Antoine Jaquet

Didier-Koumavi Ekouevi

Maiga Aboubakrine

Jules Bashi

Eugène Messou

Moussa Maiga

Hamar-Alassane Traore

Marcel Zannou

Calixte Guehi

Franck-Olivier Ba-Gomis

Eug Ne Messou

Albert Minga

G Rard Allou

Serge-Paul Eholie

Francois Dabis

Emmanuel Bissagnene

Annie-Jeanne Sasco

Tobacco use and its determinants in HIV-infected patients on antiretroviral therapy in West African countries

Keywords: Adult, Anti-Retroviral Agents, therapeutic use, Antiretroviral Therapy, Highly Active, Benin, epidemiology, Cote d'Ivoire, epidemiology, Cross-Sectional Studies, Databases as Topic, Female, HIV Infections, drug therapy, epidemiology, Humans, Logistic Models, Male, Mali, epidemiology, Marijuana Smoking, epidemiology, Middle Aged, Odds Ratio, Prevalence, Questionnaires, Smoking, epidemiology, Tuberculosis, epidemiology

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Since the introduction of Highly Active Antiretroviral therapy (HAART), the pattern of AIDS-defining and non-AIDS defining disease conditions has evolved. Chronic manifestations such as cancers or cardiovascular diseases are becoming an emerging problem among HIV-infected persons in industrialized countries . Questions have been raised concerning the relative contribution of well [START_REF] Lewden | Causes of death among human immunodeficiency virus (HIV)-infected adults in the era of potent antiretroviral therapy: emerging role of hepatitis and cancers, persistent role of AIDS[END_REF] characterized risk factors of non-AIDS related cancers such as tobacco consumption. Indeed, the prevalence of smoked tobacco has been shown to be particularly high in HIV-infected patients living in industrialized countries, ranging from 40 to 70 according to studies population , , . In sub-Saharan Africa, few data exist concerning tobacco consumption particularly in West African countries. Since [2 6 7 ] this well known cancer risk factor is a growing problem in this part of the world and is amenable to prevention strategies, there is a particular need to document its current level in this specific population. We thus aimed at estimating the prevalence of tobacco and cannabis consumption among West African HIV-infected subjects treated with HAART.

METHODS

Study population

This cross-sectional survey was conducted within the International epidemiological Database to Evaluate AIDS (IeDEA) network ( ). By collecting and harmonizing data from multiple HIV/AIDS cohorts from industrialized and resource-limited http://www.iedea-hiv.org countries, this initiative aims to address unique and evolving research questions in the field of HIV/AIDS care and treatment. In the West African region, this collaboration was initiated in July 2006 and currently involves 14 adult HIV/AIDS clinics spread over eight countries: Benin, Burkina-Faso, C te d Ivoire, Gambia, Ghana, Mali, Nigeria and Senegal (

). Of these countries, ô '

http://www.iedeawestafrica.org three of them volunteered and had the logistic capacity to perform such a survey in a total of eight HIV treatment centers. The study population was assembled by enrolling consecutively all HIV-infected patients on treatment attending for consultation at the participating HIV clinics during a four-week period. For clinics that were unable to guarantee the exhaustivity in the enrollment procedure, patients were randomly selected. Systematic sampling, a form of one-stage cluster sampling, was used. Social health workers in charge of inclusions selected every n HIV-infected patient on HAART (depending on the monthly number of attendees) after randomly selecting th the first patient through n statistical elements as the starting point.

th Patients included fulfilled the following criteria: HIV-1 or HIV-2 or HIV1 2 infected, on HAART defined as receiving three + antiretroviral drugs for a minimum of one month, aged 18 or more and willing to participate to the survey.

Study procedures

Especially trained social health workers, administered during face-to-face interviews a standardized form assessing socio-demographic characteristics, tobacco and cannabis consumption. Data collected during these interviews were secondarily linked with the IeDEA clinical database, using the identification number attributed to each patient, in order to dispose of additional information that could not be collected during the interviews. This linkage allowed the use of baseline and follow-up clinical and biological data such as CD4 cell count and duration of HAART. A written informed consent was obtained from all study participants. This study was approved by national ethic committees from C te d Ivoire, Mali and Benin.

ô '

Measures

Questions related to present and past tobacco and cannabis consumption were assessed with a standardized questionnaire adapted from several forms previously used in epidemiological studies conducted in Africa , . Regular smokers were defined as present smokers [8 9 ] who smoked at least one cigarette per day for one year or more in accordance to the World Health Organization definition (WHO) .

[10 ]

Statistical analysis

Analyses were first conducted to compare tobacco consumption characteristics according to countries using Pearson s test or Fisher

' χ 2 '
s exact test when appropriate for qualitative variables and t-test or Kruskall-Wallis test for quantitative variables. A logistic regression model was used for univariate and multivariate analyses of the determinants of smoking consumption. A stepwise descending procedure was applied to select variables that were significantly associated, using a P-value threshold of 0.20. Variables that were also identified in the literature as related to smoking tobacco were systematically included in the model. Proportion and (OR) estimates were Odds Ratio reported with their 95 Confidence Interval (95 CI). All statistical analyses were performed using SAS software, 9.1 (SAS Institute Inc, % % NC, USA).

RESULTS

General characteristics

A total of 2920 HAART-treated patients (1838 from five clinics in C te d Ivoire, 486 from one clinic in Benin and 596 from two ô ' clinics in Mali) were included in the present study. The male to female ratio from our study sample (0.41:1) did not differ significantly from the remaining 2380 HAART-treated HIV-infected patients seen in the participating clinical centers during the same study period and that could not be interviewed (0.42:1) (P 0.93).

=

The median age of these 2920 patients was 38 years (interquartile range IQR 32 45 years) and the median duration on HAART was 3 determinants of having a present or past history of regular tobacco smoking. Patients stating being single, widowed or divorced and older patients were also more likely to be regular smokers compared to married or cohabitants and younger patients. Patients with a severely impaired immunological status at HAART initiation and patients with a known history of tuberculosis were also more likely to have a present or past history of regular smoking ( ). table 2

Cannabis consumption

Overall, the prevalence of present or past cannabis use was 1.7 (95 CI 1.3 2.2). This consumption, illicit in the three countries

DISCUSSION

Smoking prevalence

The main finding of this survey is the country variability in the prevalence of smokers among this HIV-infected population. Smoking prevalence in HIV-infected patients from Mali was two-times higher than in C te d Ivoire and 10-times higher than in Benin and close to ô '

the estimates reported in industrialized countries. Little is known about smoking consumption in the general population from West African countries, especially French-speaking ones. In its third edition, the tobacco Atlas reports tobacco smoking prevalence estimates according men and 1 and 0 in women, respectively . These previous findings from the adult population of C te d Ivoire and Benin,

% % [13 ] ô '
irrespective of their HIV status, suggest that smoking prevalence is quite comparable between HIV-infected and HIV-uninfected persons.

