Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 15, 1-12; http://www.math.u-szeged.hu/ejqtde/

Strongly nonlinear problem of infinite order with L^{1} data

A. Benkirane^{∗}, M. Chrif^{∗} and S. El Manouni^{∗∗}

∗University of Fez, Faculty of Sciences DM. B.P. 1796 Atlas-Fez, Fez 30000 Morocco abd.benkirane@gmail.com & moussachrif@yahoo.fr ∗∗Al-Imam University, Faculty of Sciences Department of Mathematics P. O. Box 90950 Riyadh 11623, KSA samanouni@imamu.edu.sa & manouni@hotmail.com

Abstract

In this paper, we prove the existence of solutions for the strongly nonlinear equation of the type

Au + g(x, u) = f

where A is an elliptic operator of infinite order from a functional space of Sobolev type to its dual. g(x, s) is a lower order term satisfying essentially a sign condition on s and the second term f belongs to L^{1}(Ω).

Keywords: Strongly nonlinear problem, infinite order, L^{1} data, monotonicity condition, sign condition. Mathematics Subject Classification: 35J40, 35R30, 35R50.

1 Introduction

In a recent paper, Benkirane, Chrif and El Manouni [5] studied a class of anisotropic problems involving operators of finite and infinite higher order in the variational case. They proved the existence of solutions in generalized Sobolev spaces, also called anisotropic Sobolev spaces. The goal of this paper is to study a strongly nonlinear elliptic equation in which the operator of infinite order

Au = ∑_{|α|=0}^{∞}(−1)^{|α|}D^{α}(A_{α}(x, D^{γ}u)), |γ| ≤ |α|

is involved. Such operators includes as a special case Leray–Lions types in the usual sense. Especially, we consider the strongly nonlinear problem associated to the following differential equations

Au + g(x, u) = f,  (1.1)

for x ∈ Ω, where Ω is a bounded domain in IR^{N} and A is an operator of infinite order defined as above. The functions A_{α} are assumed to satisfy some growth and coerciveness conditions without supposing a monotonicity condition. So that, we prove the existence results and generalize the isotropic case for the problem (1.1). The nonlinear term g has to fulfil only the sign condition g(x, s)s ≥ 0, but we do not assume any growth conditions with respect to |u|. As regards the second member, we suppose that f ∈ L^{1}(Ω). If A is a Leray–Lions operator, let us recall that several studies have been devoted to the investigation of related problems with L^{1} data and a lot of papers have appeared in the classical Sobolev spaces under the additional monotonicity condition (cf. [7, 12, 14, 15]). Let us point out that in the L^{1}− case, a recent work can be found in [11] where the authors have studied some anisotropic problems of finite order of equations of (1.1) type. In this context of Leray–Lions operators, in the variational case (i.e., where f belongs to the dual space), the problem (1.1) has been extensively studied by many authors, we cite in the case of classical Sobolev spaces the works of Webb [17], Brezis and Browder [8]...etc, and the work of Benkirane and Gossez [3] in Sobolev-Orlicz spaces.

2 Preliminaries

Let Ω be a bounded domain in IR^{N}. Further a_{α} ≥ 0, p_{α} > 1 are real numbers for all multi-indices α, and ∥ · ∥_{p_{α}} is the usual norm in the Lebesgue space L^{p_{α}}(Ω). For a positive integer m, we define the following vector of real numbers:

p⃗ = {p_{α}, |α| ≤ m},

and denote p̄ = min{p_{α}, |α| ≤ m}.

Now, let us consider the generalized functional Sobolev space

W^{m,p⃗}(Ω) = {u ∈ L^{p_{0}}(Ω), D^{α}u ∈ L^{p_{α}}(Ω), 0 < |α| ≤ m}

equipped with the norm

∥u∥ = ∑_{|α|=0}^{m} ∥D^{α}u∥_{p_{α}}.  (2.1)

Here D^{α} = \frac{∂^{|α|}}{(∂x_{1})^{α_{1}} … (∂x_{N})^{α_{N}}} . We define the space W_{0}^{m,p⃗}(Ω) as the closure of

C_{0}^{∞}(Ω) in W^{m,p⃗}(Ω) with respect to the norm (2.1). Note that C_{0}^{∞}(Ω) is dense in W_{0}^{m,p⃗}(Ω). Both of W^{m,p⃗}(Ω) and W_{0}^{m,p⃗}(Ω) are separable if 1 ≤ p_{α} < ∞, reflexives if 1 < p_{α} < ∞ for all |α| ≤ m (the proof of this is an adaptation from Adams [1]). W^{−m,p⃗′}(Ω) designs its dual where p⃗′ is the conjugate of p⃗, i.e., p′_{α} = \frac{p_{α}}{p_{α}−1} for all |α| ≤ m.

