2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 49–64. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Strongly nonlinear degenerated unilateral problems with L^{1} data ^{∗}

Elhoussine Azroul, Abdelmoujib Benkirane & Ouidad Filali

Abstract In this paper, we study the existence of solutions for strongly nonlinear degenerated unilateral problems associated to nonlinear operators of the form Au + g(x, u, ∇u). Here A is a Leray-Lions operator acting from W_{0}^{1,p}(Ω, w) into its dual, while g(x, s, ξ) is a nonlinear term which has a growth condition with respect to ξ and no growth condition with respect to s, the second term belongs to L^{1}(Ω).

1 Introduction

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 on Ω, i.e. each w_{i}(x) is a measurable a.e. strictly positive on Ω. Let W_{0}^{1,p}(Ω, w) be the weighted Sobolev space associated with the vector w. Let A be a nonlinear operator from W_{0}^{1,p}(Ω, w) into its dual W^{−1,p′}(Ω, w^{∗}), i.e.

Au = −div(a(x, u, ∇u)).

In the non-degenerated case, Bensoussan, Boccardo and Murat have proved in [4], the existence of a solution for the quasilinear equation of the form,

Au + g(x, u, ∇u) = f.

They assume that g is a nonlinearity having natural growth with respect to |∇u| (of order p), and which satisfies the sign-condition and f ∈ W^{−1,p′}(Ω). Recently, in weighted case, Akdim, Azroul and Benkirane have first in [2] extended the last result to weighted Sobolev spaces and in [3] the authors have studied the following degenerated unilateral problem:

⟨Au, v − u⟩ + ∫_{Ω} g(x, u, ∇u)(v − u) dx ≥ ⟨f, v − u⟩ ∀ v ∈ K_{ψ} ∩ L^{∞}(Ω)

u ∈ W_{0}^{1,p}(Ω, w) u ≥ ψ a.e. in Ω
g(x, u, ∇u) ∈ L^{1}(Ω) g(x, u, ∇u)u ∈ L^{1}(Ω),

∗Mathematics Subject Classifications: 35J15, 35J70, 35J85.

Key words: Weighted Sobolev spaces, Hardy inequality,

quasilinear degenerated elliptic operators.

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

Published December 28, 2002.

where K_{ψ} = {v ∈ W_{0}^{1,p}(Ω, w), v ≥ ψ a.e. in Ω}, with ψ a measurable function on Ω such that ψ^{+} ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) and f ∈ W^{−1,p′}(Ω, w^{∗}). The purpose of this paper, is to study the previous problems for f ∈ L^{1}(Ω). More precisely, we prove the existence theorem for the following degenerated unilateral problem:

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) dx ≥ ⟨f, T_{k}(v − u)⟩
for all v ∈ K_{ψ} and all k > 0
u ∈ W_{0}^{1,p}(Ω, w) u ≥ ψ a.e. in Ω
g(x, u, ∇u) ∈ L^{1}(Ω).

(1.1)

For that, we assume in addition that the nonlinearity g satisfies some coercivity conditions (see (2.10)).

Concerning the existence result for the degenerated elliptic equations where the second member lies in the dual W^{−1,p′}(Ω, w^{∗}) (resp. for the quasilinear equation where the second member is in L^{1}(Ω)), we refer the reader to [6-7-8](resp. [1-2]).

Remarks

1) Note that the use of the truncation operator in (1.1) is justified by the fact that the solution does not in general belong to L^{∞}(Ω) for f ∈ L^{1}(Ω).

2) An other work in this direction can be found in [5] in non-weighted case.

The paper is organized as follows: Section 2 contains some preliminaries and is concerned with the basic assumptions and some technical lemmas. In section 3, we state and prove main results. The last section is devoted to an example which illustrates our abstract conditions.

2 Preliminaries

Let Ω be a bounded open subset of ℝ^{N} (N ≥ 1), let 1 < p < ∞, and let w = {w_{i}(x), 0 ≤ i ≤ N} be a vector of weight functions, i.e. every component w_{i}(x) is a measurable function which is strictly positive a.e. in Ω satisfying the integrability conditions

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

for 0 ≤ i ≤ N.

We define the weighted space L^{p}(Ω, γ), where γ is a weight function on Ω by

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

with the norm

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

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

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

This set of functions defines 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}.  (2.2)

Since we shall deal with the Dirichlet problem, we shall use the space

X = W_{0}^{1,p}(Ω, w)

defined as the closure of C_{0}^{∞}(Ω) with respect to the norm (2.2). Note that, C_{0}^{∞}(Ω) is dense in W_{0}^{1,p}(Ω, w) and (X, ∥.∥_{1,p,w}) is a reflexive Banach space.

