |a_{i}(x, u, ∇v)|^{p′}w_{i}^{∗} a.e. in Ω,  (3.7)

and

γ[k(x)^{p′} + |u_{n}|^{q}σ + ∑_{j=1}^{N} |\frac{∂v}{∂x_{j}} |^{p}w_{i}] → γ[k(x)^{p′} + |u|^{q}σ + ∑_{j=1}^{N} |\frac{∂v}{∂x_{j}} |^{p}w_{j}]

a.e. in Ω. Then, By Vitali’s theorem,

a_{i}(x, u_{n}, ∇v) → a_{i}(x, u, ∇v) strongly in L^{p′}(Ω, w_{i}^{∗}), as n → +∞.  (3.8)
□

Lemma 3.4 ([1]). Assume that (H1)–(H2) are satisfied, and let (u_{n}) be a sequence in W_{0}^{1,p}(Ω, w) such that u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and

∫_{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).

For v ∈ W_{0}^{1,p}(Ω, w), we associate the Nemytskii operator F with respect to f,

F (v, ∇v)(x) = f(x, v, ∇v) a.e., x ∈ Ω.

Lemma 3.5. The mapping v ↦ F (v, ∇v) is continuous from the space W_{0}^{1,p}(Ω, w) to L^{q′}(Ω, σ^{1−q′}).

Proof. By hypothesis (H3), we have

|f(x, r, ξ)| ≤ σ^{1/q}[g(x) + |r|^{η}σ^{\frac{η}{q}} + ∑_{i=1}^{N}w_{i}^{δ/p}(x)|ξ_{i}|^{δ}].

Thanks to Young’s inequality,

|r|^{η}σ^{η/q} ≤ [\frac{η}{q − 1}|r|^{q−1}σ^{(q−1)/q} + 1] ≤ [|r|^{q−1}σ^{q′} + 1],

w_{i}^{σ/p}|ξ_{i}|^{σ} ≤ [w_{i}^{1/q′}|ξ_{i}|^{p/q′} + 1],

which implies

|f(x, r, ξ)| ≤ σ^{1/q}[(N + 2) + g(x) + |r|^{q−1}σ^{1/q′} + ∑_{i=1}^{N}w_{i}^{1/q′}|ξ_{i}|^{p/q′}] .

Then

|f(x, r, ξ)|^{q′}σ^{−q′/q} ≤ c_{2}[c_{1} + g(x)^{q′} + |r|^{(q−1)q′}σ + ∑_{i=1}^{N}w_{i}|ξ_{i}|^{p}] .

Since f is a Carathéodory, and for all subset E measurable, such that |E| < η, we have

∫_{E} |f(x, v, ∇v)|^{q′}σ^{\frac{−q′}{q}} dx ≤ c_{2}[c_{3} + ∫_{E} |v|^{q}σ dx + ∫_{E} ∑_{i=1}^{N}w_{i}|\frac{∂v}{∂x_{i}} |^{p} dx] .

Then by Vitali’s theorem, we deduce the continuous of the operator F. Moreover,

(∫_{Ω} |f(x, v, ∇v)|^{q′}σ^{\frac{−q′}{q}} dx)^{1/q′} ≤ c_{2}[c + ∥|v∥|^{q/q′} + ∥|v∥|^{p/q′}].  (3.9)

□

Proof of Theorem 3.1. Step (1) We will show that the operator B : W_{0}^{1,p}(Ω, w) → W^{1,p′}(Ω, w^{∗}) defined by B(v) = A(v) − f(x, v, ∇v) is a calcul of variational. Assertion 1. Let

B(u, v) = − ∑_{i=1}^{N} \frac{∂a_{i}(x, u, ∇v)}{∂x_{i}} − f(x, u, ∇u).

Then B(v, v) = B(v) for all v ∈ W_{0}^{1,p}(Ω, w). Assertion 2. We claim that the operator v → B(u, v) is bounded for all u ∈ W_{0}^{1,p}(Ω, w). Let ψ ∈ W_{0}^{1,p}(Ω, w), we have

⟨B(u, v), ψ)⟩ = ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u, ∇v)\frac{∂ψ}{∂x_{i}} − ∫_{Ω} f(x, u, ∇u)ψ dx.