In Mali, the current smoking prevalence estimate in the general adult population is 14 in men and 0.6 in women according to the % % tobacco atlas . These results are about twice lower than the estimated prevalence in our survey suggesting a potential association [START_REF] Shafey | The Tobacco Atlas[END_REF] between smoking and HIV status.

While the few available results from the international studies on smoking prevalence in the general population place sub-Saharan Africa and particularly West African countries at a relatively low level of smoking consumption , our study highlights that smoking [14 ] habits in HIV-infected patients and probably in the general population are not homogenous in this part of the world. We thus make a plea for extending those prevalence studies in other HIV-infected populations as well as in non HIV-infected populations from African countries.

Tobacco consumption in HIV/AIDS patients from industrialized countries has been shown to be elevated and higher than in the general population with approximately half of patients being regular smokers. This high frequency of cigarettes smoking is at least partly related to the presence of a specific subpopulation of multidrug users which have a particularly high rate of smoking . The lower [START_REF] Benard | HIV infection and tobacco smoking behaviour: prospects for prevention? ANRS CO3 Aquitaine Cohort[END_REF] prevalence of smokers found in HIV-infected cohorts from West African countries could be partly explained by the fact that this population does not share the same addiction characteristics. However, the proportion of smokers in HIV-infected patients from West countries, including Africa, in order to compensate for their losses in the Northern Hemisphere at a time when the demand in these countries is decreasing . For this purpose, tobacco industry uses all kinds of strategies to market their products in these countries and [START_REF] Cherif | The tobacco problem in French-speaking Africa and regional perspectives of the French-speaking African tobacco observatory[END_REF] faces practically no barriers to their business as many of West African countries do not have any policy to restrain tobacco advertising or other promotion strategies , .

[16 17 ] ≥ % - [2 ]
consumption could act as a comfort measure against the severity of HIV infection. This association between tobacco smoking and a worse immunological status at HAART initiation could also be related to the decrease in the immune response related to tobacco use .

[

Cigarette smoking has been associated with an increased risk of developing bacterial and viral pneumonia in both HIV-negative and HIV-positive patients . The association between a low CD4 count at HAART initiation and cigarette smoking could be then [23 -27 ] related to their dual association with an increased risk of pulmonary infection. Indeed, the significant association between smoking and a history of tuberculosis infection is in accordance with this latter hypothesis. An association between tuberculosis and smoking has already been described in cross-sectional and case-control studies in low income countries such as India and Indonesia , but there is scarce [28 -30 ] information from sub-Saharan Africa where tuberculosis is highly prevalent. However, our study design did not allow us to draw any causal relation between immunological status, tuberculosis infection and smoked tobacco but this association and the occurrence of other infections surely needs further investigation.

Limitations

We acknowledge three main limitations to our survey. First, as it was conducted in a limited number of HIV/AIDS clinics volunteering and having the capacity to conduct it, these results could not be extrapolated to all the IeDEA participating centers and thus to the entire HIV/AIDS infected population currently in care in West Africa. Another limitation of this work is the lack of reliable data concerning these consumptions in the general population of comparable age and gender distribution so as to compare the impact of such consumptions according to HIV status. Consequently, we cannot actually tell if the relatively high smoking prevalence found in Mali is related to a higher tobacco use in HIV-infected patients or if it is related to the current higher prevalence of smoking in the general population of Mali compared to other countries in the region. Finally, the information related to tuberculosis infection relied on medical records from each participating site. Facing the absence of harmonized definition of tuberculosis and the relative difficulty to assess a diagnosis of certainty, we acknowledge the lack of stability concerning this association. However, facing the great burden of tuberculosis in sub-Saharan Africa and the growing evidence of its association with smoking from previous reports elsewhere in the world, this research question should become a priority.

CONCLUSION

This first description of smoking patterns in HIV-infected patients on HAART from three French-speaking countries in West Africa showed marked differences between communities. It highlighted the need for extending this survey to other clinics and regions in this continent in order to better describe these differences and to identify populations highly exposed to this cancer and cardiovascular risk factor. Finally, this survey also provides evidences concerning the association between cigarette smoking and tuberculosis in HIV-infected patients, a major public health issue in this part of the world.

Ackowledgements:

Centre

Estimated smoking prevalence (with 95 upper limit of confidence interval) according to gender and country (n 2920).

  significantly more frequently reported by men (5 ) compared to women (0.4 ) (p<0.0001). The proportions of present or % % past cannabis consumers were 1.8 in C te d Ivoire, 1.7 in Mali and 1.5 in Benin (p 0.82).

-

  Regular smokers are defined as present smokers declaring one or more cigarette per day for at least one * year. Tobacco use in HIV-infected patients in West Africa Int J Tuberc Lung Dis . Author manuscript Page /

  Being a man, reporting alcohol use during the past year and being from Mali or C te d Ivoire compared to Benin were the main

	Characteristics of smoking consumption
	Exposed patients stating a present or former smoking consumption declared smoking a median number of 8 cigarettes per day, with a
	significant difference according to country. Malian smokers declared a median number of 10 cigarettes per day while smokers from Benin
	declared a median number of three cigarettes per day (p<0.0001). The proportion of women who declared a present or past history of
	smoking was significantly higher in C te d Ivoire compared to Benin or Mali (p 0.03). Almost all smokers declared buying manufactured ô ' =
	and filtered cigarettes (	table 1	).
	Factors associated with tobacco smoking
			ô	'
	%		%

[ ]years (IQR 1 4 years). The median CD4 count at HAART initiation was 162 cells/mm (IQR 76 245). The place of living was essentially -3 urban for 90.8 of patients, 9.8 living in a rural area.

  M dical de Suivi de Donneurs de Sang/CNTS/PRIMO-CI, Abidjan, C te d Ivoire

	é	ô	'
	11)
	Service de Maladies Infectieuses et Tropicales (SMIT) de l h pital de Treichville, Abidjan, C te d Ivoire ' ô ô '
	12)
	ACONDA-MTCT Adultes, Abidjan, C te d Ivoire + ô '

Table 1

 1 Main characteristics of HIV-infected adult patients declaring present or former regular tobacco smoking according to country. IeDEA West Africa collaboration, 2007 2008 Regular tobacco smoking defined as smoking at least one cigarette per day for one year or more

	*

* † Interquartile range ‡ Kruskal-Wallis test with 2 degrees of freedom ∫ Fisher s exact test

We are indebted to the interviewers who performed the data collection and to the data clerks who entered all the data in the IeDEA/PAC-CI office in Abidjan. Special thanks to Rosemary McKaig, Caroline Williams, Gerald Sharp and Geraldina Domingez for their support in the design and conduct of these studies.

