Seydina M Ndiaye

Amadou Ndiaye

Francis Fleurat-Lessard

PLANNING AND MONITORING OF STORED MALTING BARLEY QUALITY MAINTENANCE

Keywords: Computer, Modelling, Knowledge-based control, Planning, Monitoring, Agriculture

Representing the stored grain ecosystem for quality maintenance implies the representation of all its characteristics that take part in the grain quality degradation process. A mixed qualitative and quantitative modelling is used to represent the stored grain ecosystem. The temperature, moisture content and presence of insects are used as control variables. The quality maintenance operations are represented as actions to be executed in time requiring available equipment and consumable. Our planning approach involves three consecutive stages: treatment to obtain a safe grain storage condition, storage to maintain the storage condition and dispatch to respond to the market requirements.

INTRODUCTION

Building up decision support systems (DSS) for the conservation of grain initial quality during the storage period is one of the major concerns of postharvest research. Storage proves to be the weak link in foodstuff quality chain. Modern farming technics have allowed an increase in the world production of cereals; and, consequently, an increase in storage capacities and duration. It so happens that any lengthening of storage duration means an increase in the risk of grain degradation. Grain is a living organism, and the normal outcome of its vital activity is germination and oxidation phenomena (Multon et al., 1969). A stock of grain is the natural environment of insects, mites, and micro organisms: insects and mites consume and spoil the grain (Fleurat-Lessard et al., 1994a), while microorganisms deteriorate and contaminate it (Cahagnier et al., 1993).

In the stored grain ecosystem the biotic variables (grain, insects, moulds and mites) interact with abiotic variables (temperature, moisture content, impurities, atmospheric composition) which are influenced by the environmental conditions of the storage location. Much of the knowledge required to maintain stored grain quality is fragmented and applied in different ways, with varying success. The use of a computerised knowledge based decision support system could improve the conservation of stored grain in: i/ the assessment of the grain quality and condition at delivery; ii/ the planning of the grain storage in time taking into account the equipment and consumable; iii/ the monitoring of the storage condition during plan execution in order to be able to replan the storage if something goes wrong.

The interest of building up a knowledge based system to take up this problem can be explained by the strongly experimental nature of the available knowledge. For example, the population growth rates are known only for certain insect pests, at certain temperatures and humidity and for certain grains (Beckett et al., 1994). During the last ten years, several knowledge based pest control systems have been built up to support the control of stored grain pests (insects and mites) (Flinn et al., 1990), (Wilkin et al., 1991), (Longstaff et al., 1994), (Pasqual & Mansfield, 1988), (Jones et al., 1993). These systems have confirmed the interest of DSS to help preserve grain on one hand, and, on the other hand, have shown the limits of pest control as far as grain quality maintenance is concerned. Today, current research focuses on preservation of grain initial quality (Ndiaye et al., 1997(Ndiaye et al., , 1998)).

Our aim is a prevention of grain quality degradation guided by the measured values of the environmental factors and the noises from insects. The studies on environmental factors show that they give an early information on the risks of quality degradation (Fleurat-Lessard et al., 1994b;Wrigley et al. 1994;Juliano, 1994). Our approach differs from existing systems in that we give priority to the prevention of grain initial quality degradation, and therefore to predictive signs of degradation. Our strategy is based on an assessment of grain initial quality in order to be able to determine the optimal storage technical route allowing us to monitor the quality of grain. It involves the following three stages: 1) to assess the grain initial quality and condition before storage, 2) to plan an optimal storage technical route, 3) to monitor the evolution of grain quality during storage in order to react to any event representing a foreseen or effective quality degradation by replanning the storage technical route.

In this paper we present the modelling of the planning and monitoring of malting barley initial quality maintenance during storage period.

MODELLING OF THE ECOSYSTEM OF STORED MALTING BARLEY

The representation of the ecosystem of malting barley grain stock is rather complex. This complexity is related to the fact that various events interact during storage and can be factors of stock deterioration. These events depend on the various variables composing the ecosystem and the relations which connect them. Thus, to represent the instant status of the grain ecosystem requires representing all the components of the ecosystem and the relations connecting them. Moreover, it is necessary to represent the evolution of the ecosystem during storage. In addition, in order to obtain a total modelling of the grain ecosystem throughout time, we need to represent possible actions on the ecosystem.

An overview of the ecosystem modelling is represented in the figure below. Some variables are evaluated at the beginning of storage and do not evolve through time (quantity, identity). Others evolve during storage and are modified by events and/or actions on the stock (impurities rate, quality, temperature, moisture content, mites, moulds, insects, germination). We have mathematical models which make it possible to calculate the 'natural' evolution (i.e., without unforeseen event or human intervention) of some variables during storage (moisture and temperature, insects, moulds, mites and germination). The evolution of the other variables through time, in response to the events or the actions, is given by expert knowledge. The model of the grain ecosystem that we present in this section is composed of:

Mould risks model Moulds

Mites

• a representation of state variables;

• mathematical models and expert decision rules for events and changes of some variables (control variables and deterioration variables); • a representation of actions (human interventions).

In the later sections of this paper, we present the various components of the grain ecosystem and their interactions, then we see how the model is used in the system of grain quality conservation during storage.

Representation of ecosystem's state

Representing state of the grain ecosystem, in the conservation of stored malting barley quality problem, involves representing all the characteristics of the ecosystem which take part in the quality degradation process during storage. By quality deterioration, we mean 1/ a degradation of physiological grain quality (including increase of germination, growth of moulds and mites) and 2/ a too large1 amount of impurities and insects (living or dead) in the stock. According to Multon (Multon, 1988), there are four major spoilage risks of the physiological grain quality: insects2 , microorganisms (i.e., moulds) , Maillard reactions, and oxidation. Multon stated also that these risks depend on the levels of temperature (T) and moisture content (MC) which thus define a stability zone for each risk. In a previous work (Ndiaye, 1998), we have found five polynomial equations which formalise these stability zones3 :

Insects: T=aH 4 +bH 3 +cH 2 +dH+g (1) with T 27°C or T>27°C (two equations); Micro-organisms: H=aT 3 bT 2 +cT+d

(2) with H 28% and T 75°C; Maillard reactions: T=aH 4 +bH 3 +cH 2 +dH+g (3) with T 75°C; Oxidation: H=aT 3 +bT 2 +cT+d (4) with H 5% and T 75°C.

Thus, managing temperature and moisture content is sufficient to control the physiological grain quality degradation process. The amounts of impurities and insects are managed by the variables impurity rate (IR), and amount of insects (I).