We recall that the dual space of weighted Sobolev space W_{0}^{1,p}(Ω, w) is equivalent to W^{−1,p′}(Ω, w^{∗}), where w^{∗} = {w_{i}^{∗} = w_{i}^{1−p′}, ∀i = 0, …, N}, where p′ is the conjugate of p i.e. p′ = p/(p − 1) (for more details we refer to [7]).

Now we state the following assumptions:

Assumption (H1) The expression

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

is a norm defined on W_{0}^{1,p}(Ω, w) and is equivalent to the norm (2.2). There exists a weight function σ on Ω such that σ ∈ L^{1}(Ω) and σ^{1−q′} ∈ L^{1}(Ω) for some parameter q, so that 1 < q < p + p′ such that the Hardy inequality,

( ∫_{Ω} |u(x)|^{q}σ(x) dx)^{\frac{1}{q}} ≤ c( ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx)^{1/p},  (2.3)

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

W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω, σ),  (2.4)

determined by the inequality (2.3) is compact. Note that (W_{0}^{1,p}(Ω, w), ∥|.|∥) is a uniformly convex and thus reflexive Banach space. Let A be a nonlinear operator from W_{0}^{1,p}(Ω, w) into its dual W^{−1,p′}(Ω, w^{∗}) defined by

Au = −div(a(x, u, ∇u)),

where a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory vector-function satisfying for a.e x ∈ Ω, for all s ∈ ℝ and ξ, η in ℝ^{N} with ξ ≠ η.

Assumption (H2)

|a_{i}(x, s, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + σ^{1/p′}|s|^{\frac{q}{p′}} + ∑_{j=1}^{N} w_{j}^{1/p′}(x)|ξ_{j}|^{p−1}], i = 1, …, N, (2.4)

[a(x, s, ξ) − a(x, s, η)](ξ − η) > 0,  (2.5)
a(x, s, ξ).ξ ≥ α ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p},  (2.6)

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

Assumption (H3) Let g(x, s, ξ) be a Carathéodory function satisfying

g(x, s, ξ)s ≥ 0  (2.7)
|g(x, s, ξ)| ≤ b(|s|)( ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p} + c(x)),  (2.8)

|g(x, s, ξ)| ≥ ρ_{2} ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p} for |s| > ρ_{1}  (2.9)

where b : ℝ^{+} → ℝ^{+} is a continuous increasing function and c(x) is a positive function which lies in L^{1}(Ω), c ≥ 0 and ρ_{1} > 0, ρ_{2} > 0. We consider,

f ∈ L^{1}(Ω).  (2.10)

Now we recall some lemmas which will be used later.

Lemma 2.1 (cf. [2]) Let g ∈ L^{r}(Ω, γ) and let g_{n} ∈ L^{r}(Ω, γ), with ∥g_{n}∥_{r,γ} ≤ c (1 < r < ∞). If g_{n}(x) → g(x) a.e. in Ω, then g_{n} ⇀ g weakly in L^{r}(Ω, γ), where γ is a weight function on Ω.

Lemma 2.2 (cf. [1]) Assume that (H1) holds. Let u ∈ W_{0}^{1,p}(Ω, w), and let T_{k}(u), k ∈ ℝ^{+}, be the usual truncation then T_{k}(u) ∈ W_{0}^{1,p}(Ω, w). Moreover, we have

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

Lemma 2.3 (cf. [2]) Assume that (H1) and (H2) are satisfied, and let (u_{n}) be a sequence of 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 strongly in W_{0}^{1,p}(Ω, w).

Lemma 2.4 (cf. [1]) Assume that (H1) holds, let (u_{n}) ∈ W_{0}^{1,p}(Ω, w) such that u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w),then T_{k}u_{n} ⇀ T_{k}u weakly in W_{0}^{1,p}(Ω, w).

3 Main result

Let ψ be a measurable function with values in ℝ such that,

ψ^{+} ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω).  (3.1)

and let

K_{ψ} = {v ∈ W_{0}^{1,p}(Ω, w) v ≥ ψ a.e. in Ω}.

Consider the nonlinear problem with Dirichlet boundary condition,

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) dx ≥ ⟨f, T_{k}(v − u)⟩
for all v ∈ K_{ψ} and all k > 0
u ∈ W_{0}^{1,p}(Ω, w) u ≥ ψ a.e. in Ω
g(x, u, ∇u) ∈ L^{1}(Ω),

(3.2)

where u is the solution of this problem.

Theorem 3.1 Under the assumptions (H1)-(H3), (2.10) and (3.1), there exists at least one solution of (3.2).