The Sobolev space of infinite order is the functional space defined by

W^{∞}(a_{α}, p_{α})(Ω) = {u ∈ C^{∞}(Ω) : ρ(u) = ∑_{|α|=0}^{∞} a_{α}∥D^{α}u∥_{p_{α}}^{p_{α}} < ∞}.

We denote by C_{0}^{∞}(Ω) the space of all functions with compact support in Ω with continuous derivatives of arbitrary order.

Since we shall deal with the Dirichlet problem, we shall use the functional space W_{0}^{∞}(a_{α}, p_{α})(Ω) defined by

W_{0}^{∞}(a_{α}, p_{α})(Ω) = {u ∈ C_{0}^{∞}(Ω) : ρ(u) = ∑_{|α|=0}^{∞} a_{α}∥D^{α}u∥_{p_{α}}^{p_{α}} < ∞}.

We say that W_{0}^{∞}(a_{α}, p_{α})(Ω) is a nontrivial space if it contains at least a nonzero function. The dual space of W_{0}^{∞}(a_{α}, p_{α})(Ω) is defined as follows:

W^{−∞}(a_{α}, p′_{α})(Ω) = {h : h = ∑_{|α|=0}^{∞}(−1)^{|α|}a_{α}D^{α}h_{α}, ρ′(h) = ∑_{|α|=0}^{∞} a_{α}∥h_{α}∥_{p′_{α}}^{p′_{α}} < ∞},

where h_{α} ∈ L^{p′_{α}}(Ω) and p′_{α} is the conjugate of p_{α}, i.e., p′_{α} = \frac{p_{α}}{p_{α}−1} (for more details about these spaces, see [13]).

We need the anisotropic Sobolev embeddings result.

lemma 2.1 Let Ω be a bounded open subset of IR^{N}.

If m · \underline{p} < N, then W_{0}^{m,p⃗}(Ω) ⊂ L^{q}(Ω) for all q ∈ [\underline{p}, p^{∗}[ with \frac{1}{p^{∗}} = \frac{1}{\underline{p}} − \frac{m}{N}.

If m · \underline{p} = N, then W_{0}^{m,p⃗}(Ω) ⊂ L^{q}(Ω) for all q ∈ [\underline{p}, +∞[.
If m · \underline{p} > N, then W_{0}^{m,p⃗}(Ω) ⊂ L^{∞}(Ω) ∩ C^{k}(\overline{Ω}) where k = E(m − \frac{N}{\underline{p}}).

Moreover, the embeddings are compacts.

The proof follows immediately from the corresponding embedding theorems in the isotropic case by using the fact that W^{m,p⃗}(Ω) ⊂ W^{m,\underline{p}}(Ω).

3 Main results

We denote by λ_{α} the number of multi-indices γ such that |γ| ≤ |α|. Let A be a nonlinear operator of infinite order from W_{0}^{∞}(a_{α}, p_{α})(Ω) into its dual W^{−∞}(a_{α}, p′_{α})(Ω) defined as

Au = ∑_{|α|=0}^{∞}(−1)^{|α|}D^{α}(A_{α}(x, D^{γ}u)), |γ| ≤ |α|

where A_{α} : Ω × IR^{λ_{α}} ↦ IR is a real function satisfying the following assumptions

(H_{1}) A_{α}(x, ξ_{γ}) is a Carathéodory function for all α, |γ| ≤ |α|.

(H_{2}) For a.e. x ∈ Ω, all m ∈ IN^{∗}, all ξ_{γ}, η_{α}, |γ| ≤ |α| and some constant c_{0} > 0, we assume that

| ∑_{|α|=0}^{m} A_{α}(x, ξ_{γ})η_{α}| ≤ c_{0} ∑_{|α|=0}^{m} a_{α}|ξ_{α}|^{p_{α}−1}|η_{α}|,

where a_{α} ≥ 0, p_{α} > 1 are reals numbers for all multi-indices α, and for all bounded sequence (p_{α})_{α}.

