Jean-Marc Collin
email: jeanmarc.collin@yahoo.fr

Gérard Mauvoisin

Philippe Pilvin

Rochdi El

Use of spherical indentation data changes to materials characterization based on a new multiple cyclic loading protocol

Keywords: Spherical indentation, Materials characterization, Multiple cyclic loading

An interpretation of spherical indentation experiments and the determination of hardening law parameters is the challenge in this paper. A new protocol based on multiple cyclic loading allowed models describing the indentation data changes to be determined from a numerical study. Inversion of these models, determined for two constitutive equations with isotropic hardening, combined with a sensitivity study enabled the study of the uniqueness of the solution. It is shown that although the new protocol is not necessary to determine the two Hollomon equation parameters, it is necessary to determine accurate values of the three Ludwig equation parameters.

From an experimental study, it was shown that methods based on a "reverse" analysis can be used in order to have an evaluation of the mechanical properties in a first-order approximation.

Introduction

An instrumented indentation test consists in measuring, simultaneously, the indentation load P and the indentation depth h during the penetration of an indenter into a sample. This test can be used in order to determine some parameters of the work-hardening behavior law of the tested sample. The more commonly used is the Hollomon isotropic work-hardening law defined by the following equation:

σ = σ 1-n y E n ε n (1)
This relationship gives, in the case of a monotonic solicitation in the elasto-plastic regime, the material flow stress. E, σ y , n and ε, are, respectively the Young modulus, the yield stress, the work-hardening exponent and the strain. The first identified parameter, from an indentation test, is the reduced Young modulus E* defined by

E * = [(1 -ν 2 i )/E i + (1 -ν 2 s )/E s ] -1
, where ν is the Poisson ratio, s stands for "sample," whereas i stands for "indenter". The determination of E* is based on the Hertz elastic contacts theory [1].

Sneddon [2] proposed a general solution to the problem of an elastic sample indented by any shape of indenter. In the case of spherical indentation, the contact stiffness is defined as follows:

S = dF dh = 2E * a ( 2
)
where a is the contact radius. When the elastic regime is over, Eq.

(2) can still be applied at the beginning of the unloading [3][4][5]. Following this, Doerner and Nix [6], Loubet et al. [START_REF] Loubet | Vickers in dentation curves of elastoplastic materials[END_REF] and Oliver and Pharr [START_REF] Oliver | [END_REF] proposed to deduce E* from an indentation test using relationship (2). These methods can be distinguished from one another by the way the contact radius is evaluated. Lastly, Hay and Wolff [9] proposed a correction for the application of the Hertz theory by introducing a factor in order to take into account the radial displacements of material under the indenter.

Concerning the non-elastic behavior, many methods have been proposed to deduce the mechanical parameters, usually deduced from a tensile test, from an indentation test. These methods can be distinguished from one another by the indenter they use (Vickers, sharp, spherical, etc.) and the data they use. Moreover, two kinds of methods can be used to extract mechanical properties from the indentation test. The first is based on the inversion of models established from a numerical study and is called a "reverse" analysis. The second is based on an "inverse" analysis. The greater part of established methods are based on a "reverse" analysis, however, Nakamura et al. [10] proposed the determination of the properties of graded materials by using an "inverse" analysis protocol in spherical indentation. We can also quote Bolzon et al. [11] who applied "inverse" analysis to conical indentation. Most of the "reverse" analysis methods are based on the modelling of the representative strain introduced by Tabor [START_REF] Tabor | Hardness of Metals[END_REF]. Dao et al. [START_REF] Dao | [END_REF] proposed to extract the Eq. ( 1) parameters from a unique sharp indentation. They also showed that the determined parameters are very sensitive to small variations in the indentation data. This work has been followed by Chollacoop et al. [14] who applied the previous method with two sharp indenters with different apex angles. Then, they showed an improvement in the determination of the material properties with two indenters. Moreover, the use of two sharp indenters significantly decreased the sensitivity of the predicted parameters to the perturbations in the indentation data. Bucaille et al. [15] also proposed using different sharp indenters in order to solve the problem.

The uniqueness of the solution has been fully studied by Cheng and Cheng [16], who demonstrated that in the case of sharp indentation, several sets of Eq. ( 1) parameters can lead to the same P(h) curve. However, they demonstrated that this problem does not occur in spherical indentation. Indeed, the method of Dao et al. [START_REF] Dao | [END_REF] has been extended to the spherical indentation by Cao and Lu [17] and led to a unique determination of the Eq. ( 1) parameters. We can also quote Beghini et al. [18] who determined Eq. ( 1) parameters from the inversion of a P(h) curve model in spherical indentation.

This paper deals with a study of the uniqueness of the solution in the case of spherical indentation. Moreover, we propose an experimental method based on a "reverse" analysis to evaluate the parameters of two constitutive equations by using a unique indenter and a new multiple cyclic loading protocol.

Data deduced from an indentation test

Depending on the kind of loading, different data can be deduced from an indentation test (see Fig. 1, the curves of which are deduced from numerical results). If the test consists of a unique loading, the P(h) curve and the total energy W t (h) changes can be deduced. If the test consists of a loading and an unloading cycle, in addition to P(h) and W t (h), both the elastic W e (h max ) and plastic W p (h max ) energies at the end of the loading can be deduced. Moreover, we can deduce the contact stiffness S(h max ) (defined in Eq. ( 2)) at the maximal load. Finally, if the test consists of n loading, unloading and reloading cycles, in addition to the previous data we can deduce n points of W e (h (max)i ), W p (h (max)i ) and S(h (max)i ) changes. Moreover, from the energies, we can deduce the energy ratio changes as W p /W t (h (max)i ), W e /W t (h (max)i ) and W e /W p (h (max)i ).

Among the three energies, only two are independent. Indeed, they are linked by Moreover, only one energy ratio is sufficient in order to deduce the two others. Indeed, they are linked by

W p = W t -W e (3)
W p W t = 1 - W e W t = 1 + W e W p -1 (4)
This is the reason why, for the three energies, we propose to study both the elastic and total energies. Moreover, although the W p /W t ratio is linked to the two studied energies, we propose to study it in order to limit the errors of modelling. Concerning the contact stiffness S, because it is obtained by derivating the P(h) curve, it is very sensitive to oscillations of the P(h) curve. For this reason this data is not studied in the following numerical study. Thus, this study concerns the P(h), W e , W t and W p /W t changes during a spherical indentation test which are deduced from 10 loading, unloading and reloading cycles.

Numerical study

Presentation

A numerical study was conducted in axi-symetric mode with the finite element (FE) code Cast3M. Fig. 2 shows the FE model where the axi-symetric boundary conditions are imposed on the axi-symetric axis.

The sample is modelled by a bulk divided into several areas. In the contact area, the mesh is made of 8-node elements (named QUA8 in the FE code) with quadratic interpolation. In this area, the size of the elements is less than 4 m in order to obtain P(h) curves with the lowest oscillations. The rest of the sample is Concerning the indenter, it is modelled by a quarter of a sphere divided into two areas. In the contact area, as for the sample, the mesh is made of 8-node elements with quadratic interpolation. The rest of the indenter is modelled by an area composed of 3-node elements with linear interpolation.

Because the contact is made between the superior line of the sample and the inferior line of the indenter, the middle nodes of the QUA8 elements are free. In order to solve this problem, two new lines have been created. They are composed of segment elements with linear interpolation. For n QUA8 elements, we have 2n -1 nodes in these new lines. Although the friction influence is not negligible, the contact is made without friction between the two new lines in order to focus the study to the plasticity phenomenon.

The sample is described by elasto-plastic behavior with isotropic work hardening. Two constitutive equations have been studied, the Hollomon one (Eq. ( 1)), and the Ludwig one (Eq. ( 5)):

σ = σ y + Kε m (5)
where K and m describe the work-hardening behavior. It is important to note that Eq. (1) depends on the total strain whereas Eq. ( 5) depends on the plastic strain only. In this study, the sample's Young modulus is E s = 210 GPa and the Poisson ratio ν s = 0.3 (which corresponds to that of steel). The indenter is described by an elastic behavior with E i = 600 GPa and ν i = 0.3 (which corresponds to a tungsten carbide). The indenter radius is fixed to R = 0.5 mm.

In order to follow indentation data changes, 10 loading, unloading and reloading cycles with a maximal load equal to P max = 200 N have been programmed. All the nodes of the superior line of the indenter have the same displacement and the load is applied on the first left node of this line. The indenter Fig. 3. Typical data deduced from the numerical simulations. moves down into the sample with the bottom of the sample fixed. Figs. 1 and 3 present the typical P(h) curve obtained from the numerical study from which the loading P(h) curve, W e , W t and W p /W t changes are deduced.

Sensitivity study

The aim of this paragraph is to study the influence of a variation of each behavior law parameter to the indentation data. We can write the sample's behavior law as a function f (Eq. ( 6))of nx i behavior law parameters and the strain ε:

σ = f (x 1 ,...,x i ,...,x n ,ε) (6)
Thus, indentation data D can be written as a function g (Eq. ( 7))ofthenx i behavior law parameters and the indentation depth h: D = g(x 1 ,...,x i ,...,x n ,h)

If we apply a variation to one parameter x i which becomes x ′ i , the indentation data becomes D ′ and can be written as follows:

D ′ = g(x 1 ,...,x i ,...,x n ,h) (8)
We propose to quantify the sensitivity of indentation data D to the parameter x i (noted S(D/x i )) with the following function:

S(D/x i ) = h f h 0 (D ′ (h) -D(h)) dh h f h 0 (D(h) -D(h 0 )) dh (9)
where h 0 and h f are presented in Fig. 4. Function (9) has been chosen in order to have an average sensitivity which takes into account the initial value of the indentation data.

In the present work, we have studied the sensitivity of the indentation data to Eq. ( 5) parameters, i.e. the yield stress σ y and the two work-hardening parameters K and m. However, in order to make a graphic representation of the sensitivity changes possible, we distinguish the sample behavior into a yield stress part and a work-hardening part, we then use Eq. ( 1) notations. Thus, work hardening is represented by the coefficient n of the Eq. ( 1). In other words, the evolution of the sensitivity of indentation data to σ y , K or m will be presented as a surface which depends on the yield stress and the work hardening quantified by n. Twelve sets of parameters have been chosen in order to study the evolution of the sensitivity from a low to a high yield stress (σ y = 150 MPa to σ y = 1000 MPa) and from a low to high work hardening (quantified by n = 0.02 to n = 0.4).

For each set of parameters, we have determined σ y , K and m which minimize the difference between Eqs. ( 5) and ( 1) in order to have the same flow stress curve. For example, a Hollomon sample with σ y = 150 MPa and n = 0.02 corresponds to a Ludwig sample with σ y = 150 MPa, K = 24.53 MPa and m = 0.203. The indentation behavior of these 12 samples has been simulated, and the data deduced from these simulations are considered as the initial ones (D in Eq. ( 7)).

In order to verify the symmetry of the sensitivity, a variation of +5% and -5% has been applied on σ y , K and m for each sample (which leads to 72 simulations). We deduced from these simulations the D ′ data (in Eq. ( 8)) for a variation +5% and -5% of the Eq. ( 5) parameters. We can observe in Fig. 5 that the sensitivity is symmetric data. Thus, we will consider the sensitivity as the average of the absolute values of the sensitivity to +5% and -5%.

The results of the sensitivity study are presented in Fig. A1 (Appendix A). This is composed of four lines representing the sensitivity of the four indentation data W e , W t , W p /W t and P(h), respectively. The three columns present the sensitivity to the three parameters σ y , K and m. For example the first graph in the left top corner shows the sensitivity of the W e changes to the yield stress in function of the yield stress and the work hardening (quantified by n). In the following presentation, we discuss the global trend of the sensitivity changes to the yield stress and to the work-hardening parameters.

Concerning the evolution of the sensitivity to the yield stress (first column), W e , W t and P(h) present the same trend. Moreover, the sensitivities of W t and P(h) are quite similar. It can be explained by the relationship (10) between W t and P(h):

W t = h f h 0 P(h)dh (10)
The most important result of the study of the sensitivity to the yield stress is that it increases when the work hardening decreases. We can observe that the work hardening (quantified by n) has a more important influence than the yield stress on the sensitivity changes except for W p /W t .

