Strongly nonlinear problem of infinite order with L 1 data

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 ().


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 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.

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 • pα is the usual norm in the Lebesgue space L pα (Ω).For a positive integer m, we define the following vector of real numbers: and denote p ¯= min{p α , |α| ≤ m}.Now, let us consider the generalized functional Sobolev space (2.1) . We define the space W m, p 0 (Ω) as the closure of the proof of this is an adaptation from Adams [1]).W −m, p (Ω) designs its dual where p is the conjugate of p, i.e., p α = pα pα−1 for all |α| ≤ m.
The Sobolev space of infinite order is the functional space defined by EJQTDE, 2009 No. 15, p. 2 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 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: where h α ∈ L p α (Ω) and p α is the conjugate of p α , i.e., p α = pα pα−1 (for more details about these spaces, see [13]).
We need the anisotropic Sobolev embeddings result.
where k = E(m − N 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, p (Ω) ⊂ W m,p (Ω).

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 we assume that where a α ≥ 0, p α > 1 are reals numbers for all multi-indices α, and for all bounded sequence (p α ) α .

EJQTDE, 2009
No. 15, p. 3 (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 The "sign condition" g(x, u)u ≥ 0, for a.e.x ∈ Ω and all u ∈ IR.
Finally, we assume that and we shall prove the existence result without assuming any monotonicity condition.

Example 3.1 Let us consider the operator
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 where I is a bounded open interval in IR.The corresponding Sobolev space of infinite order is W ∞ 0 ( 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].
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 where T n is the usual truncation given by 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 Define the operator of order 2n + 2 by and consider the following approximate problem with boundary Dirichlet conditions 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 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 n+1, p 0 (Ω) of the problem (P n ) in the following sense Step (2) A priori estimates.
By choosing v = u n as test function in (P n ), we have EJQTDE, 2009 No. 15, p. 5 Hence, in view of (H 3 ), (G 2 ), the Hölder inequality and by using the fact that |f n | ≤ |f |, we get the estimates with a α = c α and p α = 2 for |α| = n + 1.Consequently, 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 In fact, let n 0 be a fix number sufficiently large (n where 15, p. 6 or in another form, 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 where K is the constant given in the estimate (3.2).Since the sequence (p α ) is bounded, this implies that a α D α v pα pα is the remainder of a convergent series, therefore I 3 → 0 holds.Consequently, we have . 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 On the other hand, for all δ > 0 we have 15, p. 7 On the other hand, we have 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 = 2K ε , we obtain E |g(x, u n )| dx < ε.Then, we get by using Vitali's theorem Hence it follows that g(x, u) ∈ L1 (Ω).
By passing to the limit, we obtain Consequently, This completes the proof. 2 Remarks.

Application
Consider the following class of strongly nonlinear Dirichlet problem 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 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 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 and the corresponding space of the present strongly nonlinear Dirichlet problem is 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 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 where (H' 3 ) The N-functions φ α are such that the Sobolev-Orlicz space LW ∞ 0 (φ α , Ω) is nontrivial.
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 ).