(H_{3}) There exist constants c_{1} > 0, c_{2} ≥ 0 such that for all m ∈ IN^{∗}, for all ξ_{γ}, ξ_{α}; |γ| ≤ |α|, we have

∑_{|α|=0}^{m} A_{α}(x, ξ_{γ})ξ_{α} ≥ c_{1} ∑_{|α|=0}^{m} a_{α}|ξ_{α}|^{p_{α}} − c_{2}.

(H_{4}) The space W_{0}^{∞}(a_{α}, p_{α})(Ω) is nontrivial.

As regards to the function g, assume that g : Ω × IR ↦ IR is a Carathéodory function, that is, measurable with respect to x in Ω for every s in IR, and continuous with respect to s in IR for almost every x in Ω, satisfying the following conditions

(G_{1}) For all δ > 0,

sup_{|u|≤δ} |g(x, u)| ≤ h_{δ}(x) ∈ L^{1}(Ω).

(G_{2}) The ”sign condition” g(x, u)u ≥ 0, for a.e. x ∈ Ω and all u ∈ IR.

Finally, we assume that

f ∈ L^{1}(Ω),  (3.1)

and we shall prove the existence result without assuming any monotonicity condition.

Example 3.1 Let us consider the operator

Au = ∑_{|α|=0}^{∞}(−1)^{|α|}D^{α}(a_{α}|D^{α}u|^{p_{α}−2}D^{α}u),

where a_{α} ≥ 0, p_{α} > 1 are real numbers such that the space W_{0}^{∞}(a_{α}, p_{α})(Ω) is nontrivial. We can easily show that the conditions (H_{1}), (H_{2}) and (H_{3}) are satisfied. As second example, let us consider the differential operator of infinite order

[cosD]u(x) = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(2n)!} D^{2n}u(x) x ∈ I,

where I is a bounded open interval in IR. The corresponding Sobolev space of infinite order is W_{0}^{∞}(\frac{1}{(2n)!}, 2)(I) which is nontrivial (see [13]) and the conditions above are obviously verified. Some more explicit examples of such an operator of infinite order and of a function g that satisfies the conditions (H_{1}), (H_{2}) and (H_{3}) can be found in [5].

Theorem 3.1 Assume that the assumptions (H_{1}) − (H_{4}), (G_{1}), (G_{2}) and (3.1) hold, then there exists at least one solution u ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω) of (1.1) in the following sense

\{ \begin{matrix}g(x, u) ∈ L^{1}(Ω), g(x, u)u ∈ L^{1}(Ω) \\ ⟨Au, v⟩ + ∫_{Ω} g(x, u)v dx = ⟨f, v⟩, for all v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω). \end{matrix}

Proof. In order to prove our result, we proceed by steps.

Step (1) The approximate problem.

Consider φ ∈ C_{0}^{∞}(IR^{N}) such that 0 < φ(x) < 1 and φ(x) = 1 for x close to 0. Let f_{n} be a sequence of regular functions defined by

f_{n}(x) = φ(\frac{x}{n})T_{n}f(x),

where T_{n} is the usual truncation given by

T_{n}ξ = \{ \begin{matrix}ξ & if |ξ| < n \\ \frac{nξ}{|ξ|} & if |ξ| ≥ n. \end{matrix}

It is clear that |f_{n}| ≤ n for a.e. x ∈ Ω. Thus, it follows that f_{n} ∈ L^{∞}(Ω). Using Lebesgue’s dominated convergence theorem, since f_{n} ⟶ f a.e. x ∈ Ω and |f_{n}| ≤ |f| ∈ L^{1}(Ω), we conclude that f_{n} strongly converges to f in L^{1}(Ω). Define the operator of order 2n + 2 by

A_{2n+2}(u) = ∑_{|α|=n+1}(−1)^{n+1}c_{α}D^{2α}u + ∑_{|α|=0}^{n}(−1)^{|α|}D^{α}A_{α}(x, D^{γ}u), |γ| ≤ |α|,

and consider the following approximate problem with boundary Dirichlet conditions

A_{2n+2}(u_{n}) + g(x, u_{n}) = f_{n}.  (P_{n})

Note that c_{α} are constants small enough such that they fulfil the conditions of the following lemma introduced in [13].

lemma 3.1 (cf. [13]) . For all nontrivial space W_{0}^{∞}(a_{α}, p_{α})(Ω), there exists a nontrivial space W_{0}^{∞}(c_{α}, 2)(Ω) such that W_{0}^{∞}(a_{α}, p_{α})(Ω) ⊂ W_{0}^{∞}(c_{α}, 2)(Ω).