These four variables (T, MC, IR and I) are called control variables. However, there are other variables which are included in the representation of the ecosystem's state. There are the identity of grain (Id), the quantity of grain (qt), the quality of grain, Q (which has a qualitative value; see (Ndiaye et al, 1998) for more details), the moulds growth (Mo), the presence of mites (Mi), and the germination (G).

We use qualitative representation for managing control variables T, MC, IR and I. For T and MC, we consider three possible values: safe, risky, good and bad; and two values for IR and I: 0, in the absence of insects or if level of impurities is lower than a threshold (IRS), and 1, otherwise. After a human intervention, values of control variables are updated according to some expert rules or through an assessment 4 .

The state variables can be managed through mathematical models and decision rules given by experts.

Mathematical models and expert decision rules

These models depend on temperature and moisture content. There are three models: ♦ the insects model, developed by P.W. Flinn, is an algorithm which gives the amount of insects at the end of storage period according to an initial amount of insects, a given temperature and a given moisture content. As defined, this model is not directly usable in our problem because we would like to know when the amount of insects will be considered too large in real storage condition (evolving temperature and moisture content). Thus, we modify it in order to have the amount of insects after a short period of time. The principle is to subdivide the storage period in small time duration and to apply the model for each small time duration. At each application of the model there is a new value of temperature and moisture content. This way, we can compute iteratively the amount of insects and know when it exceeded a threshold; ♦ the micro-organisms model is a model allowing to calculate the risk of presence of moulds and the maximum duration of conservation which is associated to it. The threshold is the appearance of visible moulding. This model is represented in the form of expert decision rules depending on the activity of water (Aw). The latter can be calculated starting from the moisture content, using an equation developed in (Denne, 1988). This knowledge base integrates data relating to the growth of the principal storage moulds. The effects of the growth of storage moulds (primarily Penicillium and Aspergillus) are also modelled. The main effect is the production of ergosterol (biochemical marker). The relation between the activity of water (Aw) and the maximum storage time is also integrated in the model. ♦ the temperature and moisture content evolution model is a set of mathematical formula. These equations result from the work of Singh and Thorpe (Singh & Thorpe, 1993;Thorpe, 1995) on heat and moisture transfer in grain bulks.

All the components of the ecosystem have been defined. We foresee the evolution of the ecosystem during storage by applying to the initial state all the models and by updating state variables. Now that all the possible events during storage are modelled, it is necessary to have a model of the effects of each action on the ecosystem, mainly on the four control variables. 4 The assessment gives quantitative values which are translated in qualitative values.

Representation of actions

As quoted in Figure 1, we have five actions to control storage: drying, cooling, cleaning, select_bin and insecticide.

At a given time, the choice of the action to execute is done according to the current state of the ecosystem and the availability of the resources (instrument and consumable).

In the representation of actions, a major problem is to take into account the time: for each action, the duration of its execution and the duration of its effects. Thus, we have defined three types of actions as shown in Figure 2. In action modelling, the time is represented as a 3-uples: (time ActionDuration EffectsDuration). In this section, we present how the five actions introduced above are modelled, including conditions for application and effects of each of them, and the duration of their application and their effects.

Drying

The action of drying is carried out when: * moisture content is too high (threshold value, mcS, defined for each country) independently of the temperature (which is considered brought back to a normal value before the application of drying); * or when there is a too large amount of moulds (higher than a threshold, moS).

After drying, the moisture content variable is at a known value (which is fixed beforehand, relative with the country considered) and mould growth is inhibited.

(action drying ?t (time ε ∞)

(preconditions DA (< ε TL) (or (< mcS MC) (< moS Mo))) (effects (lower_MC) (eq Mo 0) (events (influence_moulds) (influence_insects) (influence_germination) (influence_TL)))
The drying action, as represented above, can be applied at a moment ?t with a duration ε, very small and an effect duration we note ∞. Such an action is called instantaneous action with permanent effect. The preconditions are represented as propositions which should be true before action 'drying' is selected, there are 'Dryer available' (DA), moisture content greater than threshold, growth of mould, time left before degradation (TL) greater than the action duration. Note that the time left is the minimum value of three variables: TLI (time left before a degradation related to insects), TLMo (time left before a degradation related to moulds) and TLG (time left before a degradation related to germination). The effects obtained are a reduction in moisture content and a disappearance of moulds. Some events follow which are computation of the various time left. If action drying is applied at time then the new state of the ecosystem is obtained at time ?t+ε.

Cooling

This action prevents the development of moulds and insects. Cooling has effect on temperature and the moisture content (in a less amplitude) of the stock of grains. It should be noticed that if the outside temperature is definitely lower than that of the grain, it can generate condensations on the bin walls that can cause a degradation of grain (in the case of blowing aeration, which is not true for sucking aeration). The duration of cooling depends on the outside temperature, on temperature of the stock of grains and on the air flow specific rate (m 3 air per m 3 grain per hour). The value of the temperature reached can be computed starting from the data relating to mode of cooling.

(action cooling ?t (time m n) (preconditions FA (< m TL) (< T (TO+∆T))) (effects (lower_temperature)) (events (influence_moulds) (influence_insects) (influence_germination) (influence_TL)))
Cooling is a noninstantaneous action with nonpermanent effects. At time ?t+m+n, all the effects disappear. The pre-conditions here are: fan available (FA), time left greater than application duration and temperature less than outside temperature plus a fixed value ∆T.

Cleaning

This action requires a cleaner (CA: cleaner available). It makes it possible to remove the broken grains and the various impurities. It is an instantaneous action with permanent effects. It is carried out when there is a too large impurities rate (IR = 1). Temperature and moisture content remain constant at the end of cleaning.

(action cleaning ?t (time ε ∞) (preconditions CA (= IR 1)) (effects (eq IR 0)) (events (influence_TL)))

Select-bin

This action makes it possible to fix the type of bin in which storage will be made. Thus, only some actions will be later acceptable according to the selected type of bin. This choice depends on the identity, the quantity, the initial quality of the grain and the emptiness of the bin.

(action select-bin ?t (time ε ∞) (preconditions ) (effects (eq Bin (bin_choice))) (events ))

Insecticide

The choice of insecticide to be applied depends on the species of devastating insects of the grain, which vary according to the local climates. Another parameter is the storage period considered. Of course, it is necessary to take into account the list of available and authorised insecticides. The effect of an insecticide can be instantaneous, in the case of an insecticide with shock effect (protection from 1 to 2 weeks against the insects), or a long-term effect, in the case of a long-term insecticide (protection from 4 to 6 months). Applying insecticide has no direct effect on temperature and moisture content. However, the number of insects is brought back to a null value. A negative effect of the application of insecticide is the presence of active insecticide residues which can influence the quality of the grain. Insecticide can also be applied in order to enlarge the time left by increasing the optimal duration of conservation (if no other action is possible).

