On Elliptic Equations in Orlicz Spaces Involving Natural Growth Term and Measure Data

and Applied Analysis 3 2. Preliminaries LetM : R → R be anN-function, that is,M is continuous, convex, withM t > 0 for t > 0, M t /t → 0 as t → 0 and M t /t → ∞ as t → ∞. The N-function M conjugate to M is defined byM t sup{st −M s : s > 0}. Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q; that is, for each ε > 0, P t Q εt −→ 0 as t −→ ∞. 2.1 TheN-functionM is said to satisfy the Δ2 condition if for some k > 0:M 2t ≤ kM t for all t ≥ 0; when this inequality holds only for t ≥ t0 > 0,M is said to satisfy the Δ2 condition near infinity. LetΩ be an open subset ofR . The Orlicz classLM Ω resp., the Orlicz space LM Ω is defined as the set of equivalence classes of real-valued measurable functions u onΩ such that ∫ ΩM u x dx < ∞ resp. ∫ ΩM u x /λ dx < ∞ for some λ > 0 . Note that LM Ω is a Banach space under the norm ‖u‖M,Ω inf{λ > 0 : ∫ ΩM u x /λ dx ≤ 1} and LM Ω is a convex subset of LM Ω . The closure in LM Ω of the set of bounded measurable functions with compact support in Ω is denoted by EM Ω . In general EM Ω / LM Ω and the dual of EM Ω can be identified with LM Ω by means of the pairing ∫ Ω u x v x dx, and the dual norm on LM Ω is equivalent to ‖ · ‖M,Ω. We now turn to the Orlicz-Sobolev space. WLM Ω resp. WEM Ω is the space of all functions u such that u and its distributional derivatives up to order 1 lie in LM Ω resp. EM Ω . This is a Banach space under the norm ‖u‖1,M,Ω ∑ |α|≤1 ‖Du‖M,Ω. Thus WLM Ω and WEM Ω can be identified with subspaces of the product of N 1 copies of LM Ω . Denoting this product by ΠLM, we will use the weak topologies σ ΠLM,ΠEM and σ ΠLM,ΠLM . The space W 1 0EM Ω is defined as the norm closure of the Schwartz space D Ω in WEM Ω and the space W1 0LM Ω as the σ ΠLM,ΠEM closure of D Ω in WLM Ω . We say that un converges to u for the modular convergence in WLM Ω if for some λ > 0, ∫ ΩM D un − Du /λ dx → 0 for all |α| ≤ 1. This implies convergence for σ ΠLM,ΠLM . If M satisfies the Δ2 condition on R near infinity only when Ω has finite measure , then modular convergence coincides with norm convergence. Fore more details about the Orlicz spaces and their properties one can see 11, 12 . For k > 0, we define the truncation at height k, Tk : R → R by: Tk s max −k, min k, s . 3. Main Result 3.1. Useful Results First, we give the following definitions and results which will be used in our main result. The p-capacity Cp B,Ω of any set B ⊂ Ω with respect to Ω is defined in the following classical way. The p-capacity of any compact set K ⊂ Ω is first defined as Cp K,Ω inf {∫ Ω ∣∇φ∣pdx : φ ∈ D Ω , φ ≥ χK } , 3.1 4 Abstract and Applied Analysis where χK is the characteristic function of K; we will use the convention that infφ ∞. The p-capacity of any open subsetU ⊂ Ω is then defined by Cp U,Ω sup { Cp K,Ω , K compact K ⊂ Ω } . 3.2 Finally, the p-capacity of any subset B ⊂ Ω is defined by Cp B,Ω inf { Cp U,Ω , U open B ⊂ U } . 3.3 Definition 3.1. We say that u is a weak solution of the problem ( Pμ ) if u is measurable, Tk u ∈W1 0LM Ω , ∫


