Agnieszka Korus
email: agnieszka.korus@polsl.pl

Adam Klimanek

Slawomir Sladek

Andrzej Szlęk

Airy Tilland

Airy Tilland b Stéphane Bertholin

Nils Erland

L Haugen

Sławomir Sładek

Kinetic Parameters of Petroleum Coke Gasification for Modelling Chemical-Looping Combustion Systems

Keywords: Petroleum coke, gasification, chemical-looping combustion

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

Introduction

The production of petroleum coke (petcoke), a by-product of the oil refining process, is constantly increasing due to the high demand for oil-derived fuels and chemicals [START_REF] Karasu | A new forecasting model with wrapper-based feature selection approach using multi-objective optimization technique for chaotic crude oil time series[END_REF]. This highly calorific material can, therefore, be acquired in abundance at a low cost to produce energy or gaseous fuels [START_REF] Wang | Clean and efficient use of petroleum coke for combustion and power generation[END_REF]. The advantage of using petcoke as a feedstock for thermochemical conversion processes is its high heating value, approximately 20 % higher than that of coal, and its low ash content (0.1 -0.3 %) [START_REF] Murthy | Petroleum coke J o u r n a l P r e -p r o o f gasification: A review[END_REF].

Due to the need to incorporate carbon capture and sequestration (CCS) into the thermochemical conversion of carbonaceous fuels, chemical looping combustion (CLC) technology has become an attractive alternative to conventional combustion methods. The main reason for this is that the use of oxygen carriers that are transported between the air and the fuel reactors essentially results in an oxy-fuel process without the energy penalty associated with the cryogenic oxygen separation process of a regular oxy-fuel process.

Although CLC was initially designated for gaseous fuel utilisation, the conversion of solids is also feasible, but it requires a gasifying agent, e.g. H 2 O or CO 2 , to act as the gaseous intermediate between the solids -oxygen carrier and fuel [START_REF] Leion | The use of petroleum coke as fuel in chemicallooping combustion[END_REF][START_REF] Lyngfelt | Chemical-Looping Combustion of Solid Fuels -Status and Recent Progress[END_REF]. The feasibility studies on J o u r n a l P r e -p r o o f large-scale CLC installations have confirmed the low cost of the CO 2 capture integration [START_REF] Lyngfelt | Chemical-Looping Combustion of Solid Fuels -Status and Recent Progress[END_REF], thus increasing the importance of developing this technology to achieve zero or even negative greenhouse gas emissions. To optimise the CLC reactor design, the detailed kinetic data on solid fuel gasification under the conditions characteristic for this technology must be acquired.

Therefore, the purpose of this paper is to obtain accurate kinetic parameters for petcoke gasification through kinetic modelling of experimental data.

No clear correlation between the main physicochemical properties of the petcoke and its conversion rate have been established thus far. The parameters such as the specific surface area or volatile matter content do not vary significantly between various samples and they are not responsible for the different kinetics of petcoke oxidation [START_REF] Commandré | Lack of correlation between the properties of a petroleum coke and its behaviour during combustion[END_REF]. However, the metals content in petroleum coke can be relatively diverse and these elements can have a nonnegligible, yet difficult to quantify, catalytic effect on the petcoke conversion [START_REF] Commandré | Lack of correlation between the properties of a petroleum coke and its behaviour during combustion[END_REF][START_REF] Tyler | Intrinsic reactivity of petroleum coke to oxygen[END_REF]. Among others, vanadium, iron, or alkali and alkaline earth metals have been reported to have a significant impact on petcoke gasification and combustion kinetics [START_REF] Commandré | Lack of correlation between the properties of a petroleum coke and its behaviour during combustion[END_REF][START_REF] Liu | Experimental Study on Co-gasification of Coal Liquefaction Residue and Petroleum Coke[END_REF][START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF].

The focus of this study is low-sulphur Chinese petcoke selected for testing at the 3MW CLC reactor designed in the scope of the CHEERS project 2 . Even though some studies of petcoke thermochemical reactivity already exist in the literature [START_REF] Murthy | Petroleum coke J o u r n a l P r e -p r o o f gasification: A review[END_REF][START_REF] Liu | Experimental Study on Co-gasification of Coal Liquefaction Residue and Petroleum Coke[END_REF][START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF][START_REF] Malekshahian | Kinetic Analysis of CO2 Gasification of Petroleum Coke at High Pressures. Energy &amp[END_REF][START_REF] Zou | Modeling reaction kinetics of petroleum coke gasification with CO2[END_REF], the reactivity of this particular petcoke has not previously been studied. Even more importantly, the kinetic data for CLC conditions, which require high temperatures and low oxygen concentrations, do not exist. Thus, an examination of the thermochemical conversion of the actual material selected for the pilot CLC reactor development, including the exact particle size range and gasification agent concentration, was required to provide accurate data on the apparent kinetic parameters to successfully model the reactor operation. The existing work on petcoke reactions with 2 CHEERS is jointly funded by the European Union's Horizon 2020 Research and Innovation Program (764697) and the Chinese Ministry of Science and Technology (MOST). See http://cheers-clc.eu/ for more information.

J o u r n a l P r e -p r o o f oxygen typically focuses on combustion, thus the experiments are carried out at lower temperatures (400 -600 ℃) and higher concentrations of oxidiser (>20 vol.%) [START_REF] Tyler | Intrinsic reactivity of petroleum coke to oxygen[END_REF][START_REF] Walsh | A laboratory study of petroleum coke combustion: kinetics and catalytic effects[END_REF] than the experiments performed in this work. E.g., Gajera et al. [START_REF] Gajera | Kinetics of co-gasification of rice husk biomass and high sulphur petroleum coke with oxygen as gasifying medium via TGA[END_REF] examined petcoke conversion in pure O 2 flow at temperatures up to 900 C. Gasification tests are also usually performed with undiluted oxidising agent, such as petcoke conversion in TGA in a pure flow of steam performed by Edreis et al. [START_REF] Edreis | Kinetics, thermodynamics and synergistic effects analyses of petroleum coke and biomass wastes during H2O co-gasification[END_REF]. Some petcoke gasification tests carried out at temperatures above 1000 ℃ can be found in recently published works [START_REF] Yu | Promoting effect of biomass ash additives on high-temperature gasification of petroleum coke: Reactivity and kinetic analysis[END_REF][START_REF] Kumari | Comparison of CO2 gasification reactivity and kinetics: petcoke, biomass and high ash coal[END_REF][START_REF] Wei | Effect of biomass leachates on structure evolution and reactivity characteristic of petroleum coke gasification[END_REF][START_REF] Yang | Catalytic gasification reactivity and mechanism of petroleum coke at high temperature[END_REF][START_REF] Wei | Gasification reactivity and characteristics of coal chars and petcokes[END_REF]. However, these reports are often oriented on the fundamental research on the petcoke kinetics, thus comprising thermogravimetric experiments with small sample sizes and the reaction atmosphere limited to pure CO 2 . E.g., Wei et al. [START_REF] Wei | Effect of biomass leachates on structure evolution and reactivity characteristic of petroleum coke gasification[END_REF][START_REF] Wei | High℃temperature char gasification of anthracite/petroleum coke: using biomass leachate as cheap℃effective additive[END_REF] examined the effects of the addition of biomass leachates and Yu et al. [START_REF] Yu | Promoting effect of biomass ash additives on high-temperature gasification of petroleum coke: Reactivity and kinetic analysis[END_REF] the addition of biomass ash on the petcoke gasification. Meanwhile, the conditions for gasification in a CLC reactor require presence of steam or oxygen. It is also beneficial to conduct the application-oriented research in a larger scale than the instrumental analysis such as TGA. Lulu et al. [START_REF] Lulu | Chemical looping gasification with potassiumcatalyzed petroleum coke for enhanced production of H2 and H2S[END_REF] examined petcoke gasification with O 2 and H 2 O at 900 ℃ in a fluidized bed, which represented a CLC reactor, using large sample of 0.75 g. Wei et al. [START_REF] Wei | Gasification reactivity and characteristics of coal chars and petcokes[END_REF] used a horizontal furnace with a 50 mg petcoke sample placed in a quartz boat crucible to examine CO 2 gasification at temperatures up to 1200 ℃, while Wang et al. [START_REF] Wang | Valorizing petroleum coke into hydrogen-rich syngas through K-promoted catalytic steam gasification[END_REF] carried out steam gasification tests at 650 -750 ℃ with 20 g sample in a fixed bed reactor.

Liu et al. [START_REF] Liu | Conversion characteristics of lignite and petroleum coke in chemical looping combustion coupled with an annular carbon stripper[END_REF] conducted tests in a 30 kW chemical looping combustion unit. However, the experiments carried out in the larger facilities focused mainly on the operational aspects of these reactors, i.e. monitoring the conversion efficiency or the evolved gases composition, etc., and they do not provide any kinetic data. Zhang et al. [START_REF] Zhang | Robust modeling, analysis and optimization of entrained flow co-gasification of petcoke with coal using combined array design[END_REF] presented a robust model of petcoke conversion in 15 -50 vol.% of steam and 0 -3 vol.% oxygen based on the experiments in the entrained flow gasifier with a 60 g/h carbon feeding rate. The combined array design methodology allowed for an accurate description of the process, however, these J o u r n a l P r e -p r o o f results cannot be directly incorporated into the deterministic models for petcoke conversion.

Although the existing research on petcoke conversion is extensive, there is a lack of kinetic parameters determined at the conditions relevant for gasification in chemical looping combustion units, i.e. based on the experimental work performed in the larger reactor, thus comprising more representative samples, and carried out in the oxidising agent concentrations up to 40 vol.% at the temperatures up to 1000 ℃.