Remarks

1) Theorem 3.1 generalizes to weighted case the analogous in [5].

2) In the particular case when w_{0}(x) ≡ 1, we can replace (H1) by the conditions: There exists s ∈]\frac{N}{p} , ∞[∩[\frac{1}{p−1}, ∞[ such that w_{i}^{−s} ∈ L^{1}(Ω) for all i = 1, …, N, (which is an integrability condition, stronger than (2.1)), since

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

is a norm defined on W_{0}^{1,p}(Ω, w) and equivalent to (2.2) and also the following imbeddings hold:

W_{0}^{1,p}(Ω, w) ↪ L^{q}(Ω)

for 1 ≤ q < p_{1}^{∗}, if ps < N(s + 1), and q ≥ 1 is arbitrary if ps ≥ N(s + 1), where p_{1} = ps/(s + 1) and p_{1}^{∗} is devoted the Sobolev conjugate of p_{1} (for more details see [2,7]). Hence the hypotheses (H1) is satisfied for σ ≡ 1.

Proof of Theorem 3.1 Consider the sequence of approximate problems:

⟨Au_{n}, T_{k}(v − u_{n})⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(v − u_{n}) dx ≥ ⟨f_{n}, T_{k}(v − u_{n})⟩

for all v ∈ K_{ψ} ∩ L^{∞}(Ω) and all k > 0
u_{n} ∈ W_{0}^{1,p}(Ω, w) u_{n} ≥ ψ a.e. in Ω
g(x, u_{n}, ∇u_{n}) ∈ L^{1}(Ω),

(3.3) where f_{n} is a sequence of smooth functions which converges strongly to f in L^{1}(Ω) with ∥f_{n}∥_{L^{1}(Ω)} ≤ C. For some constant C. By Theorem 6.1 and Lemma 6.2 of [1] or via Theorem 4.1 of [3] there exists at least one solution u_{n} of (3.3). In order to pass to the limit in the approximate problem (3.3), we claim that: Assertion(1)

(u_{n})_{n} is bounded in W_{0}^{1,p}(Ω, w)  (3.4)

Assertion(2)

∇T_{k}(u_{n}) → ∇T_{k}(u) strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i})  (3.5)

which implies ∇u_{n} → ∇u a.e in Ω Assertion(3)

g(x, u_{n}, ∇u_{n}) → g(x, u, ∇u) strongly in L^{1}(Ω)  (3.6)

We can pass to the limit in the approximate problems (3.3), indeed,

∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(v − u_{n}) dx + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(v − u_{n}) dx

≥ ∫_{Ω} f_{n}T_{k}(v − u_{n}) dx

For all v ∈ K_{ψ} ∩ L^{∞}(Ω) and k > 0. From (2.5) and (3.4) we deduce that a(x, u_{n}, ∇u_{n}) is bounded in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}). Using (3.5) we obtain

∇u_{n} → ∇u a.e in Ω.  (3.7)

Hence, we get

a(x, u_{n}, ∇u_{n}) → a(x, u, ∇u) a.e in Ω  (3.8)

which implies with Lemma 2.1 that

a(x, u_{n}, ∇u_{n}) ⇀ a(x, u, ∇u) weakly in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}).  (3.9)

On the other hand, let v ∈ L^{∞}(Ω) and set h = k + ||v||_{∞}, then

|\frac{∂T_{k}(v − u_{n})}{∂x_{i}}|w_{i}^{1/p} = (χ_{|v−u_{n}|≤k}|\frac{∂(v − u_{n})}{∂x_{i}}|)w_{i}^{1/p}
≤ χ_{|u_{n}|≤k+||v||_{∞}}(|\frac{∂v}{∂x_{i}} | + |\frac{∂u_{n}}{∂x_{i}} |)w_{i}^{1/p}
≤ |\frac{∂v}{∂x_{i}} |w_{i}^{1/p} + |\frac{∂T_{h}(u_{n})}{∂x_{i}}|w_{i}^{1/p}

for i = 1, …, N. Which implies by using the Vitali’s theorem with (3.5) and (3.7) that

∇T_{k}(v − u_{n}) → ∇T_{k}(v − u) strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i})  (3.10)

for any v ∈ W_{0}^{1,p}(Ω, w)∩L^{∞}(Ω). From (3.9) and (3.10) we can pass to the limit in the first term of (3.3). Since g(x, u_{n}, ∇u_{n}) → g(x, u, ∇u) strongly in L^{1}(Ω) and f_{n} → f strongly in L^{1}(Ω), then we can pass to the limit in

⟨A(u_{n}), T_{k}(v − u_{n})⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(v − u_{n}) dx ≥ ∫_{Ω} f_{n}T_{k}(v − u_{n}) dx.

