2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 25–39. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Strongly nonlinear degenerated elliptic unilateral problems via convergence of truncations ^{∗}

Youssef Akdim, Elhoussine Azroul, & Abdelmoujib Benkirane

Abstract

We prove an existence theorem for a strongly nonlinear degenerated elliptic inequalities involving nonlinear operators of the form Au+g(x, u, ∇u). Here A is a Leray-Lions operator, g(x, s, ξ) is a lower order term satisfying some natural growth with respect to |∇u|. There is no growth restrictions with respect to |u|, only a sign condition. Under the assumption that the second term belongs to W^{−1,p′}(Ω, w^{∗}), we obtain the main result via strong convergence of truncations.

1 Introduction

Let Ω be a bounded open set of ℝ^{N} and p a real number such that 1 < p < ∞. Let 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 function on Ω, satisfying some integrability conditions (see section 2). The aim of this paper, is to prove an existence theorem for unilateral degenerate problems associated to a nonlinear operators of the form Au+g(x, u, ∇u). Where A is a Leray-Lions operator from W_{0}^{1,p}(Ω, w) into its dual W^{−1,p′}(Ω, w^{∗}), defined by,

Au = − div(a(x, u, ∇u))

and where g is a nonlinear lower order term having natural growth with respect to |∇u|. With respect to |u| we do not assume any growth restrictions, but we assume a sign condition. Bensoussan, Boccardo and Murat have proved in the second part of [2] the existence of at least one solution of the unilateral problem

⟨Au, v − u⟩ + ∫_{Ω} g(x, u, ∇u)(v − u) dx ≥ ⟨f, v − u⟩ for all v ∈ K_{ψ}
u ∈ W_{0}^{1,p}(Ω) u ≥ ψ a.e.
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, variational ineqality,

strongly nonlinear degenerated elliptic operators, truncations.

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

Published December 28, 2002.

where f ∈ W^{−1,p′}(Ω) and K_{ψ} = {v ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω), v ≥ ψ a.e. Here ψ is a measurable function on Ω such that ψ^{+} ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω). For that the authors obtain the existence results by proving that the positive part u_{ε}^{+} (resp. u_{ε}^{−}) of u_{ε} strongly converges to u^{+}(resp. u^{−}) in W_{0}^{1,p}(Ω), where u_{ε} is a solution of the approximate problem. In the present paper, we study the variational degenerated inequalities. More precisely, we prove the existence of a solution for the problem (3.3) (see section 3), by using another approach based on the strong convergence of the truncations T_{k}(u_{ε}) in W_{0}^{1,p}(Ω, w). Moreover, in this paper, we assume only the weak integrability condition σ^{1−q′} ∈ L_{loc}^{1}(Ω) (see (2.11) below) instead of the stronger σ^{1−q′} ∈ L^{1}(Ω) as in [1]. This can be done by approximating Ω by a sequence of compact sets Ω_{ε}. Note that, in the non weighted case the same result is proved in [3] where f ∈ L^{1}(Ω). Let us point out that other works in this direction can be found in [6, 1].

This paper is organized as follows: Section 2 contains some preliminaries and basic assumptions. In section 3 we state and prove our main results.

2 Preliminaries and basic assumption

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. Further, we suppose in all our considerations that for 0 ≤ i ≤ N,

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

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 satisfies

\frac{∂u}{∂x_{i}} ∈ L^{p}(Ω, w_{i}) for all 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)

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

X = W_{0}^{1,p}(Ω, w)  (2.4)

defined as the closure of C_{0}^{∞}(Ω) with respect to the norm (2.3). 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 spaces 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′ = \frac{p}{p−1} (for more details we refer to [5]).

Definition 2.1 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 form

B(u) = B(u, u),  (2.5)

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

∀u ∈ Y, v → B(u, v) is bounded hemicontinuous from Y into Y^{∗}

and (B(u, u) − B(u, v), u − v) ≥ 0, (2.6)

∀v ∈ Y, u → B(u, v) is bounded hemicontinuous from Y into Y^{∗}, (2.7)

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^{∗}, ∀v ∈ Y,

(2.8)

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). (2.9)

Definition 2.2 Let Y be a reflexive Banach space, a bounded mapping B from Y to Y^{∗} is called pseudo-monotone if for any sequence u_{n} ∈ Y with u_{n} ⇀ u weakly in Y and lim sup_{n→∞}⟨Bu_{n}, u_{n} − u⟩ ≤ 0, one has

lim inf_{n→∞}⟨Bu_{n}, u_{n} − v⟩ ≥ ⟨Bu, u − v⟩ for all v ∈ Y.