To provide relevant data for the CHEERS project, measurements were conducted in a custombuilt test rig that allowed the use of a wider size fraction and higher sample mass than thermogravimetric (TGA) experiments, which typically focus on samples with narrow particles size ranges and lower sample masses <10 mg [START_REF] Liu | Experimental Study on Co-gasification of Coal Liquefaction Residue and Petroleum Coke[END_REF][START_REF] Zou | Modeling reaction kinetics of petroleum coke gasification with CO2[END_REF][START_REF] Fermoso | Kinetic models comparison for steam gasification of different nature fuel chars[END_REF][START_REF] Zhan | Catalytic effect of black liquor on the gasification reactivity J o u r n a l P r e -p r o o f of petroleum coke[END_REF][START_REF] Yoon | Thermogravimetric study of coal and petroleum coke for co-gasification[END_REF]. The unique construction of the test rig, which has a crucible with a fritted bottom, enables the gas flow through the sample. Improved sample penetration by the oxidising gases reduces the diffusional resistance at high temperatures. Therefore, the kinetic parameters for the conversion not affected by the external diffusion could be obtained for the temperatures as high as 950 ℃. The design also allows the use of a larger, more representative, sample size than conventional TGA, and the particle size range wide enough to represent the heterogenous fraction of fuel particles used in the industrial scale applications. Moreover, gasification tests for CO 2 partial pressures <0.1 MPa and steam/CO 2 mixtures were performed to meet the conditions relevant for chemical looping gasification, as opposed to the typical kinetic experiments carried out in atmospheres with a single gasifying agent [START_REF] Liu | Experimental Study on Co-gasification of Coal Liquefaction Residue and Petroleum Coke[END_REF][START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF][START_REF] Malekshahian | Kinetic Analysis of CO2 Gasification of Petroleum Coke at High Pressures. Energy &amp[END_REF][START_REF] Zou | Modeling reaction kinetics of petroleum coke gasification with CO2[END_REF][START_REF] Fermoso | Kinetic models comparison for steam gasification of different nature fuel chars[END_REF][START_REF] Zhan | Catalytic effect of black liquor on the gasification reactivity J o u r n a l P r e -p r o o f of petroleum coke[END_REF]. The combination of the mass loss measurements and analysis of the gases evolved during gasification in CO 2 , H 2 O, CO 2 /H 2 O and O 2 provide insight into the nature of petcoke conversion. Finally, the kinetic parameters are fitted using the Avrami, Random Pore, Shrinking Core, and Hybrid Model, and the most suitable approximations are identified for each gasification agent. These most common and universal models were chosen to ensure that the obtained kinetics could be easily applied to the global J o u r n a l P r e -p r o o f models of the entire gasification installations, e.g. CLC-CCS systems. E.g., the Random Pore Model and the Shrinking Core Model were successfully used to describe thermochemical conversion of petcoke in numerous thermogravimetric experiments [START_REF] Edreis | Kinetics, thermodynamics and synergistic effects analyses of petroleum coke and biomass wastes during H2O co-gasification[END_REF][START_REF] Kumari | Comparison of CO2 gasification reactivity and kinetics: petcoke, biomass and high ash coal[END_REF][START_REF] Yang | Catalytic gasification reactivity and mechanism of petroleum coke at high temperature[END_REF][START_REF] Wei | High℃temperature char gasification of anthracite/petroleum coke: using biomass leachate as cheap℃effective additive[END_REF][START_REF] Ren | Co-Gasification Mechanism and Kinetics of Petcoke with Lignite[END_REF]. The experimental work and following calculations provided the apparent kinetics of petcoke conversion under the conditions and for the particle sizes typical for chemical looping gasification, which is necessary for the design and modelling of the pilot installation yet has not been addressed in the previously published works.

Experimental

Materials

The petroleum coke (petcoke) used for the kinetic studies was a low-sulphur Chinese petcoke, which was also used as the main fuel in the CHEERS project. The reported raw petcoke composition on dry basis is: C -91.2, H -4.13, O -1.44, N -2.52, and S -0.51 wt.% and ash 0.2 wt.% (by diff.).

Particles in the size range 100 -300 µm were used in the experiments. Some preliminary tests with a larger fraction (300 -500 µm) were also carried out (as presented in the Supplement S1); although the results were similar at lower temperatures and oxidiser concentrations, under the more reactive atmospheres (e.g. 40 vol.% H 2 O at 1000 ℃) the gasification time increased by 25 % when the larger fraction was used. This means that the internal diffusion within the petcoke particles plays a nonnegligible role during conversion under more reactive conditions (high temperatures and oxidiser concentrations). The influence of the external diffusion was limited due to the construction of the test rig (a porous crucible that allows for an unrestrained gas flow through the sample bed) and limitation of the analysis to the temperatures characteristic for the kinetic regime. Thus, the obtained parameters represent the J o u r n a l P r e -p r o o f apparent kinetics, which include only internal diffusion, and they are relevant for the examined particle size that was chosen to best represent the fuel used in the CLC unit.

Therefore, for the successful utilisation of these results for other applications, external diffusion resistance should be incorporated into the presented model, adequately to the conditions in the given reactor.

To avoid contaminating the test rig with released tars, the petcoke was devolatilised prior to the experiments. The material was heated to 600 ℃ in a N 2 flow with a 10 ℃/min heating rate followed by a 30 min isotherm. It was then cooled to ambient temperature and stored in a desiccator. This pre-treatment was also recreated in a TGA instrument, where the petcoke was heated and then cooled down in a N 2 flow, followed by CO 2 gasification. The similarity in the devolatilised and raw petcoke mass loss curves during the gasification step confirmed that the devolatilization and the cooling steps introduced due to the pretreatment did not significantly affect the gasification of the material (Supplement S2). Moreover, the test with the nondevolatilised petcoke was carried out in the main test rig under the most reactive of the studied conditions, and a comparison is presented in Supplement S3. An initial, rapid loss of approximately 10% of the sample mass occurred due to the rapid release of volatiles; however, the further mass loss curve was parallel to that of the devolatilised sample, and the reactivity at 50% conversion (R 50 ) was 0.023 and 0.022 (1/min) for non-devolatilised and devolatilised petcoke, respectively. Therefore, it can be assumed that the relatively slow gasification reaction was not affected by the rapid release of lighter compounds at the beginning of the process and that the devolatilization time was negligibly small compared with the oxidation of the solid residue. Therefore, the applied petcoke pre-treatment should not affect the kinetic parameters obtained in this research.

The test rig and experimental procedure

J o u r n a l P r e -p r o o f

Petcoke gasification kinetics were determined from a series of measurements performed in the custom-built test rig for thermochemical fuel conversion studies (Fig. 1). The test rig can operate in two modes, referred to as the gravimetric and evolved gases methods. The main principle of the experiment was to perform the gasification of petcoke under isothermal conditions in a controlled flow of a gaseous mixture with a predetermined composition while registering the mass loss of the sample during the reaction. The decrease in the sample mass was determined by directly measuring the sample's weight during the reaction (gravimetric method). For some tests, the mass loss was validated by calculating the amount of carbon in the gases produced during petcoke gasification in another configuration of the test rig, where the gaseous products were monitored instead of the mass of the sample (evolved gases method).

Pictures of the test rig are shown in Fig. 1. The main part of the rig is comprised of two electrical furnaces fixed on a common panel attached to a vertical rail. An electric motor [START_REF] Karasu | A new forecasting model with wrapper-based feature selection approach using multi-objective optimization technique for chaotic crude oil time series[END_REF] allows for a rapid (ca. 300 mm/s) movement of the panel along the rail over a distance of 500 mm to quickly heat the sample (ca. 1700 ℃/min). The required gas mixture of N 2 /CO 2 /O 2 was prepared by supplying high-purity (99.999 %) gases from gas cylinders into the mixing chamber (2) using thermal mass flow controllers. Since some of the examined gasification parameters requires steam, the gaseous mixture is supplied to the separate quartz tube reactor, the evaporator (3), before entering the reaction zone. If steam is required, a constant water flow is delivered to the evaporator using a syringe pump (4) and a PTFE transfer line, inserted from the bottom of the reactor and nested in a quartz wool plug, in the middle of the heating zone of the evaporator. The temperature in the evaporator is maintained at 300 ℃ to ensure constant, complete water vaporisation. The gases from the cylinders, now mixed with the steam, are further transferred via a heated line (5) into the main reactor, enclosed in the second furnace [START_REF] Commandré | Lack of correlation between the properties of a petroleum coke and its behaviour during combustion[END_REF].

J o u r n a l P r e -p r o o f

Depending on the selected method, one of the two types of vertical quartz tube reactors can be fixed in the furnace of the test rig. For the gravimetric method, a reactor (i.d. 27 mm) sealed at the top and opened at the bottom was used. The gaseous mixture was continuously supplied to the top of the reactor. Below the open lower end of the tube, a weighing module [START_REF] Tyler | Intrinsic reactivity of petroleum coke to oxygen[END_REF] enclosed in a protective case and purged with a constant N 2 flow was placed. A quartz rod was attached to the weighing plate of the module. The shift of the panel with the furnaces to the lowest position allowed the rod to be inserted into the reactor through its open end. At the end of the rod, in the middle of the reactor heating zone, a quartz sample holder ( 8) is mounted. The holder is cylindrical with an i.d. of 15 mm and a bottom made of a G3 quartz frit disc to allow the gaseous mixture to pass through the bed of the sample. The gaseous mixture and evolved gasification products exit the reactor through the open end. The sample mass was continuously measured, with the accuracy of 0.1 mg, throughout the experiment.

The second method, involving analysis of the released gases, required an air-tight gas outlet from the reactor; thus, it is impossible to register sample mass during this measurement. In this mode, a quartz tube (i.d. 20 mm) sealed on both ends is used, and the gaseous mixture is also supplied to the top of the reactor. It passes through a fixed bed of sample that is placed in the middle of the heating zone and is supported by a quartz wool plug. Through the sealed bottom end of the reactor, a thermocouple enclosed in a protective quartz tube is inserted to monitor the temperature of the bed. The evolved gases are transferred from the reactor through a cleaning line directly to the sampling loop of a gas chromatograph with a thermal conductivity detector (GC-TCD). The cleaning line includes an isopropanol impinger to remove condensable species, a particular matter filter, and a moisture trap. Gases, which constantly purged the sampling loop of the GC-TCD, were analysed every 10 min by switching the 6-way valve, which introduced the current contents of the sampling loop into the capillary column. The analysis was performed with an Agilent 6890N gas chromatograph J o u r n a l P r e -p r o o f with a TCD detector, with the 100 ppm limit of quantitation. The samples were separated on a capillary J&W GS-CarbonPLOT 30 m x 0.53 mm x 3 µm column followed by J&W HP-PLOT 30 m x 0.53 mm x 25 µm molecular sieves. For the duration of CO 2 elution, the latter was bypassed using a 6-way valve. The measurement principles of both methods are detailed in Fig. 2. In the gravimetric method, 300 mg of the sample (20 mg in tests with O 2 to limit diffusion resistance) was weighted into the crucible attached to the rod connected to the weighing module, while the furnace panel was in the upper position, so that the reactor was above the sample holder. The experiment J o u r n a l P r e -p r o o f can be carried out with either a low or high heating rate. In the former case, the furnace panel is lowered prior to heating, and the sample is therefore enclosed inside the reactor purged by a N 2 flow. The furnace is then heated to the selected temperature at a rate of 20 ℃/min. The atmosphere is then switched to the predefined gaseous mixture and the gasification process is initiated. The mass loss in time is registered, providing the process kinetics data. A high heating rate of ca. 1700 ℃/min could be achieved by lowering the pre-heated furnace so that it rapidly enclosed the sample that was waiting in the ambient atmosphere. However, only preliminary tests are carried out using the fast heating, while the main experiments used for kinetic calculations were performed using the slow heating. The comparison of tests at both heating rates (presented in Supplement S4) revealed that, although the initial mass loss of the sample slightly increased during rapid heating, the average reactivity (R 50 ) during conversion was not significantly affected by the sample heating time, due to the relatively long total gasification time.

In the gas evolution method, the gasification products are determined using GC-TCD analysis. The test rig setup included a sealed gas-tight reactor (Fig. 2). A 500 mg sample was enclosed in the reactor prior to the experiment and purged with N 2 , while the reactor was heated with a controlled, low heating rate (20 ℃/min) up to the desired temperature. The N 2 flow was then switched to the gasification mixture and the online analysis of the gasification products, in 10 min intervals, was initiated.

Calculations

Kinetic parameters are calculated only from the mass loss curves obtained from the gravimetric method. The evolved gases mode was mainly used to evaluate the reaction products; however, the mass loss of petcoke during steam gasification is also estimated from the carbon balance, based on all detected carbon-containing species, i.e. CO, CO 2 , and CH 4 .