This allows to consider the problem

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) dx ≥ ⟨f, T_{k}(v − u)⟩
for all v ∈ K_{ψ} ∩ L^{∞}(Ω) and all k > 0
u ∈ W_{0}^{1,p}(Ω, w) u ≥ ψ a.e. in Ω
g(x, u, ∇u) ∈ L^{1}(Ω),

(3.11)

Set ϕ = T_{m}(v) as a test function, where m ≥ ∥ψ^{+}∥_{∞} and v ∈ K_{ψ}, then ϕ ∈ K_{ψ} ∩ L^{∞}(Ω). Multiplying (3.11) by ϕ, we obtain

\begin{matrix}⟨Au, T_{k}(T_{m}(v_{m}) − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(T_{m}(v_{m}) − u) dx \\ ≥ ∫_{Ω} fT_{k}(T_{m}(v_{m}) − u) dx \end{matrix}  (3.12)

From Lemma 2.2 and using the Vitali’s theorem, we have

∇T_{k}(T_{m}(v) − u) → ∇T_{k}(v − u) strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}).

Finally, passing to the limit in (3.12) as m tends to infinity, we obtain:

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) dx ≥ ∫_{Ω} fT_{k}(v − u) dx

for any v ∈ W_{0}^{1,p}(Ω, w), v ≥ ψ a.e. in Ω.

Proof of assertion 1: We consider the sequence of approximate problems (3.3). By Theorem 6.1 and Lemma 6.2 of [1], there exists at least one solution u_{n} of (3.3). Let v = ψ^{+} as test function in (3.3), then

⟨Au_{n}, T_{k}(ψ^{+} − u_{n})⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(ψ^{+} − u_{n}) dx ≥ ∫_{Ω} f_{n}T_{k}(ψ^{+} − u_{n}) dx.

Since u_{n} − ψ^{+} and u_{n} have the same sign, we obtain by using (2.8) and ∥f_{n}∥_{L^{1}(Ω)} ≤ C,

∫_{|u_{n}−ψ^{+}|≤k} a(x, u_{n}, ∇u_{n})∇(u_{n} − ψ^{+}) dx ≤ ∫_{Ω} f_{n}T_{k}(u_{n} − ψ^{+}) dx ≤ kC (3.13)

56 Strongly nonlinear degenerated unilateral problems which implies that

∫_{|u_{n}−ψ^{+}|≤k} a(x, u_{n}, ∇u_{n})∇u_{n} dx ≤ Ck + ∫_{|u_{n}−ψ^{+}|≤k} a(x, u_{n}, ∇u_{n})|∇ψ^{+}| dx using Young’s inequality, we obtain

∫ _{|u_{n}−ψ^{+}|≤k}a(x, u_{n}, ∇u_{n})∇u_{n} dx
≤Ck + ∑_{i=1}^{N} ∫_{|u_{n}−ψ^{+}|≤k} \frac{η^{p′}}{p′} |a_{i}(x, u_{n}, ∇u_{n})|^{p′}w_{i}^{1−p′} dx

+ ∑_{i=1}^{N} ∫_{|u_{n}−ψ^{+}|≤k} \frac{1}{p} \frac{1}{η^{p}} w_{i}|\frac{∂ψ^{+}}{∂x_{i}} |^{p} dx,

where η is a positive constant. From (2.5) we have

∫ _{|u_{n}−ψ^{+}|≤k}a(x, u_{n}, ∇u_{n})∇u_{n} dx
≤C_{1} + \frac{η^{p′}}{p′} β^{p′}N ∫_{Ω} k^{p′}(x) dx + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} σ|u_{n}|^{q} dx

+ \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx

≤C_{2} + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}|≤||ψ^{+}||_{∞}+k} σ|u_{n}|^{q} dx + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx
≤C_{2} + \frac{η^{p′}}{p′} β^{p′}N(||ψ^{+}||_{∞} + k)^{q} ∫_{Ω} σ dx + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx.

Consequently, using (2.7) and since σ ∈ L^{1}(Ω), we have

∫_{|u_{n}−ψ^{+}|≤k} α ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx ≤ C_{3} + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx,

we choose 0 < η < \frac{1}{β} (\frac{αp′}{N})^{1/p′}, this implies

∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx ≤ C  (3.14)

On the other hand, from (3.13) and a(x, s, ξ)ξ ≥ 0

∫Ωg(x, u_{n}, ∇u_{n})T_{k}(u_{n} − ψ^{+}) dx
≤k ∫_{Ω} |f_{n}| dx − ∫_{|u_{n}−ψ^{+}|≤k} a(x, u_{n}, ∇u_{n})∇(u_{n} − ψ^{+}) dx
≤C_{1} + ∫_{|u_{n}−ψ^{+}|≤k} a(x, u_{n}, ∇u_{n})|∇ψ^{+}| dx.

