2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE AND REGULARITY OF ENTROPY SOLUTIONS FOR SOME NONLINEAR ELLIPTIC EQUATIONS

LAHSEN AHAROUCH, ELHOUSSINE AZROUL

Abstract. This paper concerns the existence and regularity of entropy solutions to the Dirichlet problem

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

u = 0 on ∂Ω.

In particular, we show the Lq̄-regularity of the solution to this boundary-value problem.

1. Introduction

Let Ω be a bounded open subset of ℝ^{N} (N ≥ 2), and let p be a real number such that 2 − \frac{1}{N} < p ≤ N. Consider a Leray Lions operator

Au = − div(a(x, u, ∇u)),

where a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function satisfying for a.e. x ∈ Ω, all s ∈ ℝ and all ξ ≠ ξ̄ ∈ ℝ^{N} the conditions

|a(x, s, ξ)| ≤ β[c(x) + |s|^{p−1} + |ξ|^{p−1}]  (1.1)
a(x, s, ξ).ξ ≥ α|ξ|^{p}  (1.2)
⟨a(x, s, ξ) − a(x, s, ξ̄), ξ − ξ̄⟩ > 0.  (1.3)

Here α > 0, β ≥ 0 and c(x) ∈ L^{p′}(Ω). In the present paper, we study the boundaryvalue problem

Au := − div a(x, u, ∇u) = f − div ϕ(u) in Ω

u = 0 on ∂Ω, (1.4)

where the right hand side is assumed to satisfy

f ∈ L^{1}(Ω),  (1.5)
ϕ ∈ C^{0}(ℝ, ℝ^{N}).  (1.6)

2000 Mathematics Subject Classification. 35J60.

Key words and phrases. Orlicz Sobolev spaces; boundary value problems; Entropy solution.

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

Published October 15, 2004.

Recall that, since no growth hypothesis is assumed on the function ϕ, the term div ϕ(u) may be meaningless, even as a distribution for a function v ∈ W_{0}^{1,r}(Ω), r > 1 (see [4] and [7]). Definition A function u is called an entropy solution of the Dirichlet problem (1.4) if,

u ∈ W_{0}^{1,q}(Ω), 1 < q < q̄ = \frac{N(p − 1)}{N − 1},

T_{k}(u) ∈ W_{0}^{1,p}(Ω), ∀k > 0,
∫_{Ω} a(x, u, ∇u)∇T_{k}(u − φ) dx ≤ ∫_{Ω} fT_{k}(u − φ) dx + ∫_{Ω} ϕ(u)∇T_{k}(u − φ) dx,
∀φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω),

where T_{k}(s) is the truncation operator at height k > 0 defined on ℝ.

When ϕ = 0 and f is a bounded Radon measure, it is known that (1.4) admits a weak solution u in W_{0}^{1,q}(Ω) with 1 < q < \overline{q}; see for example [5, 6, 9]. It also have been shown there, that if f lies in the Orlicz space LLogL(Ω), then the critical regularity W_{0}^{1,\overline{q}}(Ω) is attained. Further contributions in this sense can be founded in the work [3] where the authors have replaced the hypotheses (1.1) and (1.2) by some general assumptions.

When ϕ ≠ 0 and f ∈ L^{1}(Ω), L. Boccardo proved in [4, Theorem 2.1] that the boundary-value problem (1.4) admits an entropy solution (in the sense of the definition 1.7) which belongs to W_{0}^{1,q}(Ω), 1 < q < \overline{q}. Moreover, the author showed that if f ∈ LLog(1 + L)(Ω), then the solution belongs to W_{0}^{1,\overline{q}}(Ω).

Our objective in this paper, is to prove the existence and Lq̄-regularity of an entropy solution to the boundary value problem (1.4), when ϕ ≠ 0 and f ∈ L^{1}(Ω). This is possible by replacing (1.1)–(1.3) by the following assumption.