The remaining relative mass of the sample is expressed as:

݉ ୰ୣ୪ ሺݐሻ = 1 - ൫݊ ை ሺݐሻ + ݊ ைଶ ሺݐሻ + ݊ ுସ ሺݐሻ൯ܯ ݉ ݔ (1)
J o u r n a l P r e -p r o o f

where ݊ is the integrated molar amount of carbon (in mmol) in the i-th reaction product released in the time interval from ݐ = 0 to ݐ = ,ݐ ܯ is the carbon molar mass (in mg/mmol), ݉ is the initial mass of petcoke (in mg), and ݔ is the mass fraction of C in petcoke (in mg C/mg).

The recorded mass losses during the gravimetric method are used to calculate the carbon conversion, which is defined as

ܺሺݐሻ = ݉ -݉ሺݐሻ ݉ -݉ ஶ (2)
where ݉, ݉ and ݉ ஶ are the instantaneous, initial and final masses of the sample, respectively (in mg). In the experiments where full conversion was not reached, the final mass was assumed based on the residual mass obtained in the tests with complete sample conversion.

A general form of the apparent rate of conversion for gas-solid reactions is given by

݀ܺ ݐ݀ = ݂݇ሺܺሻ ௫ (3)
where  ௫ is the partial pressure of the oxidizer (Pa), ݊ is the reaction order, ݇ is the apparent reaction rate coefficient (in 1/(s·Pa n )) and ݂ሺܺሻ is a model function. The apparent reaction rate coefficient ݇ takes into account the changes in temperature introduced in the Arrhenius form

݇ = ܣ exp ൬- ܧ ܴܶ ൰ ( 4
)
where ܣ is the pre-exponential factor (in 1/(s·Pa n )), ܧ is the activation energy (kJ/mol), ܴ is the universal gas constant (in kJ/(mol·K)) and ܶ is the temperature (in K). The model function

J o u r n a l P r e -p r o o f

݂ሺܺሻ takes into account variations in the physical and chemical properties of the sample as the reaction proceeds. Four model functions are tested within this study. First is the uniform conversion model [START_REF] Ye | Gasification of a South Australian low-rank coal with carbon dioxide and steam: kinetics and reactivity studies[END_REF], also known as homogeneous [START_REF] Molina | Reactivity of coal gasification with steam and CO2[END_REF] or Avrami model (AVRAMI) [START_REF] Gartner | Investigation of biomass char gasification kinetic parameters using a novel miniaturized educational system[END_REF],

which is given by

݂ሺܺሻ = ሺ1 -ܺሻ (5)
The second examined function is the Random Pore Model (RPM) [START_REF] Gartner | Investigation of biomass char gasification kinetic parameters using a novel miniaturized educational system[END_REF][32] expressed as

݂ሺܺሻ = ሺ1 -ܺሻඥ1 -߰ lnሺ1 -ܺሻ (6)
where ߰ is the pore structure parameter, which can be determined using

߰ = 2 2 lnሺ1 -ܺ ௫ ሻ + 1 (7)
where ܺ ௫ is the conversion at maximum reaction rate, which is determined using the condition

݀ሺ݀ܺ/݀ݐሻ ݀ܺ = 0 (8)
The third considered model function is the Shrinking Core Model (SCM) [START_REF] Gartner | Investigation of biomass char gasification kinetic parameters using a novel miniaturized educational system[END_REF] ݂ሺܺሻ = ሺ1 -ܺሻ

where ݉ = 2/3 for spheres is assumed. The last function analyzed is called the Hybrid Model (HM) [START_REF] Gartner | Investigation of biomass char gasification kinetic parameters using a novel miniaturized educational system[END_REF] which is identical to Eq. ( 9), but the exponent ݉ is treated as a parameter and is adjusted during data fitting.

The experimental data is analysed using a series of scripts written in Matlab. As mentioned earlier, the measurements are done for petcoke reactions with O ଶ , H ଶ O and CO ଶ . For all reactants, the reaction orders ݊ were determined first, by analysing the conversion rates J o u r n a l P r e -p r o o f obtained for various partial pressures of the reactants. Then, the obtained reaction orders were kept constant and the reaction rate coefficients were determined by fitting the conversion rates with functions ( 5)- [START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF]. Finally, the kinetic parameters ܣ and ܧ from Eq. (4) were determined using Arrhenius plots. The results of the analyses are presented in section 3.2.

Results and discussion

Results of the petcoke gasification experiments

Gas evolution profiles

Gases released during the gasification of petcoke in 10 and 40 vol.% of steam in N 2 were measured online with a gas chromatograph coupled with a thermal conductivity detector (GC-TCD). The evolution profile for the test performed at 1000 ℃ in 40 vol.% of steam in N 2 is presented in Fig. 3 as an example; the main released gases were CO and H 2 , while the CO 2 and CH 4 yields were an order of magnitude lower. The delayed increase in the CO 2 evolution profile suggests that the water-gas shift (WGS) reaction was intensified during gasification.

This could be due to either the catalytic effect of metals exposed by the initial carbon consumption with steam or as a result of a local increase in the steam concentration in the particle's surrounding, which occurred as the main steam gasification reaction, responsible for CO formation, slowed after 50 min. Another explanation for the delayed release of CO 2 might be its chemisorption on the petcoke surface at the initial stage of the process. For all examined cases, the maximum H 2 and CO yield occurred around the 0.1 conversion, and reaching even this early stage of petcoke gasification required residence times too long to be considered in operating commercial reactors. However, the experiments with steam as a sole oxidising agent were performed for the purpose of kinetic parameters determination that will be implemented in the modelling of a real gasifier. Due to the low reactivity of the sample, and J o u r n a l P r e -p r o o f the limited duration of the experiment (480 min), the carbon conversion, ܺ, in the tests with the lower steam concentration reached only 0.5 -0.75, depending on the applied temperature.

Thus, the composition of the gaseous products was averaged only for the first half of the petcoke conversion (up to ܺ = 0.5), and the result is presented in Fig. 4. As can be seen from the figure, CO and H 2 were the main gasification products, and their yields peaked at the beginning of the process and then decreased continuously with the carbon burnout. The molar ratios of the cumulative amounts of H 2 and CO (݊ ுଶ /݊ ை ) released during carbon conversion up to ܺ = 0.5 at temperatures up to 1000 ℃ were between 1.2 -1.4 for the gasification with steam. The CO 2 concentration in the gasification products (i.e. excluding N 2 and H 2 O) was below 5 vol.%. The composition of the obtained syngas was similar to the values reported by Trommer et al. [START_REF] Trommer | Hydrogen production by steam-gasification of petroleum coke using concentrated solar power -I. Thermodynamic and kinetic analyses[END_REF] for two petcoke samples gasified in 10 vol.% steam in a plug flow reactor. However, more CO 2 was formed under the most reactive conditions, i.e. 40 vol.% of steam and 1100 ℃. Wu et al. [START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF] stated that during petcoke gasification with steam in a fixedbed reactor the water-gas shift reaction (WGS) shown in equation 10 will not reach equilibrium.

CO + H

ଶ O = CO ଶ + H ଶ (10)
Since the WGS reaction rate increases rapidly above 1000 ℃, at 1100 ℃ the WGS reaction is intensified, despite its exothermic character, while the rate of the heterogenous reaction

C + H ଶ O = CO + H ଶ (11)
is constrained by the available surface of the petcoke particles, which increases CO 2 and decreases CO yield. Another possible explanation for the observed increase in the CO 2 formation is the rapid increase in the catalytic activity of the metal M-C-O conformations in petcoke, which are known to increase selectivity towards CO 2 formation, at the expense of CO yield [START_REF] Wu | Potassium-catalyzed steam gasification of petroleum coke for H2 production: Reactivity, selectivity and gas release[END_REF][START_REF] Wang | Steam gasification of coal char catalyzed by K2CO3 for enhanced production of hydrogen without formation of methane[END_REF].

J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

As can be seen from Fig. 5, the reaction rate of petcoke gasification with CO 2 was extremely slow. The measurement time was arbitrarily limited to 480 min and negligible conversion was achieved during these tests. Gasification with CO 2 gradually increased the porosity of petcoke, resulting in an increase in the CO yield. Even under the most reactive condition, i.e. 40 vol.% of CO 2 at 1000 ℃, the reaction rate did not reach its maximum even after 480 min of measurements, while, in the same concentration of steam, the petcoke bed reached full conversion in less than 240 min. Therefore, the role of CO 2 during petcoke gasification below 1000 ℃ could be considered negligible and it does not have any applicable commercial value.

Since the petcoke conversion with CO 2 was so slow, realistic kinetic parameters could not be obtained.

In contrast, petcoke gasification in 4 vol.% of O 2 was very rapid. In the temperature range of 750 -1000 ℃ total conversion was reached in less than 15 min. Since the online gas analysis with GC-TCD was performed in 10 min intervals, no evolution profiles could be determined from these tests. The kinetic parameters of oxygen conversion were, therefore, determined based on the gravimetric tests. The only detected reaction product was CO 2 , which suggest that the combustion of petcoke was complete.

J o u r n a l P r e -p r o o f in N 2 at 900, 950 and 1000 ℃

The results of the gravimetric and evolved gases experiments

Gravimetric analysis of petcoke gasification in steam and in oxygen was used to calculate kinetic rate parameters for the heterogeneous reactions. In addition, for steam gasification, gas evolution profiles were also used to determine the corresponding mass loss curves. This was done based on the amount of carbon in the gaseous reaction products (Eq. 1). The comparison of the mass loss functions obtained with both methods is presented in Fig. 6 and Fig. 7 for experiments performed in 10 vol.% and 40 vol.% of steam, respectively.

At the lower steam concentration, and thus a lower reaction rate, the curves were similar, but some discrepancies can be seen for the reactions carried out under more reactive conditions.

For those cases where full conversion was reached within the time frame of the experiment, i.e. where ݐ݀/ܺ݀ = 0 at the end of the experiment, the final conversion found from the EV

10CO 2 _1000 40CO 2 _900 40CO 2 _950 40CO 2 _1000
J o u r n a l P r e -p r o o f method is 0 -0.1 larger than for the GR method. This discrepancy may be because not all gases were correctly captured and integrated using the EV method. For these cases, the final conversion of the EV method is normalised by the one obtained using the GR method. A relatively good agreement between the mass loss plots obtained with EV and GR methods suggests that the former provides valid data on the composition of released gases and can be applied to study the petcoke gasification mechanisms and kinetics. As no solid deposits were observed in the experiments, the differences between the methods may result from the larger uncertainties of the indirect mass loss determination approach of the EV method; thus, the data from the GR method was used for the kinetic parameter calculations. This observation indicates that no such limitation occurs at lower temperatures, thus the kinetics calculated from the petcoke gasification experiments up to 950 ℃ can be attributed to the chemical reaction rates and increase in the surface area of the particles, rather than diffusional limitations of a fixed bed.