Concerning the evolution of the sensitivity to the workhardening coefficients (second and third columns), we can Fig. 8. Several sets of Eq. ( 5) can lead to the same curve. observe that the sensitivity to K and m have the same trend. However, the sensitivity to m is higher than the sensitivity to K. The most important result of the study of the sensitivity to the work-hardening coefficients is that no indentation data is sensitive to the work hardening when this work hardening is very low.

We can conclude that the sensitivity can be divided into sensitivity to the yield stress and sensitivity to the work hardening. The maximum value of the sensitivity to the yield stress is compensated by the minimum value of the sensitivity to the work hardening. Moreover, from this study, W e seems to be the most sensitive data.

Determination of models for the four studied indentation data

The area of the materials studied in the previous paragraph has been extended by considering the large domain of σ y /E s ,in order to cover a large area of metallic materials. The yield stress varies between 150 and 3000 MPa (150, 450, 720, 1000, 1400, 2200 and 3000 MPa) and the work-hardening exponent of the Eq. ( 1) varies between 0.02 and 0.4 (0.02, 0.08, 0.15, 0.22, 0.28, 0.34 and 0.4). The determination of models is then based on 49 materials which have been simulated in order to determine W e , W t , P(h) and W p /W t changes The parameters of Eq. ( 5) corresponding to these 49 materials have been determined (as in the previous paragraph) in order to determine models which depend on the parameters of both Eqs. ( 1) and (5).

Moreover, in order to make the application of the models to different samples and indenters Young modulus and different indenter radii possible, all the data are determined as functions of σ y /E*, n and h/R for Eq. ( 1) and σ y /E*, K/E*, m and h/R for Eq. (5). In order to have dimensionless data, W e and W t are divided by E*R 3 and P(h) is divided by E*R 2 .

Fig. 6 shows that the evolution of -ln(W e /E*R 3 , W t /E*R 3 and P(h)/E*R 2 ) versus (-ln(h/R)) is well described by an affine function, as for -ln(W p /W t ) versus (-ln(h/R)) which is well described by an exponential function. We can also write that Fig. 10. Comparison between experimental and numerical curves for a C22 steel.

j = h R A exp(-B) (11)
k = exp -A h R -B (12)
where A and B are quantities depending on σ y /E* and n in the case of Eq. ( 1) and on σ y /E*, K/E* and m in the case of Eq. ( 5).

From the numerical study, a data base of 49 materials leads to 49 W e , W t , P(h) and W p /W t curves and then to 49 couples (A,B) for each indentation data. Concerning Hollomon's constitutive Eq. ( 1), it is possible to represent the evolution of A and B with surfaces which are presented in Fig. B1 (Appendix B). In this figure, functions A and B (z axis) are determined with the software Table Curve 3D. They depend on σ y /E* (x axis) and n (y axis). The proposed functions are the ones which give the best correlation coefficient. The coefficients which come in these functions are given in Tables B1 andB2.

Concerning Ludwig's constitutive equation, it is impossible to represent a surface of the evolution of A and B in function of σ y /E*, K/E* and m. These functions have been determined by a minimization with several polynomial functions. If we note x 1 = σ y /E*, x 2 = K/E* and x 3 = m, the best found functions are as follows:

A = a + 3 i=1 (a i x i + b i x 2 i ) + c 1 x 1 x 2 + c 2 x 2 x 3 + c 3 (x 1 x 2 ) 2 + c 4 (x 2 x 3 ) 2 (13)
B = A + 3 i=1 (d i x 3 i ) + d 4 x 1 x 3 + d 5 x 1 x 2 x 3 -1 (14)
The coefficients which come in these functions are given in Tables B3 andB4.

In order to study the accuracy of each model, the comparison has been made between the numerical curves and the prediction of the model for each indentation data (see Fig. 7 for an example). Concerning the models which depend on Eq. ( 1) parameters, the average accuracy is 2.5%, 1.0%, 1.6% and 1.5%, respectively for W e , W t , W p /W t and P(h). For Eq. ( 5), the average accuracy becomes, respectively 3.7%, 2.6%, 1.1% and 3.4%.

Study of the uniqueness of the solution

The study of the uniqueness of the solution is based on the minimization between the numerical curves and the proposed models for each indentation data and for the two studied constitutive equations. If we note σ num as a point of a numerical curve, σ mod as a point of the curve obtained from a model and N as the number of points, the minimization is commonly made on the following function:

N (σ mod -σ num ) 2 (15)
Although function (15) gives good results on W p /W t ,itgives very inaccurate results on W e , W t and P(h). Indeed, function (15) does not give the same weight to every point of the curve. Then, we suggest minimizing with the following formula:

N σ mod -σ num σ num 2 (16)
Concerning Hollomon's constitutive equation, formula (16) has been used to inverse the functions (11) and [START_REF] Tabor | Hardness of Metals[END_REF] for each indentation data. The results of these inversions, given in Table C1 (Appendix C), show that there is a uniqueness of the solution for each indentation data. We can observe that the lower the work hardening (n = 0.02 and 0.08), the lower the precision on the work-hardening exponent n. It can be easily explained with the results of the sensitivity study. Indeed, it has been shown that if the work hardening is low, the sensitivity of the four indentation data to the work hardening is close to 0. If we observe the results of the inversion without considering the lowest values of work-hardening exponent, the average error obtained, respectively on σ y and n is 3.9% and 4.3% with W e , 2.9% and 3.4% with W t , 5.0% and 3.8% with W p /W t and 5.8% and 6.5% with P(h). Although the sensitivity study has shown that the most The energy W t could be calculated by function (10). It would be then not be necessary to program several loading, unloading and reloading cycles in order to determine the W t changes. One single loading is sufficient to calculate the W t changes and then to deduce σ y and n from inversion. However, the cyclic loading protocol proposed in this paper is very important in the case of Eq. ( 5).

Indeed, the minimization between models determined for Eq. ( 5) and numerical curves for each data does not lead to a unique set of σ y , K and m. Fig. 8 and Table C2 illustrate this fact. Several sets of Eq. ( 5) parameters lead to the same curve. This shows that one unique indentation data is not sufficient to determine the three parameters of Eq. ( 5). It is then necessary to minimize (with function ( 16)) on several indentation data simultaneously. If we consider the four studied data, only two are independent, indeed W t and P(h) are linked with function (10) and W e , W t and W p /W t are linked by Eqs. ( 3) and ( 4). Although the minimization of both W t and W e leads to a unique set of Eq. ( 5) parameters, for some cases, the results are not very accurate. As an example, for an exact solution σ y = 2200 MPa, K = 6173 MPa and m = 0.46, the minimization with W e and W t simultaneously leads to the solution σ y = 2668 MPa, K = 4241 MPa and m = 0.44 and the minimization with W e , W t and W p /W t simultaneously leads to the solution σ y = 2340 MPa, K = 6343 MPa and m = 0.47. This is the reason why we suggest minimizing, simultaneously, on three indentation data. The model of W p /W t is more accurate than the P(h) one and this is the reason why the results presented in Table C3 (middle columns) concern the minimization on W e , W t and W p /W t simultaneously. It is then shown that the use of three indentation data leads to a unique set of accurate Eq. ( 5) parameters. However, on the one hand, it can be observed that the lower the work hardening, the lower the accuracy of K.On the other hand very good accuracy is obtained on coefficient m even if the work hardening is low. This can be explained by the form of Eq. ( 5). In this function, the yield stress σ y and the work hardening (noted σ wh ) are separated. This can be written as

σ wh = Kε m ( 17
)
If an error e occurs on the work-hardening evaluation, the new work-hardening function becomes

σ ′ wh = (1 + e)Kε m = K ′ ε m ′ (18)
Function (18) has to be solved for every ε, and the only one solution is K ′ =(1+e)K and m ′ = m. In other words, if an error occurs on the work-hardening evaluation, this error will have an effect exclusively on K and not on m. Then, if we calculate the average error without considering the lowest work hardening values, we obtain 8.2%, 6.5% and 2.1%, respectively for σ y , K and m.

One can suggest that only the last points of the curves can lead to a unique set of Eq. ( 5) parameters. This suggestion is true. However, this leads to very inaccurate values of Eq. ( 5) parameters (see Table C3, last columns).

As a conclusion, although the new cyclic loading protocol is not necessary to determine Eq. ( 1) parameters, it is necessary to deduce the W e and W p /W t changes and then to deduce accurate values of Eq. ( 5) parameters.

Experimental study

Presentation

Two steels were selected (a C100 and a C22) because of their microstructure which leads to good homogeneity. For these samples, a tensile test led to the flow stress-strain curve which was fitted by both Eqs. ( 1) and ( 5).

We developed an experimental bench which allows the measurement of the P(h) curve with very good reproducibility. The two important parts of this bench are the indenter and the way the displacement is measured.

Concerning the choice of the indenter, the most common spherical indenters are composed of a sphere which is crimped into the body of the indenter. However, the way the sphere is crimped can induce non-negligible deformations between the body of the indenter and the sphere. This is the reason why we chose to use a monobloc indenter. In this indenter, both the body and the spherical part are machined in the same Tungsten carbide piece. However, despite a spherical part being asked for, it was impossible to obtain a perfectly spherical indenter. Fig. 9 shows an image of the indenter realised by electronic scanning microscope from which the true profile of the indenter was measured. It was then shown that there are non-negligible differences between the true profile and a spherical one. However, this indenter can be considered as an axi-symetric one. In order to compare numerical and experimental curves, the true profile of the indenter was introduced into the FE code as a set of coordinates defining the surface of the indenter.

The displacement was measured by using a capacitive sensor which is fixed near the indenter with an intermediate piece.

Thus our experimental bench allowed us to measure the displacement of the indenter in the sample. However, the needed displacement is the sample surface which is found under the indenter. A previous study [19] proposed a new method to deduce the wanted load-depth curve from the measured one with knowledge of the elastic properties of both the indenter and the sample. This method was based on the determination of the contact radius changes between the indenter and the sample and a modelling of indenter deformations during an indentation test. The contact radius changes were deduced from several loading, unloading and reloading cycles which lead to the determination of the contact stiff-ness S(h) changes. Moreover, due to the non-spherical indenter, we proposed an equivalent radius function which depends on the contact radius. The new cyclic loading protocol was then necessary to calculate the equivalent radius changes of our indenter and then to deduce the real sample load-depth curve.

Differences between experimental and numerical curves

In our previous work [19], it is has been shown that experimental and numerical loading curves are quite similar. However, we observed non-negligible differences between experimental and numerical unloading curves. At the end of the unloading, the experimental displacement decreased more quickly than the numerical one. Moreover, the reloading cycles sometimes led to the appearance of hysteresis on the P(h) curve. This phenomenon has been observed and studied by several authors. According to Huber and Tsakmakis [20] the hysteresis is the consequence of a kinematic hardening behavior of the tested material. Moreover, they showed that, in the case of pure kinematic behavior, or both isotropic and kinematic behavior, the displacement decreases more quickly than in the case of pure isotropic behavior during the unloading cycle. Fig. 10 presents the comparison between the studied data deduced from both the experimental and the predicted numerical curve for C22 steel. It is shown that experimental P(h) and W t changes are well predicted by numerical simulations. However, we can observe large differences between experimental W e and W p /W t changes and the predicted ones.

Experimental characterization of two steels

Although the differences between experimental and predicted W e and W p /W t changes are non-negligible, we applied function (16) between experiments and models developed for Eq. ( 1) for each indentation data. The results are presented in Table 1. They confirm that the data which give the most accurate results is the W t changes (average accuracy 13% on σ y and 4% on n). The inversion of the P(h) curve gives less accurate results than W t , which can be explained by the accuracy of the corresponding model. Concerning the indentation data which are affected by the unloading, the accuracy of results obtained from the inversion of W e is near to that obtained for P(h) inversion. This can be explained by the high sensitivity of W e which compensates the differences between experimental and numerical unloading. Lastly, W p /W t gives the least accurate results.