There exist two N-functions P, M with P << M; six positive real numbers α, δ, k_{1}, k_{2}, k_{3}, k_{4}; and a function C in E\overline{M} such that

|a(x, s, ζ)| ≤ C(x) + k_{1}\overline{P}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ζ|)  (1.7)
⟨a(x, s, ζ) − a(x, s, ξ), ζ − ξ⟩ > 0  (1.8)
a(x, s, ζ)ζ ≥ αM(\frac{|ζ|}{δ}),  (1.9)

for a.e. x ∈ Ω, for all s ∈ ℝ, and all ξ ∈ ℝ^{N}

2. Preliminaries

Let M : ℝ^{+} → ℝ^{+} be an N-function, i.e. M is continuous, convex, with M(t) > 0 for t > 0, \frac{M(t)}{t} → 0 as t → 0, and \frac{M(t)}{t} → ∞ as t → ∞. Equivalently, M admits the representation:

M(t) = ∫_{0}^{t} a(s) ds

where a : ℝ^{+} → ℝ^{+} is nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) tends to ∞ as t → ∞.

The conjugate of M is also an N-function and it is defined by \overline{M} = ∫_{0}^{t} ā(s) ds, where ā : ℝ^{+} → ℝ^{+} is the function ā(t) = sup{s : a(s) ≤ t}.

An N-function M is said to satisfy the Δ_{2}-condition if, for some k,

M(2t) ≤ kM(t) ∀t ≥ 0.  (2.1)

When (2.1) holds only for t ≥ t_{0} > 0 then M is said to satisfy the Δ_{2} condition near infinity.

We will extend these N-functions into even functions on all ℝ. Moreover, we have the following Young’s inequality

st ≤ M(t) + \overline{M}(s), ∀s, t ≥ 0.

Given two N-functions, we write P << Q to indicate P grows essentially less rapidly than Q; i.e. for each ϵ > 0, \frac{P (t)}{Q(ϵt)} → 0 as t → ∞. This is the case if and only if

lim_{t→∞} \frac{Q^{−1}(t)}{P^{−1}(t)} = 0.

Let Ω be an open subset of ℝ^{N}. The Orlicz class K_{M}(Ω) (resp. the Orlicz space L_{M}(Ω) is defined as the set of (equivalence classes of) real valued measurable functions u on Ω such that

∫_{Ω} M(u(x)) dx < +∞ (resp. ∫_{Ω} M(\frac{u(x)}{λ}) dx < +∞ for some λ > 0).

The set L_{M}(Ω) is Banach space under the norm

∥u∥_{M,Ω} = inf \{λ > 0 : ∫_{Ω} M(\frac{u(x)}{λ}) dx ≤ 1\}

and K_{M}(Ω) is a convex subset of L_{M}(Ω). The closure in L_{M}(Ω) of the set of bounded measurable functions with compact support in \overline{Ω} is denoted by E_{M}(Ω). The dual of E_{M}(Ω) can be identified with L\overline{M}(Ω) by means of the pairing ∫_{Ω} uv dx, and the dual norm of L\overline{M}(Ω) is equivalent to ∥.∥_{\overline{M},Ω}.

We now turn to the Orlicz-Sobolev space, W^{1}L_{M}(Ω) [resp. W^{1}E_{M}(Ω)] is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L_{M}(Ω) [resp. E_{M}(Ω)]. It is a banach space under the norm

∥u∥_{1,M} = ∑_{|α|≤1} ∥D^{α}u∥_{M}.

Thus, W^{1}L_{M}(Ω) and W^{1}E_{M}(Ω) can be identified with subspaces of product of N + 1 copies of L_{M}(Ω). Denoting this product by ∏ L_{M}, we will use the weak topologies σ(∏ L_{M}, ∏ E\overline{M}) and σ(∏ L_{M}, ∏ L\overline{M}). The space W_{0}^{1}E_{M}(Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in W^{1}E_{M}(Ω) and the space W_{0}^{1}L_{M}(Ω) as the σ(∏ L_{M}, ∏ E\overline{M}) closure of D(Ω) in W^{1}L_{M}(Ω).

Let W^{−1}L\overline{M}(Ω) [resp. W^{−1}E\overline{M}(Ω)] denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L\overline{M}(Ω) [resp. E\overline{M}(Ω)]. It is a Banach space under the usual quotient norm.(for more details see [1]).

We recall some lemmas introduced in [2] which will be used later.