J o u r n a l P r e -p r o o f The mass loss curves from the gravimetric tests of petcoke gasification with 2 and 4 vol.% of O 2 are presented in Fig. 8. As expected, the reaction occurs more rapidly at higher temperatures. CO 2 is the only detected product, and the higher oxygen concentration significantly shortens the reaction times. The reaction of petcoke with CO 2 was significantly slower than steam gasification; nonetheless, a possible contribution of CO 2 during petcoke conversion under the more complex gasification atmosphere was examined by performing tests in a mixture of 40 vol.% of H 2 O in CO 2 . The results were compared with the steam gasification measurements using inert N 2 as the carrier gas, as presented in Fig. 9. The mass loss registered during the 480 min J o u r n a l P r e -p r o o f conversion tests in 40 vol.% of CO 2 in N 2 are also provided as a reference. It can be noted that, up to 1000 ℃, the curves of the petcoke mass loss during reactions with 40 vol. % of steam were not affected by changing the carrier gas from N 2 to CO 2 . This finding confirmed the dominating role of H 2 O over CO 2 in the petcoke gasification process at these temperatures. However, the reaction with CO 2 became significant at 1100 ℃. At the same time, since full conversion was reached for steam gasification (after about 160 min), almost 1/3 of the sample was converted when H 2 O was substituted with CO 2 . The same nonnegligible role of CO 2 at high temperatures was confirmed by the enhanced petcoke conversion when N 2 was substituted with CO 2 for the measurement at 1100 ℃. Therefore, for high-temperature petcoke gasification the presence of CO 2 should be accounted for, while it can be disregarded under less-reactive conditions.

J o u r n a l P r e -p r o o f Petcoke gasification measurements in 20 vol.% steam in CO 2 were also carried out at temperatures up to 1000 ℃. From these results, the assumption that the type of the carrier gas does not affect the kinetics of the steam gasification of petcoke in this temperature range was confirmed by comparison with corresponding measurements in H 2 O/N 2 atmospheres.

Moreover, using 20 vol.% of steam for these additional tests allowed validation of the results of the models applied for the conversion in 10 and 40 vol.%. The mass loss of petcoke in 40, 20 and 10 vol.% of steam (in either N 2 or CO 2 ) is presented in Fig. 10. Steam concentration had a strong impact on the petcoke gasification, and when a higher steam content was used, the total conversion times were shorter, regardless of the applied temperature. At 900 ℃, the gasification reaction was significantly slower for all examined steam concentrations.

However, the offset between the mass loss curves at 950 and 1000 ℃ was less pronounced and decreased with the reaction time. Surprisingly, after reaching 0.6 and 0.9 conversions with 20 and 10 vol.% of steam, respectively, the remaining mass of petcoke was lower at 950 ℃ than at 1000 ℃, possibly due to diffusional limitations at 1000 ℃. lines were fitted to the data, as presented in Fig. 12, following the procedure described by Gartner et al. [START_REF] Gartner | Investigation of biomass char gasification kinetic parameters using a novel miniaturized educational system[END_REF]. The reaction orders were then determined from a plot of ln ሺ݀ܺ/݀ݐሻ vs.

ln ሺ ைଶ ሻ for each conversion ܺ, and based on the obtained results, a mean reaction order was calculated. For the reaction with O ଶ the determined mean reaction order is n=0.55. It should The mean reaction order was then used to calculate the apparent reaction rate coefficient ݇ (and ݉ for HM) by fitting the experimental data with the models defined in Eqs ( 5) -( 9). In Fig. 13, the ݂ሺܺሻ function models fitted to the experimental data are compared. As can be seen, the best fit was obtained for the Hybrid Model (HM), which was confirmed by comparison of the sum of squared residuals ܴܵܵ defined as

ܴܵܵ = ሺߝ ሻ ଶ ே ୀଵ ( 12
)
where ߝ is the residual (the difference between experimental data and model) and ܰ is the number of data points. The ܴܵܵ for each model for all temperatures, as presented in Fig. 13 The determined model parameters are summarised in Table 1. The obtained model parameters were then used to verify the model predictions by integrating the conversion rates )ݐ݀/ܺ݀( for the various models and comparing the results with the experimentally determined conversions. The comparison is presented in Fig. 15. The AVRAMI model was excluded, as it gave the poorest predictions. All models predicted the experimentally-determined petcoke conversion vs. time. In some instances, the models slightly overpredicted the conversion, which is associated with the observed initial time delay of the conversion due to initial sample heating up not captured by the models. In general, the best predictions were obtained for the were used to determine the reaction order. For the determination of the kinetic parameters, only data for low temperatures and high H ଶ O concentrations ݔ( ுଶை = 0.4 mol/molሻ were used. In general, large variations in the reaction orders with temperature was observed, and a mean value of ݊ = 0.9 was calculated and used. Another choice would be to take the orders for low temperatures (݊ ~ 0.86). Since the value was close to the one for all temperatures and the fitted model predictions with an order of 0.9 were satisfactory, this approach was used, and four models (AVRAMI, RPM, SCM and HM) were tested as previously. In Fig. 17,a comparison of the fitted model functions ݂ሺܺሻ to the experimental data is presented. As can be seen for the highest temperature of 1000 ℃, the reaction rate is initially the highest, however at conversions greater than 0.5, the reaction rate decreases below that of the 950 ℃ plot.

J o u r n a l P r e -p r o o f The petcoke gasification experiments in a CO ଶ /N ଶ atmosphere ݔ( ைଶ = 0.4 mol/mol) were conducted at three temperatures, i.e. 900, 950, and 1000 ℃. In Fig. 21, the conversion vs. time is presented. The reaction of petcoke with CO ଶ is an order of magnitude lower than that with H ଶ O, and decreases further as the reaction proceeds. It took approximately 6 hours to convert 10 % of the sample at 1000 ℃; thus, the conversion rate is below practical considerations under the analysed conditions. Therefore, no model was proposed here, since for practical systems operating at the temperatures considered in this analysis, the reaction rate can be assumed to be 0. Some literature reports describe the relatively rapid CO 2 gasification of petcoke with activation energies of approximately 150 kJ/mol (e.g. 142 kJ/mol reported by Kumari et al. [START_REF] Kumari | Comparison of CO2 gasification reactivity and kinetics: petcoke, biomass and high ash coal[END_REF] and 159 kJ/mol reported by Wei et al. [START_REF] Wei | Effect of biomass leachates on structure evolution and reactivity characteristic of petroleum coke gasification[END_REF]). However, those thermogravimetric experiments were performed under the atmosphere of pure CO 2 at temperatures above 1000 ℃, i.e. conditions significantly more reactive than the ones applied in this study.

J o u r n a l P r e -p r o o f

Conclusions

In this article, the kinetic parameters for petcoke gasification were calculated based on the experimental data from laboratory tests dedicated to the material selected for the CHEERS project, to enable modelling of the 3 MWth prototype of chemical looping combustion (CLC) system with inherent carbon capture.

The tests revealed that the rate of petcoke gasification at 10 and 40 vol.% of CO 2 was negligible at temperatures below 1100 ℃. The tests allowed the determination of the kinetic parameters for petcoke gasification in steam and oxygen at temperatures up to 950 ℃. At higher temperatures, the conversion was limited by diffusion. The kinetic parameters of petcoke gasification were best described by:

-

Fig. 1 .

 1 Fig. 1. Test rig for petcoke gasification in the gravimetric mode with the furnace in the lower

Fig. 2 .

 2 Fig. 2. The principle of petcoke gasification with the a) gravimetric and b) evolved gases

Fig. 3 .Fig. 4 .

 34 Fig. 3. Gas evolution profiles for petcoke gasification in 40 vol.% of steam in N 2 at 1000 ℃

Fig. 5 .

 5 Fig. 5. CO released during 480 min of petcoke gasification in 10 vol.% and 40 vol.% of CO 2

Fig. 6 .Fig. 7 .

 67 Fig. 6. Mass loss curves from gravimetric (GR) and evolved gases (EV) analysis of petcoke

Fig. 8 . 2 3. 1 . 3 .

 8213 Fig. 8. Mass loss curves from gravimetric analysis of petcoke gasification in 2 and 4 vol.% of

Fig. 9 .

 9 Fig. 9. Mass loss of petcoke during gasification in 40 vol.% of H 2 O with N 2 and CO 2 as the

Fig. 10 .℃ 3 . 2 . Calculation of the kinetic parameters of petcoke gasification 3 . 2 . 1 .

 1032321 Fig. 10. Mass loss of petcoke during gasification in 40, 20 and 10 vol.% of steam at 900, 950

40H 2 O/60CO 2 80H 2 O/20CO 2 10H 2 O/90N 2 JFig. 11 .Fig. 12 .

 2222221112 Fig. 11. Experimentally determined fuel conversion vs. time for two different mole fractions

Fig. 13 .

 13 Fig. 13. Comparison of fitted model functions ݂ሺܺሻ for the reaction with O ଶ ݔ( ைଶ = 0.04) Finally, the calculated apparent reaction rate coefficients ݇ were used to determine the preexponential factor ,ܣ and the activation energy ,ܧ by fitting the plotted lnሺ݇ሻ vs. 1/ܶ data, as presented in Fig. 14. The fitted linear functions predicted the obtained rate coefficients well,

J o u r

  n a l P r e -p r o o f Hybrid Model (HM), which is confirmed by the lowest residuals. The calculated sums of squared residuals (SSR) were: RPM ܴܵܵ = 1.3 • 10 ିହ , SCM ܴܵܵ = 2.2 • 10 ିହ , HM ܴܵܵ = 1.2 • 10 ିହ . Therefore, it is recommended to use the Hybrid Model (HM) for reactions with O ଶ . It should be also stressed that the models are valid in the kinetic regime.It was reported by Afrooz et al.[START_REF] Afrooz | A Modified Model for Kinetic Analysis of Petroleum Coke[END_REF] that the activation energy of petcoke oxidation decreased from 124 to 35.3 kJ/mol (determined with SCM), and from 124.8 to 31.3 kJ/mol (determined with RPM) when heating rate in the TGA program was increased from 10 to 20 K/min. In this work, the sample was already at the reaction temperature when the oxidising agent was introduced, thus, it is plausible, that the lack of the heating step further decreased the activation energy to the values calculated hereby.

Fig. 14 .

 14 Fig. 14. Determination of the pre-exponential factor ܣ and activation energy ܧ

Fig. 16 .Fig. 18 .Fig. 19 .Fig. 20 . 3 . 2 . 3 .

 16181920323 Fig. 16. Experimentally-determined fuel conversion vs. time

Fig. 21 .

 21 Fig. 21. Conversion vs. time for a reaction with CO ଶ

  the Hybrid Model for gasification in 2-4 % O 2 ܧ( = 15.87 kJ/mol); -the Random Pore Model for gasification in 10 -40 vol.% H 2 O ሺܧ = 103.91 kJ/molሻ. J o u r n a l P r e -p r o o f

  The same procedures for determining the kinetic parameters as described above for O ଶ were applied for the reaction with H ଶ O, where the petcoke was gasified in a H ଶ O/N ଶ mixture for the two H ଶ O concentrations of 10 and 40% vol. In Fig.16, the experimentally-determined fuel conversion vs. time is presented. As can be seen, the conversion lines cross for the high temperature tests. These phenomena are attributed to the reaction rate reduction due to diffusion at high temperature. Furthermore, not full conversion was obtained at the H 2 O concentrations of 10 vol.% due to very long reaction times. It should be however stressed that both concentrations ݔ( ுଶை = 0.1 mol/mol and ݔ ுଶை = 0.4 mol/mol) and all temperatures

	3.2.2. Reaction with H 2 O

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 764697. The project is also co-funded from National Key Research and Development Program of China by Chinese Ministry of Science and Technology (MOST) under grant agreement No 2017YFE0112500.