(action insecticide ?t (time ε k) (preconditions (or (and (< k (SD-?t)) (= I 1)) (critic))) (effects (eq IR 1) (eq I 0)

(eq Ins (ins_choice ?t)) (not critic)) (events (influence_moulds) (influence_insects) (influence_germination) (influence_TL)))

The applying of insecticide is an instantaneous action with nonpermanent effects; it is advocated when there are insects and the effects of insecticide should not remain at the exit of storage, or when no other action is applicable and TL is very low (i.e., critical situation, critic).

These five actions are used to manage the storage process. Figure 3 shows how they influence control variables and then the ecosystem as a whole. The influences of control variables on action choice are also represented. The terms ?, ↓, ↑ and -are respectively 'no information', 'decrease', 'increase' and 'not applicable'

Until now we showed how the ecosystem of the grains stock is modelled, along with a representation of all the actions. We will see in the next section an overview of the general system of generation of action plan for the control of the malting barley storage.

Overview of the control system

The aim of the system is to generate a strategy for conservation of stored malting barley quality.

Within our system, a strategy is defined in three stages:

• processing (before setting in bin);

• storage and monitoring;

• dispatch (at output of bin and sending).

At each stage, there are some applicable actions, as we can see in the figure below. We plan to use planning and monitoring technics for the generation of a storage strategy.

The general system is composed of a planning module and of a monitoring module. The former works out an initial action plan which is carried out. If an unforeseen event is detected by sensors during the execution of the plan, the planning module is reactivated.

CONCLUSION

In this paper we have presented a mixed modelling (qualitative and quantitative) of the ecosystem of stored malting barley, including the possible storage actions. This modelling based on the existing knowledge in grain storage domain, allows us to be able to integrate any new knowledge in this domain.

An implementation of the planning system is on going on a PC computer under Windows® using CLIPS and Common Lisp programming languages.

However, further experimental work is needed to expand the domain of this modelling to other kinds of cereal grain for food end-use.

Figure 1 .

 1 Figure 1. Overview of the ecosystem modelling and possible human interventions.

Figure 2 .

 2 Figure 2. Time in action representation.

Figure 3 .

 3 Figure 3. The influences actions/variables.

Figure 4 .

 4 Figure 4. The different process stages of the conservation of stored malting barley.

Figure 5 .

 5 Figure 5. Planning and Monitoring approach.

The amount of insects or impurities is considered too large when a threshold value is exceeded.

Let us notice that the insects risk here represents the risk of the degradation caused by the insects and not their presence itself.

Variables a, b, c, d and g have different values in each equation (Ndiaye, 1998).

Acknowledgements

This work has been carried out with financial support from the Commission of the European

Communities, Agriculture and Fisheries (FAIR) specific RTD programme, CT97-3648, "Quality of stored Grain (QualiGrain)". It does not necessarily reflect its views and in no way anticipates the Commission's future policy in this area. The authors wish to thank Dr Paul Flinn for the insects model provided.

Beckett S.J., Longstaff B.C., Evans D.E., (1994).

A comparison of the demography of four major stored grain coleopteran pest species and its implications for pest management. Proceedings 6th Int. Working Conf. Stored Product Protection, Canberra, April 94, 1, 4991-497. CAB International, Wallingford. Cahagnier B., Lesage B., Richard-Molard D., (1993) Hannover, Sept. 97. 219-224. Pasqual G.M., Mansfield J., (1988)


J Garc

Ía-Meli Án

A Quaas

B Sirakov
email: bsirakov@mat.puc-rio.br

ELLIPTIC EQUATIONS WITH ABSORPTION IN A HALF-SPACE

We give a necessary and sufficient condition, in the spirit of the classical works by Keller and Osserman, for the elliptic equation ∆u = f (u) to have a solution in a half-space of R N . The function f is supposed to be nondecreasing and nonnegative, and we are interested in solutions whose range is where f > 0. The possibility of obtaining such a necessary and sufficient condition has been an open question for a long time.

Introduction and results

The aim of this work is to analyze the existence and nonexistence of solutions of the problem (P) ∆u = f (u) in R N + , where the function f is continuous, nonnegative and nondecreasing on R, and R N + denotes the half-space R N + = {x ∈ R N : x N > 0}. We also study the existence of subsolutions of this problem, that is, solutions of the inequality (P -) ∆u ≥ f (u) in R N + . The classical papers by Osserman [START_REF] Osserman | On the inequality ∆u ≥ f (u)[END_REF] and Keller [START_REF] Keller | On solutions of ∆u = f (u)[END_REF] have generated a tremendous amount of work on this type of equations, most of which has been devoted to the study of entire solutions or of explosive solutions in a bounded domain. As already explained in [START_REF] Osserman | On the inequality ∆u ≥ f (u)[END_REF] (see p. 1644 in that paper) it is meaningful to search for solutions whose range is in the set where f is strictly positive, since without this assumption the solution set can indeed be very large and variable. Thus, in the following we will consider such solutions only. Note that since f is nondecreasing and nonnegative, we have {t : f (t) > 0} = (t 0 , ∞) for some t 0 ≥ -∞. Up to a simple transformation we can assume that t 0 = 0 or t 0 = -∞, in other words, either f (t) > 0 for t > 0 and u > 0, or f > 0 on R and there is no sign restriction on u.

In particular, it is known from [START_REF] Osserman | On the inequality ∆u ≥ f (u)[END_REF] that the equation ∆u = f (u) does not have solutions in the whole space R N if and only if

(KO) ∞ 1 ds F (s) < ∞, where F (s) = s 0 f (t)
dt is a primitive of f (this condition is nowadays usually referred to as the Keller-Osserman condition). On the other hand, (explosive) solutions in a bounded domain exist if and only if (KO) holds. It is outside the scope of this short paper to attempt a reasonably full bibliography for the many developments that followed [START_REF] Osserman | On the inequality ∆u ≥ f (u)[END_REF] and [START_REF] Keller | On solutions of ∆u = f (u)[END_REF], we refer for instance to the surveys [START_REF] Farina | Liouville-type theorems for elliptic problems[END_REF] and [START_REF] Rǎdulescu | Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities[END_REF], where a large set of references can be found. However, to the best of our knowledge and maybe somewhat surprisingly, a study of the problem is not available for proper unbounded subdomains of R N such as a half-space (except for a comment on nonexistence of radial solutions in Remark 1.6 of [START_REF] Marcus | Existence and uniqueness results for large solutions of general nonlinear elliptic equations[END_REF]). Elliptic equations on a half-space are both intrinsically important and appear in the study of other problems, such as the study of existence of a priori bounds for solutions through the Gidas-Spruck blow-up method. Half-spaces and cones also naturally appear in the study of boundary behaviour of solutions of elliptic equations in Lipschitz domains.