Concerning the determination of the parameters of Eq. ( 5), the minimization with W e , W t and W p /W t simultaneously or W e , W t and P(h) simultaneously gives very inaccurate results. However, the minimization on W t changes leads to a unique set of parameters. Although it has been shown in the numerical study that there is no uniqueness if the minimization is made on only one indentation data, we observed uniqueness in this experimental study. This can be explained by the use of a nonspherical indenter with an equivalent radius function. Thus, an indentation test with this kind of indenter can be assimilated to several indentations with different indenter radii. The results of inversion of W t changes for Eq. ( 5) are given in Table 2. The average accuracy is 9% on σ y , 21% on K and 3% on m.

As a conclusion, although some differences occurred between experimental and numerical results, the inversion of models proposed in the numerical study led to an evaluation of the parameters which are in Eqs. ( 1) and ( 5). These results also showed that methods based on a "reverse" analysis are limited in accuracy, the more the number of mechanical parameters the less the accuracy of the characterization. This kind of method can be used in order to have an evaluation of the mechanical behavior in a first-order approximation. However, in order to have a more accurate characterization, it will be necessary to use other kinds of methods such as the "inverse" analysis.

Conclusion

A new kind of indentation protocol is proposed in this paper. It is composed of several loading, unloading and reloading cycles which allow the determination of the indentation data changes. In this study, four indentation data are studied: the elastic and total energies W e and W t , the plastic to total energy ratio W p /W t and the loading P(h) curve. From a numerical study, models describing these data changes have been determined for two constitutive equations with isotropic hardening. These models have allowed us to study the uniqueness of the solution for the two studied constitutive equations. For the first equation with two plasticity parameters, only one indentation data is sufficient to determine the mechanical properties with good accuracy. Moreover, from the numerical results, the best data is the total energy W t which can be deduced without cyclic loading. However, for the second constitutive equation with three plasticity parameters, it has been shown that several loading, unloading and reloading cycles are necessary to determine accurate values of the mechanical parameters. The numerical study has also shown that both the sensitivity and the accuracy of models explain the accuracy of the results.

In a previous paper, we have shown that the proposed cyclic loading protocol is necessary to deduce the real load-depth curve from the measured one. It is then shown that some differences occur between experimental and numerical unloading curves. These differences justify the low accuracy obtained by inversion of data which depends on the unloading. However, inversion of the total energy which is not sensitive to these problems leads to an evaluation of the parameters of the two studied constitutive equations. In this case, the uniqueness with the second constitutive equation can be explained by the use of an indenter which can be assimilated to several indenters with different radii which are calculated using the new cyclic loading protocol.

It has thus been shown in this paper that in order to evaluate more than two mechanical parameters, it is necessary to increase the experimental data field either by using several indentation data changes or several indenter radii.

Lastly, this paper shows that methods based on a "reverse" analysis lead to a more or less accurate determination of the mechanical properties. They can, however, give an evaluation of the material's behavior in a first-order approximation and then can be used in order to have a comparison between several materials. The study of other kinds of methods like "inverse" analysis is one of our perspectives in order to increase the accuracy of the characterization.

Tables B1-B4.

Fig. 1 .

 1 Fig. 1. Different kinds of loading cycles lead to different indentation data.

Fig. 2 .

 2 Fig. 2. The FE mesh.

Fig. 4 .

 4 Fig. 4. Quantification of the sensitivity.

Fig. 5 .

 5 Fig. 5. Symmetry between +5% and -5% on the sensitivity.

Fig. 6 .

 6 Fig. 6. Determination of the functions which describe the indentation data changes.

Fig. 7 .

 7 Fig. 7. Comparison between numerical curves and prediction of the models.

Fig. 9 .

 9 Fig. 9. Our indenter and its profile.

Table 1

 1 Results of the experimental characterization for Eq.(1)

	Sample	Exact solution		Minimization on W e	Minimization on W t	Minimization on W p /W t	Minimization on P(h)
		σ y (MPa)	n	σ y (MPa)	n	σ y (MPa)	n	σ y (MPa)	n	σ y (MPa)	n
	C22	219	0.16	182	0.21	181	0.16	210	0.21	174	0.19
	C100	177	0.24	217	0.25	163	0.22	290	0.25	139	0.27
	W e /E*R 3 , W t /E*R 3 and P(h)/E*R 2 are well described by func-
	tion j (11) and W p /W t by function k (12):

Table 2

 2 Results of the experimental characterization for Eq. (5) data is W e , it can be observed that the data which gives the best results is W t . Indeed, this data is less sensitive than W e , but the model of W t is the most accurate. Moreover, it has been shown that the sensitivity of W t and P(h) are quite similar. However, the results of the inversion of W t model are more accurate (accuracy multiplied by two) than the P(h) ones. Indeed, the model of W t is more accurate than the P(h) one. It is then shown that both the accuracy of model and the sensitivity explain the accuracy of the inversion.

	Sample	Exact solution

sensitive

Table B1

 B1 Coefficients which come in function A for Eqs.(11) and[START_REF] Tabor | Hardness of Metals[END_REF]

	Data		ab		c		d		e		f		g		h	i	j
	W e		1.2337	368.7702	-16397.8780	44854.0970	5.7106	-2.6740	150.2944	-70.235686	2.4492	-2.0914
	W t		1.9315	526.6240	-45734.4050	1096080.2000	5.1262	0.7832	212.8758	-19375.6920	468759.1300	2.1066
	W p /W t	-0.0075	8.2528		40.1633		-9737.7621	0.0533	-0.0566	26.1103		-2.0304	0.0756	0.8071
	P(h)		0.8946	227.6553	-10699.6670	3.6171	0.0717	1.3472	143.5716	-6922.8572	-26221.0330	2.5028
	Table B2
	Coefficients which come in function B for Eqs. (11) and (12)
	Data		ab	c		de	f			g		hi	j	k
	W e		12.8151	367.3677	844.7441	-0.7832 -27.8083	-4794.3394	-27014.0050 -1.8007	9.0035	-253.6345	-578.8095
	W t		5.7879	405.2397	392.1503	-0.6645 -11.3579	-4573.9157	-14009.3360 -1.8652	2.2474	-215.9234	-192.3363
	W p /W t	0.6262	-23.8830	-41.0306	0.4967	-0.4412	2001.3517		2631.3604	0.5303 -0.0343	187.6590	137.6183
	P(h)		5.3303	309.8288	22.8952	-0.7235 -12.9210	-4693.6406	-7502.3214	-1.9085	6.2890	-303.3440	18.4632
	Table B3
	Coefficients of function f 1 (Eq. (13))
	Data		aa 1		a 2	a 3		b 1		b 2		b 3		c 1	c 2	c 3	c 4
	W e		0.4799	9.5799	-2.5683	5.8946	-2.8629		1.0695	-6.1122	-6.9739	9.5389	0.9933	-0.0315
	W t		1.5875	6.5688	-1.2094	2.7659	-3.3348		-2.6223	-2.9957	-7.8405	5.1887	0.9918	-1.2836
	W p /W t	0.0042	8.5243		4.2597	-0.0769	-256.8118	-8.2117	0.1465	-71.0339	-4.6614	0.9360	35.7140
	P(h)		0.5729	6.7193	-1.3417	2.8247	-3.3598		-2.5429	-3.0331	-7.9480	5.2952	0.9915	-1.2600
	Table B4
	Coefficients of function f 2 (Eq. (14))
	Data aa 1	a 2	a 3	b 1	b 2	b 3	c 1		c 2		c 3	c 4	d 1	d 2	d 3	d 4	d 5
	W e -0.0592 29.4613 18.7065 1.4148 185.1068 8.0161 -4.5846 -621.6012 -33.8156 68.1669 -275843 14.2524 28.0201 5.0787 -53.8962 1563.7558
	W t	-0.1473 62.0351 34.2116 3.2741 219.9531 24.2602 -10.5230 -473.9512 -63.7603 68.4722 -13.9860 16.2688 40.0677 11.8180 -89.7697 1641.1261
	W p /W t 0.4784 67.6387 10.8059 6.2035 188.2657 -7.8024 -3.2472 -517.1911 -27.8351 68.4045 -25.3187 15.4058 32.9577 4.5077 -321.2930 1619.2977
	P(h)	0.4072 56.1993 113.1517 7.5934 117.8745 808.9538-31.7259 -78.3279 -303.3418 68.5109 211.3025 13.7692 132.8986 44.6494 45.6690 1828.5915
	Appendix C. Study of the uniqueness of the solution
	Tables C1-C3

Table C1 (

 C1 Continued). Results of the minimization with W e , W t and W p /W t for Eq.(5)