From Hölder’s inequality, the growth condition (2.7) and the compact imbedding (2.6), we obtain

\begin{matrix}∑_{i=1}^{N} & ∫_{Ω} a_{i}(x, u, ∇v)\frac{∂ψ}{∂x_{i}} \\ ≤ ∑_{i=1}^{N}(∫_{Ω} |a_{i}(x, u, ∇v)|^{p′}wi^{\frac{−p′}{p}} dx)^{1/p′}(∫_{Ω} |\frac{∂ψ}{∂x_{i}} |^{p}w_{i} dx)^{1/p} \\ ≤ c_{4}∥|ψ∥|∑_{i=1}^{N}(∫_{Ω} k(x)^{p′} + |u|^{q}σ + ∑_{j=1}^{N}|\frac{∂v}{∂x_{j}} |^{p}w_{j} dx)^{1/p′} \\ ≤ c_{5}∥|ψ∥|[c_{6} + ∥|u∥|^{\frac{q}{p′}} + ∥|v∥|^{p−1}]. \end{matrix}  (3.10)

Similarly,

∫_{Ω} f(x, u, ∇u)ψ dx ≤ (∫_{Ω} |f(x, u, ∇u)|^{q′}σ^{\frac{−q′}{q}} dx)^{1/q′}(∫_{Ω} |ψ|^{q}σ dx)^{1/q},

by (2.5) and (3.9), we have,

∫_{Ω} f(x, u, ∇u)ψ dx ≤ c∥|ψ∥|[c_{7} + ∥|u∥|^{q−1} + ∥|u∥|^{p/q′}].  (3.11)

Since u and v belong to W_{0}^{1,p}(Ω, w) and in view of (3.10) and (3.11), we deduce that ⟨B(u, v), ψ⟩ is bounded in W_{0}^{1,p}(Ω, w) × W_{0}^{1,p}(Ω, w).

We claim that the operator v → B(u, v) is hemicontinuous for all u ∈ W_{0}^{1,p}(Ω, w), i.e., the operator λ → ⟨B(u, v_{1}+λv_{2}), ψ⟩ is continuous for all v_{1}, v_{2}, ψ ∈ W_{0}^{1,p}(Ω, w). Since a_{i} is a Carathéodory function,

a_{i}(x, u, ∇(v_{1} + λv_{2})) → a_{i}(x, u, ∇v_{1}) a.e. in Ω as λ → 0.

Further, we know from (2.7) that (a_{i}(x, u, ∇(v_{1} + λv_{2}))_{λ} is bounded in L^{p′}(Ω, w_{i}^{∗}); thus, by Lemma 3.2, we conclude

a_{i}(x, u, ∇(v_{1} + λv_{2})) ⇀ a_{i}(x, u, ∇v_{1}) weakly in L^{p′}(Ω, w_{i}^{∗}), as λ → 0 . (3.12)

Hence,

lim_{λ→0}⟨B(u, v_{1} + λv_{2}), ψ⟩

= lim_{λ→0}∑_{i=1}^{N}∫_{Ω} a_{i}(x, u, ∇(v_{1} + λv_{2}))\frac{∂ψ}{∂x_{i}} dx − ∫_{Ω} f(x, u, ∇u)ψ dx

= ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u, ∇v_{1})\frac{∂ψ}{∂x_{i}} dx − ∫_{Ω} f(x, u, ∇u)ψ dx

= ⟨B(u, v_{1}), ψ⟩ for all v_{1}, v_{2}, ψ ∈ W_{0}^{1,p}(Ω, w).

(3.13)

Similarly, we show that u → ⟨B(u, v), ψ⟩ is bounded and hemicontinuous for all v ∈ W_{0}^{1,p}(Ω, w). Indeed. By (3.9), we have f((x, u_{1} + λu_{2}, ∇(u_{1} + λu_{2})))_{λ} is bounded in L^{q′}(Ω, σ^{1−q′}) and as f is a Carathéodory function then

f(x, u_{1} + λu_{2}, ∇(u_{1} + λu_{2})) → f(x, u_{1}, ∇u_{1}) a.e. in Ω.

Hence, Lemma 3.2 gives,

f(x, u_{1} + λu_{2}, ∇(u_{1} + λu_{2})) ⇀ f(x, u_{1}, ∇u_{2}) weakly in L^{q′}(Ω, σ^{1−q′}) as λ → 0,

(3.14)

On the other hand, as in (3.12), we have

a_{i}(x, u_{1} + λu_{2}, ∇v) ⇀ a_{i}(x, u_{1}, ∇v) in L^{p′}(Ω, w_{1}^{∗}), as λ → 0.  (3.15)

Combining (3.14) and (3.15), we conclude that, u → B(u, v) is bounded and hemicontinuous. Assertion 3. From (2.8), we have,

⟨B(u, u) − B(u, v), u − v⟩ = ∑_{i=1}^{N}∫_{Ω} (a_{i}(x, u, ∇u) − a_{i}(x, u, ∇v)) (\frac{∂u}{∂x_{i}} − \frac{∂v}{∂x_{i}}) ≥ 0

Assertion 4. Assume that u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and ⟨B(u_{n}, u_{n}) − B(u_{n}, u), u_{n} − u⟩ → 0, we claim that B(u_{n}, v) ⇀ B(u, v) weakly in W^{−1,p′}(Ω, w^{∗}).