We start by stating the following assumptions:

Assumption (H1) The expression

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

is a norm on X and it is equivalent to the norm (2.3). There exist a weight function σ on Ω and a parameter q, such that

1 < q < p + p′,  (2.10)
σ^{1−q′} ∈ L_{loc}^{1}(Ω),  (2.11)

with q′ = \frac{q}{q−1}. The Hardy inequality,

( ∫_{Ω} |u(x)|^{q}σ dx)^{1/q} ≤ c( ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx)^{1/p},  (2.12)

holds for every u ∈ X with a constant c > 0 independent of u. Moreover, the imbedding

X ↪↪ L^{q}(Ω, σ),  (2.13)

expressed by the inequality (2.12) is compact.

Note that (X, ∥|.|∥_{X}) is a uniformly convex (and thus reflexive) Banach space.

Remark 2.1 If we assume that w_{0}(x) ≡ 1 and in addition the integrability condition: There exists ν ∈]\frac{N}{p} , ∞[∩[\frac{1}{p−1}, ∞[ such that

w_{i}^{−ν} ∈ L^{1}(Ω)

for all i = 1, …, N (which is stronger than (2.2)). Then

∥|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 is equivalent to (2.3). Moreover

W_{0}^{1,p}(Ω, w) ↪↪ L^{q}(Ω),

for all 1 ≤ q < p_{1}^{∗} if pν < N(ν + 1) and for all q ≥ 1 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} (see [5]). Thus the hypotheses (H1) is verified for σ ≡ 1 and for all 1 < q < min{p_{1}^{∗}, p+p′} if pν < N(ν + 1) and for all 1 < q < p + p′ if pν ≥ N(ν + 1).

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 the following assumptions:

Assumption (H2)

|a_{i}(x, s, ξ)| ≤ βw_{i}^{1/p}(x)[k(x) + σ^{\frac{1}{p′}} |s|^{\frac{q}{p′}} + ∑_{j=1}^{N} wj^{\frac{1}{p′}} (x)|ξ_{j}|^{p−1}] for i = 1, …, N, (2.14)

[a(x, s, ξ) − a(x, s, η)](ξ − η) > 0, for all ξ ≠ η ∈ ℝ^{N},  (2.15)
a(x, s, ξ).ξ ≥ α ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p},  (2.16)

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 the following assumptions:

g(x, s, ξ)s ≥ 0  (2.17)
|g(x, s, ξ)| ≤ b(|s|)( ∑_{i=1}^{N} w_{i}|ξ_{i}|^{p} + c(x)),  (2.18)

where b : ℝ^{+} → ℝ^{+} is a continuous increasing function and c(x) is a positive function which lies in L^{1}(Ω). Now we recall some lemmas introduced in [1] which will be used later.

Lemma 2.1 (cf. [1]) 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 F : ℝ → ℝ be uniformly Lipschitzian, with F (0) = 0. Let u ∈ W_{0}^{1,p}(Ω, w). Then F (u) ∈ W_{0}^{1,p}(Ω, w). Moreover, if the set D of discontinuity points of F′ is finite, then

\frac{∂(F ∘ u)}{∂x_{i}} = \begin{cases}F′(u)\frac{∂u}{∂x_{i}} & a.e. in \{x ∈ Ω : u(x) ∉ D\} \\ 0 & a.e. in \{x ∈ Ω : u(x) ∈ D\}. \end{cases}

Lemma 2.3 (cf. [1]) Assume that (H1) holds. Let u ∈ W_{0}^{1,p}(Ω, w), and let T_{k}(u), k ∈ ℝ^{+}, is 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 2.4 Assume that (H1) holds. 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). Then, T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w)

Proof. Since u_{n} ⇀ u weakly in W_{0}^{1,p}(Ω, w) and by (2.13) we have for a subsequence u_{n} → u strongly in L^{q}(Ω, σ) and a.e. in Ω. On the other hand,

∥|T_{k}(u_{n})|∥_{X}^{p} = ∑_{i=1}^{N} ∫_{Ω} |\frac{∂T_{k}(u_{n})}{∂x_{i}}|^{p}w_{i} = ∑_{i=1}^{N} ∫_{Ω} |T′_{k}(u_{n})\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i}

≤ ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u_{n}}{∂x_{i}} |^{p}w_{i} = ∥|u_{n}|∥_{X}^{p}.