		Table C3									.
	Exact solution		Minimization on W e			Minimization en W t		Minimization on W p /W t	Minimization on P(h)
	σ y (MPa) Exact solution n	σ y (MPa)		n		σ y (MPa) Minimization on data changes n		σ y (MPa)	n	σ y (MPa) Minimization on the last points	n
	450 σ y (MPa)	0.28	455 K (MPa)	m	0.28		456 σ y (MPa)	0.28 K (MPa)	m	439	0.29 σ y (MPa)	427 K (MPa)	0.29 m
	450 150	0.34	455 25	0.34 0.2031		156	446	0.34 14	458 0.2064	0.34	3	449 368	0.35 0.3381
	450 150	0.4	454 123	0.40 0.2336		143	408	0.41 143	439 0.2455	0.4	61	451 458	0.40 0.3331
	720 150 720 150 720 150 720 150 720 150 720 450 720 450 450	0.02 0.08 0.15 0.22 0.28 0.34 0.4	673 307 688 609 693 1,020 712 1,653 720 2,628 734 63 738 304 720	0.04 0.2728 0.10 0.3158 0.17 0.3554 0.23 0.3973 0.28 0.4410 0.34 0.2487 0.40 0.2772 0.3132		148 171 184 182 139 445 359 393	728 707 707 714 721 713 648	0.02 289 0.09 553 0.16 935 0.23 1,551 0.28 2,566 0.34 75 0.41 393 724	561 0.2652 615 0.3048 661 0.3431 722 0.3886 745 0.4390 759 0.2522 736 0.2658 0.3005	0.08 0.12 155 77 0.17 190 0.22 80 0.28 135 0.33 453 0.4 485 526	596 593 639 621 667 814 708 1993 724 2560 755 32 755 144 214	0.3359 0.08 0.3169 0.11 0.3257 0.17 0.4241 0.23 0.4383 0.28 0.2481 0.33 0.2588 0.39 0.2574
	1000 450	0.02	947 1,355	0.04 0.3520		444	994	0.02 1,187	877 0.3354	0.06 563	932 360	0.04 0.2661
	1000 450	0.08	935 2,159	0.10 0.3873		505	978	0.09 1,727	916 0.3662	0.11 540	952 1294	0.10 0.3382
	1000 450	0.15	957 3,317	0.17 0.4244		552	978	0.16 2,680	1004 0.4063	0.16 645	1020 1293	0.15 0.3324
	1000 450	0.22	985 4,990	0.22 0.4628		414	996	0.22 4,828	1063 0.4579	0.21 534	1056 4076	0.20 0.4409
	1000 720	0.28	1000 93	0.28 0.2737		646	1017	0.27 195	1050 0.2753	0.28 673	1074 100	0.26 0.2680
	1000 720	0.34	1023 444	0.33 0.3011		647	1023	0.33 447	1047 0.2905	0.34 702	1079 235	0.33 0.2760
	1000 720	0.4	1034 1,031	0.39 0.3353		622	956	0.40 1,086	1007 0.3257	0.40 741	1057 430	0.39 0.2873
	1400 720 1400 720 1400 720 1400 720 1400 1000 1400 1000 1400 1000 1000	0.02 0.08 0.15 0.22 0.28 0.34 0.4	1410 1,897 1386 2,961 1400 4,451 1416 6,549 1425 123 1442 577 1450 1,322 2,394	0.02 0.3719 0.08 0.4051 0.15 0.4397 0.21 0.4755 0.27 0.2936 0.33 0.3200 0.39 0.3529 0.3878		687 770 713 778 802 818 887 1084	1387 1367 1382 1419 1454 1474 1432	0.02 1,862 0.09 2,834 0.15 4,545 0.21 5,529 0.27 481 0.33 1,093 0.40 1,497 1,910	1370 0.3665 1437 0.4066 1492 0.4465 1487 0.4608 1451 0.2999 1427 0.3352 1382 0.3519 0.3744	0.03 791 0.08 785 0.14 730 0.21 739 0.28 858 0.34 884 0.40 903 868	1395 807 1462 2440 1519 4362 1534 4503 1505 264 1494 533 1440 1214 3534	0.3105 0.02 0.3903 0.07 0.4424 0.12 0.4521 0.19 0.2886 0.26 0.3036 0.32 0.3379 0.39 0.4262
	2200 1000	0.02	2350 3,685	0.00 0.4193		1150	2076	0.05 3,617	2279 0.4283	0.02 843	2205 4863	0.02 0.4500
	2200 1000	0.08	2254 5,459	0.07 0.4520		1039	2092	0.10 5,655	2269 0.4612	0.08 879	2240 5872	0.07 0.4601
	2200 1000	0.15	2192 7,910	0.15 0.4858		1013	2133	0.16 7,235	2246 0.4808	0.14 887	2232 6882	0.14 0.4638
	2200 1400	0.22	2155 162	0.23 0.3164		1404	2206	0.22 304	2190 0.3309	0.22 1092	2155 1056	0.23 0.3413
	2200 1400	0.28	2132 753	0.29 0.3416		1381	2295	0.26 973	2136 0.3523	0.28 1100	2086 1858	0.30 0.3712
	2200 1400	0.34	2101 1,699	0.36 0.3728		1422	2375	0.31 1,760	2095 0.3770	0.34 1114	1999 3146	0.37 0.4100
	2200 1400	0.4	2050 3,030	0.43 0.4059		1488	2448	0.36 2,938	2059 0.4089	0.40 1062	1806 5299	0.45 0.4548
	3000 1400 3000 1400 3000 1400 3000 2200 3000 2200 3000 2200 3000 2200 2200	0.02 0.08 0.15 0.22 0.28 0.34 0.4	2940 4,600 2836 6,716 2786 9,587 2775 233 3016 1,069 2927 2,368 2809 4,141 6,173	0.04 0.4356 0.11 0.4663 0.19 0.4979 0.25 0.3510 0.27 0.3744 0.34 0.4030 0.44 0.4335 0.4605		1572 1674 1449 2405 2271 2184 2210 2340	2891 3018 2938 3012 2994 2990 2695	0.06 4,416 0.08 6,535 0.17 9,598 0.22 214 0.28 1,084 0.33 2,404 0.45 4,210 6,344	2953 0.4422 2950 0.4796 2932 0.4990 2891 0.3689 2882 0.3790 2813 0.4010 3214 0.4343 0.4717	0.03 1001 0.09 931 0.16 3415 0.23 1488 0.29 2791 0.35 2911 0.45 2896 2865	2851 7167 3093 8960 2844 174 2904 3502 2978 0 2972 363 2954 1185 2258	0.4755 0.05 0.4843 0.06 0.4010 0.18 0.4173 0.24 0.3823 0.2S 0.3885 0.33 0.3962 0.41 0.4084
		2200		8,847		0.4883		2391		9,481	0.5114	2823	4443	0.4399
	2200 Table C2 3000 Several sets of parameters can lead to the same curve 12,386 0.5168 298 0.3775 3000 1,353 0.3997	2185 2985 2868		12,313 572 1,463	0.5134 0.3833 0.3914	3026 2329 2419	4718 3226 3918	0.4372 0.4156 0.4270
	Exact solution 3000		2,962	Minimization on W e 0.4270	2764	Minimization on W t 3,046	Minimization on W p /W t 0.4123 2615	Minimization on P(h) 4366 0.4334
		Table C1 Results of inversion of models for each indentation data 12 3000 5,111 0.4549 2788 3000 7,326 0.4720 2950	12 5,446 8,176	1 0.4485 0.4870	2	2791 2971	1 4643 4900	2	0.4345 0.4328
	Exact solution σ y (MPa) 450 3000 K (MPa) 2159 3000	10,674 14,074	Minimization on W e 1253 1426 0.5146 391 0 0.5039	2832 2810	Minimization en W t 1180 1058 11,619 0 356 15,005	Minimization on W p /W t 920 1398 0.5121 3111 998 0 0.5084 3359	Minimization on P(h) 900 5173 0.4322 200 0 3864 5787 0.4307
	m	σ y (MPa)	n 0.39	σ y (MPa) 0.36	n	0.36		σ y (MPa) 0.35	n 0.36		σ y (MPa) 0.39	n 0.41	σ y (MPa) 0.34	n 0.49
		150	0.02	144		0.02		153	0.01		204	0.03	230	0.00
		150	0.08	137		0.10		124	0.11		148	0.11	161	0.10
		150	0.15	172		0.13		150	0.14		148	0.16	165	0.13
		150	0.22	159		0.21		150	0.22		134	0.23	132	0.22
		150	0.28	147		0.28		154	0.28		131	0.29	131	0.28
		150	0.34	136		0.34		157	0.34		136	0.34	139	0.34
		150	0.4	117		0.41		149	0.4		127	0.41	147	0.4
		450	0.02	453		0.00		487	0.00		433	0.00	446	0.00
		450	0.08	471		0.07		461	0.08		394	0.10	411	0.09
		450	0.15	451		0.15		450	0.15		393	0.17	392	0.18
		450	0.22	459		0.23		455	0.22		410	0.24	404	0.24

Acknowledgment

This research was sponsored by the Brittany Region.


Marie-Françoise Bidaut-Veron

Marta Garcia-Huidobro

Laurent Veron

Estimates of solutions of elliptic equations with a source reaction term


Marie-Françoise Bidaut-Véron

Quoc-Hung Nguyen
email: quochung.nguyen@sns.it

Laurent Véron
email: veronl@lmpt.univ-tours.fr

Quasilinear and Hessian Lane-Emden type systems with measure data

Keywords: Mathematics Subject Classification. 35J70, 35J60, 45G15, 31C15 p-Laplacian, k-Hessian, Bessel and Riesz capacities, measures, maximal functions

We study nonlinear systems of the form -∆pu = v q 1 + µ, -∆pv = u q 2 + η and

η in a bounded domain Ω or in R N where µ and η are nonnegative Radon measures, ∆p and F k are respectively the p-Laplacian and the k-Hessian operators and q1, q2, s1 and s2 positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities. 2010

Introduction and Main results

Let Ω ⊂ R N be either a bounded domain or the whole R N , p > 1 and k ∈ {1, 2, ..., N }. We denote by ∆ p u := div |∇u| p-2 ∇u the p-Laplace operator and by

F k [u] = 1≤j1<j2<...<j k ≤N λ j1 λ j2 ...λ j k
the k-Hessian operator where λ 1 , ..., λ N are the eigenvalues of the Hessian matrix D 2 u. In the work [START_REF] Phuc | Quasilinear and Hessian equations of Lane-Emden type[END_REF], Phuc and Verbitsky obtained necessary and sufficient conditions for existence of nonnegative solutions to the following equations

-∆ p u = u q + µ in Ω u = 0 on ∂Ω, (1.1)
and

F k [-u] = u q + µ in Ω u = 0 on ∂Ω. (1.2)
Their conditions involve the continuity of the measures with respect to Bessel or Riesz capacities and Wolff potentials estimates. For example, if Ω is bounded and µ has compact support in Ω, they proved that it is equivalent to solve (1.1), or to have µ(E) ≤ c 1 Cap Gp, q q+1-p (E) for all compact set E ⊂ Ω, (1.3) for some constant c 1 > 0 where Cap Gp, q q+1-p is a Bessel capacity, or to have

ˆB W R 1,p [µ B ](x) q dx ≤ c 2 µ(B) for all ball B s.t. B ∩ suppµ = ∅, (1.4)
for some constant c 2 > 0, where R = 2 diam(Ω) and W R 1,p [µ B ] denotes the R-truncated Wolff potential of the measure µ B = χ B µ. Concerning the k-Hessian operator in a bounded (k -1)-convex domain Ω, they proved that if µ has compact support, the problem (1.2) with q > k admits a nonnegative solution if and only if

µ(E) ≤ c 3 Cap G 2k , q q-k (E) for all compact set E ⊂ Ω, (1.5)
for some c 3 . In turn this condition is equivalent to ˆB W R for some c 4 > 0. The results concerning the linear case p = 2 and k = 1, can be found in [START_REF] Adams | Capacitary strong type estimates in semilinear problems[END_REF][START_REF] Baras | Critère d'existence des solutions positives pour des équations semilinéaires non monotones[END_REF][START_REF] Véron | Stationary Partial Differential Equations[END_REF].

The natural counterpart of equation (1.1) and (1.2) for systems:

-∆ p u = v q1 + µ in Ω -∆ p v = u q2 + η in Ω u = v = 0 on ∂Ω, (1.7)
and

F k [-u] = v s1 + µ in Ω F k [-v] = u s2 + η in Ω u = v = 0 on ∂Ω, (1.8)
where q 1 , q 2 > p -1, s 1 , s 2 > k and µ, η are Radon measures. If Ω = R N , we consider the same equations, except that the boundary conditions are replaced by inf R N u = inf R N v = 0 and our statements involve the Riesz potentials and their associated capacities Cap I α,β . Our main results are the following.

Theorem A Let 1 < p < N , q 1 , q 2 > 0 and q 2 q 1 > (p -1) 2 . Let µ, η be nonnegative Radon measures in R N . If the following system

-∆ p u = v q1 + µ in R N -∆ p v = u q2 + η in R N , (1.9)
admits a nonnegative p-superharmonic solution (u, v) then there exists a positive constant c 5 depending on N, p, q 1 , q 2 such that η(E) + ˆE (W 1,p [µ](x)) q2 dx ≤ c 5 Cap I p(q 1 +p-1) q 1 , q 1 q 2 q 1 q 2 -(p-1) 2 (E) for all Borel sets E. (1.10) Conversely, if µ and η are bounded, there exists c 6 > 0 depending on N, p, q 1 , q 2 such that if 0 < q 1 < N (p-1)

N -p and (1.10) holds with c 5 replaced by c 6 , then (1.9) admits a nonnegative p-superharmonic solution (u, v) satisfying

v ≤ c 8 W 1,p [ω], u ≤ c 9 W 1,p [(W 1,p [ω]) q1 ] + c 7 W 1,p [µ] (1.11)
in R N for some c 7 , c 8 , c 9 > 0 where dω = (W 1,p [µ]) q2 dx + dη .

We notice that the left-hand side in (1.10) is not symmetric in η and µ and the capacity in the right-hade side is not symmetric in q 1 and q 2 . Hence the following symmetrized inequality holds

µ(E) + ˆE (W 1,p [η](x))
q1 dx ≤ c 5 Cap I p(q 2 +p-1) q 2 , q 1 q 2 q 1 q 2 -(p-1) 2 (E) for all Borel sets E. (1.12) It is known that Cap I α,β (K) = 0 ∀K compact, if αβ ≥ N , the first part of above implies the following Liouville theorem, obtained by another method in [START_REF] Bidaut-Véron | Nonexistence results and estimates for some nonlinear elliptic problems[END_REF]].

Corollary B Assume that p(q 1 q 2 + (p -1) max{q 1 , q 2 }) q 1 q 2 -(p -1) 2 ≥ N.

Any nonnegative p-superharmonic solution (u, v) of inequalities

-∆ p u ≥ v q1 in R N -∆ p v ≥ u q2 in R N , (1.13)
is trivial, i.e. u = v = 0.