We can write ⟨B(u_{n}, u_{n}) − B(u_{n}, u), u_{n} − u⟩ → 0 as n → ∞,

⟨∑_{i=1}^{N} − [\frac{∂}{∂x_{i}} a_{i}(x, u_{n}, ∇u_{n}) − \frac{∂}{∂x_{i}} a_{i}(x, u_{n}, ∇u)], u_{n} − u⟩

= ∑_{i=1}^{N}∫_{Ω} [a_{i}(x, u_{n}, ∇u_{n}) − a_{i}(x, u_{n}, ∇u)] \frac{∂}{∂x_{i}} (u_{n} − u) dx → 0

Then, by Lemma 3.4, we have u_{n} → u strongly in W_{0}^{1,p}(Ω, w) and it follows from Lemma 3.5 that

f(x, u_{n}, ∇u_{n}) → f(x, u, ∇u) in L^{q′}(Ω, σ^{1−q′}).  (3.16)

Since u_{n} → u in L^{p}(Ω, w) and by (2.7) and W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, σ), we can obtain from Lemma 3.3 that

a_{i}(x, u_{n}, ∇v) → a_{i}(x, u, ∇v) in L^{p′}(Ω, w_{i}^{∗}) .  (3.17)

This implies

∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂ψ}{∂x_{i}} dx → ∫_{Ω} a_{i}(x, u, ∇v)\frac{∂ψ}{∂x_{i}} dx.  (3.18)

On the other hand, by Hölders inequality,

∫_{Ω} |f(x, u_{n}, ∇u_{n})ψ| dx ≤ (∫_{Ω} |f(x, u_{n}, ∇u_{n})|^{q′}σ^{1−q′} dx)^{1/q′}(∫_{Ω} |ψ|^{q}σ dx)^{1/q}.

Thanks to (3.16), (2.5), and Lebegue’s dominated convergence theorem, we obtain

∫_{Ω} f(x, u_{n}, ∇u_{n})ψ dx → ∫_{Ω} f(x, u, ∇u)ψ dx .  (3.19)

Then, we have

lim_{n→∞}⟨B(u_{n}, v), ψ⟩ = lim_{n→∞}∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂ψ}{∂x_{i}} dx − ∫_{Ω} f(x, u_{n}, ∇u_{n})ψ dx

= ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u, ∇v)\frac{∂ψ}{∂x_{i}} dx − ∫_{Ω} f(x, u, ∇u)ψ dx

= ⟨B(u, v), ψ⟩, for all ψ ∈ W_{0}^{1,p}(Ω, w).

Assertion 5. Assume u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and B(u_{n}, v) ⇀ ψ weakly in W^{−1,p′}(Ω, w). We claim that ⟨B(u_{n}, v), u_{n}⟩ → ⟨ψ, u⟩. Thanks to u_{n} ⇀ u in W_{0}^{p}(Ω, w), we obtain by Lemma 3.3,

a_{i}(x, u_{n}, ∇v) → a_{i}(x, u, ∇v) strongly in L^{p′}(Ω, w_{i}^{∗}) as n → +∞.  (3.20)

And so

∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂u_{n}}{∂x_{i}} dx → ∫_{Ω} a_{i}(x, u, ∇v)\frac{∂u}{∂x_{i}} dx.  (3.21)

Hence together with

∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂u}{∂x_{i}} dx − ∫_{Ω} f(x, u_{n}, ∇v)u dx → ⟨ψ, u⟩,  (3.22)

we have

⟨B(u_{n}, v), u_{n})⟩ = ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂u_{n}}{∂x_{i}} dx − ∫_{Ω} f(x, u_{n}, ∇u_{n})u_{n} dx

= ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx + ∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)\frac{∂u}{∂x_{i}} dx

− ∫_{Ω} f(x, u_{n}, ∇u_{n})u dx − ∫_{Ω} f(x, u_{n}, ∇u_{n})(u_{n} − u) dx.