As in the proof of (3.14) we can show that

∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(u_{n} − ψ^{+}) dx

≤ C_{2} + \frac{η^{p′}}{p′} β^{p′}N ∫_{|u_{n}−ψ^{+}|≤k} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx ≤ C_{3},

where C_{1}, C_{2} and C_{3} are positive constants. When |u_{n} − ψ^{+}| > k, we have T_{k}(u_{n}−ψ^{+}) = +k(or −k), and since T_{k}(u_{n}−ψ^{+}), u_{n}−ψ^{+}, u_{n} and g(x, u_{n}, ∇u_{n}) have the same sign, we obtain

∫_{|u_{n}−ψ^{+}|≤k} g(x, u_{n}, ∇u_{n})T_{k}(u_{n} − ψ^{+}) dx

+ ∫_{|u_{n}−ψ^{+}|>k} g(x, u_{n}, ∇u_{n})T_{k}(u_{n} − ψ^{+}) dx ≤ C_{3}

which gives

k ∫_{|u_{n}−ψ^{+}|>k} |g(x, u_{n}, ∇u_{n})| dx ≤ C_{3}.

From (2.10), we have

|g(x, u_{n}, ∇u_{n})| ≥ ρ_{2} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} for |u_{n}| ≥ ρ_{1}

Choosing k > ρ_{1} + ||ψ^{+}||_{∞}, then |u_{n} − ψ^{+}| > k implies |u_{n}| > ρ_{1}. We deduce that

∫_{|u_{n}−ψ^{+}|>k} ρ_{2} ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p} dx ≤ C_{4}.  (3.15)

Finally, combining (3.14) and (3.15) we have |||u_{n}||| ≤ C.

Proof of assertion 2 Let k ≥ ∥ψ^{+}∥ and δ = (\frac{b(k)}{2α})^{2}. Set φ(s) = se^{δs^{2}}, z_{n} = T_{k}(u_{n}) − T_{k}(u), η = e^{−4δk^{2}}, and v_{n} = u_{n} − ηφ(z_{n}). By the choice of k, the above test function is admissible for (3.3). Multiplying (3.3) by v_{n}, for h > 0, we obtain

⟨A(u_{n}), T_{h}(ηφ(z_{n}))⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{h}(ηφ(z_{n})) dx ≤ ∫_{Ω} f_{n}T_{h}(ηφ(z_{n})) dx.

Choosing h > 2k, we have |ηφ(z_{n})| ≤ |z_{n}| ≤ 2k < h and

⟨A(u_{n}), φ(z_{n})⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})φ(z_{n}) dx ≤ ∫_{Ω} f_{n}φ(z_{n}) dx.

As n → ∞, we have f_{n} → f strongly in L^{1}(Ω), and φ(z_{n}) ⇀ 0 weak^{∗} in L^{∞}(Ω). Hence we have

∫_{Ω} f_{n}φ(z_{n}) dx → 0.

58 Strongly nonlinear degenerated unilateral problems Since g(x, u_{n}, ∇u_{n})φ(z_{n}) ≥ 0 on the subset |u_{n}(x)| > k, this implies

⟨A(u_{n}), φ(z_{n})⟩ + ∫_{|u_{n}|≤k} g(x, u_{n}, ∇u_{n})φ(z_{n}) dx ≤ ε(n),  (3.16)

where ε(n) is a real number which converge to zero when n tends to infinity. On the other hand

⟨A(u_{n}), φ(z_{n})⟩ = ∫_{|u_{n}|≤k} a(x, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx
+ ∫_{|u_{n}|>k} a(x, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx
= ∫_{Ω} a(x, u_{n}, ∇u_{n})(∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx
− ∫_{|u_{n}|>k} a(x, u_{n}, ∇u_{n})∇T_{k}(u)φ′(z_{n}) dx
= ∫_{Ω}(a(x, u_{n}, ∇T_{k}(u_{n})) − a(x, u_{n}, ∇T_{k}(u)))(∇T_{k}(u_{n})
− ∇T_{k}(u))φ′(z_{n}) dx
+ ∫_{Ω} a(x, u_{n}, ∇T_{k}(u)))(∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx
− ∫_{|u_{n}|>k} a(x, u_{n}, ∇u_{n})∇T_{k}(u)φ′(z_{n}) dx.