Classical Liouville results for one equation or inequality, are proved in [START_REF] Bidaut-Véron | Local and global behavior of solutions of quasilinear equations of Emden-Fowler type[END_REF], [START_REF] Bidaut-Véron | Necessary conditions of existence for an elliptic equation with source term and measure data involving p-Laplacian[END_REF], [START_REF] Birindelli | Some Liouville theorems for the p-Laplacian[END_REF], [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF].

When Ω is bounded domain, we have a similar result in which we denote by d the distance function to the boundary x → d(x) = dist (x, ∂Ω).

Theorem C Let 1 < p < N , q 1 , q 2 > 0 and q 2 q 1 > (p -1) 2 . Let Ω ⊂ R N be a bounded domain and µ, η nonnegative Radon measures in Ω. If the following problem

-∆ p u = v q1 + µ in Ω -∆ p v = u q2 + η in Ω u = v = 0 on ∂Ω, (1.14)
admits a nonnegative renormalized solution (u, v), then then for any compact set K ⊂ Ω, there exists a positive constant c 10 depending on N, p, q 1 , q 2 and dist(K, ∂Ω) such that

η(E)+ ˆE W d(x) 4 1,p [µ](x) q2 dx ≤ c 10 Cap G p(q 1 +p-1) q 1 , q 1 q 2 q 1 q 2 -(p-1) 2 (E) for all Borel sets E ⊂ K. (1.15)
Conversely, let µ and η be bounded with the property that there exists c 11 > 0 depending on N, p, q 1 , q 2 and R = 2diam (Ω) such that if 0 < q 1 < N (p-1)

N -p and

η(K) + ˆK W 2R 1,p [µ] q2 dx ≤ c 11 Cap G p(q 1 +p-1) q 1 , q 1 q 2 q 1 q 2 -(p-1) 2 (K), (1.16)
for all compact set K ⊂ Ω, then (1.14) admits a nonnegative renormalized solution (u, v) satisfying

v ≤ c 13 W R 1,p [ω], u ≤ c 14 W R 1,p [ W R 1,p [ω] q1 ] + c 12 W R 1,p [µ] (1.17) in Ω, where dω = W R 1,p [µ] q2 dx + dη.
It is known that

Cap G α,β ({x 0 }) > 0
if and only if αβ > N . Thus, as an application in a partially subcritical case we have,

Corollary D Let the assumptions on p, q 1 , q 2 , Ω and R of Theorem C be satisfied, x 0 ∈ Ω, a > 0 and µ be a nonnegative Radon measures in Ω. If the following problem

-∆ p u = v q1 + µ in Ω -∆ p v = u q2 + aδ x0 in Ω u = v = 0 on ∂Ω, (1.18)
admits a nonnegative renormalized solution (u, v), then there exist positive constants c 15 = c 15 (N, p, q 1 , q 2 , d(x 0 )) and, for any compact subset K of Ω, c 16 = c 16 (N, p, q 1 , q 2 , dist (K, ∂Ω)), such that

(i) N < pq 2 (q 1 + p -1) q 1 q 2 -(p -1) 2 , (ii) a ≤ c 15 , (iii)
ˆK W 2R 1,p [µ] q2 dx ≤ c 16 .
(1.19)

Conversely, assuming that µ is bounded, there exist positive constants c 17 = c 17 (N, p, q 1 , q 2 , d(x 0 )), c 18 = c 18 (N, p, q 1 , q 2 ) such that if 0 < q 1 < N (p-1) N -p and (1.19) holds with c 15 and c 16 replaced respectively by c 17 and c 18 , then there exists a nonnegative renormalized solution (u, v) of (1.18)

satisfying v ≤ c 21 W R 1,p [ω], u ≤ c 22 W R 1,p [ W R 1,p [ω] q1 ] + c 20 W R 1,p [µ] (1.20)
in Ω, where

W R 1,p [ω] = W R 1,p W R 1,p [µ] q2 + a 1 p-1 |x -x 0 | -N -p p-1 -R -N -p p-1 + .
Concerning the k-Hessian operator we recall some notions introduced by Trudinger and Wang [START_REF] Trudinger | Hessian measures[END_REF][START_REF] Trudinger | Hessian measures II[END_REF][START_REF] Trudinger | Hessian measures III[END_REF], and we follow their notations. For k = 1, ..., N and u ∈ C 2 (Ω) the k-Hessian operator F k is defined by

F k [u] = S k (λ(D 2 u)),
where λ(D 2 u) = λ = (λ 1 , λ 2 , ..., λ N ) denotes the eigenvalues of the Hessian matrix of second partial derivatives D 2 u and S k is the k-th elementary symmetric polynomial that is

S k (λ) = 1≤i1<...<i k ≤N λ i1 ...λ i k . Since D 2 u is symmetric, it is clear that F k [u] = D 2 u k ,
where we denote by [A] k the sum of the k-th principal minors of a matrix A = (a ij ). In order that there exists a smooth k-admissible function which vanishes on ∂Ω, the boundary ∂Ω must satisfy a uniformly (k-1)-convex condition, that is

S k-1 (κ) ≥ c 0 > 0 on ∂Ω,
for some positive constant c 0 , where κ = (κ 1 , κ 2 , ..., κ n-1 ) denote the principal curvatures of ∂Ω with respect to its inner normal. We also denote by Φ k (Ω) the class of uppersemicontinuous functions Ω → [ -∞, ∞) which are k-convex, or subharmonic in the Perron sense (see Definition 5.1). In this paper we prove the following theorem (in which expression E[q] is the largest integer less or equal to q)

Theorem E Let 2k < N, s 1 , s 2 > 0, s 1 s 2 > k 2 . Let Ω be a bounded uniformly (k-1)-convex domain in R N with diameter R. Let µ = µ 1 + f and η = η 1 + g be nonnegative Radon measures where µ 1 , η 1 has compact support in Ω and f, g ∈ L l (Ω) for some l > N 2k . If the following problem F k [-u] = v s1 + µ in Ω F k [-v] = u s2 + η in Ω u = v = 0 on ∂Ω, (1.21)
admits a nonnegative solutions (u, v), continuous near ∂Ω, with -u and -v elements of Φ k (Ω), then for any compact set K ⊂ Ω, there exists a positive constant c 23 depending on N, k, s 1 , s 2 and dist(K, ∂Ω) such that there holds

η(E) + ˆE W d(x) 4 2k k+1 ,k+1 [µ](x) s2 dx ≤ c 23 Cap G 2k(s 1 +k) s 1 , s 1 s 2 s 1 s 2 -k 2 (E) ∀E ⊂ K, E Borel.
(1.22) Conversely,, if µ and η are bounded, there exist a positive constant c 24 depending on N, k, s 1 , s 2 and diam (Ω) such that, if k ≤ s 1 < N k N -2k and

η(K) + ˆK W 2R 2k k+1 ,k+1 [µ] s2 dx ≤ c 24 Cap G 2k(s 1 +k) s 1 , s 1 s 2 s 1 s 2 -k 2 (K) (1.23)
for all Borel set K ⊂ Ω, then (1.21) admits a nonnegative solution (u, v), continuous near

∂Ω, with -u, -v ∈ Φ k (Ω) satisfying v ≤ c 28 W R 2k k+1 ,k+1 [ω], u ≤ c 29 W R 2k k+1 ,k+1 [ W R 2k k+1 ,k+1 [ω] s1 ] + c 27 W R 2k k+1 ,k+1 [µ] (1.24)
in Ω for some constants c j (j = 27, 28, 29) depending on N, k, s 1 , s 2 , and diam (Ω).

If Ω is replaced by the whole space we prove,

Theorem F Let 2k < N, s 1 , s 2 > 0, s 1 s 2 > k 2 . Let µ, η be a nonnegative Radon measures in R N . If the following problem F k [-u] = v s1 + µ in R N F k [-v] = u s2 + η in R N , (1.25)
admits a nonnegative solutions (u, v) with -u and -v belonging to Φ k (R N ), then there exists a positive constant c 30 depending on N, k, s 1 , s 2 such that there holds

η(E) + ˆE W 2k k+1 ,k+1 [µ](x) s2 dx ≤ c 30 Cap G 2k(s 1 +k) s 1 , s 1 s 2 s 1 s 2 -k 2 (E) ∀E Borel. (1.
v in Φ k (R N ) satisfying v ≤ c 33 W 2k k+1 ,k+1 [ω], u ≤ c 34 W 2k k+1 ,k+1 [ W 2k k+1 ,k+1 [ω] s1 ] + c 32 W 2k k+1 ,k+1 [µ] (1.27)
in R N , where the c j (j = 32, 33, 34) depend on N, k, s 1 , s 2 .

As in the p-Laplace case, we have a Liouville property for Hessian systems.

Corollary G Assume that 2k(s 2 s 1 + k max{s 1 , s 2 }) s 1 s 2 -k 2 ≥ N. (1.28)
Any nonnegative solution (u,v) of inequalities

F k [-u] ≥ v s1 in R N F k [-v] ≥ u s2 in R N , (1.29)
with -u and -v in Φ k (R N ) is trivial.

Estimates on potentials

Throughout this article c j , j=1,2,..., denote structural positive constants and c N is the volume of the unit ball in R N . The following inequality will be used several times in the sequel.

Lemma 2.1 Let κ, γ, θ ∈ R, such that κ, γ > 0. Let h : (0, ∞) → (0, ∞) be nondecreasing. Then,

ˆR 0 t κ ˆR t h(r)r θ dr r γ dt t ≤ c 35 ˆ2R 0 t κ+θγ h γ (t) dt t ∀R ∈ (0, ∞], (2.1)
for some c 35 > 0 depending on κ, γ, θ.

Proof. Case 1: γ ≤ 1. Since there holds

  ∞ j=0 a j   γ ≤ ∞ j=0 a γ j ∀a j ≥ 0,
we deduce

ˆR t h(r)r θ dr r γ ≤ c γ,θ   j0 j=0 h(2 j+1 4 t)(2 j 4 t) θ   γ ≤ c γ,θ j0 j=0 h γ (2 j+1 4 t) (2 j 4 t) θγ ≤ c γ,θ ˆ2R t h γ (r)r θγ dr r ,
where c γ,θ = 2 γ 4 max{1, 2 -γθ 4 } and 2

j 0 4 t < R ≤ 2 j 0 +1 4 t if R < ∞ and j 0 = ∞ if R = ∞. By Fubini's theorem, ˆR 0 t κ ˆR t h(r)r θ dr r γ dt t ≤ c γ,θ ˆR 0 t κ ˆ2R t h γ (r)r θγ dr r dt t ≤ c γ,θ κ ˆ2R 0 t κ+θγ h γ (t) dt t ,
which is (2.1).

Case 2: γ > 1. Since ˆR t h(r)r θ dr r γ ≤ ˆR t r -γ γ-1 dr r γ-1 ˆR t h γ (r)r γ(1+θ) dr r ,
we obtain

ˆR 0 t κ ˆR t h(r)r θ dr r γ dt t ≤ c γ,κ ˆ2R 0 t κ+θγ h γ (t) dt t ,
by Fubini's theorem, which completes the proof.

We recall that if α > 0, 1 < β < N α and µ belongs to the set of positive Radon measures in R N that we denote M + (R N ), the Wolff potential of µ is defined by

W α,β [µ](x) = ˆ∞ 0 µ(B r (x)) r N -αβ 1 p-1 dr r , (2.2)
and if R > 0, the R-truncated Wolff potential of µ is W R α,β [µ](x) = ˆR 0 µ(B r (x)) r N -αβ 1 p-1 dr r . (2.3)
If µ is a Radon measure on a Borel set G, it's Wolff potential (or truncated Wolff potential) is the potential of its extension by 0 in G c . We start with the following composition estimate on Wolff potentials. Lemma 2.2 Let 1 < β < N/α. Then for any q > 0 and µ ∈ M + (R N ) we have

W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ] ≤ c 36 W α,β [(W α,β [µ]) q ] , (2.4)
in R N for some c 36 > 0 depending on α, β, N, q. Moreover, if 0 < q < N (β-1) N -αβ , there holds

W α,β [(W α,β [µ]) q ] (x) ≤ c 37 W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ], (2.5)
in R N , where c 37 > 0 depends on α, β, N, q.