But in view of (3.20) and (3.21), we obtain

∑_{i=1}^{N}∫_{Ω} a_{i}(x, u_{n}, ∇v)(\frac{∂u_{n}}{∂x_{i}} − \frac{∂u}{∂x_{i}}) dx → 0.  (3.23)

On the other hand, by Hölder’s inequality,

∫_{Ω} |f(x, u_{n}, ∇u_{n})(u_{n} − u)| dx
≤ (∫_{Ω} |f(x, u_{n}, ∇u_{n})|^{q′}σ^{1−q′} dx)^{1/q′}(∫_{Ω} |u_{n} − u|^{q}σ dx)^{1/q}
≤ c∥u_{n} − u∥_{L^{q}(Ω,σ)} → as n → ∞

i.e.,

∫_{Ω} f(x, u_{n}, ∇u_{n})(u_{n} − u) dx → 0 as n → ∞.  (3.24)

Thanks to (3.22), (3.23) and (3.24), we obtain

⟨B(u_{n}, v), u_{n}⟩ → ⟨ψ, u⟩.

Step 2. We claim that the operator B satisfies the coercivity condition

lim_{∥|v∥|→+∞} \frac{⟨B(v), v⟩}{∥|v∥|} = ∞.  (3.25)

Since

⟨Bv, v⟩ = ∑_{i=1}^{N}∫_{Ω} a_{i}(x, v, ∇v)\frac{∂v}{∂x_{i}} dx − ∫_{Ω} f(x, v, ∇v)v dx.

Then, using (2.9), we have

⟨Bv, v⟩ ≥ α∑_{i=1}^{N}w_{i}|\frac{∂v}{∂x_{i}} |^{p} − ∫_{Ω} f(x, v, ∇v)v dx.  (3.26)

Moreover,

\begin{matrix}∫_{Ω} f(x, v, ∇v)v dx \\ ≤ ∫_{Ω} σ^{1/q}g(x)v dx + ∫_{Ω} |v|^{η+1}σ^{(η+1)/q} dx + ∫_{Ω} ∑_{i=1}^{N}w_{i}^{δ/p}|\frac{∂v}{∂x_{i}} |^{δ}σ^{1/q}|v| dx. \end{matrix} (3.27)

Thanks to Hölder’s inequality and (2.5), we have

∫_{Ω} σ^{1/q}g(x)v dx ≤ (∫_{Ω} |g(x)|^{q′} dx)^{1/q′}(∫_{Ω} |v|^{q}σ dx)^{1/q} ≤ c∥|v∥|.  (3.28)

On the other hand, by Hölder’s inequality,

∑_{i=1}^{N}w_{i}^{δ/p}|\frac{∂v}{∂x_{i}} |^{δ}σ^{1/q}|v| ≤ c∑_{i=1}^{N}(∫_{Ω} wi^{\frac{δq′}{p}} |\frac{∂v}{∂x_{i}} |^{δq′} dx)^{1/q′}(∫_{Ω} |v|^{q}σ dx)^{1/q}.

In view of (2.5), we have

∑_{i=1}^{N}w_{i}^{δ/p}|\frac{∂v}{∂x_{i}} |^{δ}σ^{1/q}|v| ≤ c∑_{i=1}^{N}(∫_{Ω} wi^{\frac{δq′}{p}} |\frac{∂v}{∂x_{i}} |^{δq′} dx)^{1/q′}∥|v∥|.  (3.29)

Since 0 ≤ \frac{δq′}{p} < 1, hence by Hölder’s inequality, we deduce

(∫_{Ω} w_{i}^{δ/q′}p|\frac{∂v}{∂x_{i}} |^{δq′} dx)^{1/q′} ≤ (∫_{Ω} w_{i}|\frac{∂v}{∂x_{i}} |^{p} dx)^{δ/p},  (3.30)

remark that,

(a + b)^{r} ≥ c(a^{r} + b^{r}) if 0 ≤ r < 1.  (3.31)

Combining (3.29), (3.30) and (3.31), we conclude that

∑_{i=1}^{N}w_{i}^{δ/p}|\frac{∂v}{∂x_{i}} |^{δ}σ^{1/q}|v| ≤ c∥|v∥|(∑_{i=1}^{N}∫_{Ω} w_{i}|\frac{∂v}{∂x_{i}} |^{p} dx)^{δ/p} ≤ c∥|v∥| ∥|v∥|^{δ}. (3.32)

Further, 0 ≤ \frac{η+1}{q} < 1, then by Hölder’s inequality and (2.6), we deduce

∫_{Ω} |v|^{η+1}σ^{(η+1)/q} dx ≤ c∥|v∥|^{η+1}.  (3.33)