Lemma 2.1. A domain Ω has the segment property if for every x ∈ ∂Ω there exists an open set G_{x} and a nonzero vector y_{x} such that x ∈ G_{x} and if z ∈ \overline{Ω} ∩ G_{x}, then z + ty_{x} ∈ Ω for all 0 < t < 1.

Lemma 2.2. Let F : ℝ → ℝ be uniformly Lipschitzian, with F (0) = 0. Let M be an N-function and let u ∈ W^{1}L_{M}(Ω) (resp. W^{1}E_{M}(Ω)). Then F (u) ∈ W^{1}L_{M}(Ω) (resp. W^{1}E_{M}(Ω)). Moreover, if the set D of discontinuity points of F′ is finite, then

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

Lemma 2.3. Let F : ℝ → ℝ be uniformly Lipschitzian, with F (0) = 0. We suppose that the set of discontinuity points of F′ is finite. Let M be an N-function, then the mapping F : W^{1}L_{M}(Ω) → W^{1}L_{M}(Ω) is sequentially continous with respect to the weak* topology σ(∏ L_{M}, ∏ E\overline{M}).

We give now the following lemma which concerns operators of the Nemytskii type in Orlicz spaces (see [2]).

Lemma 2.4. Let Ω be an open subset of ℝ^{N} with finite measure. Let M, P, Q be N-functions such that Q << P, and let f : Ω × ℝ → ℝ be a Carathéodory function such that, for a.e. x ∈ Ω and all s ∈ ℝ:

|f(x, s)| ≤ c(x) + k_{1}P^{−1}M(k_{2}|s|),

where k_{1}, k_{2} are real constants and c(x) ∈ E_{Q}(Ω). Then the Nemytskii operator N_{f} defined by N_{f}(u)(x) = f(x, u(x)) is strongly continuous from P(E_{M}(Ω), \frac{1}{k_{2}} ) = \{u ∈ L_{M}(Ω) : d(u, E_{M}(Ω)) < \frac{1}{k_{2}} \} into E_{Q}(Ω).

3. Main results

In the sequel we assume that Ω is an open bounded subset of ℝ^{N}, N ≥ 2, with the segment property, and that M is an N-functions satisfying the Δ_{2}-condition near infinity. We shall prove the following existence theorems.

Theorem 3.1. Assume that (1.7)–(1.9) hold, 2 − \frac{1}{N} < p < N, f ∈ L^{1}(Ω), ϕ ∈ C^{0}(ℝ, ℝ^{N}), \frac{t^{p}}{M(t)} is nondecreasing near infinity and ∫_{.}^{∞} \frac{t^{p−1}}{M(t)} dt < ∞. Then the problem,

\begin{matrix}T_{k}(u) ∈ W_{0}^{1}L_{M}(Ω), ∀k > 0 \\ ∫_{Ω} a(x, u, ∇u)∇T_{k}(u − φ) dx ≤ ∫_{Ω} fT_{k}(u − φ) dx + ∫_{Ω} ϕ(u)∇T_{k}(u − φ) dx, \\ ∀ φ ∈ W_{0}^{1}L_{M}(Ω) ∩ L^{∞}(Ω) \end{matrix} (3.1)

admits at least one solution u ∈ W_{0}^{1,\overline{q}}(Ω).

When p = N we assume, in addition, that There exists an N-function H such that H(t^{N}) is equivalent to M(t).

Theorem 3.2. Assume that for p = N the above hypothesis hold, (1.7)–(1.9) hold, f ∈ L^{1}(Ω), ϕ ∈ C^{0}(ℝ, ℝ^{N}), ∫_{.}^{∞} \frac{t^{N−1}}{M(t)} dt < ∞ and \frac{t^{N}}{\overline{H}^{−1}(e^{t^{N′}})} remains bounded near

infinity. Then (3.1) admits at least one solution in W_{0}^{1,N}(Ω).