We observe that the solvability of (P) has been extensively studied in the particular case when the function is supposed to satisfy a boundary condition at ∂R N + = {x ∈ R N : x N = 0}, such as u = 0 on this set. We refer for instance to [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF], [START_REF] Bandle | On positive solutions of Emden equations in cone-like domains[END_REF], [START_REF] Berestycki | Superlinear indefinite elliptic problems and nonlinear Liouville theorems[END_REF], [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF], [START_REF] Chen | Monotonicity and nonexistence results to cooperative systems in the half space[END_REF], [START_REF] Kondratiev | Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains[END_REF], and the references in these works.

In this paper we provide a necessary and sufficient condition for (P) to have a solution. This is the content of the following theorem. Throughout the paper, an existence statement will mean that (P) has a strong (or even classical, if f is supposed to be Hölder continuous) solution, whereas a non-existence statement will mean (P) does not even have solutions in any weak (such as weak-Sobolev or viscosity) sense which leaves valid the comparison principle.

Note that since the existence of a solution in R N implies existence in any subdomain of R N , we only need to study the problem under the hypothesis (KO).

           1 0 ds F (s) = ∞ if t 0 = 0, or -1 -∞ f (s) ds < ∞ if t 0 = -∞
If this hypothesis is not satisfied, even subsolutions do not exist.

We stress that this theorem completely characterizes the existence of solutions and subsolutions of (P), and thus represents a counterpart for a half-space of the results of Keller and Osserman for bounded domains or the whole space.

At this point a discussion on the possibility of extending the above theorem to more general unbounded domains is in order. The situation turns out to be very different depending on whether t 0 = 0 or t 0 = -∞.

First, if t 0 = 0 then the first condition in (HS) actually guarantees the existence of a positive solution in any subdomain of R N whose closure is not the whole space. Note that this condition is what ensures the validity of the strong maximum principle for nonnegative solutions of ∆u = f (u), by the well-known result of Vázquez ( [START_REF] Vázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF], see also [START_REF] Pucci | The maximum principle[END_REF]). On the other hand, if (HS) fails when t 0 = 0 then positive solutions do not exist in any sufficiently "fat" subdomain of R N . Thus, the result in Theorem 1 remains the same if t 0 = 0 and the half-space is replaced by any strictly proper subdomain of R N which contains balls of arbitrary radius -such as, for instance, a nontrivial cone in R N (see Theorem 2 below).

Second, if t 0 = -∞ then the condition in (HS) appears strongly related to the specific geometry of the half-space. If a different cone in R N is considered, then this condition no longer characterizes the solvability of ∆u = f (u). For instance, the function u = -x α 1 x α 2 is a solution of the inequality ∆u ≥ c|u| β in the quarter-space {x 1 > 0, x 2 > 0} for any β ∈ (-1, 0) (and so f (u) = |u| β does not satisfy (HS)), provided α = 1 1-β and c = 2α(1 -α). Hence if f : R → (0, ∞) is increasing and f (u) ≤ |u| β for u < 0 then a solution of ∆u = f (u) in the quarterspace existssince u is a subsolution, ū = 0 is a supersolution, and they are ordered. Currently we do not know how to obtain a necessary and sufficient condition for existence of solutions of (P) in a cone different from R N or R N + , in the case t 0 = -∞. We believe this is an interesting open problem.

It is to be stressed that, while the existence and nonexistence of (sub-)solutions of (P) in the whole space depends only on the behavior at infinity of f , the result above shows that the behavior at t 0 = inf{ t : f (t) > 0} of f turns out to be important when a proper unbounded subdomain is considered.

Proofs

In this section we will prove Theorem 1. The proofs depend on the comparison principle, together with the resolution of some auxiliary problems. We recall that the monotonicity of f implies the equation is proper, thus satisfies the weak comparison principle, in the sense that a subsolution and a supersolution which are ordered on the boundary of any bounded domain are also ordered throughout the domain. Also, Perron's method applies and yields the existence of a solution between any given ordered subsolution and a supersolution (see for instance [START_REF] Du | Order structure and topological methods in nonlinear partial differential equations[END_REF] for classical or weak-Sobolev solutions and [START_REF] Crandall | User's guide to viscosity solutions of second-order partial differential equations[END_REF] for viscosity solutions).

Before giving the proof we briefly recall the construction of an entire (radial) solution of ∆u = f (u) when the condition (KO) is not satisfied. Fix a real number v 0 such that f (v 0 ) > 0 and consider the following ordinary differential equation for a function v = v(r), (2.1)

   v + N -1 r v = f (v), r > 0 v(0) = v 0 , v (0) = 0. Since f is continuous, problem (2.1) has at least one C 2 solution v defined in a maximal interval of existence [0, R) for some R > 0. Since (r N -1 v ) = r N -1 f (v) ≥ 0 we have r N -1 v ≥ 0 so that v is increasing and therefore v ≤ f (v)
, by (2.1). We can multiply this inequality by v , write the result as ((v ) 2 ) ≤ (2F (v)) , and integrate to get

v(r) v 0 dt 2F (t) ≤ r
for every r ∈ (0, R), which implies, since (KO) does not hold, that R is infinite. Thus for any

x 0 ∈ R N the function u(x) = v(|x -x 0 |) is a positive solution of ∆u = f (u) in R N .
The minimum of u is attained at x 0 and its value is v 0 .

On the other hand, when (KO) is satisfied, the solution of (2.1) blows up at a finite R; indeed, since the equation in (2.1) can also be written as

(2.2) v (r) = r 0 s r N -1 f (v(s))ds,
from the monotonicity of f and v we have v (r) ≤ r N f (v(r)), so that

v ≥ 1 N f (v).
Multiplying by v as above we get

v(r) v 0 dt F (t) -F (v 0 ) ≥ 1 √ N r
for every r ∈ (0, R), which implies that R is finite, by (KO). Thus

u(x) = v(|x -x 0 |) solves ∆u = f (u) in B R (x 0 ) with v = +∞ on ∂B R (x 0 ) (throughout the paper, B r (x 0
) stands for the ball with radius r centered at x 0 ; we write B r when x 0 = 0).