Introduction
We deal with boundary value problems Ω is a bounded domain of R N , with the segment property.a : Ω × R × R N → R N is a Carathéodory function i.e., measurable with respect to x in Ω for every s, ξ in R × R N , and continuous with respect to s, ξ in R × R N for almost every x in Ω such that there exist two N-functions P M and for all ξ, ξ * ∈ R N , ξ / ξ * , the following hypotheses are true where c x belongs to E M Ω , c ≥ 0, k i i 1, 2, 3, 4 to R , and α, λ to R * .For the sake of simplicity, we suppose in 1.2 that α λ 1.This paper is devoted to study the Dirichlet problem for some nonlinear elliptic equations whose simplest model is −Δ p u g u |∇u| p μ. 1.8This kind of problems has been widely studied.Many authors have proved results for secondorder elliptic problems with lower-order terms depending on the gradient; these works include, for instance, 1-6 .After the classical example by Kazdan and Kramer see 7 , which shows that 1.8 cannot always have solutions, two different kind of questions have been considered.On the one hand, in some papers, the existence of solutions when the source f is small in a suitable norm are proved.On the other hand, conditions on which the function g have been considered in order to get a solution for all f in a given Lebesgue space.This is the way chosen in 1, 5, 6 under the hypothesis g ∈ L 1 .In 8 , the authors present some results concerning existence, nonexistence, multiplicity, and regularity of positive solutions for two elliptic quasilinear problems with Dirichlet data in a bounded domain.The first problem is similar to 1.8 and the other is 1.9 where λ, f, b, g required some specified conditions.The first one, of unknown u, involves a gradient term with natural growth.The second one, of unknown v, presents a source term of order 0. They gave and established a precise connection between problems in u and v. Also, they proved a result of existence for the problem in v with general bounded Radon measures data and obtained some results for the problem in u by using the connection between these two problems.Other authors have established this connection between the two problems 1.8 and 1.9 ; one can see, for example, 9 .
Many researchers have investigated the possibility to find solutions of 1.8 under the sign condition g s s ≥ 0, in which case the term g u |∇u| p is said to be an absorption term.In 5 , the author treated the problem 1.8 without using the above sign condition but by supposing the summability of the function g.The principal tools used is the Lebesgue decomposition theorem see 10 and the cut functions with respect to the measure μ.The result given is optimal.
In the spirit of the work 5 , our purpose in this paper is to prove existence results in the setting of the Orlicz Sobolev space W 1 L M Ω when the operator does not satisfy the classical polynomial growth.

Preliminaries
Let M : R → R be an N-function, that is, M is continuous, convex, with M t > 0 for t > 0, M t /t → 0 as t → 0 and M t /t → ∞ as t → ∞.The N-function M conjugate to M is defined by M t sup{st − M s : s > 0}.Let P and Q be two N-functions.P Q means that P grows essentially less rapidly than Q; that is, for each ε > 0, The N-function M is said to satisfy the Δ 2 condition if for some k > 0: M 2t ≤ kM t for all t ≥ 0; when this inequality holds only for t ≥ t 0 > 0, M is said to satisfy the Δ 2 condition near infinity.
Let Ω be an open subset of R N .The Orlicz class L 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 u x /λ dx < ∞ for some λ > 0 .
Note that L M Ω is a Banach space under the norm u M,Ω inf{λ > 0 : In general E M Ω / L M Ω and the dual of E M Ω can be identified with L M Ω by means of the pairing Ω u x v x dx, and the dual norm on L M Ω is equivalent to • 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 .This is a Banach space under the norm u 1,M,Ω |α|≤1 D α u M,Ω .Thus W 1 L M Ω and W 1 E M Ω can be identified with subspaces of the product of N 1 copies of L M Ω .Denoting this product by ΠL M , we will use the weak topologies σ ΠL M , ΠE M and σ ΠL M , ΠL M .The space W 1 0 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 1 0 L M Ω as the σ ΠL M , ΠE M closure of D Ω in W 1 L M Ω .We say that u n converges to u for the modular convergence in W 1 L M Ω if for some λ > 0, Ω M D α u n − D α u /λ dx → 0 for all |α| ≤ 1.This implies convergence for σ ΠL M , ΠL M .If M satisfies the Δ 2 condition on R near infinity only when Ω has finite measure , then modular convergence coincides with norm convergence.
Fore more details about the Orlicz spaces and their properties one can see 11, 12 .
For k > 0, we define the truncation at height k, T k : R → R by: T k s max −k, min k, s .

Useful Results
First, we give the following definitions and results which will be used in our main result.
The p-capacity C p B, Ω of any set B ⊂ Ω with respect to Ω is defined in the following classical way.The p-capacity of any compact set K ⊂ Ω is first defined as where χ K is the characteristic function of K; we will use the convention that inf φ ∞.The p-capacity of any open subset U ⊂ Ω is then defined by Finally, the p-capacity of any subset B ⊂ Ω is defined by We say that u is a weak solution of the problem P μ if

3.4
We define M b Ω as the space of all Radon measures on Ω with bounded total variation, and C b Ω as the space of all bounded, continuous functions on Ω, so that Ω ϕ dμ is defined for We say that a sequence for every ϕ ∈ C b Ω .If this convergence holds only for all the continuous functions ϕ with compact support in Ω, then we have the usual weak * convergence in M b Ω .
Lemma 3.2.Under the hypotheses 1.2 -1.6 , f ∈ L ∞ Ω and f ≥ 0, there exists at least one positive weak solution of the problem For the proof see 12 .
Let M be a fixed N-function, we define K as the set of N functions D satisfying the following conditions: iii there exists an N-function H such that HoM −1 oM ≤ D and H ≤ D near infinity.