As in [5], the operator A_{2n+2} is clearly monotone since the term of higher order of derivation is linear and satisfies the monotonicity condition. Moreover from assumptions (H_{1}), (H_{2}) and (H_{3}), we deduce that A_{2n+2} satisfies the growth, the coerciveness and the monotonicity conditions. Thanks to [[5], Theorem 3.2], there exists at least one solution u_{n} ∈ W_{0}^{n+1,p⃗}(Ω) of the problem (P_{n}) in the following sense

\{ \begin{matrix}g(x, u_{n}) ∈ L^{1}(Ω), g(x, u_{n})u_{n} ∈ L^{1}(Ω) \\ ⟨A_{2n+2}(u_{n}), v⟩ + ∫_{Ω} g(x, u_{n})v dx = ⟨f_{n}, v⟩ v ∈ W_{0}^{n+1,p⃗}(Ω) \end{matrix}

Step (2) A priori estimates.

By choosing v = u_{n} as test function in (P_{n}), we have

⟨A_{2n+2}(u_{n}), u_{n}⟩ + ∫_{Ω} g(x, u_{n})u_{n} dx = ⟨f_{n}, u_{n}⟩,

which implies that

∑_{|α|=n+1} c_{α} ∫_{Ω} |D^{α}u_{n}|^{2} dx + ∑_{|α|=0}^{n} ∫_{Ω}A_{α}(x, D^{γ}u_{n})D^{α}u_{n} dx ≤ ∫_{Ω} f_{n}u_{n} dx.

Hence, in view of (H_{3}), (G_{2}), the Hölder inequality and by using the fact that |f_{n}| ≤ |f|, we get the estimates

∑_{|α|=n+1} c_{α}∥D^{α}u_{n}∥_{2}^{2} + ∑_{|α|=0}^{n} a_{α}∥D^{α}u_{n}∥_{p_{α}}^{p_{α}} ≤ K  (3.2)

∫_{Ω} g(x, u_{n})u_{n} dx ≤ K  (3.3)

for some constant K = K(f) > 0. The estimate (3.2) is equivalent to

∑_{|α|=0}^{n+1} a_{α}∥D^{α}u_{n}∥_{p_{α}}^{p_{α}} ≤ K  (3.4)

with a_{α} = c_{α} and p_{α} = 2 for |α| = n + 1. Consequently,

∥u_{n}∥_{W_{0}^{n+1,p⃗}(Ω)} ≤ K.  (3.5)

By using the same approach as in [5] and via a diagonalization process, there exists a subsequence still, denoted by u_{n}, which converges uniformly to an element u ∈C_{0}^{∞}(Ω), also for all derivatives there holds D^{α}u_{n} → D^{α}u (for more details we refer to [13]). So that,

u ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω),

since (p_{α})_{α} is a bounded sequence.

Step (3) Convergence of the approximate problem (P_{n}).

There exists a solution u_{n} of problem (P_{n}), n = 1, 2, . . .. Then by passing to the limit, we have

lim_{n→+∞}⟨A_{2n+2}(u_{n}), v⟩ + lim_{n→+∞} ∫_{Ω} g(x, u_{n})v dx = lim_{n→+∞}⟨f_{n}, v⟩,

for v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω). Since f_{n} ⟶ f strongly in L^{1}(Ω), it is clear that

lim_{n→+∞}⟨f_{n}, v⟩ = ⟨f, v⟩

for all v ∈W_{0}^{∞}(a_{α}, p_{α})(Ω).

Now, we shall prove that

lim_{n→+∞}⟨A_{2n+2}(u_{n}), v⟩ = ⟨Au, v⟩, for all v ∈W_{0}^{∞}(a_{α}, p_{α})(Ω).