Then (T_{k}(u_{n})) is bounded in W_{0}^{1,p}(Ω, w), hence by using (2.13), T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w).

Lemma 2.5 (cf. [1]) 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).

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)

Set

K_{ψ} = {v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) v ≥ ψ a.e.}.  (3.2)

Note that (3.1) implies K_{ψ} ≠ ∅. Consider the nonlinear problem with Dirichlet boundary conditions,

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

(3.3)

Then, the following result can be proved for a solution u of this problem.

Theorem 3.1 Assume that the assumptions (H1)–(H3) and (3.1) hold and let f ∈ W^{−1,p′}(Ω, w^{∗}). Then there exists at least one solution of (3.3).

Remark 3.1 1) Theorem 3.1 can be generalized in weighted case to an analogous statement in [2].

2) Note that in [1] the authors have assumed that σ^{1−q′} ∈ L^{1}(Ω) which is stronger than (2.11).

In the proof of theorem 3.1 we need the following lemma.

Lemma 3.1 Assume that f lies in W^{−1,p′}(Ω, w^{∗}). If u is a solution of (P), then, u is also a solution of the variational inequality

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) dx ≥ ⟨f, T_{k}(v − u)⟩ ∀k > 0,

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

(3.4)

Conversely, if u is a solution of (3.4) then u is a solution of (3.3).

The proof of this lemma is similar to the proof of [3, Remark 2.2] for the non weighted case.

Proof of theorem 3.1 Step (1) The approximate problem and a priori estimate. Let Ω_{ε} be a sequence of compact subsets of Ω such that Ω_{ε} increases to Ω as ε → 0. We consider the sequence of approximate problems,

\begin{matrix}⟨Au_{ε}, v − u_{ε}⟩ + ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})(v − u_{ε}) dx ≥ ⟨f, v − u_{ε}⟩ \\ v ∈ W_{0}^{1,p}(Ω, w) v ≥ ψ a.e. \\ u_{ε} ∈ W_{0}^{1,p}(Ω, w) u_{ε} ≥ ψ a.e. \end{matrix}  (3.5)

where,

g_{ε}(x, s, ξ) = \frac{g(x, s, ξ)}{1 + ε|g(x, s, ξ)|}χ_{Ω_{ε}}(x),

and where χ_{Ω_{ε}} is the characteristic function of Ω_{ε}. Note that g_{ε}(x, s, ξ) satisfies the following conditions,

g_{ε}(x, s, ξ)s ≥ 0, |g_{ε}(x, s, ξ)| ≤ |g(x, s, ξ)| and |g_{ε}(x, s, ξ)| ≤ \frac{1}{ε}.

We define the operator G_{ε} : X → X^{∗} by,

⟨G_{ε}u, v⟩ = ∫_{Ω} g_{ε}(x, u, ∇u)v dx.

Thanks to Hölder’s inequality we have for all u ∈ X and v ∈ X,

\begin{matrix}| ∫_{Ω} g_{ε}(x, u, ∇u)v dx| ≤ & ( ∫_{Ω} |g_{ε}(x, u, ∇u)|^{q′}σ^{− \frac{q′}{q}} dx)^{1/q′}( ∫_{Ω} |v|^{q}σ dx)^{1/q} \\ ≤ & \frac{1}{ε}( ∫_{Ω_{ε}} σ^{1−q′} dx)^{1/q′}∥v∥_{q,σ} ≤ c_{ε}∥|v|∥. \end{matrix}

(3.6)

The last inequality is due to (2.11) and (2.13).

Lemma 3.2 The operator B_{ε} = A + G_{ε} from X into its dual X^{∗} is pseudomonotone. Moreover, B_{ε} is coercive, in the sense that: There exists v_{0} ∈ K_{ψ} such that

\frac{⟨B_{ε}v, v − v_{0}⟩}{∥|v|∥} → +∞ as ∥|v|∥ → ∞, v ∈ K_{ψ}.