Proof. For any x ∈ R N , using the fact if y ∈ B t (x) then B t (x) ⊂ B 2t (y), we have

W α,β [(W α,β [µ]) q ] (x) = ˆ∞ 0 1 t N -αβ ˆBt(x) ˆ∞ 0 µ(B r (y)) r N -αβ 1 β-1 dr r q dy 1 β-1 dt t ≥ c 38 ˆ∞ 0 1 t N -αβ ˆBt(x) µ(B 2t (y)) t N -αβ q β-1 dy 1 β-1 dt t ≥ c 36 ˆ∞ 0 t αβ(β-1) q µ(B t (x)) t N -αβ q (β-1) 2 dt t = c 36 W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x),
where c 38 = c 38 (α, β, N, q) > 0, which proves (2.4).

In order to prove (2.4) we recall the following estimate on Wolff potentials [START_REF] Bidaut-Véron | Quasilinear Lane-Emden equations with absorption and measure data[END_REF] ||W

α,β [ω]|| L (β-1)N N -αβ ,∞ ≤ c 39 ω(R N ) 1 β-1 ∀ ω ∈ M + b (R N ), (2.6)
where

L (p-1)N N -αβ ,∞ denotes the weak-L (p-1)N N -αβ space. In particular, since 0 < q < N (β-1) N -αβ , ˆBr(x) (W α,β [ω]) q dy ≤ c 40 r N ω(R N ) r N -αβ q β-1 ∀x ∈ R N , ∀r > 0.
(2.7)

Applying this inequality to ω = χ B2r(x) µ yields

ˆBr(x) W r α,β [µ] q dy ≤ c 40 r N µ(B 2r (x)) r n-αβ q β-1 ∀x ∈ R N , ∀r > 0.
(2.8)

We claim that

I := ˆ∞ 0 1 t N -αβ ˆBt(x) ˆ∞ t µ(B r (y)) r N -αβ 1 β-1 dr r q dy 1 β-1 dt t ≤ c 37 W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x).
(2.9)

Since B r (y) ⊂ B 2r (x) for any y ∈ B t (x), r ≥ t, we have ˆBt(x) ˆ∞ t µ(B r (y)) r N -αβ 1 β-1 dr r q dy ≤ ˆBt(x) ˆ∞ t µ(B 2r (x)) r N -αβ 1 β-1 dr r q dy ≤ c N t N ˆ∞ t µ(B 2r (x)) r N -αβ 1 β-1 dr r q .
Hence,

I ≤ c 1 β-1 N ˆ∞ 0 t αβ β-1 ˆ∞ t µ(B 2r (x)) r N -αβ 1 β-1 dr r q β-1 dt t .
Using Lemma 2.1, we infer

I ≤ c 37 ˆ∞ 0 r αβ β-1 µ(B r (x)) r N -αβ q (β-1) 2 dr r = c 37 W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x),
which completes the proof.

The following is a version of Lemma 2.2 for truncated Wolff potentials,

Lemma 2.3 Let 1 < β < N/α and q > 0. If δ ∈ (0, 1) there holds for any µ ∈ M + (R N ) W δd 2 αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x) ≤ c 42 W δd α,β W δd(.) α,β [µ] q (x) (2.10) in Ω. Moreover, if 0 < q < N (β-1) N -αβ , there holds for any µ ∈ M + (R N ), W R α,β W R α,β [µ] q (x) ≤ c 43 W 4R αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x) (2.11) in R N . Proof. For any x ∈ Ω, W δd(x) α,β W δd(.) α,β [µ](.) q (x) = ˆδd(x) 0 1 t N -αβ ˆBt(x) ˆδd(y) 0 µ(B r (y)) r N -αβ 1 β-1 dr r q dy 1 β-1 dt t .
Since δd(y) ≥ 7δ 8 d(x) for all y ∈ B t 8 (x), provided 0 < t < δd(x), ˆBt(x) ˆδd(y)

0 µ(B r (y)) r N -αβ 1 β-1 dr r q dy ≥ ˆBt/8 (x) ˆ7δ 8 d(x) 0 µ(B r (y)) r N -αβ 1 β-1 dr r q dy ≥ ˆBt/8 (x) ˆ7t 8 0 µ(B r (y)) r N -αβ 1 β-1 dr r q dy ≥ c 44 ˆBt/8 (x) µ(B 3t 4 (y)) t N -αβ q β-1 dy ≥ c 44 ˆBt/8 (x) µ(B 3t 4 -t 8 (x)) t N -αβ q β-1 dy ≥ c 45 t N µ(B t 2 (x)) t N -αβ q β-1 . Hence W δd(x) α,β W δd(.) α,β [µ](.) q (x) ≥ c 46 ˆδd(x) 0   t αβ µ(B t 2 (x)) t N -αβ q β-1   1 β-1 dt t ,
which implies (2.10). Because of (2.8), it is sufficient to prove that there holds

ˆR 0 1 t N -αβ ˆBt(x) ˆR t µ(B r (y)) r N -αβ 1 β-1 dr r q dy 1 β-1 dt t ≤ c 47 W 4R αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [µ](x), (2.12)
in order to obtain (2.11). Since B ρ (y) ⊂ B 2ρ (x) for any y ∈ B r (x), ρ ≥ r, we have

ˆBt(x) ˆR t µ(B r (y)) r N -αβ 1 β-1 dr r q dy ≤ ˆBt(x) ˆR t µ(B 2r (x)) r N -αβ 1 β-1 dr r q dy ≤ c N t N ˆR t µ(B 2r (x)) r N -αβ 1 β-1 dr r q . Therefore ˆR 0 1 t N -αβ ˆBt(x) ˆR t µ(B r (y)) r N -αβ 1 β-1 dr r q dy 1 β-1 dt t ≤ c N ˆR 0 t αβ ˆR t µ(B 2r (x)) r N -αβ 1 β-1 dr r q 1 β-1 dt t .
We infer (2.12) by Lemma 2.1, which completes the proof.

The next two propositions link Wolff potentials of a measure with Riesz capaciticies (in the case of whole space) and truncated Wolff potentials with Bessel capaciticies (in the bounded domain case). Their proof can be found in [START_REF] Phuc | Quasilinear and Hessian equations of Lane-Emden type[END_REF][START_REF] Phuc | Singular quasilinear and Hessian equation and inequalities[END_REF] (and [START_REF] Bidaut-Véron | Quasilinear elliptic equations with source mixed term and measure data[END_REF] with a different method).

Proposition 2.4 Let 1 < β < N/α, q > β -1, ν ∈ M + (R N ).
Then, the following statements are equivalent: (a) The inequality

ν(K) ≤ c 48 Cap I αβ , q q-β+1 (K) (2.13) holds for any compact set K ⊂ R N , for some c 48 > 0. (b) The inequality ˆRN W α,β [χ Bt(x) ν](y) q dy ≤ c 49 ν(B t (x)) (2.14)
holds for any ball B t (x) ⊂ R N , for some c 49 > 0.

(c) The inequality

W α,β [(W α,β [ν]) q ] ≤ c 50 W α,β [ν] < ∞ a.e in R N (2.15)
holds for some c 50 > 0.

Proposition 2.5 Let 1 < β < N/α, q > β -1, R > 0 and ν ∈ M + b (B R (x 0 )) for some x 0 ∈ R N .
Then, the following statements are equivalent: (a) The inequality

ν(K) ≤ c 51 Cap G αβ , q q-β+1 (K) (2.16
)

holds for any compact set K ⊂ R N , for some c 51 = c 51 (R) > 0. (b) The inequality ˆRN W 4R α,β [χ Bt(x) ν](y) q dy ≤ c 52 ν(B t (x)) (2.17)
holds for any ball B t (x) ⊂ R N , for some c 52 = c 52 (R) > 0.

(c) The inequality

W 4R α,β W 4R α,β [ν] q ≤ c 53 W 4R α,β [ν] a.e in B 2R (x 0 ) (2.18)
holds for some c 53 = c 53 (R) > 0.

In the following statement we obtain capacitary estimates on combination of measures.

Proposition 2.6 Let η, µ be in M + (R N ). Assume that 0 < q < N (β-1) N -αβ and qs > (β -1) 2 .

(i) If there holds

η(K) + ˆK (W α,β [µ]) s dx ≤ Cap I αβ(q+β-1) q , qs qs-(β-1) 2 (K), (2.19)
for any compact set K ⊂ R N , then

W α,β (W α,β [(W α,β [ω]) q ]) s ≤ c 54 W α,β [ω] < ∞ a.e in R N , (2.20)
where ω = (W α,β [µ]) s + η.

(ii) If there holds

η(K) + ˆK W 2R α,β [µ] s dx ≤ Cap G αβ(q+β-1) q , qs qs-(β-1) 2 (K), (2.21)
for any compact set K ⊂ R N , then

W 2R α,β W 2R α,β W 2R α,β [ω] q s ≤ c 55 W 2R α,β [ω] < ∞ a.e in B R (x 0 ), (2.22)
where

ω = χ B R (x 0 ) W 2R α,β [µ] s + χ B R (x 0 ) η.
Proof

(B 2ρ (x)) = ρ N - αβ(q+β-1)s qs-(β-1) 2 Cap I αβ(q+β-1) q , qs qs-(β-1) 2 (B 2 (0)), we deduce from (2.19) ω(B ρ (x)) ≤ c 55 ρ N - αβ(q+β-1)s qs-(β-1) 2 ∀ ρ > 0, which is equivalent to ρ αβ β-1 ω(B ρ (x)) ρ N -αβ(q+β-1) q qs (β-1) 3 ≤ c 56 ω(B ρ (x)) ρ N -αβ 1 β-1 ∀ ρ > 0.
(

We apply Proposition 2.4 to ν = ω with (α, β, q) = αβ(q+β-1) q+(β-1) 2 , (β-

W α,β W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω] s ≤ c 58 W α,β [ω] < ∞ a.e R N . (2.25)
Therefore, it is enough to show that (2.23) and (2.24) imply (2.25). In fact, since for t > 0 ˆBt(x)

W t αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q

+1

[ω](y)

s dy = ˆBt(x) W t αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [χ B 2t (x) ω](y) s dy,
we apply (2.24) and obtain ˆBt(x)

W t αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q

+1

[ω](y)

s dy ≤ c 57 ω(B 2t (x)).
So, it is enough to show that

I := ˆ∞ 0 1 t N -αβ ˆBt(x) ˆ∞ t ω(B r (y)) r N -αβ(q+β-1) q q (β-1) 2 dr r s dy 1 β-1 dt t ≤ c 58 W α,β [ω](x).
(2.26)

Since B r (y) ⊂ B 2r (x) for any y ∈ B t (x), r ≥ t, we have

I ≤ c N ˆ∞ 0 t αβ ˆ∞ t ω(B 2r (x)) r N -αβ(q+β-1) q q (β-1) 2 dr r s 1 β-1 dt t = c N ˆ∞ 0 t αβ β-1 ˆ∞ t ω(B 2r (x)) r N -αβ(q+β-1) q q (β-1) 2 dr r s β-1 dt t .
It follows from Lemma 2.1 and (2.23) that

I ≤ c 59 ˆ∞ 0 t αβ β-1 ω(B 2t (x)) t N -αβ(q+β-1) q qs (β-1) 3 dt t ≤ c 56 c 59 ˆ∞ 0 ω(B 2t (x)) t N -αβ 1 β-1 dt t , which is (2.26). Statement (ii): We assume that (2.21) holds. Put dω = χ Ω (W α,β [µ]) s + χ Ω η, then ω(B ρ (x)) ≤ c 60 ρ N - αβ(q+β-1)s qs-(β-1) 2 ∀ 0 < ρ < 2R.
As in the proof of statement (i), the above inequality is equivalent to

ρ αβ β-1 ω(B ρ (x)) ρ N -αβ(q+β-1) q qs (β-1) 3 ≤ c 61 ω(B ρ (x)) ρ N -αβ 1 β-1 ∀ 0 < ρ < 2R.
(2.27)