Source of support

This work was funded by the following institutes: the National Cancer Institute (NCI), the Eunice Kennedy Shriver National Institute of Child Health & Human Development (NICHD) and the National Institute of Allergy and Infectious Diseases (NIAID).

Smoking prevalence

Among the 2920 enrolled patients, the prevalence of present or past history and regular smoking consumption was 46.2 (95 CI % % 42.8 49.5) and 15.6 (95 CI 13.2 18.0) in men and 3.7 (95 CI 2.9 4.5) and 0.6 (95 CI 0.3 0.9) in women, respectively. In men,

the prevalence of regular smokers in Mali (29.7 (95 CI 22.9 36.5)) was twice higher than in C te d Ivoire (14.

Factors associated with smoking

The important variation of smoking prevalence according to gender is in accordance with most previous studies conducted in HIV-negative populations on the African continent , . There is some evidence that this particularly unbalanced in smokers

in girls, respectively . Although no reliable comparisons to our study population can be drawn from these results, these [START_REF] Warren | Global youth tobacco surveillance, 2000 2007[END_REF] population-based prevalence estimates already showed higher figures for girls. A report from Simen-Kapeu found a smoking et al prevalence of 1.4


Daniela Giachetti

François Murat

An elliptic problem with a lower order term having singular behaviour

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution u in a singular way and on its gradient Du with quadratic growth. The prototype of the problem under consideration is

we prove the existence of a solution for both the "+" and the "-" signs, while if k ≥ 1, we prove the existence of a solution for the "+" sign only.

Introduction

Second order quasilinear elliptic problems involving a first order term b(x, u, Du) depending on the solution u and on its gradient Du with a quadratic growth with respect to Du have been studied by many authors. Let us just quote [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique[END_REF], [START_REF] Boccardo | Résultats d'existence pour certains problèmes elliptiques quasilinéaires[END_REF], [START_REF] Boccardo | Existence of bounded solutions for nonlinear elliptic unilateral problems[END_REF], [START_REF] Boccardo | L ∞ -estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF], [START_REF] Dall'aglio | Nonlinear elliptic equations with natural growth in general domains[END_REF], [START_REF] Ferone | Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, in Equations aux dérivées partielles et applications[END_REF], [START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal[END_REF] and [START_REF] Porretta | Nonlinear elliptic equations having a gradient term with natural growth[END_REF] and references therein.

The first order term b(x, u, Du) appears in a natural way when one considers the Euler equations of functionals of the type

1 2 Ω a(x, u)|Du| 2 - Ω f u , (1.1)
which are perturbations of the classical energy functional

1 2 Ω |Du| 2 - Ω f u ,
since the Euler equation of (1.1) reads as -div (a(x, u)Du) + 1 2

∂a ∂u (x, u)|Du| 2 = f in Ω.
More in general, the literature deals with the case where b(x, s, ξ) is a Carathéodory function, which implies, in particular, that b(x, s, ξ) is continuous in the s variable.

In contrast, in the present paper, we want to study the case where b(x, s, ξ) is singular in s = 0. In particular, since we are interested in homogeneous Dirichlet problems in a bounded open set Ω of R N , the function b(x, u, Du) is singular at each point of the boundary of Ω.

As far as we know, the only results in this case have been obtained in [START_REF] Arcoya | A quasilinear approach for semilinear problems with blow-up at the boundary[END_REF], [START_REF] Arcoya | Quadratic quasilinear equations with general singularities[END_REF], [START_REF] Arcoya | Quasilinear equations with natural growth[END_REF] and [START_REF] Boccardo | Dirichlet problems with singular and gradient quadratic lower order terms[END_REF]. In the first paper, the authors consider the equation

-∆u + g(u)|Du| 2 = f (x) in Ω, u = 0 on ∂Ω, (1.2)
with g ≥ 0, lim sup s→0 sg(s) < +∞ and a datum f which is supposed to satisfy

f ∈ L ∞ (Ω), inf{f (x) : x ∈ ω} > 0 ∀ ω ⋐ Ω.
In [START_REF] Arcoya | Quasilinear equations with natural growth[END_REF], a slight variation on the hypothesis g ≥ 0 is considered. In [START_REF] Arcoya | Quadratic quasilinear equations with general singularities[END_REF] existence and nonexistence results are given for the previous problem. In [START_REF] Boccardo | Dirichlet problems with singular and gradient quadratic lower order terms[END_REF], the datum f is assumed to belong to L q (Ω) with q > N 2 and to satisfy f ≥ 0: existence results of strictly positive solutions are then proved for

-α∆u + 1 u k |Du| 2 = f (x) in Ω, u = 0 on ∂Ω,
with α > 0 and 0 < k < 1 or α > 1 and k = 1.

The problem that we consider in the present paper is -div(a(x, u, Du)) + λu = b(x, u, Du) + f (x) in Ω, u = 0 on ∂Ω,

where the principal part -div(a(x, u, Du)) of the equation is a Leray-Lions operator acting on H 1 0 (Ω) and where λ > 0. The datum f is supposed to satisfy f ∈ L ∞ (Ω), f (x) ≥ 0, so that, in view the assumptions made on the term b(x, u, Du), it results that u ≥ 0. As for the nonlinear term b(x, u, Du), we will assume that

|b(x, s, ξ)| ≤ C 2 |s| k |ξ| 2 ;
if 0 < k < 1, this will be the sole assumption on b, and in particular no sign condition will be imposed to b. We make a more restrictive assumption if k ≥ 1, assuming in this case that

C 1 |s| k |ξ| 2 ≤ b(x, s, ξ) ≤ C 2 |s| k |ξ| 2 . (1.4)
In both cases, we will prove the existence of a problem (1.2), namely of a function u which satisfies

u ∈ H 1 loc (Ω) ∩ L ∞ (Ω) , u ≥ 0 , |Du| 2 u k χ u>0 ∈ L 1 (Ω), Ω a(x, u, Du)DΦ + λ Ω uΦ = Ω f Φ + Ω b(x, u, Du)χ u>0 Φ ∀ Φ ∈ C ∞ c (Ω) .
Moreover, when 0 < k < 1, the function u satisfies u ∈ H 1 0 (Ω) (and not only u ∈ H 1 loc (Ω)), and satisfies the boundary condition u = 0 in the usual weak sense, while in the case k ≥ 1, the function u satisfies ψ(u) ∈ H 1 0 (Ω), where

ψ(s) = s 0 e γ(σ) dσ, γ(s) ∼ - 1 s k-1 when k > 1,
which also expresses the homogeneous Dirichlet boundary condition, but not in the usual weak sense.