The proof of this lemma will be presented below. In view of lemma 3.2, (3.5) has a solution by the classical result (cf. Theorem 8.1 and Theorem 8.2 chapter 2 [7]).

With v = ψ^{+} as test function in (3.5), we deduce that ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})(u_{ε} − ψ^{+}) ≥ 0, then, ⟨Au_{ε}, u_{ε}⟩ ≤ ⟨f, u_{ε} − ψ^{+}⟩ + ⟨Au_{ε}, ψ^{+}⟩, i.e.,

∫_{Ω} a(x, u_{ε}, ∇u_{ε})∇u_{ε} dx ≤ ⟨f, u_{ε} − ψ^{+}⟩ + ∑_{i=1}^{N} ∫_{Ω} a_{i}(x, u_{ε}, ∇u_{ε})\frac{∂ψ^{+}}{∂x_{i}} dx,

then,

α ∑_{i=1}^{N} ∫_{Ω} w_{i}|\frac{∂u_{ε}}{∂x_{i}} |^{p} dx

= α∥|u_{ε}|∥^{p}
≤ ∥f∥_{X^{∗}}(∥|u_{ε}|∥ + ∥|ψ^{+}|∥) +

+ ∑_{i=1}^{N} ( ∫_{Ω} |a_{i}(x, u_{ε}, ∇u_{ε})|^{p′}w_{i}^{1−p′} dx)^{\frac{1}{p′}} ( ∫_{Ω} |\frac{∂ψ^{+}}{∂x_{i}} |^{p}w_{i} dx)^{1/p}

≤ ∥f∥_{X^{∗}}(∥|u_{ε}|∥ + ∥|ψ^{+}|∥) +

+c ∑_{i=1}^{N} ( ∫_{Ω}(k^{p′} + |u_{ε}|^{q}σ + ∑_{j=1}^{N} |\frac{∂u_{ε}}{∂x_{j}} |^{p}w_{j}) dx)^{1/p′}∥|ψ^{+}|∥.

Using (2.13) the last inequality becomes,

α∥|u_{ε}|∥^{p} ≤ c_{1}∥|u_{ε}|∥ + c_{2}∥|u_{ε}|∥^{\frac{q}{p′}} + c_{3}∥|u_{ε}|∥^{p−1} + c_{4},

where c_{i} are various positive constants. Then, thanks to (2.10) we can deduce that u_{ε} remains bounded in W_{0}^{1,p}(Ω, w), i.e.,

∥|u_{ε}|∥ ≤ β_{0},  (3.7)

where β_{0} is some positive constant. Extracting a subsequence (still denoted by u_{ε}) we get

u_{ε} ⇀ u weakly in X and a.e. in Ω.

Note that u ≥ ψ a.e. Step (2) Strong convergence of T_{k}(u_{ε}). Thanks to (3.7) and (2.13) we can extract a subsequence still denoted by u_{ε} such that

u_{ε} ⇀ u weakly in W_{0}^{1,p}(Ω, w,)

u_{ε} → u a.e. in Ω. (3.8)

Let k > 0 by lemma 2.4 we have

T_{k}(u_{ε}) ⇀ T_{k}(u) weakly in W_{0}^{1,p}(Ω, w) as ε → 0.  (3.9)

Our objective is to prove that

T_{k}(u_{ε}) → T_{k}(u) strongly in W_{0}^{1,p}(Ω, w) as ε → 0.  (3.10)

Fix k > ∥ψ^{+}∥_{∞}, and use the notation z_{ε} = T_{k}(u_{ε}) − T_{k}(u). We use, as a test function in (3.5),

v_{ε} = u_{ε} − ηφ_{λ}(z_{ε})  (3.11)

where φ_{λ}(s) = se^{λs^{2}} and η = e^{−4λk^{2}}. Then we can check that v_{ε} is admisible test function. So that

−⟨Au_{ε}, ηφ_{λ}z_{ε}⟩ − ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})ηφ_{λ}(z_{ε}) dx ≥ −⟨f, ηφ_{λ}(z_{ε})⟩

which implies that

⟨Au_{ε}, φ_{λ}(z_{ε})⟩ + ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})φ_{λ}(z_{ε}) dx ≤ ⟨f, φ_{λ}(z_{ε})⟩.  (3.12)