Moreover, at the temperatures up to 1000 ℃, conversion in CO 2 was negligible and it did not affect the reaction rate, when CO 2 was added to the steam gasification of petcoke.

The apparent kinetic parameters determined in this work include the internal diffusion within the particles; thus, to make the obtained results more universal, the continuation of this research, to account for different particles sizes, will be considered in the future.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:


Ahmed Attar

Rachid Bentifour

El-Haj Laamri

NONLINEAR ELLIPTIC SYSTEMS WITH COUPLED GRADIENT TERMS

Keywords: . 2010 Mathematics Subject Classification. 35J55, 35D10, 35J60 Nonlinear elliptic systems, Hamilton Jacobi equation, fixed point theorem, apriori estimates

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type :

.

Introduction

The main goal of this work is to analyze the issue of existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems with gradient terms. More precisely, we deal with the following model :

(1.1)

       -∆u = |∇v| q + λf in Ω, -∆v = |∇u| p + µg in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω,
where Ω is a bounded domain of IR N , (p, q) ∈ [1, +∞[ 2 , (λ, µ) ∈ (0, +∞) 2 and f, g are nonnegative measurable functions that satisfy additional assumptions that will be specified later. The purpose of this paper is to find the conditions between (f, g) and (λ, µ) which allow us to establish existence of solutions to System (1.1). Moreover, we will prove some non-existence results that show, in some sense, the optimality of our assumptions. In this article, we focus our attention on the proof of the existence of weak solutions ; by weak solution, we mean solution in the sense of distributions (see Definition 2.5).

Let us mention that a class of nonlinear elliptic systems with gradient terms has come to the fore in some electrochemical models of engineering, and in some other models in fluid dynamics. More recently, another class was introduced in [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et controle optimal[END_REF] in the theory of mean-field games ; see also [START_REF] Boccardo | Strongly coupled elliptic equations related to mean-field games systems[END_REF] for an in-depth analysis. Ù Let us recall some previously known results related to the present study. First Case : One Equation. In the case where the gradient term acts as source term, the problem (1.2)

-∆w = |∇w| q + λh in Ω, w ∈ W 1,q 0 (Ω), is known as a stationary Hamilton-Jacobi equation, or the stationary Kardar-Parisi-Zhang equation. This problem has been extensively investigated. Contributions in this topic include [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 quasi-linéaires[END_REF][START_REF] Grenon | Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms[END_REF][START_REF] Hansson | Criteria of solvability for multidimensional Riccati equations[END_REF][START_REF] Abdellaoui | Some remarks on elliptic problems with critical growth in the gradient[END_REF][START_REF] Ferone | Quasilinear problems having quadratic growth in the gradient : an existence result when the source term is small[END_REF] and the references therein. Needless to say, these references do not exhaust the rich literature on the subject. To be more precise, if q ≤ 2 (sub-quadratic and quadratic growth), existence of solutions is obtained using comparison arguments and a priori estimates. The superquadratic case (q > 2) has been studied, for instance, in the references [START_REF] Hansson | Criteria of solvability for multidimensional Riccati equations[END_REF], [START_REF] Cong | Morrey global bounds and qusilinear Riccarti type equation bellow the natural exponent[END_REF] and [START_REF] Mengesha | Quasilinear Riccati type equations with distributional data in Morrey space framework[END_REF] where the existence of solutions is obtained using potential theory and some fixed-point arguments.

Concerning the case of one equation where the gradient term acts as an absorption term, the associated problem appears in theory of mean-field games, we refer to the pioneering works [START_REF] Lions | Résolution de problèmes elliptiques quasilinéaires[END_REF], [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF] and [START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et controle optimal[END_REF] for more details. Second Case : Systems. In contrast to single equations, only few results are available in the literature for systems where the source terms depend on the gradient. Moreover, only partial results have been obtained, see for instance [START_REF] Abdellaoui | On the existence of positive solutions to semilinear elliptic systems involving gradient term[END_REF][START_REF] Attar | Existence of positive solutions to nonlinear elliptic systems involving gradient term and reaction potential[END_REF][START_REF] Boccardo | A quasilinear elliptic system with natural growth terms[END_REF][START_REF] Boccardo | Existence of finite energy solutions for elliptic systems with L 1 -value nonlinearities[END_REF][START_REF] Diaz | Large solutions for a system of elliptic equation arising from fluid dynamics[END_REF]. We recall the results of [START_REF] Diaz | Large solutions for a system of elliptic equation arising from fluid dynamics[END_REF] and [START_REF] Boccardo | Existence of finite energy solutions for elliptic systems with L 1 -value nonlinearities[END_REF] where the dependence on the gradient appears in one equation. In [START_REF] Attar | Existence of positive solutions to nonlinear elliptic systems involving gradient term and reaction potential[END_REF] and [START_REF] Abdellaoui | On the existence of positive solutions to semilinear elliptic systems involving gradient term[END_REF], nonlinear systems with potential and gradient terms are studied. Existence of solutions is obtained under some optimal conditions on the data.

The main objective of this work is to analyze System (1.1), where the dependence on the gradient appears in both equations, without any restriction on the value of p and q. To the best of our knowledge, no result was previously obtained under these general assumptions.

Moreover, in order to show that (1.1) has a solution, it is necessary to have (minimal) regularity assumptions on the data f, g and smallness conditions on λ and µ. Indeed, such conditions are necessary even in the case of a single equation. More precisely, for the problem (1.2), we have the following existence result proved in [START_REF] Hansson | Criteria of solvability for multidimensional Riccati equations[END_REF],

Theorem 1.1. Assume that h ∈ L 1 (Ω) is a nonnegative function. Define (1.3) m(h) := inf {φ∈C ∞ 0 (Ω) ; φ =0} Ω |∇φ| q dx Ω h|φ| q dx .
Then, Problem (1.2) has a nonnegative solution u ∈ W 1,q 0 (Ω) for λ > 0 if and only if m(h) > 0. Moreover, there exists λ * (m(h)) > 0 such that Problem (1.2) has a solution for all λ < λ * (m(h)).

In term of elliptic capacity, the positivity of m(h) is equivalent to the following fact : denoting by cap 1,q , the elliptic capacity defined by cap 1,q (E) := inf

Ω |∇φ| q dx with φ ∈ C ∞ 0 (Ω) and φ ≥ χ E , then m(h) > 0 if and only if (1.4) |E| h ≤ C cap 1,q (E),
where E ⊂ Ω is a measurable set and |E| h = E h(y)dy. We refer to [START_REF] Hansson | Criteria of solvability for multidimensional Riccati equations[END_REF]Theorems 1.1,1.2,3.1] for more details.

The paper is organized as follows. In section 2, we introduce some useful tools such as regularity results for elliptic problems, the sense in which solutions are defined, and the Schauder fixed-point Theorem used in this paper. In Section 3, we state and prove the main existence results of this work. Under suitable conditions on the data, we prove the existence of a nonnegative solution in suitable Sobolev spaces. Some examples and non-existence results showing the optimality of our conditions are also presented in Section 4. Finally, in the last section we treat more general systems where the interaction between u and v appears in both gradient and potential terms.

Throughout this paper, for a, b ≥ 0, we use the notation a b, to say that,

C 1 a ≤ b ≤ C 2 a, with C 1 , C 2 > 0 are independent of a, b.

Preliminaries and useful results

For the convenience of the reader and for the sake of completeness, we recall in this short section some classical results we will use in our proofs.

In order to prove the existence results of this paper we will invoke the following Proposition and the famous Schauder fixed-point Theorem.

Proposition 2.1. Let h ∈ L 1 (Ω) and {h n } n ⊂ L r (Ω) with 1 < r < +∞. Assume that h n → h strongly in L 1 (Ω) and {h n } n is bounded in L r (Ω). Then, h ∈ L r (Ω) and for any a ∈ [1, r), we have the following interpolation inequality

(2.1) h n -h L a (Ω) ≤ h n -h θ L 1 (Ω) h n -h 1-θ L r (Ω) where θ := r -a a(r -1)
In particular, h n → h strongly in L a (Ω). Theorem 2.2 (Schauder fixed-point Theorem). Assume that E is a closed convex set of a Banach space X. Let T be a continuous and compact mapping from E into itself. Then, T has a fixed point in E.

Systematically, we will use the following existence and compactness result proved in [START_REF] Baras | Singularités éliminables pour des équations semi-linéaires[END_REF]Appendix].

Theorem 2.3. Assume that h ∈ L 1 (Ω), then the problem (2.2) -∆z = h in Ω, z = 0 on ∂Ω, has a unique weak solution z ∈ W 1,s 0 (Ω) for all s ∈ [1, N N -1 ). Moreover, for s ∈ [1, N N -1 ) fixed, there exists a positive constant C = C(Ω, N, s) such that (2.3) ||∇z|| L s (Ω) ≤ C||h|| L 1 (Ω) .
and the operator Φ :

h → z is compact from L 1 (Ω) into W 1,s 0 (Ω).
Finally, let us recall the following classical regularity result that we will use in several proofs below.

Theorem 2.4. Let z be the unique weak solution to Problem (2.2) with h ∈ L m (Ω), m > 1. Then, there exists a positive constant C = C(Ω, N, m) independent of h such that

(1) If 1 < m < N , then

(2.4) ||∇z|| L m * (Ω) ≤ C||h|| L m (Ω) where m * = mN N -m .
(

) If m = N , |∇z| ∈ L s (Ω) for all s ∈ [1, +∞). ( 2
) If m > N , z ∈ C 1,γ (Ω) 3
for some γ ∈ (0, 1).

For a proof see for instance [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] or [START_REF] Boccardo | Elliptic Partial Differential Equations : existence and regularity of distributional solutions[END_REF].

We now define what we mean by solution to our system (1.1).

Definition 2.5. Let (f, g) ∈ (L 1 (Ω) + ) 2 be nonnegative functions. We say that (u, v) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω) is a weak solution to System (1.1) if u, v ≥ 0 and for all ϕ, ψ ∈ C ∞ 0 (Ω), we have

(2.5) Ω ∇ϕ∇u dx = Ω |∇v| q ϕ dx + λ Ω f ϕ dx,
and

Ω ∇ψ∇v dx = Ω |∇u| p ψ dx + µ Ω gψ dx.
In what follows, we denote by C 1 , C 2 , ... any positive constants that depend only on the datum of the problem and that can be changed from one line to next one.

Existence results

Recall that we are considering the following nonlinear system (3.1)

       -∆u = |∇v| q + λf in Ω, -∆v = |∇u| p + µg in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω,
where Ω is a bounded domain of IR N and p, q ≥ 1.

Let us begin by stating the first existence result of this paper.

Theorem 3.1. Assume that p, q ≥ 1 are such that pq > 1.