In fact, let n_{0} be a fix number sufficiently large (n > n_{0}) and let v ∈W_{0}^{∞}(a_{α}, p_{α})(Ω). Set ⟨A(u) − A_{2n+2}(u_{n}), v⟩ = I_{1} + I_{2} + I_{3}, where

I_{1} = ∑_{|α|=0}^{n_{0}}⟨A_{α}(x, D^{γ}u) − A_{α}(x, D^{γ}u_{n}), D^{α}v⟩

I_{2} = ∑_{|α|=n_{0}+1}^{∞}⟨A_{α}(x, D^{γ}u), D^{α}v⟩

I_{3} = − ∑_{|α|=n_{0}+1}^{n+1}⟨A_{α}(x, D^{γ}u_{n}), D^{α}v⟩

or in another form,

I_{3} = − ∑_{|α|=n_{0}+1}^{n}⟨A_{α}(x, D^{γ}u_{n}), D^{α}v⟩ − ∑_{|α|=n+1} c_{α}⟨D^{α}u_{n}, D^{α}v⟩

with A_{α}(x, ξ_{γ}) = c_{α}ξ_{α} and c_{α} ≥ 0 for |α| = n + 1.

We will pass to the limit as n ⟶ +∞ to prove that I_{1}, I_{2} and I_{3} tend to 0. Starting by I_{1}, we have I_{1} → 0 since A_{α}(x, ξ_{γ}) is of Carathéodory type. The term I_{2} is the remainder of a convergent series, hence I_{2} → 0. For what concerns I_{3}, for all ε > 0, there holds k(ε) > 0 (see [9]) such that

| ∑_{|α|=n_{0}+1}^{n+1}⟨A_{α}(x, D^{γ}u_{n}), D^{α}v⟩| ≤ ∑_{|α|=n_{0}+1}^{n+1} ∫_{Ω} |A_{α}(x, D^{γ}u_{n})D^{α}v| dx

≤ c_{0} ∑_{|α|=n_{0}+1}^{n+1} a_{α} ∫_{Ω} |D^{α}u_{n}|^{p_{α}−1}|D^{α}v| dx

≤ c_{0} ∑_{|α|=n_{0}+1}^{n+1} a_{α}∥D^{α}u_{n}∥_{p_{α}}^{p_{α}−1}∥D^{α}v∥_{p_{α}}

≤ εc_{0} ∑_{|α|=n_{0}+1}^{n+1} a_{α}∥D^{α}u_{n}∥_{p_{α}}^{p_{α}}

+ k(ε)c_{0} ∑_{|α|=n_{0}+1}^{n+1} a_{α}∥D^{α}v∥_{p_{α}}^{p_{α}}

≤ εc_{0}K + k(ε)c_{0} ∑_{|α|=n_{0}+1}^{∞} a_{α}∥D^{α}v∥_{p_{α}}^{p_{α}},

where K is the constant given in the estimate (3.2). Since the sequence (p_{α}) is

bounded, this implies that ∑_{|α|=n_{0}+1}^{∞}a_{α}∥D^{α}v∥_{p_{α}}^{p_{α}} is the remainder of a convergent

series, therefore I_{3} → 0 holds. Consequently, we have

⟨A_{2n+2}(u_{n}), v⟩ → ⟨A(u), v⟩ as n → +∞

for all v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω).

It remains to show, for our purposes, that

lim_{n→+∞} ∫_{Ω} g(x, u_{n})v dx = ∫_{Ω} g(x, u)v dx.

for v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω). Indeed, we have u_{n} → u uniformly in Ω, hence g(x, u_{n}) → g(x, u) for a.e. x ∈ Ω. In view of (3.3), we deduce by Fatou’s lemma as n → ∞ that

∫_{Ω} g(x, u)u dx ≤ lim_{n→+∞} ∫_{Ω} g(x, u_{n})u_{n} dx ≤ K,

this implies g(x, u)u ∈L^{1}(Ω). On the other hand, for all δ > 0 we have

|g(x, u_{n})| ≤ sup_{|t|≤δ} |g(x, t)| + δ^{−1}|g(x, u_{n}).u_{n}|
≤ h_{δ}(x) + δ^{−1}|g(x, u_{n})u_{n}|

On the other hand, we have

∫_{E} |g(x, u_{n})| dx ≤ ∫_{E} h_{δ}(x) dx + δ^{−1}K,

for some measurable subset E of Ω and for some ε > 0. Here, K is the constant of (3.3) which is independent of n. For |E| sufficiently small and δ = \frac{2K}{ε} , we

obtain ∫_{E} |g(x, u_{n})| dx < ε. Then, we get by using Vitali’s theorem

g(x, u_{n}) → g(x, u) in L^{1}(Ω).