Since φ_{λ}(z_{ε}) is bounded in X and converges a.e. in Ω to zero and using (2.13), we have φ_{λ}(z_{ε}) ⇀ 0 weakly in X as ε → 0. Then

η_{1}(ε) = ⟨f, φ_{λ}(z_{ε})⟩ → 0,  (3.13)

and since g_{ε}(x, u_{ε}, ∇u_{ε})φ_{λ}(z_{ε}) ≥ 0 in the subset {x ∈ Ω : |u_{ε}(x)| ≥ k} hence (3.12) and (3.13) yield

⟨Au_{ε}, φ_{λ}(z_{ε})⟩ + ∫_{{|u_{ε}|≤k}} g_{ε}(x, u_{ε}, ∇u_{ε})φ_{λ}(z_{ε}) dx ≤ η_{1}(ε).  (3.14)

We study each term in the left hand side of (3.14). We have,

⟨Au_{ε}, φ_{λ}(z_{ε})⟩ = ∫_{Ω} a(x, u_{ε}, ∇u_{ε})∇(T_{k}(u_{ε}) − T_{k}(u))φ′_{λ}(z_{ε}) dx
= ∫_{Ω} a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε}))∇(T_{k}(u_{ε}) − T_{k}(u))φ′_{λ}(z_{ε}) dx
− ∫_{{|u_{ε}|>k}} a(x, u_{ε}, ∇u_{ε})∇T_{k}(u)φ′_{λ}(z_{ε}) dx
= ∫_{Ω} (a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε})) − a(x, T_{k}(u_{ε}), ∇T_{k}(u))) ∇(T_{k}(u_{ε})
− T_{k}(u))φ′_{λ}(z_{ε}) dx + η_{2}(ε),
(3.15)

where,

\begin{aligned}η_{2}(ε) = & ∫_{Ω} a(x, T_{k}(u_{ε}), ∇T_{k}(u))∇(T_{k}(u_{ε}) − T_{k}(u))φ′_{λ}(z_{ε}) dx \\ & − ∫_{\{|u_{ε}|>k\}} a(x, u_{ε}, ∇u_{ε})∇T_{k}(u)φ′_{λ}(z_{ε}) dx, \end{aligned}

34 Strongly nonlinear degenerated elliptic unilateral problems which converges to 0 as ε → 0. On the other hand,

| ∫_{{|u_{ε}|≤k}} g_{ε}(x, u_{ε}, ∇u_{ε})φ_{λ}(z_{ε}) dx|
≤ ∫_{\{|u_{ε}|≤k\}} b(k)[c(x) + ∑_{i=1}^{N} |\frac{∂u_{ε}}{∂x_{i}} |^{p}w_{i}]|φ_{λ}(z_{ε})| dx
≤b(k) ∫_{\{|u_{ε}|≤k\}} c(x)|φ_{λ}(z_{ε})| dx + \frac{b(k)}{α} ∫_{\{|u_{ε}|≤k\}} a(x, u_{ε}, ∇u_{ε})∇u_{ε}|φ_{λ}(z_{ε})| dx
=η_{3}(ε) + \frac{b(k)}{α} ∫_{Ω} a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε}))∇T_{k}(u_{ε})|φ_{λ}(z_{ε})| dx
=\frac{b(k)}{α} ∫_{Ω} (a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε})) − a(x, T_{k}(u_{ε}), ∇T_{k}(u))) ∇(T_{k}(u_{ε})
− T_{k}(u))|φ_{λ}(z_{ε})| dx + η_{4}(ε)
(3.16)

where

η_{3}(ε) = b(k) ∫_{{|u_{ε}|≤k}} c(x)|φ_{λ}(z_{ε})| dx → 0 as ε → 0

and

η_{4}(ε) = η_{3}(ε) + \frac{b(k)}{α} ∫_{Ω} a(x, T_{k}(u_{ε}), ∇T_{k}(u))∇(T_{k}(u_{ε}) − T_{k}(u))|φ_{λ}(z_{ε})| dx

+\frac{b(k)}{α} ∫_{Ω} a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε}))∇T_{k}(u)|φ_{λ}(z_{ε})| dx → 0 as ε → 0.