The fact that nonlinear functions of u appear in the formulation of the solution of the problem is not really surprising. It is indeed well known that, in this kind of problems, functions which are related to the behaviour of the nonlinearity b(x, s, ξ) in the s variable play an essential role. In particular test functions of the type e γ(u) ϕ(u) (with ϕ a convenient function) are used to get a priori estimates.

Let us emphasize that there is therefore an important difference between the case 0 < k < 1 and the case k ≥ 1. In particular, the stronger hypothesis (1.4) made in the case k ≥ 1 is probably a crucial and not only a technical hypothesis. Indeed it has recently been proved in [START_REF] Arcoya | Quadratic quasilinear equations with general singularities[END_REF] that there is no solution u ∈ H 1 0 (Ω) of (1.2) in the case g(u) = 1

u k and k ≥ 2, see Remark 2.6 below. To conclude this introduction, let us note that, as far as we know, the case where the datum f (and therefore the solution u) takes both positive and negative values is an open problem. Also, as far as we know, the problem with f positive but where the singularity in u takes place in a point m > 0 (and is therefore of the type ) is an open problem, which seems to exhibit difficulties similar to the previous one.

The plan of the present paper as follows: In Section 2 we give the precise hypotheses and statements of our results. In Section 3 we define the approximating problems. In Section 4 we prove Theorem 2.2 (case 0 < k < 1), while in Section 5 we prove Theorem 2.3 (case k ≥ 1). Let us explicitly note that the first two steps are the same in the proofs of Theorems 2.2 and 2.3.

Hypotheses and results

In the present paper we consider the problem -div(a(x, u, Du))

+ λu = b(x, u, Du) + f (x) in Ω, u = 0 on ∂Ω, (2.1)
where Ω is an open bounded set of R N , where λ > 0, (2.2)

f (x) ∈ L ∞ (Ω), f (x) ≥ 0, (2.3)
where the function a(x, s, ξ)

: Ω × R × R N → R N
is a Carathéodory function which satisfies for some α > 0

a(x, s, ξ)ξ ≥ α|ξ| 2 , (2.4) |a(x, s, ξ)| ≤ ν|ξ|, (2.5)
(a(x, s, ξ)) -a(x, s, η))(ξ -η) > 0, ∀ ξ = η, (2.6)
a.e. x ∈ Ω, ∀s ∈ R, ∀ξ ∈ R N , and where the function As far as the behaviour of b(x, s, ξ) near s = 0 (for x and ξ fixed) is concerned, it is worth to distinguish two cases, the case 0 < k < 1 and the case k ≥ 1, that present different features and that will be treated separately.

b(x, s, ξ) : Ω × (R -{0}) × R N → R
We will suppose either that

   0 < k < 1, |b(x, s, ξ)| ≤ C 2 |s| k |ξ| 2 , a.e. x ∈ Ω, ∀ s = 0, ∀ξ ∈ R N , (2.7)
or that for some

C 1 > 0,    k ≥ 1, C 1 |s| k |ξ| 2 ≤ b(x, s, ξ) ≤ C 2 |s| k |ξ| 2 , a.e. x ∈ Ω, ∀ s = 0, ∀ξ ∈ R N .
(2.8)

Note that (2.8) is much more restrictive than (2.7), since (2.8) is a growth condition for b(x, s, ξ) both from above and from below, while (2.7) is only a growth condition from above. In particular, when (2.7) holds true, b(x, s, ξ) is not assumed to have a specified sign, while b(x, s, ξ) has in particular to be strictly positive (for ξ = 0) when (2.8) holds true. Let M > 0 and β : (0, M ] → R be defined by

M = f ∞ λ , β(s) = 1 s k .
(2.9)

In the case where 0 < k < 1, the function β belongs to L 1 (0, M ), while in the case where k ≥ 1, the function β is not integrable in 0.

Let us introduce the following function γ(s), defined for s ∈ (0, M ], which is a primitive function of the function C2 α β(s), defined by

γ(s) =                  C 2 α s 0 1 σ k dσ = C 2 α s 1-k 1 -k , if 0 < k < 1, C 2 α s M 1 σ dσ = C 2 α log s M , if k = 1, C 2 α s M 1 σ k dσ = C 2 α 1 k -1 1 M k-1 - 1 s k-1 , if k > 1 .
(2.10) Let us finally define, for s ∈ [0, M ], the function ψ by

ψ(s) = s 0 e γ(σ) dσ, (2.11)
and, for m > 0 and s ∈ R, the function S m by

S m (s) = m if s ≤ m, s if s ≥ m .
(2.12)

Remark 2.1 Let us point out that, in the case where 0 < k < 1 (i.e. when (2.7) holds), γ(s) is an increasing, non negative bounded function on [0, M ], while in the case where k ≥ 1 (i.e. when (2.8) holds), γ(s) is an increasing, non positive function on (0, M ] with lim s→0 + γ(s) = -∞.

In both cases e γ(s) is a bounded function on [0, M ] and, therefore, the function ψ(s) is well defined by (2.11).

Our results are the following: Theorem 2.2 Suppose that (2.2)-(2.7) hold true. Then there exists at least a function u such that

u ∈ H 1 0 (Ω) ∩ L ∞ (Ω), u ≥ 0, (2.13) ψ(u) ∈ H 1 0 (Ω), |Du| 2 u k χ u>0 ∈ L 1 (Ω), (2.14) Ω a(x, u, Du)DΦ + λ Ω uΦ = Ω b(x, u, Du)χ u>0 Φ + Ω f Φ, ∀ Φ ∈ C ∞ c (Ω). (2.
u ∈ H 1 loc (Ω) ∩ L ∞ (Ω), u ≥ 0, (2.16)    ψ(u) ∈ H 1 0 (Ω), S m (u) ∈ H 1 (Ω), ∀ m > 0, |Du| 2 u k χ u>0 ∈ L 1 loc (Ω),
(2.17)