Proof of Theorem 1, case t 0 = 0. Existence. Assume that (HS) holds, that is, the Vázquez condition for the strong maximum principle 0 F -1/2 = ∞ is verified. Take any x 0 ∈ R N and r > 0 such that B r (x 0 ) ⊂ {x N < 0}. Since f (0) = 0 and f (δ) > 0 for some small δ > 0, the functions u = 0 and u = δ are respectively a subsolution and a supersolution for the problem

(2.3)    ∆u = f (u) in B R (x 0 ) \ B r (x 0 ) u = δ on ∂B r (x 0 ) u = 0 on ∂B R (x 0 ).
Hence there exists a solution u R of this problem such that 0 ≤ u R ≤ δ. By standard elliptic estimates (see for instance [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], Theorem 9.13) the sequence u R W 2,p (K) is bounded independently of R in any fixed compact set K of R N \ B r (x 0 ), for any p < ∞. Hence we can extract a subsequence which converges strongly in C 1,α (K) and weakly in W 2,p (K) to a function u. Thus, by a diagonal argument, taking larger and larger compacts K, we get a sequence R n → ∞ such that u Rn → u in W 2,p loc (R N \ B r (x 0 )). Passing to the limit we see that u is a nonnegative solution to the equation ∆u = f (u) in R N \ B r (x 0 ). This solution does not vanish identically by the boundary condition u = δ on ∂B r (x 0 ). Since the strong maximum principle holds (see [START_REF] Vázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF]) we obtain that u is strictly positive in R N \ B r (x 0 ) ⊃ R N + . Nonexistence. As we recalled above, the Keller-Osserman condition (KO) implies that for each v 0 = ε > 0 there is a finite number R ε such that the problem ∆w = f (w) has a solution in each ball with radius R ε , which blows up at the boundary of the ball, and has value ε at the center of the ball.

Assume problem (P) has a positive solution u. Take a (small) number δ > 0 and fix any ball B = B Rε+δ (x 0 ) ⊂ R N + . We claim that u < 2ε in B δ (x 0 ). In fact, if we had u(x) ≥ 2ε for some x ∈ B δ (x 0 ), we could define w(x) = v(|x -x|), where v is the solution of (2.1) with v(0) = ε, and use the comparison principle in the ball B Rε (x) ⊂ R n + to get u ≤ w in this ball. However, this is a contradiction since u(x) ≥ 2ε while w(x) = v(0) = ε.

Finally, we claim that the failure of (HS) implies that there exists a supersolution z of (P) in B δ (x 0 ) with z = 2ε on ∂B δ (x 0 ) and z = 0 on B δ/2 (x 0 ) (a dead core). By using the comparison principle again, this time in B δ (x 0 ), we find that u ≤ 2ε = z on ∂B δ (x 0 ) implies u ≤ z = 0 in B δ/2 (x 0 ), contradicting the fact that u is positive.

Hence to conclude the proof of the theorem only this last claim remains to be proved. We will find a radial super-solution z such that z(x) = β(|x -x 0 |), and β is a real function which satisfies the ODE (2.4)

β + N -1 r β ≤ f (β) in (0, δ),
and is such that β = 0 on (0, δ/2) and β(δ) = 2ε. For this purpose it suffices to find a solution in (δ/2, δ) such that β(δ/2) = 0, β (δ/2) = 0 and β(δ) = 2ε, and then extend this solution as zero in (0, δ/2).

To find β we assume N ≥ 3 and perform the change of variables s = 1 N -2 r 2-N , β(s) = β(r) (a similar argument can be made in the case N = 2 with the change of variables s = log r; we will skip the details) and obtain the one-dimensional problem

(2.5) β ≤ cs -γ f ( β) in (s 0 , s 1 ) β(s 0 ) = ε, β(s 1 ) = β (s 1 ) = 0, where γ = 2(N -1) N -2 , c = (N -2) -γ , s 0 = δ 2-N /(N -2) and s 1 = 2 N -2 s 0 . Observe that γ > 2.
Since the integral 0 F -1/2 is convergent we can find adequately small ε > 0 and δ > 0 such that

ε 0 ds 2F (s) = cs -γ 1 (s 1 -s 0 ) = cδ γ .
where c, γ are positive constants which depend only on the dimension N . Thus we can define implicitly the function β through the equality

ε β(t) ds 2F (s) = cs -γ 1 (t -s 0 ) for t ∈ (s 0 , s 1 ). Therefore β satisfies β = cs -γ 1 f ( β), β(s 0 ) = ε, β(s 1 ) = β (s 1 ) = 0 and β ≤ ε. Using s -γ 1 ≤ s -γ
for s ∈ (s 0 , s 1 ) we infer (2.5) and the claim follows. This concludes the proof.

Observe that in the above proof we did not actually use the fact that the domain in which the equation ∆u = f (u) is set is a half-space. In the existence part of the proof we only used that we can find a ball outside the domain, whereas in the non-existence part we only used that the domain contains a sufficiently large ball. In other words, we have actually proved the following result.

Theorem 2. Suppose (KO) holds, and f (0) = 0, f (t) > 0 for t > 0 is monotone increasing. Assume Ω is such that Ω = R N and for each R > 0 there exists a ball with radius R included in Ω. Then the equation ∆u = f (u) has a positive solution in Ω if and only if Proof of Theorem 1, case t 0 = -∞.

Existence. Assume (HS) holds, that is, -∞ f is convergent. Fix a positive constant ∞). We define implicitly v as follows:

C * such that C * > 2 0 -∞ f (t)dt = -2F (-
0 v(t) ds 2F (s) + C * = t.
Observe that v is well defined by (HS) and the choice of C * . Moreover, by differentiating -v (t) = 2F (v(t)) + C * , and therefore v is strictly negative and bounded for t ≥ 0. By differentiating once more we see that v satisfies v = f (v) in (0, +∞) and thus u(x) = v(x N ) is a (one-dimensional and negative) solution of (P) in the half-space.

Nonexistence. Let us assume that (KO) holds, -∞ f is divergent and there exists a sub-solution u of (P) in R N + . First we observe that (2.6) lim

x N →+∞ u(x) = uniformly for x ∈ R N -1 ,
where x = (x , x N ).