Note that, when λ ≥ (\frac{b(k)}{2α})^{2} we have

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

Which combining with (3.14),(3.15) and (3.16) one obtains

∫_{Ω} (a(x, T_{k}(u_{ε}), ∇T_{k}(u_{ε})) − a(x, T_{k}(u_{ε}), ∇T_{k}(u)))∇(T_{k}(u_{ε}) − T_{k}(u)) dx

≤ η_{5}(ε) = 2(η_{1}(ε) − η_{2}(ε) + η_{4}(ε)) → 0 as ε → 0.

Finally lemma 2.5 implies (3.10) for any fixed k ≥ ∥ψ∥_{∞}. Step (3) Passage to the limit. In view of (3.10) we have for a subsequence,

∇u_{ε} → ∇u a.e. in Ω,  (3.17)

which with (3.8) imply, a(x, u_{ε}, ∇u_{ε}) → a(x, u, ∇u) a.e. in Ω,

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

(3.18)

On the other hand, thanks to (2.14) and (3.7) we have a(x, u_{ε}, ∇u_{ε}) is bounded in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}) then by lemma 2.1 we obtain

a(x, u_{ε}, ∇u_{ε}) ⇀ a(x, u, ∇u) weakly in ∏_{i=1}^{N} L^{p′}(Ω, w_{i}^{∗}).  (3.19)

We shall prove that,

g_{ε}(x, u_{ε}, ∇u_{ε}) → g(x, u, ∇u) strongly in L^{1}(Ω).  (3.20)

By (3.18), to apply Vitali’s theorem it suffices to prove that g_{ε}(x, u_{ε}, ∇u_{ε}) is uniformly equi-integrable. Indeed, thanks to (2.17), (3.6) and (3.7) we obtain,

0 ≤ ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})u_{ε} dx ≤ c_{0},  (3.21)

where c_{0} is some positive constant. For any measurable subset E of Ω and any m > 0 we have,

∫_{E} |g_{ε}(x, u_{ε}, ∇u_{ε})| dx = ∫_{E∩X_{m}^{ε}} |g_{ε}(x, u_{ε}, ∇u_{ε})| dx + ∫_{E∩Y_{m}^{ε}} |g_{ε}(x, u_{ε}, ∇u_{ε})| dx

where,

X_{m}^{ε} = {x ∈ Ω, |u_{ε}(x)| ≤ m}, Y_{m}^{ε} = {x ∈ Ω, |u_{ε}(x)| > m}.

From these expressions, (2.18), and (3.21), we have

\begin{matrix}_{∫E} & |g_{ε}(x, u_{ε}, ∇u_{ε})| dx \\ = & ∫_{E∩X_{m}^{ε}} |g_{ε}(x, u_{ε}, ∇T_{m}(u_{ε}))| dx + ∫_{E∩Y_{m}^{ε}} |g_{ε}(x, u_{ε}, ∇u_{ε})| dx \\ ≤ & ∫_{E∩X_{m}^{ε}} |g_{ε}(x, u_{ε}, ∇T_{m}(u_{ε}))| dx + \frac{1}{m} ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})u_{ε} dx \\ ≤ & b(m) ∫_{E}(∑_{i=1}^{N} w_{i}|\frac{∂T_{m}(u_{ε})}{∂x_{i}}|^{p} + c(x)) + \frac{c_{0}}{m}. \end{matrix}  (3.22)

Since the sequence (∇T_{m}(u_{ε})) strongly converges in ∏_{i=1}^{N} L^{p}(Ω, w_{i}), then (3.22) implies the equi-integrability of g_{ε}(x, u_{ε}, ∇u_{ε}).