Then from (3.26), (3.28), (3.32) and (3.33), we deduce that

⟨Bv, v⟩ ≥ α∥|v∥|^{p−1} − c_{1} − c_{2}∥|v∥|^{η} − c_{3}∥|v∥|^{δ−1}

and since p − 1 > η and p > δ, we conclude that \frac{⟨Bv,v⟩}{∥v∥} → +∞. Finally, the proof of Theorem is complete thanks to the classical Theorem in [7]. □

4. Examples

Let us consider the Carathéodory functions

a_{i}(x, r, ξ) = w_{i}|ξ_{i}|^{p−1} sgn(ξ_{i})

Where w_{i}(x)(i = 1, …, N) are a given weight functions strictly positive almost everywhere in Ω. We shall assume that the weight function satisfies w_{i}(x) = w(x), x ∈ Ω for i = 0, …, N. It is easy to show that the a_{i}(x, s, ξ) are Carathéodory function satisfying the growth condition (2.7) and the coercivity (2.9). On the other side, the monotonicity condition (2.8) is verified. In fact,

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

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

for almost all x ∈ Ω and for all ξ, ξ̂ ∈ ℝ^{N} with ξ ≠ ξ̂, since w > 0 a.e. in Ω. We consider the Hardy inequality (2.5) in the form

(∫_{Ω} |u(x)|^{q}σ(x) dx)^{1/q} ≤ c(∫_{Ω} |∇u(x)|^{p}w(x) dx)^{1/p},

where σ and q are defined in (2.5). In particular, let us use a special weight functions w and σ expressed in terms of the distance to the bounded ∂Ω. Denote d(x) = dist(x, ∂Ω) and set

w(x) = d^{λ}(x), σ(x) = d^{μ}(x).

In this case, the Hardy inequality reads

(∫_{Ω} |u(x)|^{q}d^{μ}(x) dx)^{1/q} ≤ c(∫_{Ω} |∇u(x)|^{p}d^{λ}(x) dx)^{1/p}.

The corresponding imbedding is compact if:

(i) For, 1 < p ≤ q < ∞,

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.  (4.1)
(ii) For 1 ≤ q < p < ∞,
λ < p − 1, \frac{μ}{q} − \frac{λ}{p} + \frac{1}{q} − \frac{1}{p} + 1 > 0.  (4.2)

Remarks. 1. Condition (4.1) or Condition (4.2) is sufficient for the compact imbedding (2.6) to hold; see for example [4, example 1], [5, example 1.5], and [6, Theorems 19.17, 19.22].

Let us consider the Carathéodory function

f(x, r, ξ) = d^{\frac{μ}{q}} (x)(d^{\frac{μδ}{q}} (x)|r|^{η} + ∑_{i=1}^{N}d^{\frac{λδ}{p}} (x)|ξ_{i}|^{δ} + g(x)),

with g ∈ L^{q′}(Ω), σ(x) is weight function and 0 ≤ η < min(p−1, q−1), 0 ≤ δ < \frac{p−1}{q′}. Because of its definition, f(x, r, ξ) satisfies the growth condition (2.10). Also the hypotheses of Theorem 3.1 are satisfied. Therefore, the problem

∑_{i=1}^{N} ∫_{Ω} (d^{λ}(x)|\frac{∂u}{∂x_{i}} |^{p−2} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{i}} ) dx

= ∫_{Ω} d^{μ/q}(x)(d^{μδ/q}(x)|u|^{η} + ∑_{i=1}^{N}d^{λδ/p}(x)|ξ_{i}|^{δ} + g(x))v dx ,

for all v ∈ W_{0}^{1,p}(Ω, w), has at last one solution.

References

[1] Y. Akdim, E. Azroul, and A. Benkirane, Existence of Solution for Quasilinear Degenerated Elliptic Equations, Electronic J. Diff. Equ., Vol. 2001 No. 71, (2001) pp 1-19.

[2] L. Boccardo, F. Murat, and J. P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Math. Pur. Appl. 152 (1988) 183-196.

[3] L. Boccardo, F. Murat, and J. P. Puel, Résultats d’existences pour certains problèmes elliptiques quasilinéaires, Ann. Scuola. Norm. Sup. Pisa 11 (1984) 213-235.

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

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

[6] A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, (1985).

[7] J. Lions, Quelques méthodes de résolution des problèmes aux limites non linénaires, Dunod, Paris (1969).

[8] T. Kuo and C. Tsai, On the solvability of solution to some quasilinear elliptic problems, Taiwanese Journal of Mathematics Vol. 1, No. 4, pp 547-553, (1997).

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

E-mail address, Y. Akdim: akdimyoussef@yahoo.fr

  E-mail address, E. Azroul: azroul elhoussine@yahoo.fr

E-mail address, M. Rhoudaf: rhoudaf mohamed@yahoo.fr