Proof of Theorems 3.1 and 3.2. Step 1 The approximate problem and a priori estimate. Let f_{n} be a sequence in W^{−1}E\overline{M}(Ω) ∩ L^{1}(Ω) such that f_{n} → f in L^{1}(Ω), and ∥f_{n}∥_{1} ≤ ∥f∥_{1}. Consider the approximate problem

\begin{matrix}Au_{n} = f_{n} − div ϕ_{n}(u_{n}) \\ u_{n} ∈ W_{0}^{1}L_{M}(Ω) \end{matrix}  (3.2)

where ϕ_{n}(x) = ϕ(T_{n}(x)). From the work [8], there exists at least one solution u_{n} of the approximate problem (3.2). Moreover, as in [3], there exists a constant C = C(p, α, ∥f∥_{1}) such that

∥∇u_{n}∥_{L\overline{q}(Ω)} ≤ C,

which implies that u_{n} is bounded in W_{0}^{1,\overline{q}}(Ω). Then there exists u ∈ W_{0}^{1,\overline{q}}(Ω) and a subsequence still denoted by u_{n} such that

u_{n} ⇀ u weakly in W_{0}^{1,\overline{q}}(Ω)
u_{n} → u strongly in L\overline{q}(Ω) and a.e. in Ω.

(3.3)

Moreover, the use of T_{k}(u_{n}) as test function in (3.2) implies that the sequence T_{k}(u_{n}) is bounded in W_{0}^{1}L_{M}(Ω), then there exists a subsequence of T_{k}(u_{n}) still denoted by T_{k}(u_{n}) such that

T_{k}(u_{n}) ⇀ T_{k}(u) weakly in W_{0}^{1}L_{M}(Ω) for σ(∏ L_{M}, ∏ E\overline{M})
T_{k}(u_{n}) → T_{k}(u) strongly in E_{M}(Ω) and a.e. in Ω.

(3.4)

Step 2 Convergence of the gradient. Let Ω_{r} = {x ∈ Ω : |∇T_{k}(u(x))| ≤ r} and denote by χ_{r} the characteristic function of Ω_{r}. Clearly, Ω_{r} ⊂ Ω_{r+1} and meas(Ω\Ω_{r}) → 0 as r → ∞. Fix r and let s ≥ r. We have,

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

≤ ∫_{Ω_{s}}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u))][∇T_{k}(u_{n}) − ∇T_{k}(u)] dx

= ∫_{Ω_{s}}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})][∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx

≤ ∫_{Ω}[a(x, T_{k}(u_{n}), ∇T_{k}(u_{n})) − a(x, T_{k}(u_{n}), ∇T_{k}(u)χ_{s})][∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}] dx.

On the other hand, let h > k and M = 4k + h. If one takes w_{n} = T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) as test function in (3.2), it is easy to see that ∇w_{n} = 0 when |u_{n}| > M. We can write

∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))∇w_{n} dx = ∫_{Ω} f_{n}w_{n} dx + ∫_{Ω} ϕ_{n}(u_{n})∇w_{n} dx.

We have

∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx
≥ ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇T_{k}(u)) dx
− ∫_{|u_{n}|>k} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)| dx

= ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}) dx
− ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u) − ∇T_{k}(u)χ_{s}) dx
− ∫_{|u_{n}|>k} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{s} dx
− ∫_{|u_{n}|>k} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))|(|∇T_{k}(u)| − |∇T_{k}(u)|χ_{s}) dx .

Then

∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx
≥ ∫_{Ω} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))(∇T_{k}(u_{n}) − ∇T_{k}(u)χ_{s}) dx
− ∫_{Ω\Ω_{s}} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u) dx
− ∫_{|u_{n}|>k} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{s} dx
− ∫_{Ω\Ω_{s}} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)| dx .

From this inequality, it follows

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

≤ ∫_{|u_{n}|>k} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)|χ_{s} dx

+ ∫_{Ω\Ω_{s}} a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))∇T_{k}(u) dx

+ ∫_{Ω\Ω_{s}} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)| dx

+ ∫_{Ω} f_{n}T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx

+ ∫_{Ω} ϕ_{n}(u_{n}))∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx

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

(3.5)

Now, we study each term of the right hand side of the above inequality. We denote by ε_{i}(t) (i = 1, 2, 3, …) various sequences of real numbers which tends to 0 when t tends to infinity. Remark that a(x, T_{μ}(u_{n}), ∇T_{μ}(u_{n})) is bounded in L\overline{M}(Ω) for all μ > 0. Let ε > 0, we have

M(\frac{|∇T_{k}(u)|χ_{s}χ_{\{|u_{n}|>k\}}}{ε}) ≤ M(\frac{s}{ε}) ∈ L^{1}(Ω)

and

|∇T_{k}(u)|χ_{s}χ_{{|u_{n}|>k}} → 0 a.e.