Applying Proposition 2.5 with ν = ω and (α,

β, q) = αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q + 1, s , ˆRN W 4R αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [χ B t (x) ω] s dy ≤ c 62 ω(B t (x)). ( 2

+1

[ω](y)

s dy = ˆBt(x) W t αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [χ B2t(x) ω](y)
s dy for all 0 < t < 4R, thus applying (2.28), we obtain ˆBt(x)

W t αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q

+1

[ω](y)

s dy ≤ c 64 ω(B 2t (x)).
So, it is sufficient to show that for any x ∈ B R (x 0 )

II := ˆ4R 0 1 t N -αβ ˆBt(x) ˆ4R t ω(B r (y)) r N -αβ(q+β-1) q q (β-1) 2 dr r s dy 1 β-1 dt t ≤ c 65 W 4R α,β [ω](x).
(2.30) Since B r (y) ⊂ B 2r (x) for any y ∈ B t (x) with r ≥ t, we have

II ≤ c N ˆ4R 0 t αβ β-1 ˆ4R t ω(B 2r (x)) r n-αβ(q+β-1) q q (β-1) 2 dr r s β-1 dt t .
Combining this with Lemma 2.1 and (2.27) yields

II ≤ c 66 W 16R α,β [ω](x).
Therefore, (2.29) follows since

W 16R α,β [ω] ≤ c 67 W 4R α,β [ω] in B R (x 0 ).
Proposition 2.7 Let η, µ be in M + (R N ). Assume that 0 < q < N (β-1) N -αβ and qs > (β -1) 2 . Let (u m , v m ) be nonnegative measurable funtions in R N verifying, for all m ≥ 0,

u m+1 ≤ c * W α,β [v q m + µ], v m+1 ≤ c * W α,β [u s m + η] a.e. in R N ,
for some c * > 0 and (u 0 , v 0 ) = 0. Then, there exists a constant M * > 0 depending only on N, α, β, q, s, c * such that if the measure dω

= (W α,β [µ]) s dx + dη satisfies ω(K) ≤ M * Cap I αβ(q+β-1) q , qs qs-(β-1) 2 (K),
(2.31)

for any compact set K ⊂ R N , then v m ≤ c 69 W α,β [ω], u m ≤ c 70 W α,β [(W α,β [ω]) q ] + c 68 W α,β [µ] ∀ m ≥ 0, (2.32)
for some constants c 68 , c 69 , c 70 depending only on N, α, β, q, s and c * .

Proof. By Proposition 2.6, (2.31) implies

W α,β (W α,β [(W α,β [ω]) q ]) s ≤ c 71 M qs (β-1) 3 W α,β [ω] < ∞ a.e in R N . (2.33)
We set

c 68 = c * 2 1 β-1 , c 69 = c * 2 1+ 1 β-1 (c s 68 2 s-1 + 1) 1 β-1 , c 70 = c * 2 1 β-1 c q β-1 69
, and choose M * > 0 such that

c * 2 1 β-1 c s 70 2 s-1 1 β-1 c 71 M * qs (β-1) 3 = c 69 2 .
We claim that

v m ≤ c 69 W α,β [ω], u m ≤ c 70 W α,β [(W α,β [ω]) q ] + c 68 W α,β [µ] ∀ m ≥ 0. (2.34)
Clearly, by definition of c 68 , c 69 and c 70 , we have (2.34) for m = 0, 1. Next we assume that (2.34) holds for all integer m ≤ l for some l ∈ N * + , then

u l+1 ≤ c * W α,β [v q l + µ] ≤ c * 2 1 β-1 c q β-1 69 W α,β [(W α,β [ω]) q ] + c * 2 1 β-1 W α,β [µ] = c 70 W α,β [(W α,β [ω]) q ] + c 68 W α,β [µ],
and

v l+1 ≤ c * W α,β [(c 70 W α,β [(W α,β [ω]) q ] + c 68 W α,β [µ]) s + η] ≤ c * W α,β [c s 70 2 s-1 (W α,β [(W α,β [ω]) q ]) s + c s 68 2 s-1 (W α,β [µ]) s + η] ≤ c * 2 1 β-1 c s 70 2 s-1 1 β-1 W α,β [(W α,β [(W α,β [ω]) q ]) s ] + c * 2 1 β-1 (c s 68 2 s-1 + 1) 1 β-1 W α,β [(W α,β [µ]) s + η] ≤ c * 2 1 β-1 c s 70 2 s-1 1 β-1 c 71 M * qs (β-1) 3 W α,β [ω] + c * 2 1 β-1 (c s 68 2 s-1 + 1) 1 β-1 W α,β [ω] = c 69 2 W α,β [ω] + c 69 2 W α,β [ω] = c 69 W α,β [ω].
Thus, (2.34) holds true for m = l + 1. Hence, (2.34) is valid for all l ≥ 0.

The next result is an adaptation of Proposition 2.7 to truncated Wolff potentials.

Proposition 2.8 Let η, µ be in M + b (B R (x 0 )). Assume that 0 < q < N (β-1) N -αβ and qs > (β -1) 2 . Let (u m , v m ) be nonnegative measurable funtions in R N such that for all m ≥ 0

u m+1 ≤ c * W R α,β [χ B R (x 0 ) v q m + µ], v m+1 ≤ c * W R α,β [χ B R (x 0 ) u s m + η] a.e. in B R (x 0 ), and
(u 0 , v 0 ) = 0. If we set dω = W 2R α,β [µ]
s dx + dη, there exists a constant M * > 0 depending only on N, α, β, q, s, R and c * such that if

ω(K) ≤ M * Cap G αβ(q+β-1) q , qs qs-(β-1) 2 (K), (2.35) for any compact set K ⊂ R N , then v m ≤ c 73 W 2R α,β [ω], u m ≤ c 74 W 2R α,β [ W 2R α,β [ω] q ] + c 72 W 2R α,β [µ] ∀ k ≥ 0 (2.36) in B R (x 0 )
for some constants c 72 , c 73 , c 74 depending only on N, α, β, q, s, R and c * .

Proof. The proof is similar to the one of Proposition 2.7 and we omit the details.

Proposition 2.9 Let 1 < β < N/α and q, s > 0 such that qs > (β -1) 2 . (i) Assume that η and µ belong to M + b (R N ) and (u, v) are nonnegative measurable functions satisfying

(i) W α,β [v q ] + W α,β [µ] ≤ c 75 u, (ii) W α,β [u s ] + W α,β [η] ≤ c 75 v a.e. in R N , (2.37)
for some c 75 > 0. Then there exists a constant c 76 > 0 depending only on N, α, β, q, s and c 75 such that

η(K) + ˆK (W α,β [µ](x)) s dx ≤ c 76 Cap I αβ(q+β-1) q , qs qs-(β-1) 2 (K), (2.38)
for any compact set K ⊂ R N .

(ii) Assume that η and µ belong to M + b (Ω) and (u, v) are nonnegative functions satisfying

(i) W δd(.) α,β [v q ] + W δd α,β [µ] ≤ c 77 u, (ii) W δd(.) α,β [u s ] + W δd α,β [η] ≤ c 77 v a.e. in Ω, (2.39)
for some c 77 > 0. Then for any Ω ⊂⊂ Ω, there exists a constant c 78 > 0 depending only on n, α, β, q, s, c 77 and dist(Ω , ∂Ω) such that

η(K) + ˆK W δd(x) α,β [µ](x) s dx ≤ c 78 Cap G αβ(q+β-1) q , qs qs-(β-1) 2 (K), (2.40)
for any compact set K ⊂ Ω .

Proof. (i): Set

ω = u s + η, then ω ≥ u s ≥ (W α,β [v q ]) s ≥ c 79 (W α,β [(W α,β [ω]) q ]) s .
By (2.4) in Lemma 2.2, we get

ω ≥ c 80 W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω] s , which implies ˆRN W αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [χ B t (x) ω] s dy ≤ c 81 ω(B t (x)) ∀ x ∈ R N , ∀ t > 0.
Applying Proposition 2.4 to µ = ω with (α, β, q) = αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q + 1, s , we get (2.38).

(ii) We define ω as above and we have

ω ≥ u s ≥ W δd α,β [v q ] s ≥ c 82 W δd α,β W δd(.) α,β [ω] q s
a.e. in Ω, which leads to

ω ≥ c 83 W δ 2 d αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω]
M ω f (x) = sup t>0 1 ω(B t (x)) ˆBt(x) |f |dω. Let K ⊂ Ω be compact. Set r K = dist(K, ∂Ω) and Ω K = {x ∈ Ω : d(x, K) < r K /2}. Then, for any Borel set E ⊂ K, c 84 ˆΩ (M ω χ E ) sq (β-1) 2 W δ 2 d(x) αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω] s dx ≤ ˆΩ (M ω χ E ) sq (β-1) 2 dω.
Since M ω is a bounded linear map on L p (R N , dω) for any p > 1 and

(M ω χ E ) sq (β-1) 2 W δ 2 d(x) αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω] s ≥ ˆδ 2 d(x) 0 ω(B t (x) ∩ E) ω(B t (x)) ω(B t (x) t N -αβ(q+β-1) q sq (β-1) 2 dt t , we obtain ˆΩ W δ 2 d(x) αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω E ] s dx ≤ c 85 ω(E),
where ω E = χ E ω. Note that if x ∈ Ω and d(x) ≤ r K /8, then B t (x) ⊂ Ω\Ω K for all t ∈ (0, δd(x) 2 ); indeed, for all y ∈ B t (x)

d(y, ∂Ω) ≤ d(x, ∂Ω) + |x -y| < (1 + δ)d(x, ∂Ω) < 1 4 r K , thus d(y, K) ≥ d(K, ∂Ω) -d(y, ∂Ω) > 3 4 r K > 1 2 r K , which implies y / ∈ Ω K . We deduce that W δ 2 d(x,∂Ω) αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω E ](x) ≥ W δ 16 r K αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω E ](x) ∀x ∈ Ω,
and

W δ 16 r K αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω E ](x) = 0 ∀x ∈ Ω c .
Hence we obtain ˆRN W

δ 16 r K αβ(q+β-1) q+(β-1) 2 , (β-1) 2 q +1 [ω E ] s dx ≤ c 85 ω(E) ∀E ⊂ K, E Borel. (2.41)
Applying Proposition 2.5 with µ = χ K∩B 2 -6 δr K (x) ω we get (2.40), which completes the proof.

Quasilinear Dirichlet problems

Let Ω be a bounded domain in R N . If µ ∈ M b (Ω), we denote by µ + and µ -respectively its positive and negative parts in the Jordan decomposition. We denote by M 0 (Ω) the space of measures in Ω which are absolutely continuous with respect to the c Ω 1,p -capacity defined on a compact set K ⊂ Ω by

c Ω 1,p (K) = inf ˆΩ |∇ϕ| p dx : ϕ ≥ χ K , ϕ ∈ C ∞ c (Ω) .
We also denote M s (Ω) the space of measures in Ω with support on a set of zero c Ω 1,pcapacity. Classically, any µ ∈ M b (Ω) can be written in a unique way under the form

µ = µ 0 + µ s where µ 0 ∈ M 0 (Ω) ∩ M b (Ω) and µ s ∈ M s (Ω) ∩ M b (Ω). It is well known that any µ 0 ∈ M 0 (Ω) ∩ M b (Ω) can be written under the form µ 0 = f -div g where f ∈ L 1 (Ω) and g ∈ L p (Ω, R N ).
For k > 0 and s ∈ R we set T k (s) = max{min{s, k}, -k}. If u is a measurable function defined in Ω, finite a.e. and such that T k (u) ∈ W 1,p loc (Ω) for any k > 0, there exists a measurable function v : Ω → R N such that ∇T k (u) = χ {|u|≤k} v a.e. in Ω and for all k > 0. We define the gradient a.e. ∇u of u by v = ∇u. We recall the definition of a renormalized solution given in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF].