Moreover, since g_{ε}(x, u_{ε}, ∇u_{ε})u_{ε} ≥ 0 a.e. in Ω, then by (3.18), (3.21) and Fatou’s lemma, we have g(x, u, ∇u)u ∈ L^{1}(Ω). On the other hand, for v ∈ L^{∞}(Ω), set h = k + ∥v∥_{∞}, then

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

which implies, using Vitali’s theorem with (3.10) and (3.17) that

∇T_{k}(v − u_{ε}) → ∇T_{k}(v − u) strongly in ∏_{i=1}^{N} L^{p}(Ω, w_{i})  (3.23)

for any v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω). Thanks to lemma 3.1 and from (3.19), (3.20) and (3.23) we can pass to the limit in

⟨Au_{ε}, T_{k}(v − u_{ε})⟩ + ∫_{Ω} g_{ε}(x, u_{ε}, ∇u_{ε})T_{k}(v − u_{ε}) ≥ ⟨f, T_{k}(v − u_{ε})⟩

and we obtain,

⟨Au, T_{k}(v − u)⟩ + ∫_{Ω} g(x, u, ∇u)T_{k}(v − u) ≥ ⟨f, T_{k}(v − u)⟩  (3.24)

for any v ∈ W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) and for all k > 0.

Taking for any v ∈ W_{0}^{1,p}(Ω, w) and v ≥ ψ the test function T_{m}(v) which belongs to W_{0}^{1,p}(Ω, w) ∩ L^{∞}(Ω) for m ≥ ∥ψ^{+}∥_{∞} and passing to the limit in (3.24) as m → ∞, then u is a solution of (3.4). Using again lemma 3.1 we obtain the desired result, i.e., u is a solution of (3.3).

Proof of lemma 3.2 By proposition 2.6 chapter 2 [7], it is sufficient to show that B_{ε} is of the calculus of variations type in the sense of definition 2.1. Indeed put,

b_{1}(u, v, w̃) = ∑_{i=1}^{N} ∫_{Ω} a_{i}(x, u, ∇v)∇w̃ dx, b_{2}(u, w̃) = ∫_{Ω} g_{ε}(x, u, ∇u)w̃ dx.

Then the mapping w̃ ↦ b_{1}(u, v, w̃) + b_{2}(u, w̃) is continuous in X. Then

b_{1}(u, v, w̃) + b_{2}(u, w̃) = b(u, v, w̃) = ⟨B_{ε}(u, v), w̃⟩, B_{ε}(u, v) ∈ W^{−1,p′}(Ω, w^{∗})

and we have

B_{ε}(u, u) = B_{ε}u.

Using (2.14) and Hölder’s inequality we can show that A is bounded as in [4], and thanks to (3.6) B_{ε} is bounded. Then, it is sufficient to check (2.6)-(2.9).

Next we show that (2.6) and (2.7) are true. By (2.15) we have,

(B_{ε}(u, u) − B_{ε}(u, v), u − v) = b_{1}(u, u, u − v) − b_{1}(u, v, u − v) ≥ 0.

The operator v → B_{ε}(u, v) is bounded hemicontinuous. Indeed, we have

a_{i}(x, u, ∇(v_{1}+λv_{2})) → a_{i}(x, u, ∇v_{1}) strongly in L^{p′}(Ω, w_{i}^{∗}) as λ → 0. (3.25)

On the other hand, (g_{ε}(x, u_{1} + λu_{2}, ∇(u_{1} + λu_{2})))_{λ} is bounded in L^{q′}(Ω, σ^{1−q′}) and g_{ε}(x, u_{1} + λu_{2}, ∇(u_{1} + λu_{2})) → g_{ε}(x, u_{1}, ∇u_{1}) a.e. in Ω, hence lemma 2.1 gives

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

(3.26)

Using (3.25) and (3.26) we can write

b(u, v_{1} + λv_{2}, w̃) → b(u, v_{1}, w̃) as λ → 0 ∀u, v_{i}, w̃ ∈ X.

Similarly we can prove (2.7). Proof of assertion (2.8). Assume that u_{n} ⇀ u weakly in X and (B(u_{n}, u_{n}) − B(u_{n}, u), u_{n} − u) → 0. We have,

(B(u_{n}, u_{n}) − B(u_{n}, u), u_{n} − u)

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

then, by lemma 2.5, u_{n} → u strongly in X, which gives

b(u_{n}, v, w̃) → b(u, v, w̃) ∀w̃ ∈ X,

i.e., B_{ε}(u_{n}, v) ⇀ B_{ε}(u, v) weakly in X^{∗}. It remains to prove (2.9). Assume that

u_{n} ⇀ u weakly in X  (3.27)

and that

B(u_{n}, v) ⇀ ψ weakly in X^{∗}.  (3.28)