Since ∇T_{k}(u)χ_{{|u_{n}|>k}} → 0 strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}), and from (2.3), (2.5) and (3.4) we have (a(x, u_{n}, ∇u_{n})φ′(z_{n}))_{n} is bounded in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}), then

− ∫_{|u_{n}|>k} a(x, u_{n}, ∇u_{n})∇T_{k}(u)φ′(z_{n}) dx = ε(n) → 0 as n → ∞. Moreover, since u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w), by Lemma 2.3 we have

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

Then

∇T_{k}(u_{n}) ⇀ ∇T_{k}(u) weakly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}).

Since the sequence (a(x, u_{n}, ∇T_{k}(u))φ′(z_{n}))_{n} converges strongly in the space ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}),

∫_{Ω} a(x, u_{n}, ∇T_{k}(u))(∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx = ε(n) → 0 as n → ∞, and

\begin{aligned}⟨A(u_{n}), φ(z_{n})⟩ = & ∫_{Ω}(a(x, u_{n}, ∇T_{k}(u_{n})) − a(x, u_{n}, ∇T_{k}(u))) \\ & × (∇T_{k}(u_{n}) − ∇T_{k}(u))φ′(z_{n}) dx + ε(n). \end{aligned}  (3.17)

Elhoussine Azroul, Abdelmoujib Benkirane & Ouidad Filali 59 On the other hand

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

− a_{i}(x, u_{n}, ∇T_{k}(u))](\frac{∂T_{k}(u_{n})}{∂x_{i}} − \frac{∂T_{k}(u)}{∂x_{i}})|φ(z_{n})|\frac{b(k)}{α} dx

+ ∫_{|u_{n}|≤k} ∑_{i=1}^{N} a_{i}(x, u_{n}, ∇T_{k}(u))(\frac{∂T_{k}(u_{n})}{∂x_{i}} − \frac{∂T_{k}(u)}{∂x_{i}})|φ(z_{n})|\frac{b(k)}{α} dx

+ ∫_{|u_{n}|≤k} ∑_{i=1}^{N} a_{i}(x, u_{n}, ∇T_{k}(u_{n}))\frac{∂T_{k}(u)}{∂x_{i}}|φ(z_{n})|\frac{b(k)}{α} dx.

Since ∇T_{k}(u_{n}) ⇀ ∇T_{k}(u) weakly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}) and

a(x, u_{n}, ∇T_{k}(u))|φ(z_{n})| → 0 strongly in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}),

it follows that

∫_{Ω} ∑_{i=1}^{N} a_{i}(x, u_{n}, ∇T_{k}(u))(\frac{∂T_{k}(u_{n})}{∂x_{i}} − \frac{∂T_{k}(u)}{∂x_{i}})|φ(z_{n})|\frac{b(k)}{α} dx = ε(n).

From (2.5) and (3.4), (a(x, u_{n}, ∇T_{k}(u_{n})))_{n} converges weakly in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}). Since ∇T_{k}(u)|φ(z_{n})|\frac{b(k)}{α} → 0 strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}),

∫_{Ω} ∑_{i=1}^{N} a_{i}(x, u_{n}, ∇T_{k}(u_{n}))\frac{∂T_{k}(u)}{∂x_{i}})|φ(z_{n})|\frac{b(k)}{α} dx = ε(n).

Moreover, since ∫_{|u_{n}|≤k} b(k)c(x)|φ(z_{n})| dx = ε(n), we have

| ∫ _{|u_{n}|≤k}g(x, u_{n}, ∇u_{n})φ(z_{n}) dx|

≤ ∫_{Ω} ∑_{i=1}^{N} [a_{i}(x, u_{n}, ∇T_{k}(u_{n})) − a_{i}(x, u_{n}, ∇T_{k}(u))]

× (\frac{∂T_{k}(u_{n})}{∂x_{i}} − \frac{∂T_{k}(u)}{∂x_{i}})|φ(z_{n})|\frac{b(k)}{α} dx + ε(n)

which with (3.16) and (3.17) gives

∫_{Ω} [a(x, u_{n}, ∇T_{k}(u_{n})) − a(x, u_{n}, ∇T_{k}(u))]

× (∇T_{k}(u_{n}) − ∇T_{k}(u))(φ′(z_{n}) − \frac{b(k)}{α}|φ(z_{n})|) dx ≤ ε(n).

Choosing, δ ≥ (b(k)/(2α))^{2}, we obtain for all s ∈ ℝ

φ′(s) − \frac{b(k)}{α}|φ(s)| ≥ \frac{1}{2};

thus,

\frac{1}{2} ∫_{Ω}[a(x, u_{n}, ∇T_{k}(u_{n})) − a(x, u_{n}, ∇T_{k}(u))](∇T_{k}(u_{n}) − ∇T_{k}(u)) dx ≤ ε(n).