Let (m, σ) ∈ [1, +∞) 2 . Suppose that (f, g) ∈ L m (Ω) + × L σ (Ω) +
where (m, σ) satisfies one of the following conditions :

(3.2) m, σ ∈ (1, N ), pσ < m * = mN N -m and qm < σN N -σ = σ * ; (3.3) or m ≥ N and σ > qmN N + qm ; (3.4) or σ ≥ N and m > pσN N + pσ .
Then, there exists Λ * > 0 such that for all (λ, µ) ∈ Π where

(3.5) Π := {(λ, µ) ∈ [0, +∞) × [0, +∞) | λ p f p L m (Ω) + µ g L σ (Ω) ≤ Λ * }, System (3.1) has a nonnegative solution (u, v). Moreover (u, v) ∈ W 1,θ1 0 (Ω) × W 1,θ2 0 (Ω) for all θ 1 < mN (N -m) + and θ 2 < σN (N -σ) + .
Proof. We will give the proof under the assumption (3.2). The other cases follow using similar approach. The proof will be given in several steps.

Step 1 : For s ≥ 0, we define the function

Υ(s) := s 1 pq -Cs,
where C is a positive constant depending only on the data, which we will specify later.

As pq > 1, then there exists s 0 > 0 such that Υ(s 0 ) = 0, Υ(s) > 0, ∀s ∈ (0, s 0 ), Υ(s) < 0, ∀s ∈ (s 0 , +∞). Hence, there exist two positive constants > 0 and Λ * > 0 such that

max s≥0 Υ(s) = Υ( ) = Λ * .
Thus

1 pq = C( + Λ * C
). Fix > 0 as above, and define the set

Π:= (λ, µ) ∈ [0, +∞) × [0, +∞) ; λ p ||f || p L m (Ω) + µ||g|| L σ (Ω) ≤ Λ * C .
Obviously, Π is a non-empty, bounded and closed set of IR 2 . Moreover, for all (λ, µ) ∈ Π, we have

(3.6) C + λ p ||f || p L m (Ω) + µ||g|| L σ (Ω) ≤ 1 pq .
For a fixed r ∈ (qm, σN N -σ ), let

(3.7) H := ϕ ∈ W 1,r 0 (Ω) and ||∇ϕ|| L r (Ω) ≤ 1 pq . It is clear that H is a convex set. We claim that H is a closed subset of W 1,1 0 (Ω). To see that, let {u n } n ⊂ H be such that u n → u strongly in W 1,1 0 (Ω). Then ||∇u n || L r (Ω) ≤ 1
pq for all n and ||∇u n -∇u|| L 1 (Ω) → 0. Thus, up to a subsequence, ∇u n → ∇u a.e in Ω. Using Fatou's lemma we obtain that |∇u| ∈ L r (Ω) and

||∇u|| L r (Ω) ≤ lim inf n→∞ ||∇u n || L r (Ω) ≤ 1 pq .
Thus u ∈ H and the claim follows.

Let ϕ ∈ H, then |∇ϕ| a ∈ L 1 (Ω) for all a ≤ r ; in particular |∇ϕ| q ∈ L 1 (Ω). Thus, we can define u as the unique weak solution to the problem

(3.8) -∆u = |∇ϕ| q + λf in Ω, u = 0 on ∂Ω.
In addition |∇ϕ| q ∈ L m (Ω), thus using Theorem 2.4, it holds that

||∇u|| L β (Ω) ≤ C 1 |∇ϕ| q + λf L m (Ω)
, for all β ≤ m * = mN N -m , and then

(3.9) ||∇u|| L β (Ω) ≤ C 2 ||∇ϕ|| q L qm (Ω) + λ||f || L m (Ω) , for all β ≤ m * .
Recall that qm ≤ r, hence setting β = p ≤ m * , it follows that

(3.10) ||∇u|| L p (Ω) ≤ C 3 ||∇ϕ|| q L r (Ω) + C 4 ||f || L m (Ω)
. Thanks to Theorem 2.3, we define v as the unique weak solution to the problem

(3.11) -∆v = |∇u| p + µg in Ω, v = 0 on ∂Ω.
By the same theorem, v ∈ W 1,a 0 (Ω) for all a < N N -1 . Therefore the operator

T : H -→ W 1,1 0 (Ω) ϕ -→ T (ϕ) = v, is well defined. Moreover, if v is a fixed point of T , then (u, v
) is a weak solution to System (3.1). So we just have to show that T has a fixed point in H.

Step 2 : We show that T (H) ⊂ H. According to (3.9) and since pσ < m * , then |∇u| p ∈ L σ (Ω). Hence by using Theorem 2.4, we obtain that for all θ ≤ σ * = σN N -σ , (3.12)

||∇v|| L θ (Ω) ≤ C 5 |∇u| p + µg L σ (Ω) ≤ C 6 ||∇u|| p L pσ (Ω) + µ||g|| L σ (Ω) .
Using again pσ < m * , qm < r and (3.9), we get

||∇v|| L θ (Ω) ≤ C 7 ||∇ϕ|| pq L r (Ω) + µ||g|| L σ (Ω) + λ p ||f || p L m (Ω) .
Since r < σ * , by choosing θ = r in the previous inequality it holds that

||∇v|| L r (Ω) ≤ C 8 ||∇ϕ|| pq L r (Ω) + µ||g|| L σ (Ω) + λ p ||f || p L m (Ω) .
Recall that ϕ ∈ H, thus

||∇v|| L r (Ω) ≤ C 9 + µ||g|| L σ (Ω) + λ p ||f || p L m (Ω) .
By choosing C = C 9 and taking into consideration the definition of , we conclude that ||∇v|| L r (Ω) ≤ 1 pq . Thus v ∈ H and then T (H) ⊂ H.

Step 3 : T is a continuous and compact operator on H endowed with the topology of W 1,1 0 (Ω). • We begin by proving the continuity of T . Let

{ϕ n } n ⊂ H, ϕ ∈ H be such that ϕ n → ϕ in W 1,1 0 (Ω) and define v n = T (ϕ n ), v = T (ϕ). Then (u n , v n ) and (u, v) satisfy (3.13)    -∆u n = |∇ϕ n | q + λf in Ω, -∆u = |∇ϕ| q + λf in Ω, u n = u = 0 on ∂Ω, and
(3.14)    -∆v n = |∇u n | p + µg in Ω, -∆v = |∇u| p + µg in Ω, v n = v = 0 on ∂Ω.
Thanks to Theorem 2.3, for all s < N N -1 ,

∇u n -∇u L s (Ω) ≤ C |∇ϕ n | q -|∇ϕ| q L 1 (Ω)
.

Recall that ϕ n → ϕ in W 1,1 0 (Ω) and {ϕ n } n ⊂ H, thus ||∇ϕ n || L r (Ω) ≤ 1
pq for all n. Hence for a ∈ (1, r) fixed, setting θ = r-a a(r-1) and using interpolation inequality (2.1), we obtain

∇ϕ n -∇ϕ L a (Ω) ≤ ∇ϕ n -∇ϕ θ L 1 (Ω) ∇ϕ n -∇ϕ 1-θ L r (Ω) ≤ C ∇ϕ n -∇ϕ θ L 1 (Ω) → 0 as n → ∞.
Thus ∇ϕ n → ∇ϕ strongly in (L a (Ω)) N for all a < r. Since r > qm, we can choose a = q to conclude that

|∇ϕ n | q -|∇ϕ| q L 1 (Ω) → 0 as n → ∞.
Hence, for all s < N N -1 , ∇u n -∇u L s (Ω) → 0 as n → ∞.

In particular u n → u strongly in W 1,1 0 (Ω). Now going back to (3.9), we deduce (3.15) ∇u n L pσ (Ω) ≤ C 10

1 p + λ f L m (Ω) .
Thus {u n } n is bounded in W 1,pσ 0 (Ω). Since p < pσ, we again apply Proposition 2.1 to obtain

∇u n -∇u L p (Ω) → 0 as n → ∞.
Hence, we deduce from Theorem 2.3 that v n → v strongly in W 1,1 0 (Ω) and then the continuity of T follows.

• Now, we prove the compactness of T . Let

{ϕ n } n ⊂ H be such that ||ϕ n || W 1,1 0 (Ω) ≤ C 11 and define v n = T (ϕ n ). Since {ϕ n } n ⊂ H, then ||∇ϕ n || L r (Ω) ≤ C 12 . So, |∇ϕ n | q + λf is bounded in L 1 (Ω)
, then according to Theorem 2.3, we deduce that, up to a subsequence, u n → u strongly in W 1,s 0 (Ω) for all s < N N -1 . On the other hand, ||∇u n || L pσ (Ω) ≤ C 13 for all n. Hence, by using again Proposition 2.1, we conclude that u n → u strongly in W 1,a 0 (Ω) for all a < pσ. In particular, |∇u n | p + µg → |∇u| p + µg strongly in L 1 (Ω). Thus, by setting v as the unique solution to the problem

-∆v = |∇u| p + µg in Ω, v = 0 on ∂Ω,
we deduce from Theorem 2.3 that v n → v strongly in W 1,a 0 (Ω) for all a < N N -1 . Hence we conclude.

Therefore using Schauder fixed-point Theorem, we get the existence of v ∈ H such that

T (v) = v. Thus (u, v) is a solution to System (3.1). Moreover, (u, v) ∈ W 1,θ1 0 (Ω) × W 1,θ2 0 (Ω) for all θ 1 < mN N -m
and θ 2 < σN N -σ . Remark 1. Using the same arguments as above and under the hypotheses of Theorem 3.1, we can treat the more general system

(3.16)        -∆u = α|∇v| q + λf in Ω, -∆v = β|∇u| p + µg in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω,
where α, β ∈ {-1, 1}. Existence is proved under suitable smallness condition on λ and µ. Notice that for α = β = -1, we obtain a generalization of the well-known system appearing in mean-field game theory. See for instance [START_REF] Boccardo | Strongly coupled elliptic equations related to mean-field games systems[END_REF] and the references therein.

It is clear that the above concavity argument does not hold if p = q = 1. In this case, we have to take into consideration the loss of homogeneity ; then, we will consider the following modified system (3.17)

       u -∆u = |∇v| + f in Ω, v -∆v = |∇u| + g in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω.
Hence we have the next existence result.

Theorem 3.2. Let N ≥ 2. Assume that f, g are nonnegative functions such that (f, g) ∈ L 2N N +2 (Ω) × L 2N N +2 (Ω). Then, System (3.17) has a nonnegative solution (u, v) such that u, v ∈ W 1,2
0 (Ω). Proof. We proceed by approximation. By [START_REF] Leray | Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder[END_REF], the following system

(3.18)                    u n -∆u n = |∇v n | 1 + 1 n |∇v n | + f 1 + 1 n f in Ω, v n -∆v n = |∇u n | 1 + 1 n |∇u n | + g 1 + 1 n g in Ω, u n = v n = 0 on ∂Ω, u n , v n ≥ 0 in Ω, has a nonnegative solution (u n , v n ) ∈ (W 1,2 0 (Ω)) 2 .
Using u n as a test function in the first equation, v n as a test function in the second equation and applying Young's inequality, we obtain (3.19)

Ω u 2 n dx+ Ω |∇u n | 2 dx ≤ Ω |∇v n |u n dx+ Ω f u n dx ≤ Ω |∇v n |u n dx+ε Ω |∇u n | 2 dx+C(ε)||f || 2 L 2N N +2 (Ω)
, and (3.20)

Ω v 2 n dx+ Ω |∇v n | 2 dx ≤ Ω |∇u n |v n dx+ Ω gv n dx ≤ Ω |∇v n |u n dx+ε Ω |∇u n | 2 dx+C(ε)||g|| 2 L 2N N +2 (Ω)
, where ε > 0 will be chosen later.