Thanks to (2.13), (2.14) and (3.27) we obtain,

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

then,

b_{1}(u_{n}, v, u_{n}) → b_{1}(u, v, u).  (3.29)

On the other hand, by Hölder’s inequality,

|b_{2}(u_{n}, u_{n} − u)| ≤ (∫_{Ω} |g_{ε}(x, u_{n}, ∇u_{n})|^{q′}σ^{\frac{−q′}{q}} dx)^{1/q′} (∫_{Ω} |u_{n} − u|^{q}σ dx)^{1/q}

≤\frac{1}{ε} (∫_{Ω_{ε}} σ^{\frac{−q′}{q}} dx)^{1/q′} ∥u_{n} − u∥_{L^{q}(Ω,σ)} → 0 as n → ∞,

i.e.,

b_{2}(u_{n}, u_{n} − u) → 0 as n → ∞,  (3.30)

but in view of (3.28) and (3.29) we obtain

b_{2}(u_{n}, u) = (B_{ε}(u_{n}, v), u) − b_{1}(u_{n}, v, u) → (ψ, u) − b_{1}(u, v, u)

and from (3.30) we have b_{2}(u_{n}, u_{n}) → (ψ, u) − b_{1}(u, v, u). Then,

(B_{ε}(u_{n}, v), u_{n}) = b_{1}(u_{n}, v, u_{n}) + b_{2}(u_{n}, u_{n}) → (ψ, u).

Now show that B_{ε} is coercive. Let v_{0} ∈ K_{ψ}. From Hölder’s inequality, the growth condition (2.14) and the compact imbedding (2.13) we have

⟨Av, v_{0}⟩ = ∑_{i=1}^{N} ∫_{Ω} a_{i}(x, v, ∇v)\frac{∂v_{0}}{∂x_{i}} dx

≤ ∑_{i=1}^{N} ( ∫_{Ω} |a_{i}(x, v, ∇v)|^{p′}wi^{\frac{−p′}{p}} dx)^{\frac{1}{p′}} ( ∫_{Ω} |\frac{∂v_{0}}{∂x_{i}} |^{p}w_{i} dx)^{1/p}

≤ c_{1}∥|v_{0}|∥( ∫_{Ω} k(x)^{p′} + |v|^{q}σ + ∑_{j=1}^{N} |\frac{∂v}{∂x_{j}} |^{p}w_{j} dx)^{\frac{1}{p′}}
≤ c_{2}(c_{3} + ∥|v|∥^{\frac{q}{p′}} + ∥|v|∥^{p−1}),

where c_{i} are various constants. Thanks to (2.16), we obtain

\frac{⟨Av, v⟩}{∥|v|∥} − \frac{⟨Av, v_{0}⟩}{∥|v|∥} ≥ α∥|v|∥^{p−1} − ∥|v|∥^{p−2} − ∥|v|∥^{\frac{q}{p′} −1} − \frac{c}{∥|v|∥}.

In view of (2.10) we have p − 1 > \frac{q}{p′} − 1. Then,

\frac{⟨Av, v − v_{0}⟩}{∥|v|∥} → ∞ as ∥|v|∥ → ∞.

Since ⟨G_{ε}v, v⟩ ≥ 0 and ⟨G_{ε}v, v_{0}⟩ is bounded, we have

\frac{⟨B_{ε}v, v − v_{0}⟩}{∥|v|∥} ≥ \frac{⟨Av, v − v_{0}⟩}{∥|v|∥} − \frac{⟨G_{ε}v, v_{0}⟩}{∥|v|∥} → ∞ as ∥|v|∥ → ∞.

Remark 3.2 Assumption (2.10) appears to be necessary to prove the boundedness of (u_{ε})_{ε} in W_{0}^{1,p}(Ω, w) and the coercivity of the operator B_{ε}. While Assumption (2.11) is necessary to prove the boundedness of G_{ε} in W_{0}^{1,p}(Ω, w). Thus, when g ≡ 0, we don’t need to assume (2.11).

References

[1] 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.

[2] 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.

[3] 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.

[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] P. Drabek and F. Nicolosi, Existence of Bounded Solutions for Some Degenerated Quasilinear Elliptic Equations, Annali di Mathematica pura ed applicata (IV), Vol. CLXV (1993), pp. 217-238.

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

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