Lemma 3.3. Let u n be a sequence of solutions of the problem
where F n is a sequence of functions bounded in For the proof one can see that the technique used in 13 and adapted to the elliptic case gives the result, but for the simplicity we give a sketched proof.
Proof.Let denote by X N : NC 1/N N , C N the measure of the unit ball of R N , and μ θ |{|u| > θ}|.
Let ϕ be a truncation defined by for all θ, h > 0.
Using v ϕ u n as a test function, we obtain after tending h to zero Following the same way as in 14 , we have for D ∈ K,

3.9
We obtain ds.

3.11
Then the sequence u n is bounded in Abstract and Applied Analysis Lemma 3.4.Let μ be a nonnegative Radon measure which is concentrated in a set E of zero p-capacity.Then there exists a sequence ψ δ of D Ω functions such that For the proof see 15 .
Remark 3.5.By the above lemma, we have that ψ δ converge to zero both strongly in W 1,p 0 Ω , a.e. in Ω, and in the weak * topology of L ∞ Ω .

Existence Result
In what follows, we suppose that the set K is nonempty.Theorem 3.6.Let x 0 ∈ Ω, under the hypotheses 1.2 -1.6 and N ≥ 2, there exists at least one positive weak solution of the problem The condition N ≥ 2 is very important to ensure the existence of cut functions for the measure δ x 0 .The case N 1 is easily treated, since we come back to the variational case.
1.9 The condition ∞ is supposed to guarantee that we are not in the variational case and the study has a sense see 16 .
Remark 3.8.The conditions i , ii , and iii permit us to determine the regularity of the solutions of P δ x 0 and improve the one given by Porretta in 5 .If M t t p , one can find a solution u such that Ω |∇u| N p−1 / N−1 /Log σ e |∇u| < ∞, for some σ > 1.
Remark 3.9 see 5 .The condition g ∈ L 1 is optimal in the sense introduced by Porretta in 5 .Indeed, if μ is the Dirac mass, there is no solution which can be obtained by approximation.In particular, in the reaction case g s s ≤ 0 , if μ is approximated by a sequence of smooth functions, the sequence of approximating solutions converges to a solution of P μ if g ∈ L 1 , while it blows up everywhere if g / ∈ L 1 .Finally, let us say a few words on how positive constant will be denoted hereafter.If no otherwise specified, we will write C to denote any positive constant possibly different which only depends on the data, that is on quantities that are fixed in the assumptions N, Ω, and so on. . .; in any case such constants never depend on the different indexes having a limit.In the sequel and throughout the paper, we will omit for simplicity the dependence on x in the function a x, s, ξ and denote n, j, δ, s, m all quantities possibly different such that 13 and this will be in the order in which the parameters we use will tend to infinity, that is, first n, then j, δ, s, and finally m.Similarly, we will write only n , or n, j , . . . to mean that the limits are made only on the specified parameters.Moreover, for the sake of simplicity, in what follows, the convergence, even if not explicitly stressed, may be understood to be taken possibly up to a suitable subsequence extraction.

A Sequence of Approximating Problems
Consider the approximate problem where f n is a smooth sequence of functions such that f n 1 ≤ C and f n δ x 0 in M b Ω .The existence of solutions of the above problem u n ∈ W 1 0 L M Ω was ensured by Lemma 3.2.

A Priori Estimates
Lemma 3.10.There exists a subsequence of u n (also denoted u n ); there exists a measurable function u such that strongly in E M Ω , and a.e. in Ω.

Proof. Let v T k u n exp
u n 0 g s ds as test function in P f n .One has There exist a subsequence also denoted u n and a measurable function ω k such that T k u n ω k , weakly in W 1 0 L M Ω , a.e. in Ω, and strongly in E M Ω .

3.16
By an easy argument we can see that, there exists a measurable function u such that 3.17

Almost Everywhere Convergence of Gradients
Lemma 3.11.The subsequence u n obtained in Lemma 3.10 satisfies ∇u n −→ ∇u a.e. in Ω.