Ω a(x, u, Du)DΦ + λ Ω uΦ = Ω b(x, u, Du)χ u>0 Φ + Ω f Φ, ∀ Φ ∈ C ∞ c (Ω). (2.18) Remark 2.4 When u ≥ 0 and S m (u) ∈ H 1 loc (Ω)
for every m > 0, the assertion It is likely that the more restrictive condition (2.8) is necessary in order to have the existence of a solution of (2.18) if k ≥ 2 . Indeed, it has been proved in [START_REF] Arcoya | Quadratic quasilinear equations with general singularities[END_REF] that for λ > 0, f strictly positive on every compactly embedded subset of Ω and k ≥ 2, there is no solution of the problem

|Du| 2 u k χ u>0 is equivalent to the assertion |D(u 1-k 2 )| ∈ L 2 loc (Ω) for every k = 2,
   u ∈ H 1 0 (Ω), u ≥ 0, -∆u + 1 u k |Du| 2 = f (x) in D ′ (Ω).
(R -{0}) × R N ) by b(x, s, ξ) = b(x, s, ξ) if s = 0, 0 if s = 0 .
Note that b is not a Carathéodory function since it is not continuous at the point s = 0. Nevertheless, for u and v measurable functions with values in R and R N , the function

x ∈ Ω → b(x, u(x), v(x)) is measurable 1 under the assumption made on b(x, s, ξ) that b is a Carathéodory function on Ω × (R -{0}) × R N .
1 Consider indeed a sequence (un, vn) of step functions which converge almost everywhere on Ω to (u, v) and which satisfy un(x) = 0 for every x and every n (it is always possible to build such a sequence un from a given sequence of step functions ûn by defining the functions un defined by un(x) = ûn(x) if ûn(x) = 0, un(x) = 1 n if ûn(x) = 0). Then b(x, un(x), vn(x)) is a measurable function and b(x, un(x), vn(x))χ u>0 converges a.e. to b(x, u, v), which is therefore a measurable function.

We could therefore have replaced the first terms of the right-hand sides of (2.15) and (2.18) by Ω b(x, u, Du)Φ, but we chose not to do so, in order to emphasize the fact that b is not a Carathéodory function.

On the other hand, in the case where hypothesis (2.8) holds, it is natural to

consider the function b(x, s, ξ) defined by b(x, s, ξ) =      b(x, s, ξ) if s = 0, 0 if s = 0, ξ = 0, +∞ if s = 0, ξ = 0,

Approximating problems

In order to prove Theorem 2.2 and 2.3, we introduce in this Section a sequence of approximating problems.

For n ∈ N, we consider the problems

u n ∈ H 1 0 (Ω) ∩ L ∞ (Ω), -div(a(x, u n , Du n )) + λu n = b n (x, u n , Du n ) + f (x) in D ′ (Ω), (3.1)
where, for almost every x × Ω, for every s ∈ R and every ξ ∈ R N , the function

b n : Ω × R × R N → R is defined by b n (x, s, ξ) =      b(x, s, ξ) if s > 1 n , b x, 1 n , ξ if s ≤ 1 n .
We also define, for s ∈ R,

β n (s) =      β(s) if s ≥ 1 n , β 1 n if s ≤ 1 n , γ n (s) =          C 2 α s 0 β n (σ)dσ if 0 < k < 1, C 2 α s M β n (σ)dσ if k ≥ 1, (3.2)
ψ n (s) = s 0 e γn(σ) dσ . (3.3)
In contrast with the function b, which is defined on Ω × (R -{0}) × R N , the function b n is a Carathéodory function defined on Ω × R × R N and we have

|b n (x, s, ξ)| ≤ C 2 β n (s)|ξ| 2 , a.e. x ∈ Ω, ∀ (s, ξ) ∈ R × R N , (3.4)
which implies in particular that

|b n (x, s, ξ)| ≤ C 2 β 1 n |ξ| 2 , a.e. x ∈ Ω, ∀ (s, ξ) ∈ R × R N . (3.5)
For n fixed, the function b n is now a classical Carathéodory function with quadratic growth with respect to ξ, and since f ∈ L ∞ (Ω), it is well known that problem (3.1) has at least one solution u n (see e.g. [START_REF] Boccardo | Résultats d'existence pour certains problèmes elliptiques quasilinéaires[END_REF]). Moreover, since f ≥ 0, this solution satisfies u n ≥ 0 a.e. in Ω. Indeed, denoting for every s ∈ R (s) + = max{s, 0}, (s) -= max{-s, 0}, the result u n ≥ 0 is easily obtained, in the case where b(x, s, ξ) ≥ 0 (which implies b n (x, s, ξ) ≥ 0), by using -(u n ) -as test function in (3.1); in the general case, the result u n ≥ 0 is proved by using in (3.1) the test function

-(u n ) -e -knun , with αk n ≥ C 2 β( 1 n ), since k n a(x, u n , Du n )Du n e -knun (u n ) -+ b n (x, u n Du n )e -knun (u n ) -≥ 0.
This proves the existence of a function u n such that

u n ∈ H 1 0 (Ω) ∩ L ∞ (Ω), u n ≥ 0, -div(a(x, u n , Du n )) + λu n = b n (x, u n , Du n ) + f (x) in D ′ (Ω). (3.6)
4 Proof of Theorem 2.2

We begin by proving Theorem 2.2 and divide the proof in several steps. Let us explicitly note that the first two steps of the proof of Theorem 2.3 will be identical to the first two steps of the present proof.

Step 1 Uniform estimate of (u n ) n∈N in L ∞ (Ω).

Let us use as test function in (3.1) the function v n = e γn(un) (u n -M ) + which belongs to

H 1 0 (Ω) ∩ L ∞ (Ω) since u n belongs to H 1 0 (Ω) ∩ L ∞ (Ω)
, where M is defined by (2.9). Using (2.4), (3.2) and (3.4) we get

α Ω |D(u n -M ) + | 2 e γn(un) + + C 2 Ω |Du n | 2 β n (u n )e γn(un) (u n -M ) + + Ω λu n e γn(un) (u n -M ) + ≤ ≤ C 2 Ω β n (u n )|Du n | 2 e γn(un) (u n -M ) + + Ω f e γn(un) (u n -M ) + .
We simplify the two terms which are equal, forget the first (non negative) term in the left-hand side and add to both sides -λM Ω e γn(un) (u n -M ) + , getting

λ Ω (u n -M ) 2 + e γn(un) ≤ Ω (f -λM )e γn(un) (u n -M ) + ≤ 0, which implies u n ≤ M . Therefore we have 0 ≤ u n ≤ M ∀n ∈ N . (4.1)
Step 2 Uniform estimate of (ψ(u n )) n∈N in H 1 0 (Ω). Let us take as test function in (3.1) the function v n = ψ(u n )e γn(un) which belongs to H 1 0 (Ω). Using (2.4), (3.3) and (3.4), we get

α Ω |Du n | 2 e 2γn(un) +C 2 Ω |Du n | 2 e γn(un) β n (u n )ψ n (u n )+λ Ω u n ψ n (u n )e γn(un) ≤ ≤ C 2 Ω β n (u n )|Du n | 2 e γn(un) ψ n (u n ) + Ω f e γn(un) ψ(u n ) .
Recalling that, by Step 1, f e γn(un) ψ n (u n ) is uniformly bounded in n, we get

α Ω |Dψ n (u n )| 2 = α Ω |Du n | 2 e 2γn(un) ≤ cst ∀ n ∈ N . (4.2)
Note that, in these first two steps, only the growth condition of hypothesis (2.7) (and never the growth condition from below of hypothesis (2.8)) is used.