Then

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

as n → ∞. Moreover, T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w) and in view of Lemma 2.3, we have T_{k}(u_{n}) → T_{k}(u) as n → ∞ strongly in W_{0}^{1,p}(Ω, w); hence,

∇T_{k}(u_{n}) → ∇T_{k}(u) strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i}).

Consequently, there exists a subsequence still denoted by (u_{n})_{n} such that, ∇u_{n} → ∇u a.e in Ω.

Proof of assertion 3 From (3.5) we deduce that

g(x, u_{n}, ∇u_{n}) → g(x, u, ∇u) a.e in Ω.  (3.18)

For any measurable subset E of Ω and any m > 0 we have

\begin{matrix}∫_{E} |g(x, u_{n}, ∇u_{n})| dx \\ ≤ ∫_{E∩\{|u_{n}|≤m\}} |g(x, u_{n}, ∇u_{n})| dx + ∫_{E∩\{|u_{n}|>m\}} |g(x, u_{n}, ∇u_{n})| dx. \end{matrix} (3.19)

By (2.9), (3.5) and by using Vitali’s theorem, we have for ε > 0 there exists ρ(ε, m) > 0 such that for ρ(ε, m) > |E| we have

∫_{E∩\{|u_{n}|≤m\}} |g(x, u_{n}, ∇u_{n})| dx ≤ \frac{ε}{2} ∀n.  (3.20)

Now let v_{n} = u_{n} − S_{m}(u_{n}) where for m > 1,

S_{m}(s) = \begin{cases}0 & |s| ≤ m − 1 \\ \frac{s}{|s|} & |s| ≥ m \\ S′_{m}(s) = 1 & m − 1 ≤ |s| ≤ m. \end{cases}

Note that: If u_{n} ≤ m − 1, we have S_{m}(u_{n}) ≤ 0 and v_{n} ≥ u_{n} ≥ ψ; if u_{n} ≥ m − 1, we have 0 ≤ S_{m}(u_{n}) ≤ 1 and

u_{n} − S_{m}(u_{n}) ≥ u_{n} − 1 ≥ m − 2 ≥ ψ for m ≥ 2 + ||ψ^{+}||_{∞}.

Then, v_{n} is admissible for (3.3). So, multiplying (3.3) by v_{n} we obtain

⟨A(u_{n}), T_{k}(S_{m}(u_{n}))⟩ + ∫_{Ω} g(x, u_{n}, ∇u_{n})T_{k}(S_{m}(u_{n})) dx ≤ ∫_{Ω} f_{n}T_{k}(S_{m}(u_{n})) dx.

Which by choosing k ≥ 1 implies

∫_{Ω} a(x, u_{n}, ∇u_{n})∇u_{n}s′_{m}(u_{n}) dx + ∫_{Ω} g(x, u_{n}, ∇u_{n})S_{m}(u_{n}) dx

≤ ∫_{Ω} f_{n}S_{m}(u_{n}) dx,

i.e,

∫_{m−1≤|u_{n}|≤m} a(x, u_{n}, ∇u_{n})∇u_{n} dx + ∫_{|u_{n}|>m−1} g(x, u_{n}, ∇u_{n})S_{m}(u_{n}) dx

≤ ∫_{|u_{n}|>m−1} |f_{n}| dx.

Since a(x, u_{n}, ∇u_{n})∇u_{n} ≥ 0,

∫_{|u_{n}|>m−1} g(x, u_{n}, ∇u_{n})S_{m}(u_{n}) dx ≤ ∫_{|u_{n}|>m−1} |f_{n}| dx.

Since S_{m}(u_{n}) and u_{n} have the same sign,

∫_{|u_{n}|>m−1} |g(x, u_{n}, ∇u_{n})||S_{m}(u_{n})| dx ≤ ∫_{|u_{n}|>m−1} |f_{n}| dx

and

∫_{|u_{n}|>m} |g(x, u_{n}, ∇u_{n})| dx ≤ ∫_{|u_{n}|>m−1} |f_{n}| dx.

Since f_{n} → f strongly in L^{1}(Ω) and since |{|u_{n}| > m − 1}| → 0 uniformly in n when m → ∞ (due to the fact that σ^{1−q′} ∈ L^{1}(Ω)), there exists m(ε) > 1 such that

∫_{|u_{n}|>m−1} |f_{n}| dx ≤ \frac{ε}{2} ∀n.