Using again Young's inequality, with a particular choice of the constant, we get

Ω |∇v n |u n dx ≤ Ω u 2 n dx + 1 4 Ω |∇v n | 2 dx and Ω |∇u n |v n dx ≤ Ω v 2 n dx + 1 4 Ω |∇u n | 2 dx.
Now, going back to (3.19) and (3.20), we obtain

(1 -ε) Ω |∇u n | 2 dx ≤ 1 4 Ω |∇v n | 2 dx + C(ε)||f || 2 L 2N N +2 (Ω)
, and

(1 -ε)

Ω |∇v n | 2 dx ≤ 1 4 Ω |∇u n | 2 dx + C(ε)||g|| 2 L 2N N +2 (Ω)
.

Thus (3.21) (1 -ε) Ω |∇u n | 2 dx ≤ 1 16(1 -ε) Ω |∇u n | 2 dx + C(ε) ||f || 2 L 2N N +2 (Ω) + ||g|| 2 L 2N N +2 (Ω) . Choosing ε small enough such that (1 -ε) > 1 16(1 -ε) , it follows (3.22) Ω |∇u n | 2 dx ≤ C(ε) ||f || 2 L 2N N +2 (Ω) + ||g|| 2 L 2N N +2 (Ω)
.

In the same way we obtain that

Ω |∇v n | 2 dx ≤ C(ε) ||f || 2 L 2N N +2 (Ω) + ||g|| 2 L 2N N +2 (Ω)
.

Hence setting

D n (x) := |∇v n | 1 + 1 n |∇v n | + f 1 + 1 n f and Dn (x) := |∇u n | 1 + 1 n |∇u n | + g 1 + 1 n g , it holds that {D n } n , { Dn } n are bounded in L 2N N +2 (Ω)
. By the compactness result of Theorem 2.3, we get that, up to a subsequence, u n → u and v n → v strongly in W 1,s 0 (Ω) for all s < N N -1 . Thus, (u, v) is a weak solution to System (3.17). From (3.21) and (3.22) and by Fatou's Lemma, we conclude that u, v ∈ W 1,2 0 (Ω) and then the proof is finished.

Remark 2. It would be interesting to consider the case m = σ = 1. However, it seems complicated to adapt the argument based on the choice of test functions.

Non-existence results

In this section, we show that the size of λ, µ and regularity conditions on the data found in the previous section are optimal in order to get the existence results. Let us begin with the following result.

Theorem 4.1. Let p, q > 1. Assume that f, g are nonnegative functions such that (f, g) = (0, 0) and (f, g) ∈ L m (Ω) × L σ (Ω) where m, σ satisfy the conditions of Theorem 3.1. For φ ∈ C ∞ 0 (Ω) with φ ≥ 0, we define

F (φ) = Ω φ 1-q |∇φ| q dx + Ω φ 1-p |∇φ| p dx.
Let (λ, µ) ∈ (0, +∞) 2 be such that System (3.1) has a nonnegative solution (u, v). Then there exists a positive constant C(p, q) (depending only) on p and q such that λ ≤ Σ(f ) = C(p, q)λ * (f ) and µ ≤ Σ(g) = C(p, q)µ * (g) where Thanks to Young's inequality, we get

λ * (f ) := inf F (φ) ; 0 ≤ φ ∈ C ∞ 0 (Ω) and
Ω (|∇v| q + λf ) φ dx ≤ Ω |∇φ||∇u|φ 1/p φ -1/p dx ≤ Ω φ|∇u| p dx + C(p) Ω φ 1-p |∇φ| p dx.
In the same way, we obtain

Ω (|∇u| p + µg) φ dx ≤ Ω φ|∇v| q dx + C(q) Ω φ 1-q |∇φ| q dx.
Thus, combining the previous inequalities gives (4.1)

Ω (|∇v| q + λf ) φ dx ≤ Ω φ|∇v| q dx + C(q) Ω φ 1-q |∇φ| p dx + C(p) Ω φ 1-p |∇φ| p dx. Hence (4.2) λ Ω f φ dx ≤ C(q) Ω φ 1-q |∇φ| q dx + C(p) Ω φ 1-p |∇φ| p dx.
Setting

F (φ) := Ω φ 1-q |∇φ| q dx + Ω φ 1-p |∇φ| p dx, then λ ≤ Σ(f ) = C(p, q)λ * (f ) where (4.3) λ * (f ) := inf F (φ) ; 0 ≤ φ ∈ C ∞ 0 (Ω)
and

Ω f φ dx = 1 .
Conversely, if λ > Σ(f ), System (3.1) does not have any nonnegative solution.

In a similar way, we obtain that µ ≤ Σ(g) = C(p, q)µ * (g) with

µ * (g) := inf F (φ) ; 0 ≤ φ ∈ C ∞ 0 (Ω)
and

Ω gφ dx = 1 .
Therefore, if µ > Σ(g), System (3.1) does not have any nonnegative solution.

To complete our proof, we just have to show that λ * (f ), µ * (g) > 0. Notice that for φ ∈ C ∞ 0 (Ω) with φ 0, we have

F (φ) = (p ) p Ω |∇φ 1 p | p dx + (q ) q Ω |∇φ 1 q | q dx.
Hence, using Sobolev's inequality to obtain

F (φ) ≥ C 16 Ω φ N N -p dx N -p N + Ω φ N N -q dx N -q N
. Now, according to the values of m and σ, we will distinguish two cases m > σ and m ≤ σ.

• Case σ < m. According to condition (3.2) in Theorem 3.1, we have qm < σ * . Thus qm < m * and then m < N N -q . By using Hölder and Sobolev inequalities, we get

Ω f φ dx ≤ ||f || L m (Ω) ||φ|| L m (Ω) ≤ C 17 ||f || L m (Ω) Ω φ N N -q dx N -q N ≤ C 18 ||f || L m (Ω) Ω |∇φ 1 q | q dx ≤ C 19 ||f || L m (Ω) F (φ).
• Case σ ≥ m. In this case and since pσ < m * , it holds that m < N N -p . Thus

Ω f φ dx ≤ ||f || L m (Ω) ||φ|| L m (Ω) ≤ C 20 ||f || L m (Ω) Ω |∇φ 1 p | p dx ≤ C 21 ||f || m F (φ).
In the two cases, we obtain that

F (φ) Ω f φ ≥ C 22 ||f || L m (Ω) and then λ * (f ) ≥ C 22 ||f || L m (Ω) > 0.
By the same argument, we show that F (φ)

Ω gφ ≥ C 23 ||g|| L σ (Ω)
, thus µ * (g) > 0.

To assess the optimality of the conditions on f and g, we prove the following Theorem.

Theorem 4.2. Let p, q > 1. Assume that f, g are positive functions such that (f, g)

∈ L m (Ω) × L σ (Ω) where m, σ ≥ 1. Define Λ(f ) := inf φ∈C ∞ 0 (Ω), φ 0 Ω |∇φ| p dx + Ω |∇φ| q |φ| p -q dx Ω f |φ| p dx , and
Γ(g) := inf φ∈C ∞ 0 (Ω), φ 0 Ω |∇φ| q dx + Ω |∇φ| p |φ| q -p dx Ω g|φ| q dx .
Assume that System (3.1) has a nonnegative solution for some λ, µ > 0. Then Λ(f ), Γ(g) > 0.

Proof. We follow closely the arguments used in [START_REF] Baras | Critère d'existence des solutions positives pour des équations semi-lineaires non monotones[END_REF]. Let (u, v) be a nonnegative solution to (3.1). Let φ ∈ C ∞ 0 (Ω) such that φ ≥ 0. Using |φ| p as a test function in the equation for v, it follows that (

Ω ∇v∇|φ| p dx = p Ω ∇v∇φ|φ| p -2 φ dx = Ω |∇u| p |φ| p dx + µ Ω g|φ| p dx. 4.4)
Using Young's inequality we get

p Ω ∇v∇φ|φ| p -2 φ dx ≤ p Ω |∇v| |∇φ| |φ| |φ| p dx ≤ ε Ω |∇v| q |φ| p dx + C(ε) Ω |∇φ| q |φ| p -q dx. Thus (4.5) Ω |∇u| p ||φ| p dx + µ Ω g|φ| p dx ≤ ε Ω |∇v| q |φ| p dx + C(ε) Ω |∇φ| q |φ| p -q dx.
On the other hand, taking |φ| p as a test function in the equation in u, it holds (4.6)

Ω ∇u∇|φ| p dx = p Ω ∇u∇φ|φ| p -2 φ dx = Ω |∇v| q |φ| p dx + λ Ω f |φ| p dx.
As above, using again Young's inequality to obtain

p Ω ∇u∇φ|φ| p -2 φ dx ≤ ε Ω |∇u| p |φ| p dx + C(ε) Ω |∇φ| p dx. Hence (4.7) Ω |∇v| q |φ| p dx + λ Ω f |φ| p dx ≤ ε Ω |∇u| p |φ| p dx + C(ε) Ω |∇φ| p dx.
Now combining (4.5), (4.7) and choosing ε small enough, we obtain :

(1 -ε 2 ) Ω |∇v| q |φ| p dx + λ Ω f |φ| p dx ≤ C(ε) Ω |∇φ| q |φ| p -q dx + C(ε) Ω |∇φ| p dx. So, for ε < 1, λ Ω f |φ| p dx ≤ C (ε) Ω |∇φ| q |φ| p -q dx + C(ε) Ω |∇φ| p dx,
and then Λ(f ) > 0. Now, using |φ| q as a test function and following the same computations as above, we obtain :

(4.8) µ Ω g|φ| q dx ≤ C (ε) Ω |∇φ| p |φ| q -p dx + C(ε) Ω |∇φ| q dx.
Then the result follows.

Remark 3.

(1) The result of Theorem 4.2 means that we have the same kind of conditions as in the case of one equation. See condition

(1.3) in Theorem 1.1. (2) For φ 0, setting Q(φ) := Ω |∇φ| p dx + Ω |∇φ| q |φ| p -q dx Ω f |φ| p dx , L(φ) := Ω |∇φ| q dx + Ω |∇φ| p |φ| q -p dx Ω g|φ| q dx .
Then, Q(ηφ) = Q(φ) and L(ηφ) = L(φ) for all η = 0.

(3) Notice that Σ(f ) Λ(f ). This follows from the fact that if we set ϕ = |φ| p -1 φ, then

Ω |∇φ| p dx + Ω |∇φ| q |φ| p -q dx Ω |ϕ| 1-p |∇ϕ| p dx + Ω |ϕ| 1-q |∇ϕ| q dx = F (ϕ).
In the same way, we have Σ(g) Γ(g).

We are now able to prove the next non-existence result.

Theorem 4.3. Suppose that p, q > 1 are such that qm > σ * and pσ > m * . Then, there exist f ∈ L m (Ω) and g ∈ L σ (Ω) with f, g 0 such that System (3.1) does not have any nonnegative solution for any λ, µ > 0.