In fact, if this were not the case, there would exist γ < 0 and a sequence x (n) with x (n) N → ∞ such that u(x (n) ) ≥ γ as n → ∞. As we observed before the proof of Theorem 1, we may use the condition (KO) to ensure that there exists a radial solution w n of the equation ∆w = f (w) in B R (x (n) ), with w n (x (n) ) = 2γ and w n = +∞ on ∂B R (x (n) ), for some sufficiently large and fixed R > 0. Observe that B R (x (n) ) ⊂ R N + for large n, therefore we can apply the comparison principle and deduce u(x) ≤ w n (x) in B R (x (n) ) for large n. We obtain a contradiction since

w n (x (n) ) = 2γ < γ ≤ u(x n ).
By the above argument, and replacing if necessary the half-space {x N > 0} by the half-space {x N > A} for some conveniently chosen A > 0, we can assume that u < 0 in R N + . Moreover, for a given M > 0 there exists

R = R(M ) > 0 such that u ≤ -M on R N -1 × {R} and R(M ) → +∞ as M → +∞.
For any R, T > 0 consider the finite cylinder

D T,R := {x = (x , x N ) ∈ R N + : |x | < T, x N < R}.
Let Ω n be a sequence of smooth bounded domains such that D n,R ⊂ Ω n ⊂ D n+1,R , and let h n be a continuous function defined on

∂Ω n such that -M ≤ h n ≤ 0 on ∂Ω n , h n = -M on ∂Ω n ∩ {|x | < n, x N = R} and h n = 0 on ∂Ω n ∩ {x N < R}. Consider the problem (2.7) ∆w = f (w) in Ω n w = h n on ∂Ω n .
By the construction of h n we see that u is a subsolution of this problem, while u = 0 is a supersolution (recall that f (0) > 0), and they are ordered. Therefore there exists a solution w n of (2.7) verifying u ≤ w n ≤ 0 in Ω n . Again by the usual elliptic estimates the sequence w n is bounded in W 2,p (K) for each compact K ⊂ {0 ≤ x N ≤ R}. Therefore we can pass to the limit by using a diagonal argument, to obtain a solution w M of the following problem in a strip (2.8)

   ∆w = f (w) in {0 ≤ x N ≤ R} w = 0 on x N = 0 w = -M on x N = R
Of course we also have u ≤ w M ≤ 0. We claim that actually w M is the maximal solution of (2.8) in the order interval [u, 0]. To show this, let w be any solution of (2.8) such that u ≤ w ≤ 0. Therefore w is a subsolution of (2.7), so by comparison w ≤ w n in Ω n for every n. At the limit w ≤ w M and maximality follows. By maximality, w M depends only on the variable x N (since any translation of w M in a direction parallel to {x N = 0} is still a solution of (2.8)). This means that v M (x N ) = w M (x) is a classical solution of v M = f (v M ) in (0, R) with v M (0) = 0 and v M (R) = -M . We can pass to the limit as M → +∞ as before to obtain that v Mn → v in C 2 loc [0, +∞) for some sequence M n → +∞, where v = f (v) in (0, +∞) with v(0) = 0. Now w(x) = v(x N ) is itself a solution of (P), and hence, by (2.6), we have v(x N ) → -∞ as x N → ∞. However, as we already observed, (v ) 2 = 2F (v) + v (0) 2 and so F (v(x N )) is bounded from below as x N → ∞, while on the other hand the failure of (HS) means precisely that F (t) = -0 t f tends to -∞ as t → -∞. The proof is concluded.

Theorem 1 .

 1 Suppose (KO) holds. Then equation (P) has a solution if and only if (HS)

  If this hypothesis is not satisfied, even positive subsolutions do not exist in Ω.

Acknowledgements. J. G-M and A. Q. were partially supported by Ministerio de Ciencia e Innovación under grant MTM2011-27998 (Spain), A. Q. was partially supported by Fondecyt Grant No. 1110210 and CAPDE, Anillo ACT-125 Chile). All three authors were partially supported by Programa Basal CMM, U. de Chile.


Boyan Sirakov

Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient

We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for the solutions is proved through an argument which uses a boundary growth lemma, and consequences such as boundary "half"-Harnack inequalities, which are of independent interest. Our results are new even for linear equations.

Introduction and Main Results

This note is an account of some recent results on solvability and multiplicity of solutions of the Dirichlet problem for a uniformly elliptic operator in which the first order term has the same scaling with respect to dilations as the second order term. This work can be considered as a continuation of the paper [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF], and is also strongly motivated by a number of very recent papers on a particular equation of our type, [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], [START_REF] Arcoya | Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF]. The purpose of this note is to state our main theorems in a particular but typical case, and give a sketch of their proofs; the full presentation will appear in the forthcoming work [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE with critical growth in the gradient[END_REF].

Apart from their appearance in a number of applications, this type of PDE is of theoretical importance, since it represents a class of equations which is invariant with respect to diffeomorphic changes of dependent and independent variable. An additional difficulty in the study of these equations is the "critical" behaviour of the gradient, in the sense that the first-order term has the same scaling with respect to dilations as the second-order term, and thus does not scale out after a "zoom" in a point.

Even though the results we obtain are valid for general Isaacs operators as in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF], here for simplicity we will state them only for the equation

-L 0 u = c(x)u + < M (x)∇u, ∇u > + h(x) in Ω u = g(x)
on ∂Ω

where Ω is a C 2 -smooth bounded domain in R N , and

L 0 u = a ij (x)∂ ij u + b i (x)∂ i u
1 e-mail : bsirakov@mat.puc-rio.fr 1 is a linear uniformly elliptic operator whose coefficients have the necessary regularity to ensure satisfactory theory for the linear problem (M = 0). For simplicity here we will assume all coefficients in (1) are bounded functions, A is continuous and λI

≤ A(x) ≤ ΛI, |M (x)| ≤ M 0 , 0 < λ ≤ Λ, M 0 ≥ 0.
The equation ( 1) can be thought of as the "closure" of the class of linear elliptic equations with respect to diffeomorphic changes of the dependent variable (u → v = ψ(u)) and of the independent variable (x → y = Ψ(x)).

We have separated the zero order term in the elliptic operator, since the sign and size of c(x) matter, both for the solvability and for the uniqueness of solutions.

When reduced to (1), the main theorem in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF] states the following.

1. The problem (1) has a unique solution if c(x) ≤ -c 0 < 0.

There exists

δ 0 > 0 depending on λ, Λ, b n , diam(Ω), such that if max { M 0 h L n , c + L n , M 0 max ∂Ω g c + L n } ≤ δ 0
then (1) has a solution. This hypothesis cannot be improved in general.

The solution is unique if c + = 0.

3. If c ≡ c 0 ∈ (0, δ 0 ), L 0 = ∆, M = µ 0 I, µ 0 > 0, g = h = 0, then (1)
has at least two solutions.

The possibility of extending the result in 3. above, that is, to show nonuniqueness in general when c is not nonpositive was left as an open question in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF]. This question was taken up in a number of recent works, which we now describe in a little more detail. In the following we also assume for simplicity that the boundary data g = 0 on ∂Ω, and that c(x) 0 in Ω. Of course we always need to assume that the problem with c ≡ 0 has a solution (as we know, such a solution is unique, and its existence is implied by an upper bound on M 0 h L n ).