Hence it follows that g(x, u) ∈ L^{1}(Ω).

By passing to the limit, we obtain

⟨Au, v⟩ + ∫_{Ω} g(x, u)v dx = ⟨f, v⟩, for all v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω).

Consequently,

g(x, u) ∈ L^{1}(Ω), g(x, u)u ∈ L^{1}(Ω)
⟨Au, v⟩ + ∫_{Ω} g(x, u)v dx = ⟨f, v⟩ for all v ∈ W_{0}^{∞}(a_{α}, p_{α})(Ω)

This completes the proof. □

Remarks.

1. Note that the existence result is given without assuming a monotonicity condition on the operator. Also, no restrictions to the parameters p_{α} are imposed.

2. In general, some methods of approximation need at least the segment property of the domain, here we assume no regularity condition of Ω.

4 Application

Consider the following class of strongly nonlinear Dirichlet problem

B(u) + u|u|^{r+1}h(x) = f in Ω,  (4.1)

where r > 0 is a real number, h ∈ L^{1}(Ω) with h(x) ≥ 0 a.e x ∈ Ω and the operator B is defined as

B(u) = (−\sqrt{I + Δ})u.

Our technique here consists to exploit certain result in the setting of functional spaces of infinite order. Thus as in [13], we can write the operator B as follows

B(u) = (−\sqrt{I + Δ})u = ∑_{k=0}^{∞} a_{k}(−Δ)^{k}u,  (4.2)

where a_{k} > 0, k = 0, 1, ... are real numbers which guarantee the nontriviality of the corresponding functional space, (for more details see [13]). The characteristic function of our operator is defined by

φ(x) = ∑_{k}^{∞} a_{k}ξ^{2k} with ξ^{2} = ξ_{1}^{2} + ... + ξ_{N}^{2},

and the corresponding space of the present strongly nonlinear Dirichlet problem is

W_{0}^{∞}(a_{k}, 2)(Ω) = {u ∈ C_{0}^{∞}(Ω) : ρ(u) = ∑_{k=0}^{∞} a_{k} ∥∇^{k}u∥_{2}^{2} < ∞}.

Definition 4.1 A function u ∈ W_{0}^{∞}(a_{k}, 2)(Ω) is a solution of the Dirichlet problem (4.1), if for any v ∈ W_{0}^{∞}(a_{k}, 2)(Ω) the equality

∑_{k=0}^{∞} a_{k} ∫_{Ω} ∇^{k}u ∇^{k}v dx + ∫_{Ω} u|u|^{r+1}h(x)v dx = ⟨f, v⟩

is valid.

The explicit form of the operator B given in (4.2), shows simply that it satisfies the assumptions (H_{1}), (H_{2}) and (H_{3}). Now, taking into account that B is monotone and the term u|u|^{r+1}h(x) fulfils the sign condition, then by the same argument as in the proof of Theorem 3.1 we prove the following existence result.

Theorem 4.1 For all f ∈ L^{1}(Ω), there exists at least one nontrivial solution u ∈ W_{0}^{∞}(a_{k}, 2)(Ω) such that

⟨Bu, v⟩ + ∫_{Ω} u|u|^{r+1}h(x)v dx = ⟨f, v⟩ for all v ∈ W_{0}^{∞}(a_{α}, 2)(Ω).

Remark 4.1 In his book, Dubinskii [13] considered the Sobolev spaces of infinite order corresponding to boundary value problem for differential equations of infinite order and obtained the solvability of these problems in the case where the coefficients of the equation grow polynomially with respect to the derivatives. Dyk Van extends the results of Dubinskii to include the case of operators with rapidly or slowly increasing coefficients ( see Dyk Van[10]). In their works, Tran Duk Van et al.[16] introduced Sobolev-Orlicz spaces of infinite order and investigated their principal properties. They also established the existence and uniqueness of solutions of some Dirichlet problems for nonlinear differential equations of infinite order. In particular, let Ω a bounded domain in IR^{N}, N ≥ 1, with boundary ∂Ω. Consider the Dirichlet problem defined by

(P b) \{ \begin{matrix}Au(x) = ∑_{|α|=0}^{∞}(−1)^{|α|}D^{α}A_{α}(x, ..., D^{α}u) = f(x) x ∈ Ω, \\ D^{ω}u(x) = 0, x ∈ ∂Ω, |ω| = 0, 1, .... \end{matrix}