3.18
Proof.Let us recall that since the N-capacity C N {x 0 }, Ω 0, there exists a sequence ψ δ satisfying the Lemma 3.4 with μ δ x 0 and p N.
Step 1.In this step we will show the following: Let m > 0, k > 0 such that m > k.Let ρ m be a truncation defined by

3.20
We have

10 Abstract and Applied Analysis
About I 2 : it is obvious since

3.24
About there exists a measurable function h k such that up to a subsequence also denoted

3.27
About I 5 : since ρ u n 0 on the set {|u n | ≥ m 1}, then and as for I 4 , one has 0 g s ds as test function, in one hand we obtain

3.31
On the other hand

3.32
Let us come back to our main estimation i .We have

3.33
Then we deduce that

3.34
Since we have 7 i 1 I i J 1 J 2 , I 3 ≥ 0 and using 1.2 , we deduce that

3.35
For r ≤ s, one has

3.36
Abstract and Applied Analysis 13 We have T m 1 u n is bounded in we easily deduce i .
For ii we proceed as in i by using the fact that |u n |≥m f n ψ δ dx n, δ, m .Then we conclude that

3.38
Step 2. In this step we prove that ∇u n → ∇u a.e. in Ω. Since

3.41
Therefore, we have ∇T k u n −→ ∇T k u a.e. in Ω.

3.42
Lemma 3.12.For all k > 0, ∇T k u n −→ ∇T k u for the modular convergence in L M Ω N .

3.43
For the proof we can adopt the same way as in 13 .

The Convergence of the Problems P f n and the Completion of the Proof of Theorem 3.6
In one hand by using as test function ϕ u n 0 g s χ {s>h} exp u n 0 g s ds in P f n , we can deduce as in 5 that g u n M |∇u n | converge strongly in L 1 Ω to g u M |∇u| .
In the other hand, by Lemma 3.3, we deduce that u n is bounded in W 1 0 L D Ω for every D ∈ K. Then a •, u n , ∇u n is bounded in L H Ω and the passage to the limit is an easy task.
Corollary 3.13.Under the hypotheses 1.2 -1.7 , μ is a convex combination of Dirac measures (μ J i 1 α j δ x j x j ∈ Ω and J i 1 α j 1) and the problem P μ admits at least one weak solution.
The proof is a simple adaptation of the one of Theorem 3.6 by taking in z j,δ n,m , J i 1 α j 1− ψ δ x j in the place of 1−ψ δ x j , where ψ δ x j is the cut functions corresponding to the measure δ x j .

General Case
Before giving the general case of Theorem 3.6, let's recall that the set of finite convex combination of Dirac measures is dense in the set of measures probability.
In the following theorem we denote by μ n the sequence defined by μ n n i 1 α i δ x i for some x i ∈ Ω, and α i ∈ R with n i 1 α i 1.Since Ω is bounded and by using the approximation of μ/μ Ω which is a measure of probability, the existence of solutions can be obtained as consequence of the last theorem.Remark 3.17.Let us recall that if we suppose that the N-functions M and M satisfy the Δ 2 condition, we can prove with the same way as in Theorem 3.6 that the problem P μ has a weak solution for all singular measure μ in the sense that it is concentrated in some Borel set with zero M-capacity.

3.52
Here p : Ω → 1, ∞ is a measurable function such that 1 < ess inf where k is a nonnegative function in L p • Ω , p x p x / p x − 1 , and C is a positive real.

Remark 3 . 14 .
The technique used in the proof of Theorem 3.6 allows us to prove that the problem A u g u a •, u, ∇u ∇u δ x 0 in Ω,

Ω ρ m u n 1 − ψ δ exp u n 0 Corollary 3 . 16 .
g s ds * J η dμ n η, n, δ, m .By using the above estimation, we obtain a similar equation to 3.20 and we follow the same technique used in the step of almost everywhere convergence of gradients in Theorem 3.6 to prove the existence result.Under the hypotheses 1.2 -1.7 , μ is positive Radon measure and the problem P μ admits at least one weak solution.

Remark 3 .
18.One can see that the technique used in this paper can be adopted to prove the existence of solutions of the following problem− div a •, u, ∇u g u |∇u| p x μ in Ω, u 0 on ∂Ω, 3.51where a satisfies, for almost every x in Ω, for every s in R, for every ξ and ξ * in R N ,a x, s, ξ ξ ≥ |ξ| p x , a x, s, ξ − a x, s, ξ * ξ − ξ * > 0, ξ / ξ * , |a x, s, ξ | ≤ C k x |s| p x −1 |ξ| p x −1 .
is finite a.e. in Ω. Hence there exists a measurable set E such that |E| 0 and|∇T k u n | < ∞ in Ω − E. Let η xlim n ∇T k u n , then by using Fatou lemma, we get Moreover, there exists a measurable function u such that T k u ∈ W 1 0 L M Ω and u is a weak solution of P μ .
Theorem 3.15.Let μ n be a sequence of Radon measures convergent to a Radon measure μ, and let u n , a sequence of weak solutions of P μ n .Then there exists M > 0 such that Ω M |∇T k u n | dx ≤ Mk 3.45 for every n and every k > 0. Ω a x, u n , ∇u n ∇u n g u n T k u n exp M Ω , a.e. in Ω, and strongly in E M Ω .3.48 Almost everywhere convergence of gradients: Let us consider z j,δ n,m * J η as test function in P μ n , we have