In [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] it was shown that if the second order operator L 0 is the Laplacian, and M (x) = µ(x)I, where the function µ is such that |µ(x)| ≥ µ 1 > 0, then (1) has at least two solutions if c(x) ≤ C 0 , for some explicit constant C 0 . That paper extends the earlier work [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] and further develops the topological degree method used in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE with critical growth in the gradient[END_REF] and [START_REF] Arcoya | Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]. The behaviour of the solutions with respect to a parameter which measures the size of c(x) is also studied in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], through a Rabinowitz-type bifurcation method.

The following diagrams from [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] are worth reproducing. Consider the model problem for a given parameter λ ∈ R

-∆u = λc(x)u + µ(x)|∇u| 2 + h(x) in Ω u = 0 on ∂Ω (2) mu>>0 mu=0 mu<0
Here we have mapped the parameter λ on the horizontal axis, and the value of the solution of (2) for some point in the domain on the vertical axis. These diagrams correspond to the cases µ >> 0, µ = 0 and µ << 0 respectively, and are valid for instance if h 0. The main difficulty in establishing their validity is to obtain uniform a priori bounds for the solutions, for all λ in any interval of the form [0, A] if µ < 0, and in any interval of the form

[a, A], 0 < a ≤ A < ∞, if µ > 0.
As can be seen from the above pictures, it might be complicated to understand the behaviour of solutions if we assume only µ(x) 0 (instead of µ(x) ≥ µ 1 > 0). This question for the model equation ( 2) was studied in the very recent work [START_REF] Souplet | A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient[END_REF], where it was shown that a necessary condition for an a priori bound when µ(x) ≥ 0 is that the intersection of the supports of c(x) and µ(x) contains a ball. It was also shown that this condition is sufficient if (i) the spatial dimension n = 2; (ii) or if we assume n = 3 and a growth assumption on c and µ close to ∂Ω; (iii) or if n ≤ 4 and µ(x) ≥ µ 1 > 0 on supp(c).

In all previous works the results were restricted to the model equation ( 2), since the obtention of the a priori bounds crucially depended on the fact that the second order operator in the equation is the Laplacian. In [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] a variational argument of "mountain-pass" type was used, in [START_REF] Arcoya | Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] and [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] the weak-Sobolev formulation of the system (3)-( 4) below is tested by the first eigenfunction and the solution itself, and in [START_REF] Souplet | A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient[END_REF] interpolation and elliptic estimates for the Laplacian in weighted Lebesgue spaces were used.

It is our goal here to present another method for obtaining a priori bounds, which depends only on the uniform ellipticity of the operator, and thus gives the above described results for any operator L 0 . We prove the upper bound by using a growth lemma (quantitative strong maximum principle) combined with a weak Harnack inequality and a local maximum principle. A supplementary difficulty is that boundary versions of these results do not appear to be available, and need to be proved. This can be done by a modification of the original Krylov's argument for proving boundary regularity for elliptic equations. Furthermore, when µ(x) ≥ 0 we prove the uniform bound under the hypothesis µ(x) ≥ µ 1 > 0 on supp(c) in any dimension, thus extending and joining the results for the Laplacian from [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] and [START_REF] Souplet | A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient[END_REF] (see (iii) above).

It is also interesting that the bound in the case µ ≤ 0 turns out to be equivalent to the validity of the strong maximum principle for a coercive elliptic inequality with a superlinear nonlinearity which satisfies the Vazquez condition.

Theorem 1 Assume M (x) ≥ µ 1 I or M (x) ≤ -µ 1 I on supp(c), µ 1 > 0.
If the problem (1) with c = 0 has a solution u 0 , then (1) has at least two solutions for 0 c(x) ≤ δ 0 , where δ 0 depends on the ellipticity, the upper bounds for the coefficients of the equation, and on the domain.

If M (x) ≥ µ 1 I on supp(c) and u 0 0 then the first diagram in Fig. 1 is valid. If M (x) ≤ -µ 1 I on Ω and u 0 0 then the third diagram is valid.

In the proof of this theorem we use the following boundary estimates which are clearly of independent interest. We assume we have a domain with a flat portion of the boundary {x n = 0}, and denote with B + R a half-ball in the domain centered at a point on this portion. We set

B 0 R = ∂B + R ∩{x n = 0}.
Proposition 1.1 (boundary quantitative strong maximum principle, BQSMP) Assume u is a solution of -L 0 u ≥ 0, u ≥ 0 in B + 2 , and u = 0 on B 0 2 . Then there exists ε > 0 depending on λ, Λ and b L p such that inf

B + 1 u x n ≥ c B + 1 (-L 0 u) ε 1/ε . Proposition 1.2 (boundary weak Harnack inequality, BWHI) Assume that -L 0 u ≥ f , u ≥ 0 in B + 2 , u = 0 on B 0 2 .
Then there exists ε > 0 such that

B + 3/2 u x n ε 1/ε ≤ C inf B + 3/2 u x n + f - L n (B + 2 ) . Proposition 1.3 (boundary local maximum principle, BLMP) Assume that -L 0 u ≤ d(x)u + f in B + 2 , u = 0 on B 0 2 , d ∈ L q ,
for some q > n. Then for each p > 0 sup

B + 1 u + x n ≤ C   B + 3/2 (u + ) p 1/p + f + L n (B + 2 )  
The first two of these propositions can be proved by using Krylov's original idea for proving boundary regularity, which consists in writing a equation for u/x n that degenerates in a special way, and for which the fundamental "growth lemma" can still be proved. See [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE with critical growth in the gradient[END_REF] for details. An interior QSMP in this convenient form can be found in [START_REF] Krylov | Some Lp-estimates for elliptic and parabolic operators with measurable coefficients[END_REF], the interior versions of WHI and LMP for instance in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. Of course the BLMP above is a consequence of the boundary Lipschitz estimate and the usual LMP (u + is a subsolution in a full ball if we extend u = 0 outside Ω).

It is interesting to notice that the combination of BWHI and BLMP yields the full boundary Harnack inequality, a result which is very well known, and goes back to [START_REF] Caffarelli | Boundary behavior of nonnegative solutions of elliptic operators in divergence form[END_REF], [START_REF] Bauman | Positive solutions of elliptic equations in nondivergence form and their adjoints[END_REF]. However, in all texts where this inequality has appeared it has been proved by a method different from the above splitting into separate results for supersolutions and subsolutions.