Proof. To get the non-existence result, we will construct f, g satisfying the hypotheses of the Theorem and such that Λ(f ) = 0 or Γ(g) = 0. Without loss of generality, we assume that m, σ < N . In what follows, we will assume that Ω = B 1 (0). For technical reasons, we distinguish two cases. Case 1 : m ≥ σ and p ≥ q . Using Hölder's inequality, it holds that

Ω |∇φ| q |φ| p -q dx ≤ Ω |∇φ| p dx q p Ω |φ| p dx p -q p .
According to Poincaré's inequality, we obtain that

Ω |∇φ| q |φ| p -q dx ≤ C P Ω |∇φ| p dx.
Thus

Ω |∇φ| p dx + Ω |∇φ| q φ p -q dx Ω |∇φ| p dx.
Then

Q(φ) Ω |∇φ| p dx Ω f |φ| p dx . Notice that pm ≥ pσ > m * , then p > N N -m
. Thus, we can fix α > 0 such that p < α < N m . Define f (x) = 1 |x| α , then f ∈ L m (Ω). Now, introduce the function φ defined by Hence, using a suitable approximation argument, we conclude that Λ(f ) = 0 and the non-existence result follows in this case.

φ(x) =    1 |x| θ if |x| ≤ 1 4 (1 -|x|) γ if 1 2 ≤ |x| ≤ 1,
Case 2 : m ≥ σ and p < q . In the same way as above, we obtain qσ > m * > σ * . Thus q > N N -σ , hence q < N σ . As in the proof of the first case, since p < q , by Hölder's inequality, we obtain

Ω |∇φ| q dx + Ω |∇φ| p φ q -p dx Ω |∇φ| q dx. Thus L(φ) Ω |∇φ| q dx Ω g|φ| q dx . Setting g(x) = 1
|x| β where q < β < N σ , then g ∈ L σ (Ω). Choosing φ as in the first case with N -β q < θ < N -q q , then Ω |∇φ| p dx < +∞ and

Ω g|φ| q dx Ω 1 |x| q θ+β dx = +∞.
Hence Γ(g) = 0 and then we conclude.

In a similar way, we can treat the case m < σ. Hence the result follows.

A system with coupled gradient-potential terms

This section is devoted to the slightly different system given as :

(5.1)

       -∆u = u|∇v| q + λf in Ω, -∆v = v|∇u| p + µg in Ω, u = v = 0 on ∂Ω, u, v ≥ 0 in Ω.
In this case, we have the following existence result.

Theorem 5.1. Let p, q ≥ 1 and m, σ ≥ 1. Assume that f, g are nonnegative functions such that (f, g) ∈ L m (Ω) × L σ (Ω) where m, σ satisfy (5.2) m > pN p + 2 , and σ > qN q + 2 .

Then, there exists A * > 0 such that, if λ + µ < A * , System (5.1) has a nonnegative solution

(u, v) ∈ W 1,α 0 (Ω) × W 1,β 0 (Ω) for all α < N m (N -m) + and β < N σ (N -σ) + .
Proof. We will give the proof in the case 1 ≤ m, σ < N . The other cases follow using similar apprach.

As in the proof of Theorem 3.1, we define the function

Υ 1 (s) := s -C(s 1+q + s 1+p ),
with C being a positive constant depending only on the data. Since p, q ≥ 1, we deduce the existence of a unique value such that

max s≥0 Υ 1 (s) = Υ 1 ( ) = Λ * . Thus, = C( 1+q + 1+p + Λ * C ). (5.3) As > 0, there exists A * > 0 such that, if λ + µ < A * , then (5.4) λ||f || L m (Ω) + µ||g|| L σ (Ω) ≤ Λ * C . Hence (5.5) C 1+q + 1+p + λ||f || L m (Ω) + µ||g|| L σ (Ω) ≤ . Now, we define the set H r,θ ⊂ W 1,1 0 (Ω) × W 1,1 0 (Ω) as : H r,θ = (ϕ, ψ) ∈ W 1,r 0 (Ω) × W 1,θ 0 (Ω) and ||ϕ|| W 1,r 0 (Ω) + ||ψ|| W 1,θ 0 (Ω) ≤
, where r and θ are chosen such that (5.6)

         pσN θ θ(N + σ) -N σ < r < mN N -m , qmN r r(N + m) -N m < θ < N σ N -σ .
Notice that, the existence of (r, θ) follows from (5.2). As above, H r,θ is a closed convex subset of

W 1,1 0 (Ω) × W 1,1 0 (Ω). We claim that if (ϕ, ψ) ∈ H r,θ (Ω), then (5.7) ϕ + |∇ψ| q L m (Ω) ≤ C 24 ||ϕ|| W 1,r 0 (Ω) × ||ψ|| q W 1,r 0 (Ω)
. Indeed, by definition:

ϕ + |∇ψ| q m L m (Ω) = Ω ϕ m + |∇ψ| qm dx ≤ Ω |ϕ| m |∇ψ| qm dx.
Hence, using the Sobolev and Hölder inequalities, it follows that

ϕ + |∇ψ| q m L m (
ϕ + |∇ψ| q m L m (Ω) ≤ C 26 ||ϕ|| m W 1,r 0 (Ω) × ||ψ|| qm W 1,θ 0 (Ω)
, and the claim (5.7) follows.

In the same way, using pσ N θ θ(N + σ) -σN < r, we obtain (5.9)

ψ + |∇ϕ| p L σ (Ω) ≤ C 27 ||ψ|| W 1,θ 0 (Ω) × ||ϕ|| p W 1,r 0 (Ω)
. Therefore, we can define u and v as the unique weak solutions to the problems (5.10)

-∆u = ϕ + |∇ψ| q + λf in Ω, u = 0 on ∂Ω , and
(5.11)

-∆v = ψ + |∇ϕ| p + µg in Ω, v = 0 on ∂Ω.
According to Theorem 2.4 and estimates (5.7), (5.9), it holds that (5.12)

||∇u|| L m * (Ω) ≤ C 28 ||ϕ|| W 1,r 0 (Ω) × ||ψ|| q W 1,θ 0 (Ω) + λ||f || L m (Ω) , and
(5.13) ||∇v|| L σ * (Ω) ≤ C 29 ||ψ|| W 1,θ 0 (Ω) × ||ϕ|| p W 1,r 0 (Ω) + µ||g|| L σ (Ω) . So the operator T : H r,θ -→ W 1,1 0 (Ω) × W 1,1 0 (Ω) (ϕ, ψ) -→ L(ϕ, ψ) = (u, v)
is well defined. Moreover, any fixed point of T in H r,θ is a solution to System (5.1). Since r < m * and θ < σ * , then (u, v) ∈ W 1,r 0 (Ω) × W 1,θ 0 (Ω) and By choosing C = C 30 , we deduce from (5.5) that ||u|| W 1,r 0 (Ω) + ||v|| W 1,r 0 (Ω) ≤ So we conclude that T (H r,θ ) ⊂ H r,θ .

||u|| W 1,r 0 (Ω) + ||v|| W 1,r 0 (Ω) ≤ C 30 ||ϕ|| W 1,r 0 (Ω) ||ψ|| q W 1,θ 0 (Ω) + ||ψ|| W 1,θ 0 (Ω) ||ϕ|| p
The continuity and compactness of T follow as in the proof of Theorem 3.1. We conclude thanks to the Schauder fixed-point Theorem.

Remark 4. Let us review some particular, explicit cases, where the existence conditions hold.

• If m, σ > N , the existence result holds for all p, q > 1.

• If p = q = 1 and m = σ, a sufficient condition on m is that m = σ > N 3 .

• If p = q = 2 and m = σ, condition (5.2) implies that m = σ > N 2 .

• If m = σ = 2, condition (5.2) implies that

p < 4 N -2 , q < 4 N -2 .
which imply p, q < 1, if N ≥ 6. In this case, Theorem 5.1 fails.

To show that condition (5.2) is in some sense optimal, we state and prove the following nonexistence result. Proposition 5.2. Assume that m = σ = 2. For any p, q > 1, we can find f, g ∈ L 2 (Ω) and n 0 ∈ IN such that if N > n 0 , System (5.1) does not have any nonnegative solution.

Proof. Without loss of generality, we assume that 0 ∈ Ω. Set f (x) = g(x) = 1 |x| α where 2 < α < N 2

. If (u, v) is a solution to System (5.1), then

-∆u ≥ λ |x| α , -∆v ≥ µ |x| α in Ω.
Then, using a suitable comparison principle it holds that u(x), v(x) ≥ C 31 |x| α-2 in B R (0) ⊂⊂ Ω, for a suitable R. 1 |x| q * (α-2)+ q * q (α-2) dx ≤ C(Ω).

Thus

p * (α -2) + p * p (α -2) < N and q * (α -2) + q * q (α -2) < N.

For p, q > 1 fixed, choosing N > n 0 := max{ 2p+4 p-1 , 2q+4 q-1 }, we get the existence of α ∈ (2, N 2 ) such that p * (α -2) + p * p (α -2) ≥ N and q * (α -2) + q * q (α -2) ≥ N, and we get a contradiction with the existence hypothesis.

Acknowledgments :

• The authors would like to thank -the referee for the very careful reading ; -Prof. Boumediene Abdellaoui for his helpful suggestions and fruitful discussions during the preparation of this work.

• Part of this work was realized while the first author was visiting the Institut Elie Cartan, Université de Lorraine. He would like to thank the Institute for its warm hospitality.

Ω f φ dx = 1 ,

 1 and µ * (g) := inf F (φ) ; 0 ≤ φ ∈ C ∞ 0 (Ω) and Ω gφ dx = 1 . Proof. Fix λ, µ > 0 be such that System (3.1) has a nonnegative solution (u, v). Without loss of generality, we can assume that m, σ < N . Let φ ∈ C ∞ 0 (Ω) such that φ 0. Using φ as a test function in System (3.1), we obtain Ω (|∇v| q + λf ) φ dx = Ω ∇u∇φ dx and Ω (|∇u| p + µg) φ dx = Ω ∇v∇φ dx.

where N -α p < θ < N -p p and γ > 1

 1 and such that φ ∈ C 1 (B 1 (0) \ {0}) and φ > 0 in B 1 (0). Since θ < N -p p , then Ω |∇φ| p dx < +∞. However Ω f |φ| p dx Ω 1 |x| p θ+α dx = +∞.

8 )

 8 On the other hand, qm r * r * -m = qm rN r(N + m) -mN . Hence, by the definition of r and θ, it holds that qm r * r * -m < θ. Finally,

W 1,r 0

 0 (Ω) + λ||f || L m (Ω) + µ||g|| L σ (Ω) ≤ C 30 1+q + 1+p + λ||f || L m (Ω) + µ||g|| L σ (Ω) .

Ω

  |x| -(α-2) |∇u| p dx + Ω |x| -(α-2) |∇v| q dx < ∞.According to Caffarelli-Kohn-Nirenberg inequality ([START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF]), we obtain Ω into account the behavior of u, v near the origin, we find :B R (0) 1 |x| p * (α-2)+ p * p (α-2) dx + B R (0)