Step 3 Uniform estimate of (u n ) n∈N in H 1 0 (Ω), of (a(x,

u n , Du n )) n∈N in L 2 (Ω) N and of (b n (x, u n , Du n )) n∈N in L 1 (Ω).
In this step we use for the first time the fact that k < 1. Since here 0 < k < 1, the functions γ and γ n defined by (2.10) and (3.2) are non negative. Therefore γ n (u n ) ≥ 0 and (4.2) implies that

Ω |Du n | 2 ≤ cst ∀n ∈ N, (4.3)
and from (2.5) and (4.3) it follows that

Ω |a(x, u n , Du n )| 2 ≤ ν 2 Ω |Du n | 2 ≤ cst ∀ n ∈ N . (4.4)
We then use as test function in (3.1) the function v n = e γn(un) -1 which belongs to H 1 0 (Ω). Since γ n (u n ) ≥ 0, we have v n ≥ 0, and using (3.4) we get

C 2 α Ω a(x, u n , Du n )Du n β n (u n ) e γn(un) + λ Ω u n (e γn(un) -1) ≤ ≤ C 2 Ω β n (u n )|Du n | 2 (e γn(un) -1) + Ω
f (e γn(un) -1) .

Using (2.4), simplifying the two terms which are equal, then using the fact the last term of the left-hand side is non negative, and the fact that e γn(un) is uniformly bounded in n, we get

C 2 Ω β n (u n )|Du n | 2 ≤ cst ∀ n ∈ N, (4.5)
which by (3.4) gives

Ω |b n (x, u n , Du n )| ≤ cst ∀n ∈ N. (4.6)
Step 4 Weak convergence of (u n ) n∈N in H 1 0 (Ω) and strong convergence of (DS m (u n )) n∈N in (L 2 (Ω)) N for any fixed m > 0.

By estimate (4.3), we deduce that, up to a subsequence, there exists a func-

tion u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) such that u n ⇀ u weakly in H 1 0 (Ω) , u n → u a.e. in Ω .
Therefore, for every m > 0,

S m (u n ) ⇀ S m (u) weakly in H 1 0 (Ω), S m (u n ) → S m (u) a.e. in Ω
where S m is defined by (2.12).

We want now to prove that

lim n→+∞ Ω |D(S m (u n ) -S m (u))| 2 = 0 ∀ m > 0. (4.7)
We first use as test function in (3.1) the function

v n = Φ((S m (u n ) -S m (u))
+ )e γn(un)-γn(Sm(un)) , where Φ(s) = e µs -1, for some µ ≥ 2β(m). Using (3.2) and (3.4), we get

Ω a(x,u n ,Du n )D(S m (u n )-S m (u)) + Φ ′ ((S m (u n )-S m (u)) + )e γn(un)-γn(Sm(un)) + + C 2 α Ω a(x, u n , Du n )Du n Φ((S m (u n ) -S m (u)) + )β n (u n )e γn(un)-γn(Sm(un)) + - C 2 α Ω a(x,u n ,Du n )DS m (u n )Φ((S m (u n )-S m (u)) + )β n (S m (u n ))e γn(un)-γn(Sm(un)) + + λ Ω u n Φ((S m (u n ) -S m (u)) + )e γn(un)-γn(Sm(un)) ≤ ≤ C 2 Ω β n (u n )|Du n | 2 Φ((S m (u n ) -S m (u)) + )e γn(un)-γn(Sm(un)) + + Ω f Φ((S m (u n ) -S m (u)
) + e γn(un)-γn(Sm(un)) .

Using (2.4), then simplifying the two terms which are equal and forgetting the last term of the left-hand side, which is non negative, we get

Ω a(x,u n ,Du n )D(S m (u n )-S m (u)) + Φ ′ ((S m (u n )-S m (u)) + )e γn(un)-γn(Sm(un)) - - C 2 α Ω a(x,u n ,Du n )D(S m (u n ))Φ((S m (u n )-S m (u)) + )β n (S m (u n ))e γn(un)-γn(Sm(un)) ≤ ≤ Ω f Φ((S m (u n ) -S m (u))
+ )e γn(un)-γn(Sm(un)) .

Due to (4.1) and to the almost everywhere convergence of (u n ) n∈N to u, we have

Ω f Φ((S m (u n ) -S m (u)) + )e γn(un)-γn(Sm(un)) = ω(n),
where here and in the sequel ω(n) is a sequence of real numbers for which

lim n→+∞ ω(n) = 0. Therefore Ω a( )D( ) + Φ ′ (( ) + )e γn(un)-γn(Sm(un)) + - C 2 α Ω a( )DS m (u n )Φ(( ) + )β n (S m (u n ))e γn(un)-γn(Sm(un)) = = I + II = ω(n) . (4.8)
Let us split the term I as

I = un≤m + un≥m = I 1 + I 2 . ( 4.9)
Now

I 1 = 0 I 2 = un≥m [a(x,u n ,Du n )-a(x,u n ,DS m (u))]D(u n -S m (u)) + Φ ′ ((u n -S m (u)) + )+ + un≥m a(x, u n , DS m (u))D(u n -S m (u)) + Φ ′ ((u n -S m (u)) + ) .
The last term goes to zero as n → +∞ by the fact that

DS m (u n ) ⇀ DS m (u) weakly in (L 2 (Ω)) N while a(x, u n , DS m (u))Φ ′ ((u n -S m (u)) + )χ un≥m χ u>m → 0 strongly in (L 2 (Ω)) N .
Therefore, by (4.8), we have

un≥m [a(x,u n ,Du n )-a(x,u n ,DS m (u))] D(u n -S m (u)) + Φ ′ ((u n -S m (u)) + )+II = = H + II = ω(n) (4.10)
where we have defined

H = un≥m [a(x,u n ,Du n ) -a(x,u n ,DS m (u))]D(u n -S m (u)) + Φ ′ ((u n -S m (u)) + ).
II = - C 2 α un≥m a(x, u n , Du n )Du n Φ((u n -S m (u)) + )β n (u n ) = = - C 2 α un≥m [a(x, u n , Du n ) -a(x, u n , DS m (u))] [1ex] D(u n -S m (u))Φ((u n -S m (u)) + )β n (u n )+ [1ex] - C 2 α un≥m a(x, u n , DS m (u))D(u n -S m (u))Φ((u n -S m (u)) + )β n (u n ) = = - C 2 α un≥m a(x, u n , Du n )DS m (u)Φ((u n -S m (u)) + )β n (u n ) = = I 1 + II 2 + II 3 .
(4.12) We have