Then by the Lebesgue dominated convergence theorem we deduce that

|∇T_{k}(u)|χ_{s}χ_{{|u_{n}|>k}} → 0 in L_{M}(Ω),

which implies that the first term in the right hand side of (3.5) tends to 0 as n tends to ∞. Concerning the second and third terms on the right hand side of (3.5), since |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))| and |a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))| are bounded in L\overline{M}(Ω) then there exist two functions φ and ψ in L\overline{M}(Ω) such that

\begin{matrix}|a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))| → φ for σ(L\overline{M}, E_{M}) \\ |a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))| → ψ for σ(L\overline{M}, E_{M}) . \end{matrix}  (3.6)

This implies

∫_{Ω\Ω_{s}} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇T_{k}(u)| dx → ∫_{Ω\Ω_{s}} φ|∇T_{k}(u)| dx  (3.7)

and

∫_{Ω\Ω_{s}} |a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))||∇T_{k}(u)| dx → ∫_{Ω\Ω_{s}} ψ|∇T_{k}(u)| dx .  (3.8)

On the other hand,

lim_{n→∞} ∫_{Ω} f_{n}T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx = ∫_{Ω} fT_{2k}(u − T_{h}(u)) dx = ε_{3}(h)

and, for n large enough, one can write.

∫_{Ω} ϕ_{n}(u_{n})∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx
= ∫_{Ω} ϕ(T_{4k+h}(u_{n}))∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx,

which yields,

lim_{n→∞} ∫_{Ω} ϕ_{n}(u_{n})∇T_{2k}(u_{n} − T_{h}(u_{n}) + T_{k}(u_{n}) − T_{k}(u)) dx

= ∫_{Ω} ϕ(u)∇T_{2k}(u − T_{h}(u) dx = 0.

The right-most term in (3.5) tends to 0: Since a(x, T_{k}(u_{n}), ∇T_{k}(u)χ_{s}) converges strongly to a(x, T_{k}(u), ∇T_{k}(u)χ_{s}) in (E\overline{M}(Ω))^{N}, using Lemma 2.4 while ∇T_{k}(u_{n}) tends weakly to ∇T_{k}(u) by (3.3). We conclude then that

0 ≤ lim sup_{n→∞} ∫_{Ω_{r}} [a(x, T_{k}(u_{n}), ∇T_{k}(u_{n}))
− a(x, T_{k}(u_{n}), ∇T_{k}(u)][∇T_{k}(u_{n}) − ∇T_{k}(u)] dx
≤ ∫_{Ω\Ω_{s}} φ|∇T_{k}(u)| dx + ∫_{Ω\Ω_{s}} ψ|∇T_{k}(u)| dx + ∫_{Ω} fT_{2k}(u − T_{h}(u)) dx .

Letting s and h approach infinity we get,

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

as n → ∞. Passing to a subsequence if necessary, we can assume that

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

a.e. in Ω_{r}. As in [2], we deduce that there exists a subsequence still denoted by u_{n} such that ∇u_{n} → ∇u a.e. in Ω. Step 3 Passage to the limit. Let φ ∈ W_{0}^{1}L_{M}(Ω)∩L^{∞}(Ω), and set M = k+∥φ∥_{∞} with k > 0. We shall prove that

lim inf_{n→∞} ∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx ≥ ∫_{Ω} a(x, u, ∇u)∇T_{k}(u − φ) dx.

We have: If |u_{n}| > M then |u_{n} − φ| > k which implies

a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ)
= a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))(∇u_{n} − ∇φ)χ_{{|u_{n}−φ|≤k}}
= a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))(∇T_{M}(u_{n}) − ∇φ)χ_{{|u_{n}−φ|≤k}}.