Then

∫_{|u_{n}|>m} |g(x, u_{n}, ∇u_{n})| dx ≤ \frac{ε}{2} ∀n.  (3.21)

From (3.19), (3.20) and (3.21), we have

∫_{E} |g(x, u_{n}, ∇u_{n})| dx ≤ ε ∀n.  (3.22)

Then (g(x, u_{n}, ∇u_{n}))_{n} is equi-integrable. Thanks to (3.18), (3.22) and Vitali’s theorem yields,

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

4 Example

The following example is closely inspired from the one used in [1,2]. Let Ω be a bounded domain of ℝ^{N}(N ≥ 1) satisfying the cone condition and let ψ be a real valued measurable function such that ψ^{+} ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω). Let us consider the Carathéodory functions

a_{i}(x, s, ξ) = w_{i}|ξ_{i}|^{p−1}sgn(ξ) for i = 0, …, N

and

g(x, s, ξ) = ρs|s|^{r} ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p} with ρ > 0,

where w_{i}(x) are a given weight functions on Ω satisfying:

w_{i}(x) ≡ some weight function w(x) in Ω for all i = 0, …, N.

Then, we consider the Hardy inequality (2.3) in the form,

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

It is easy to show that the functions a_{i}(x, s, ξ) satisfy the growth condition (2.5) and the coercivity (2.7). Also the Carathéodory function g(x, s, ξ) satisfies the conditions (2.8), (2.9) and (2.10), in fact, concerning (2.10) we have,

|g(x, s, ξ)| = ρ|s|^{r+1} ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p}.

Then

|g(x, s, ξ)| ≥ ρ|ρ|^{r+1} ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p} for |s| > ρ_{1} ≥ 1.

Choosing for example ρ_{1} = 1 and ρ_{2} = ρ > 0. On the other hand, the monotonicity condition is satisfied. In fact,

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

= w(x) ∑_{i=1}^{N}(|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. In particular, let us use the special weight functions w and σ expressed in terms of the distance to the boundary ∂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)

(iii) For q > 1,

\frac{1}{q′ − 1} > μ > −1.  (4.3)

Corollary 4.1 If f ∈ L^{1}(Ω), the following problem:

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

+ ∫_{Ω} ρu|u|^{r} ∑_{i=1}^{N} d^{λ}(x)|\frac{∂u}{∂x_{i}} |^{p}T_{k}(v − u) dx ≥ ∫_{Ω} fT_{k}(v − u) dx.

u ∈ W_{0}^{1,p}(Ω, d^{λ}), u ≥ ψ, ρu|u|^{r} ∑_{i=1}^{N} d^{λ}(x)|\frac{∂u}{∂x_{i}} |^{p} ∈ L^{1}(Ω)

v ∈ W_{0}^{1,p}(Ω, d^{λ}), for all v ≥ ψ and all k > 0,

has at least one solution.

Remarks

1) Note that conditions (4.1) and (4.2) are sufficient to show the compact imbedding (2.4) (cf. [6, example 1], [8, example 1.5] and [10, theorem 19.17, 19.22]).

2) Condition (4.3) is sufficient for (2.3) to hold (cf. [9 p.p 40-41]).

References

[1] Y. Akdim, Sur certains problèmes quasi-linéaires et unilatéraux dégénérés ou singuliers, Thèse de Doctorat, Université de Fès, (2002).

[2] Y. Akdim, E. Azroul and A. Benkirane, Existence of solutions for quasilinear degenerated elliptic equations, Electronic J. Diff. Eqns., vol. 2001 N 71 (2001) 1-19.

[3] Y. Akdim, E. Azroul and A. Benkirane, Existence of solutions for quasilinear degenerated elliptic unilateral problem, to appear in J. Math. Blaise-Pascal.

[4] A. Bensoussan, L. Boccardo and F. Murat, On a non linear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. Henri Poincaré 5 N 4 (1988), 347-364.

[5] A. Benkirane and A. Elmahi, Strongly nonlinear elliptic unilateral problems having natural growth terms and L^{1} data, Rendiconti di Matematica, Serie VII vol. 18, (1998), 289-303.

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

[7] P. Drabek, A. Kufner and F. Nicolosi, Quasilinear elliptic equations with degenerations and singularities, De Gruyter Series in Nonlinear Analysis and Applications, Berlin New York, (1997).

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

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

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

Elhoussine Azroul (e-mail: elazroul@caramail.com) Abdelmoujib Benkirane (e-mail: abdelmoujib@iam.net.ma ) Ouidad Filali (e-mail: ouidadf@hotmail.com) Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P. 1796 Atlas, Fès, Maroc.