II 1 ≥ -c(m) un≥m [ ]D(u n -S m (u))Φ((u n -S m (u)) + ) ≥ - 1 2 H (4. 13
)
where H is defined in (4.11) and

c(m) = max s∈[m,k] β n (s) .
Here we used the fact that we can choose λ sufficiently large that

c(m)Φ(s) ≤ 1 2 Φ ′ (s) .
Moreover it is easy to check that

II 2 = ω(n) (4.14) |II 3 | ≤ c(m) Ω Φ 2 ((S m (u n ) -S m (u)) + )|DS m (u)| 2 1 2 = ω(n) (4.15)
(here we used estimate (4.4)). By (4.10)-(4.15), by the fact that Φ ′ (s) ≥ 1 ∀s ≥ 0 and by [?], we have

lim n→+∞ Ω |D(S m (u n ) -S m (u)) + | 2 = 0 ∀ m > 0 .
In a similar way, using as test function in (3.1)

v = Φ(-(S m (u n ) -S m (u)) -)e γn(Sm(un))-γn(un)
we get

lim n→+∞ Ω |D(S m (u n ) -S m (u)) -| 2 = 0 ∀ m > 0
which concludes the proof of Step 4.

Step 5 In this step we prove that for every C ⋐ Ω, we have

v n = -(e γn(m)-γn(un) -1) + ϕ 2 ϕ ∈ C ∞ c (Ω) . By (2.4) -2 Ω a(x, u n , Du n )Dϕ (e γn(m)-γn(un) -1) + ϕ + + C 2 {un≤m} |Du n | 2 ϕ 2 e γn(m)-γn(un) β n (u n )-λ {un≤m} u n (e γn(m)-γn(un) -1)ϕ 2 ≤ ≤ {un≤m} |b n (x, u n , Du n )|(e γn(m)-γn(un) -1)ϕ 2
where we used the fact that f (x) ≥ 0 and the test function is non positive. Using (2.7), cancelling similar terms, we have, by Holder inequality, observing that in this case γ n (m) ≤ γ(m): we are going to prove that we have also

{un≤m} |b n (x, u n , Du n )|ϕ 2 ≤ λ {un≤m} u n e γ(m) ϕ 2 + + c {un≤m} |a(x, u n , Du n )| 2 ϕ 2 1 2 Ω |Dϕ| 2 ϕ 2 1 2 = A + B .
B ≤ cst m . (4.19)
Indeed, to see the last inequality, it is sufficient to take

-(u n -m) -ϕ 2 , ϕ ∈ C ∞ c (Ω), as test function in (3.1): {un≤m} a(x, u n , Du n )Du n ϕ 2 -2 {un≤m} a(x, u n , Du n )Dϕ (u n -m) -ϕ ≤ ≤ λ un≤m u n (m -u n )ϕ 2 + {un≤m} |b(x, u n , Du n )|(m -u n )ϕ 2 + + {un≤m} f (u n -m)ϕ 2 .
(4.20)

The last integral is non positive, while the second one at the right-hand side is bounded as

{un≤m} |b( )|(m -u n )ϕ 2 ≤ cst m ,
taking into account (4.5).

The same holds true for the first integral in (4.20) by Step 1. Let us finally estimate:

{un≤m} |a(x, u n , Du n )| |ϕ| |Dϕ|(m -u n ) ≤ ≤ m Ω |Dϕ| 2 1 2 Ω |a(x, u n , Du n )| 2 ϕ 2 1 2 ≤ ≤ cstm by (4.4).
Therefore, by (4.20), {un≤m} |Du n | 2 ϕ 2 ≤ cstm ∀n ∈ N which implies, by (2.5)

{un≤m} |a(x, u n , Du n )| 2 ϕ 2 ≤ cstm ∀ n ∈ N (4.21)
and, consequently, also (4.19). Therefore (4.17) and (4.19) imply (4.16), taking ϕ ≡ 1 on the compact set C.

Step 6 Equintegrability of the sequence (b n (x, u n , Du n )) n∈N on C ⋐ Ω.

In this step we are going to prove that for any fixed C ⋐ Ω

∀ ε > 0, ∃ δ ε > 0 : ∀ E ⊆ C, |E| < δ ε sup n E |b n (x, u n , Du n )| < ε . (4.22) Indeed E |b n (x, u n , Du n )| = E∩{un≤m} |b n (x, u n , Du n )|+ + E∩{un≥m} |b n (x, u n , Du n )| = I 1 + I 2 .
By (4.16) in Step 5, we have lim m→0 I 1 = 0 uniformly in n, so that

∀ ε > 0 ∃m 0 : ∀ m ≤ m 0 I 1 < ε 2 ∀ n ∈ N .
Therefore, for such an m, due to the strong convergence of |DS m (u n )| in L 2 (Ω) (see Step 4), we can choose |E| so small that

I 2 ≤ C 2 β(m) E |DS m (u n )| 2 < ε 2 ∀ n ∈ N .
Step 7 Passage to the limit. By (4.22), taking into account the a.e. convergence of (Du n ) n∈N and of (u n ) n∈N (up to a subsequence), we have, for any compact set

C ⋐ Ω b n (x, u n , Du n ) → b(x, u, Du) strongly in L 1 (C ∩ {u > 0}) . It remains to prove that b n (x, u n , Du n ) → 0 strongly in L 1 (C ∩ {u = 0}) .
To this aim, for any ε > 0, we have

C∩{u=0} |b n (x, u n , Du n )| = = C ε ∩{u=0} |b n (x, u n , Du n )| + (C-C ε )∩{u=0} |b n (x, u n , Du n )| = J 1 + J 2
where C ε is a subset of C (existing by Severivi-Egoroff theorem) such that |C ε | < δ ε and in C -C ε the sequence (u n ) n∈N converges uniformly. Here δ ε is the number defined in (4.22), and, in fact, by this condition it follows

J 1 < ε 2 ∀ n ∈ N,
while, for m sufficiently small, m = m(ε), by (4.16), we get

J 2 ≤ C∩{un≤m} |b n (x, u n , Du n )| < ε 2 ∀ n ≥ n 0 (m(ε)) = n 0 (ε) . So that we proved lim n→+∞ b n (x, u n , Du n ) = 0 strongly in L 1 (C ∩ {u = 0}),
and we can pass to the limit in the term Ω b n (x, u n , Du n )ϕ in the weak formulation for (3.1).