It is also worth noting that inequalities called "boundary weak Harnack inequalities" have appeared in a different form, which is sufficient for a proof of Hölder regularity up to the boundary (see for instance [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], Theorems 8.26 and 9.27). However, that inequality is void for a function which vanishes on the boundary and does not imply the full boundary Harnack inequality.

Sketch of the proof of the theorem. First, we observe that if c < λ 1 (L 0 , Ω) then we can assume without loss of generality that M (x) ≥ 0, h ≥ 0, u 0 ≥ 0 and we can search for positive solutions of our equation. This is done by observing that the difference of u and the solution of L 0 v = c(x)v +h satisfies an equation (with a modified operator L0 ) in which these hypotheses are verified.

Next, we observe that if there exists an unbounded sequence of solutions, then there also exists a sequence of solutions u k and points x k such that u k (x k ) → ∞, and x k → x 0 ∈ Ω is such that for any ball B centered at x 0 we have B∩Ω c > 0. This is because for any domain G ⊂ {c ≡ 0} we can apply the comparison principle in G to the functions u -sup ∂G u and u 0 -inf Ω u 0 .

So we see we need to prove an a priori bound in a ball B (resp. a halfball if x 0 ∈ ∂Ω) such that c 0 in B and 0 < µ 1 ≤ µ(x) ≤ µ 2 in Ω, by the hypothesis of the theorem.

We make a classical exponential change of the dependent variable u, set-

ting v i = µ -1 i (e µ i u -1), i = 1, 2. In B, for large values of u -L 1 v 1 ≥ f 1 (x, v 1 ) ∼ c 0 c(x) v 1 log(v 1 ) (3) -L 2 v 2 ≤ f 2 (x, v 2 ) ∼ C 0 c(x) v 2 log(v 2 ) (4) v 2 ∼ v A 1 , A = µ 2 /µ 1 v i = 0 on ∂Ω ∩ B.
To simplify, and since we only want to give the main idea of the proof here, we do not write the full expression of f i ; what actually matters is the above asymptotic behaviour for large values of u (resp. v i ). We observe at this point that a priori bounds for this system of inequalities do not seem to be provable by classical methods such as the Gidas-Spruck rescaling method. This is because we may have c(x 0 ) = 0, and, on the other hand, if we write this system as a system in only one of the functions v i , then one of the operators will become degenerate.

Let us assume we are in the more difficult case, when B is a half-ball, with ∂Ω ∩ B ⊂ {x n = 0} (after a "flattening" diffeomorphic change of the independent variable, which only changes the coefficients of the equation).

Then the first inequality (3) and the BQSMP imply inf

B + 1 v 1 x n ≥ c 0 inf B + 1 v 1 x n log inf B + 1 v 1 x n B + 1 c ε x 1+ε n
This implies that inf

B + 1 v 1 x n ≤ C. Then -L 1 v 1 ≥ 0 (which follows from (3))
and the BWHI imply

B + 1 (v 1 ) ε 1/ε ≤ C B + 1 v 1 x n ε 1/ε ≤ C.
Now we apply the BLMP to the second inequality (4), in which d(x) = C 0 c(x) log(v 2 ) (note log(v 2 ) ∼ A log(v 1 )). Since for large values of v 1 we have (log(v 1 )) n+1 ≤ Cv ε 1 , we obtain from LMP and what we already proved sup

B + 1 v 2 ≤ B + 1 (v 2 ) ε/A A/ε ≤ C B + 1 (v 1 ) ε A/ε ≤ C.
which establishes the required a priori bound, in terms of the ellipticity, bounds on the coefficients, and a lower bound on B c ε . We remark it would have been enough to use instead of the BQSMP a "growth lemma" which says that a positive supersolution is uniformly bounded below by a constant which depends on the measure of the set where the right-hand side of the inequality is strictly positive. Observe we did not need to reason by contradiction in order to obtain the a priori bound, nor had we to rescale the solutions or use any passage to the limit. We also do not need to assume any continuity of the coefficients of the equation, including the leading terms a ij .

It should be noted that interior Harnack inequalities (but not QSMP) have been used for proving a priori bounds on compact subsets of a domain, for instance in [START_REF] Amann | A priori bounds and multiple solutions for superlinear indefinite elliptic problems[END_REF], [START_REF] Ruiz | A priori estimates and existence of positive solutions for strongly nonlinear problems[END_REF], in conjunction with variational structure or rescaling arguments, which require more regularity and that an equation (as opposed to two one-sided inequalities) be satisfied by the solution.

It is worth observing that the results (i) and (ii) from the paper [START_REF] Souplet | A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient[END_REF] which we quoted above can be understood differently, by applying the above argument in which one uses the weak Harnack inequality applied to the function u (which is itself superharmonic) instead of v 1 , and the exact best exponent ε in this inequality (which is known for the Laplacian), in order to see what is the best regularity one can get for d(x) = c(x)u(x) in the inequality -∆v 2 ≤ f 2 (x, v 2 ) ∼ Cc(x)u(x)v 2 , so that d ∈ L q with q > n/2. This leads us to the conjecture that (i) and (ii) from [START_REF] Souplet | A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient[END_REF], as quoted above, cannot be improved.

Finally, let us sketch the proof of the a priori bound in the "negative" case M (x) ≤ -µ 1 I in Ω. In other words, we want to prove an upper bound for subsolutions of

-L 0 u ≤ c(x)u -µ 1 |∇u| 2 + h(x) in Ω u = 0 on ∂Ω (5)
By setting v = µ -1 1 (1 -e µ i u ) we see that v < 1 is a subsolution of

-L 1 v ≤ c(1 -v)| log(1 -v)| + h(1 -v) =: f (x, v) in Ω v = 0 on ∂Ω
Here the function f (x, v(x)) is bounded in Ω for v 0 ≤ v ≤ 1, where v 0 is any fixed subsolution of the above inequality. So by the standard Lipschitz bound at ∂Ω we have v ≤ Cdist(x, ∂Ω). Denote with v the supremum of all subsolutions v ∈ (v 0 , 1) of the last inequality. Then by Perron's method v is also a subsolution. By the Lipschitz bound v ≡ 1. Now, the existence of an unbounded sequence of subsolutions of ( 5) means precisely that z = 1 -v ≡ 0 is a nonnegative supersolution of -L 2 z ≥ c 0 | log(z)|z and z vanishes somewhere in Ω. This is impossible by Vazquez's strong maximum principle [START_REF] Vázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF] (and its extension to nonlinear inequalities, see for instance [START_REF] Pucci | The maximum principle[END_REF], [START_REF] Felmer | Solvability of nonlinear elliptic equations with gradient terms[END_REF]), since one over the square root of the primitive of t log(t) is not integrable at zero.