Here A_{α} : Ω × IR^{λ_{α}} ↦ IR is a real function, λ_{α} denotes the number of multiindices γ such that |γ| ≤ |α|, satisfying the following assumptions:

(H’_{1}) There exist an N-function ϕ_{α}, a function a_{α} ∈ L\{\overline{ϕ}_{α}, Ω\}, a continuous bounded function c_{α}^{1}, (1 ≤ c_{α}^{1}(|t|) ≤ const) and a constant b > 0 such that

|A_{α}(x, ξ)| ≤ a_{α}(x) + b\overline{ϕ}_{α}^{−1}\overline{ϕ}_{α}(c_{α}^{1}(|ξ_{α}|)ξ_{α})

where

∑_{|α|=0}^{∞} ∥a_{α}∥_{ϕ_{α}} < +∞.

(H’_{2}) There exist functions b_{m} ∈ L_{1}(Ω), g_{α} ∈ E{ϕ_{α}, Ω}, a continuous bounded function c_{α}^{2}, (c_{α}^{2}(|t|) ≥ c_{α}^{1}(|t|)) and a constant d > 0 such that

∑_{|α|=m}(A_{α}(x, ξ) − g_{α}(x))ξ_{α} ≥ d ∑_{|α|=m} ϕ_{α}(c_{α}^{2}(|ξ_{α}|)ξ_{α}) − b_{m}(x),

where

∑_{|α|=0}^{∞} ∥g_{α}∥_{ϕ_{α}} < +∞ and ∑_{|α|=0}^{∞} ∫_{Ω} |b_{m}(x)| dx < +∞.

(H’_{3}) The N-functions ϕ_{α} are such that the Sobolev-Orlicz space LW_{0}^{∞}(ϕ_{α}, Ω) is nontrivial.

(H’_{4}) For all ξ = (ξ_{0}, ..., ξ_{α}) and ξ′ = (ξ′_{0}, ..., ξ′_{α}) such that ξ ≠ ξ′ we have the inequality

∑_{|α|=0}^{m}(A_{α}(x, ξ) − A_{α}(x, ξ′))(ξ_{α} − ξ′_{α}) ≥ 0.

The corresponding functional setting is the Sobolev-Orlicz of infinite order given by

LW_{0}^{∞}(ϕ_{α}, Ω) = {u ∈ C_{0}^{∞}(Ω) : ∥u∥_{∞} < +∞},

where

∥u∥_{∞} = inf\{k > 0 : ∑_{|α|=0}^{∞} ∫_{Ω} ϕ_{α}(\frac{D^{α}u}{k}) dx ≤ 1\}.

The dual space of LW_{0}^{∞}(ϕ_{α}, Ω) is defined by

EW^{−∞}(ϕ_{α}, Ω) = {f : h(x) = ∑_{|α|=0}^{∞}(−1)^{|α|}D^{α}f_{α}(x)},

where

f_{α} ∈ E_{ϕ_{α}}(Ω) for all multi-indice α,

and

ρ′(f) = ∑_{|α|=0}^{∞} ∥f_{α}∥_{ϕ_{α}} < +∞.

The duality of the spaces LW_{0}^{∞}(ϕ_{α}, Ω) and EW^{−∞}(ϕ_{α}, Ω) is determined by the expression

⟨f, v⟩ = ∑_{|α|=0}^{∞} ∫_{Ω} f_{α}(x)D^{α}v(x) dx,

which is obviously correct. ( For more details about the definition of N-function and Sobolev-Orlicz spaces of infinite order we refer to Tran Duk Van et al.[16]). When the data f belongs to the dual, under assumptions (H′_{1})-(H′_{4}) the authors in Tran Duk Van et al.[16] proved the existence and uniqueness of the solution of the nonlinear problem (P b). In our case, with a suitable choice of the N-functions ϕ_{α}, we can deal with the boundary value problem (1.1) in the more general class of Sobolev-Orlicz spaces of infinite order using operators satisfying (H′_{1}) − (H′_{4}).

Acknowledgements. The authors are grateful to the anonymous referee for valuable comments, useful suggestions and help in the presentation of this paper. The research of the third author is supported by Al-Imam University project No. 281206, Riyadh, KSA.

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(Received March 11, 2008)