As far as the term Ω a n (x, u n , Du n )Dϕ, we can use (4.21) and Step 4 and repeat exactly the same arguments used for the term b n (x, u n , Du n ), getting

a n (x, u n , Du n ) → a(x, u, Du) strongly in L 1 (C ∩ {u > 0}), a n (x, u n , Du n ) → 0 strongly in L 1 (C ∩ {u = 0}) .
Therefore we obtained a distributional solution u for (2.1), in the sense that (2.15) holds true.

Proof of Theorem 2.3

As observed at the beginning of Section 4, the first two steps of the proof of Theorem 2.2 remain valid here, since these steps only the fact that |b(x, S, ξ

)| ≤ C2 |s k | |ξ| 2 .
Let us emphasize that the main difference in the proof below consists in the fact that we cannot achieve global estimates on the main terms, but only local ones.

Step

3 Uniform L 1 loc -estimate on (b n (x, u n , Du n )) n∈N . Let η(x) ∈ C ∞ c (
Ω). We use v = (e γn(un) -1)η 2 ∈ H 1 0 (Ω) as test function in (3.1). Let us explicitly point out that we need to use a cut-off function η 2 (x), since, in this case, by the definition (2.10) of γ n (s), e γn(un) -1 does not vanish on ∂Ω. Note also that v ≤ 0. (5.26)

Step 4 Weak convergence of (u n ) n∈N in H 1 loc (Ω) and strong convergence of (DS m (u n )) n∈N in (L 2 loc (Ω)) N for any m > 0. By the previous steps we deduce that there exists u ∈ H 1 loc (Ω) ∩ L ∞ (Ω) such that, up to a subsequence, To prove this, it is sufficient to take the same test functions used in Theorem 2.2, multiplied by a cut-off function η 2 ∈ C ∞ c (Ω) and use the local estimates available from the previous Step 3 (estimates (5.25), (5.26)).

Step 5 For any compact set C ⋐ Ω:

  is a Carathéodory function on Ω × (R -{0}) × R N , i.e. a function which is, for every (s, ξ) ∈ (R -{0}) × R N , a measurable function x ∈ Ω → b(x, s, ξ) ∈ R and which is, for almost every x ∈ Ω, a continuous function (s, ξ) ∈ (R-{0})×R N → → b(x, s, ξ) ∈ R for almost every x ∈ Ω (see Remark 2.7 below concerning the definition of b(x, s, ξ) when s = 0).

Remark 2 . 7

 27 The first terms of the right-hand sides of equations (2.15) and (2.18) involve the function b(x, u, Du)χ u>0 , which can also be written as b(x, u, Du), where b(x, s, ξ) is the function defined on Ω × R × R N (and no more on Ω ×

  since the function b is continuous for every s ≥ 0 and every ξ ∈ R N , except in the point s = 0 and ξ = 0. With this definition, we have b(x, s, ξ) = b(x, s, ξ) + (+∞)χ {s=0}∩{ξ =0} . But for u ∈ H 1 loc (Ω), we have b(x, u, Du) = b(x, u, Du) a.e. in Ω, since, when u ∈ H 1 loc (Ω), one has Du = 0 almost everywhere on the set where u = 0. Therefore, since Theorem 2.3 asserts that u ∈ H 1 loc (Ω), we could also have replaced the integral Ω b(x, u, Du)ξ u>0 φ by the integral Ω b(x, u, Du)φ in the first term of the right-hand side of (2.18).

(4. 11 )

 11 We now estimate term II in(4.10)

  lim m→0 C∩{un≤m} |b n (x, u n , Du n )| = 0 uniformlyin n. (4.16) Let us take as test function in (3.1)

C 2 Ωη 2 Ω( 1 -u n η 2 + + C 2 Ωη 2

 221222 |Du n | 2 e γn(un) β n (u n ) ≤ ≤ ν e γn(un) )2|η| |Dη| |Du n | + λ Ω u n (1e γn(un) )η 2 + + Ω b n (x, u n , Du n )(e γn(un) -1)η 2 + Ω f (e γn(un) -1)η 2 ≤ ≤ 2ν Ω |η| |Dη| |Du n | + λ Ω β n (u n )|Du n | 2 e γn(un) η 2 -Ω b n (x, u n , Du n )η 2 .(5.23) This follows by the fact that Ω f (e γn(un) -1)η 2 ≤ 0 since γ n (u n ) ≤ 0.We absorb the term C 2 Ω β n (u n )|Du n | 2 e γn(un) η 2 by the term at the lefthand side of (5.23).Moreover, for any ε > 0, there exists C(ε) such that 2νΩ |η| |Dη| |Du n | ≤ C(ε) |Du n | 2 ≤ C(ε) Ω |Dη| 2 + εM k Ω η 2 β n (u n )|Du n | 2 .We have also, by Step 1,λ Ω u n η 2 ≤ cst ∀ n ∈ N .Using the last two estimates in (5.23), recalling (2.8) and taking ε = C12M k , we getΩ β n (u n )|Du n | 2 η 2 (x) ≤ cst ∀ n ∈ N (5.24)which, by (2.8) and the fact thatβ n (u n ) ≥ C > 0, gives also, for η ∈ C ∞ c (Ω) Ω b n (x, u n , Du n )η 2 (x) ≤ cst ∀ n ∈ N (5.25) Ω |Du n | 2 η 2 (x) ≤ cst ∀ n ∈ N .

  u n ⇀ u in H 1 loc (Ω) u n → u a.e. in Ω .Moreover, with minor modifications, we can prove, as in Step 4 of Theorem 2.2, that limn→+∞ Ω |D(S m (u n ) -S m (u))| 2 η 2 = 0 ∀ η ∈ C ∞ c (Ω) .

  lim m→0 C∩{un≤m} b n (x, u n , Du n ) = 0 uniformly in n .Note that we cannot use the same arguments of Step 5 in Theorem 2.2 since now we do not have γ(s) ≥ 0 and bounded. We choose as test function in (3.1)v = -(e γn(un)-γn(m) -1) -ϕ 2 ∈ H 1 0 (Ω), ϕ ∈ C ∞ c (Ω) .

1

  |s-m| k and no more of the type1 |s| k

  and to |D log u| ∈ L 2 loc (Ω) for k = 2.

Remark 2.5 Remark 2.6 Let us observe that in Theorem 2.2, where (2.7) is assumed to hold true, we can consider a general term b(x, u, Du) without any sign condition, while hypothesis (2.8), assumed in Theorem 2.3, obliges b(x, s, ξ) to be strictly positive.

By (2.4), noticing that v ≤ 0:

We use condition (2.8) to cancel similar terms and note that (e γn(un)-γn(m) -1) -≤ 1, so that, by Holder inequality, we get

The last inequality follows by (4.21) which still holds true.

This concludes Step 5. The next Step 6 and Step 7 can be achieved in the same way of the corresponding ones in Theorem 2.2, which concludes the proof.