Let Ω_{s} = {x ∈ Ω : |∇φ| ≤ s} and denote by χ_{s} the characteristic function of Ω_{s}. Then

∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx
= ∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))(∇T_{M}(u_{n}) − ∇φ)χ_{{|u_{n}−φ|≤k}} dx
= ∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))(∇T_{M}(u_{n}) − ∇φχ_{s})χ_{{|u_{n}−φ|≤k}} dx
− ∫_{Ω} a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))(∇φ − ∇φχ_{s})χ_{{|u_{n}−φ|≤k}} dx,

and

\begin{matrix}∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx \\ ≥ − ∫_{Ω\Ω_{s}} |a(x, T_{M}(u_{n}), ∇T_{M}(u_{n}))||∇φ| dx \\ + ∫_{Ω} [a(x, T_{M}(u_{n}), ∇T_{M}(u_{n})) − a(x, T_{M}(u_{n}), ∇φχ_{s})] \\ × [∇T_{M}(u_{n}) − ∇φχ_{s}]χ_{\{|u_{n}−φ|≤k\}} dx \\ + ∫_{Ω} a(x, T_{M}(u_{n}), ∇φχ_{s})[∇T_{M}(u_{n}) − ∇φχ_{s}]χ_{\{|u_{n}−φ|≤k\}} dx. \end{matrix}  (3.9)

Similarly to the proof of (3.7), the first term in the right hand side of (3.9) is greater than a value ε_{6}(s), which implies

lim inf_{n→∞} ∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx

≥ lim_{n→∞} ∫_{Ω} a(x, T_{M}(u_{n}), ∇φχ_{s})[∇T_{M}(u_{n}) − ∇φχ_{s}]χ_{{|u_{n}−φ|≤k}} dx + ε_{6}(s)

+ ∫_{Ω}[a(x, T_{M}(u), ∇T_{M}(u)) − a(x, T_{M}(u), ∇φχ_{s})]

× [∇T_{M}(u) − ∇φχ_{s}]χ_{{|u−φ|≤k}} dx.

(3.10)

From Lemma 2.4, the first term in the right hand side of (3.10) is equal to

∫_{Ω} a(x, T_{M}(u), ∇φχ_{s})[∇T_{M}(u) − ∇φχ_{s}]χ_{{|u−φ|≤k}} dx + ε_{6}(s),

then

lim inf_{n→∞} ∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx

≥ ∫_{Ω} a(x, T_{M}(u), ∇T_{M}(u))[∇T_{M}(u) − ∇φχ_{s}]χ_{{|u−φ|≤k}} dx + ε_{6}(s).

By letting s → +∞, we obtain

lim inf_{n→∞} ∫_{Ω} a(x, u_{n}, ∇u_{n})∇T_{k}(u_{n} − φ) dx

≥ ∫_{Ω} a(x, T_{M}(u), ∇T_{M}(u))[∇T_{M}(u) − ∇φ]χ_{{|u−φ|≤k}} dx

= ∫_{Ω} a(x, u, ∇u)∇T_{k}(u − φ) dx.

Now taking T_{k}(u_{n} − φ) as test function in (3.7) and passing to the limit we deduce the desired statement. □

Remark 3.3. If M and \overline{M} satisfy the Δ_{2} condition, instead of (1.7) we can assume the condition:

|a(x, s, ξ)| ≤ c(x) + k_{1}\overline{M}^{−1}M(k_{2}|s|) + k_{3}\overline{M}^{−1}M(k_{4}|ξ|).  (3.11)

Then we prove the same result as in Theorems 3.1 and 3.2.

Remark 3.4. If wf belongs to W^{−1}L\overline{M}(Ω) the statements of Theorems 3.1 and 3.2 still hold.

Example. Let 2 − \frac{1}{N} < p ≤ N, (N ≥ 2), and let the N-function be M(t) = t^{p} log^{αp}(e + t) with αp > 1. Then it is easy to verify that M(t) satisfies the condition of Theorems 3.1 and 3.2.

References

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[4] L. Boccardo, Some nonlinear Dirichlet problem in L^{1} involving lower order terms in divergence form, Progress in elliptic and parabolic partial differential equations (Capri, 1994), Pitman Res. Notes Math. Ser., 350, p. 43-57, Longman, Harlow, 1996.

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

E-mail address: lahrouche@caramail.com

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

E-mail address: elazroul@caramail.com