Definition 3.1 Let µ = µ 0 + µ s ∈ M b (Ω).
A measurable function u defined in Ω and finite a.e. is called a renormalized solution of

-∆ p u = µ in Ω u = 0 on ∂Ω, (3.1)
if T k (u) ∈ W 1,p 0 (Ω) for any k > 0, |∇u| p-1 ∈ L r (Ω) for any 0 < r < N N -1 ,
+ ˆΩ ϕdλ + k -ˆΩ ϕdλ - k , for every ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). Remark 3.2 We recall that if u is a renormalized solution to problem (3.1), then |∇u| p (|u|+1) r ∈ L 1 (Ω) for all r > 1. Furthermore, u ≥ 0 a.e. in Ω if µ ∈ M + b (Ω).
The following general stability result has been proved in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF]Th 4.1].

Theorem 3.3 Let µ = µ 0 + µ + s -µ - s , with µ 0 = F -div g ∈ M 0 (Ω) and µ + s , µ - s belonging to M + s (Ω). Let µ n = F n -div g n + ρ n -η n with F n ∈ L 1 (Ω), g n ∈ (L p (Ω)) N and ρ n , η n belonging to M + b (Ω). Assume that {F n } converges to F weakly in L 1 (Ω), {g n } converges to g strongly in (L p (Ω)) N and (div g n ) is bounded in M b (Ω); assume also that {ρ n } converges to µ +
s and {η n } to µ - s in the narrow topology. If {u n } is a sequence of renormalized solutions of (3.1) with data µ n , then, up to a subsequence, it converges a.e. in Ω to a renormalized solution u of problem (3.1). Furthermore, T k (u n ) converges to T k (u) in W 1,p 0 (Ω) for any k > 0.

We also recall the following estimate [20, Th 2.1].

Proposition 3.4

Let Ω be a bounded domain of R N . Then there exists a constant C > 0, depending on p and N such that if µ ∈ M + b (Ω) and u is a nonnegative renormalized solution of problem (3.1) with data µ, there holds

1 c 86 W d(x,∂Ω) 3 1,p [µ](x) ≤ u(x) ≤ c 86 W
Proof of Theorem C. The condition is necessary. Assume that (1.14) admits a nonnegative renormalized solutions (u, v). By Proposition 3.4 there holds

u(x) ≥ c 87 W d(x,∂Ω) 3 1,p [v q1 + µ](x) v(x) ≥ c 87 W d(x,∂Ω) 3 1,p [u q2 + µ](x)
a.e. in Ω.

Hence, we infer (1.15) from Proposition 2.9-(ii).

Sufficient conditions. Let {(u m , v m )} m∈N be a sequence of nonnegative renormalized solutions of the following problems for m ∈ N,

-∆ p u m+1 = v q1 m + µ in Ω -∆ p v m+1 = u q2 m + η in Ω u m+1 = v m+1 = 0 on ∂Ω, (3.3)
with initial condition (u 0 , v 0 ) = 0. The sequences {u m } and {v m } can be constructed in such a way that they are nondecreasing (see e.g. [START_REF] Phuc | Singular quasilinear and Hessian equation and inequalities[END_REF]). By Proposition 3.4 we have

u m+1 ≤ c 86 W R 1,p [v q1 m + µ](x) v m+1 ≤ c 86 W R 1,p [u q2 m + η](x) a.e. in Ω,
where R = 2 diam (Ω). Thus, by Proposition 2.8 there exists a constant M * > 0 depending only on N, p, q 1 , q 2 , R such that if

ω(K) ≤ M * Cap G p(q 1 +p-1) q 1 , q 1 q 2 q 1 q 2 -(p-1) 2 (K) (3.4) for any compact set K ⊂ R N with dω = W R 1,p [µ] q2 dx + dη, then v m ≤ c 73 W R 1,p [ω], u m ≤ c 74 W R 1,p [ W R 1,p [ω] q1 ] + c 72 W R 1,p [µ] ∀ k ≥ 0 (3.5)
in Ω, and

W R 1,p [ω] ∈ L q2 (Ω), W R 1,p [ W R 1,p [ω] q1 ] + W R 1,p [µ] ∈ L q1 (Ω). (3.6)
This implies that {u m }, {v m } m∈N are well defined and nondecreasing. Thus {(u m , v m )} converges a.e in Ω to some functions (u, v) which satisfies (1.17) in Ω. Furthermore, we deduce from (3.6) and the monotone convergence theorem that u q1 m → u q1 and v q2 m → u q2 in L 1 (Ω). Finally we infer that u is a renormalized solution of (1.14) by Theorem 3.3.

p-superharmonic functions and quasilinear equations in R N

We recall some definitions and properties of p-superharmonic functions (see e.g. [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF], [START_REF] Kilpelainen | Degenerate elliptic equation with measure data and nonlinear potentials[END_REF], [START_REF] Kilpelainen | The Wiener test and potential estimates for quasilinear elliptic equations[END_REF] for general properties and [START_REF] Véron | Local and global aspects of quasilinear degenerate elliptic equations. Quasilinear elliptic singular problems[END_REF] for a simple presentation).

Definition 4.1 A function u is said to be p-harmonic in R N if u ∈ W 1,p loc (R N ) and -∆ p u = 0 in D (R N ); it is always C 1 . A function u is called a p-supersolution in R N if u ∈ W 1,p loc (R N ) and -∆ p u ≥ 0 in D (R N ). Definition 4.2 A lower semicontinuous (l.s.c) function u : R N → (-∞, ∞] is called p- super-harmonic if u is not identically infinite and if, for all open D ⊂⊂ R N and all v ∈ C(D), p-harmonic in D, v ≤ u on ∂D implies v ≤ u in D.
Let u be a p-superharmonic in R N . It is well known that u ∧ k := min{u, k} ∈ W 1,p loc (R N ) is a p-supersolution for all k > 0 and u < ∞ a.e in R N , thus, u has a gradient (see the previous section). We also have

|∇u| p-1 ∈ L q loc (R N ), |∇u| p (|u|+1) r ∈ L 1 loc (R N ) and u ∈ L s loc (R N ) for 1 ≤ q < N N -1 and r > 1, 1 ≤ s < N (p-1)
N -p (see [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]Theorem 7.46]). Thus for any 0 ≤ ϕ ∈ C 1 c (Ω), by the dominated convergence theorem,

-∆ p u, ϕ = ˆRN |∇u| p-2 ∇u.∇ϕdx = lim k→∞ ˆRN |∇(u ∧ k)| p-2 ∇(u ∧ k).∇ϕ ≥ 0.
Hence, by the Riesz Representation Theorem, there is a nonnegative Radon measure denoted by µ[u], called the Riesz measure, such that

-∆ p u = µ[u] in D (R N ).
The following weak convergence result for Riesz measures proved in [START_REF] Trudinger | On the weak continuity of elliptic operators and applications to potential theory[END_REF] will be used to obtain the existence of p-superharmonic solutions to quasilinear equations. Proposition 4.3 Suppose that {u n } is a sequence of nonnegative p-superharmonic functions in R N that converges a.e to a p-superharmonic function u. Then the sequence of measures {µ[u n ]} converges to µ[u] in the weak sense of measures.

The proof of the next result can be found in [START_REF] Phuc | Quasilinear and Hessian equations of Lane-Emden type[END_REF].

Proposition 4.4 Let µ be a measure in M + (R N ). Suppose that W 1,p [µ] < ∞ a.e. Then there exists a nonnegative p-superharmonic function u in R N such that -∆ p u = µ in D (R N ), inf R N u = 0 and 1 c 86 W 1,p [µ](x) ≤ u(x) ≤ c 86 W 1,p [µ](x), (4.1)
for almost all x in R N , where the constant c 86 is the one of Proposition 3.4. Furthermore any p-superharmonic function u in R N , such that inf R N u = 0 satisfies (4.1) with µ = -∆ p u.

Proof of Theorem A. The condition is necessary. Assume that (1.14) admits a nonnegative p-superharmonic functions (u, v). By Proposition 4.4 there holds

u(x) ≥ c 87 W 1,p [v q1 + µ](x), v(x) ≥ c 87 W 1,p [u q2 + η](x)
for almost all x ∈ Ω.

Hence, we obtain (1.10) from Proposition 2.9-(i).

The condition is sufficient. Let {(u m , v m )} m∈N be a sequence of nonnegative p-superharmonic solutions of the following problems for m ∈ N,

-∆ p u m+1 = v q1 m + µ in R N -∆ p v m+1 = u q2 m + η in R N inf R N u m+1 = inf R N v m+1 = 0, (4.2)
with (u 0 , v 0 ) = (0, 0). As in the proof of Theorem C we can assume that {u m } and {v m } are nondecreasing. By Proposition 4.4 we have

u m+1 ≤ c 86 W 1,p [v q1 m + µ](x) v m+1 ≤ c 86 W 1,p [u q2 m + η](x) for all x ∈ Ω.
Thus, by Proposition 2.7 there exists a constant c > 0 depending only on N, p, q 1 , q 2 such that, if

ω(K) ≤ M * Cap I p(q 1 +p-1) q 1 , q 1 q 2 q 1 q 2 -(p-1) 2 (K) (4.3)
for any compact set K ⊂ R N with dω = (W 1,p [µ]) q2 dx + dη, then there holds in Ω,

v m ≤ c 69 W 1,p [ω], u m ≤ c 70 W 1,p [(W 1,p [ω]) q1 ] + c 68 W 1,p [µ] for all m ≥ 0, (4.4)
and

W 1,p [ω] ∈ L q2 loc (R N ), W 1,p [(W 1,p [ω]) q1 ] + W 1,p [µ] ∈ L q1 loc (R N ). (4.5)
This implies that {u m }, {v m } are well defined and nondecreasing. Thus {(u m , v m )} converges a.e in R N to some functions (u, v) which satisfies (1.17) in R N . Furthermore, we infer from (3.6) and the monotone convergence theorem that

u q1 m → u q1 , v q2 m → u q2 in L 1 loc (R N )
. By Proposition 4.3 we deduce that (u, v) are nonnegative p-superharmonic solutions of (1.9).

Hessian equations

In this section Ω ⊂ R N is either a bounded domain with a C 2 boundary or the whole R N . For k = 1, ..., N and u ∈ C 2 (Ω) the k-hessian operator F k is defined by

F k [u] = S k (λ(D 2 u)),
where λ(D 2 u) = λ = (λ 1 , λ 2 , ..., λ N ) denotes the eigenvalues of the Hessian matrix of second partial derivative D 2 u and S k is the k-th elementary symmetric polynomial that is

S k (λ) = 1≤i1<...<i k ≤N λ i1 ...λ i k .
We can see that

F k [u] = D 2 u k ,
where for a matrix A = (a ij ), [A] k denotes the sum of the k-th principal minors. We assume that ∂Ω is uniformly (k-1)-convex, that is

S k-1 (κ) ≥ c 0 > 0 on ∂Ω,
for some positive constant c 0 , where κ = (κ 1 , κ 2 , ..., κ n-1 ) denote the principal curvatures of ∂Ω with respect to its inner normal.

Definition 5.1 An upper-semicontinuous function u

: Ω → [-∞, ∞) is k-convex (k-subharmonic) if, for every open set Ω ⊂ Ω ⊂ Ω and for every function v ∈ C 2 (Ω ) ∩ C(Ω ) satisfying F k [v] ≤ 0 in Ω , the following implication is true u ≤ v on ∂Ω =⇒ u ≤ v in Ω .
We denote by Φ k (Ω) the class of all k-subharmonic functions in Ω which are not identically equal to -∞.

The following weak convergence result for k-Hessian operators proved in [START_REF] Trudinger | Hessian measures II[END_REF] is fundamental in our study.

Proposition 5.2 Let Ω be either a bounded uniformly (k-1)-convex in R N or the whole R N . For each u ∈ Φ k (Ω), there exists a nonnegative Radon measure

µ k [u] in Ω such that 1 µ k [u] = F k [u] for u ∈ C 2 (Ω).
2 If {u n } is a sequence of k-convex functions which converges a.e to u, then µ k [u n ] µ k [u] in the weak sense of measures.

As in the case of quasilinear equations with measure data, precise estimates of solutions of k-Hessian equations with measures data are expressed in terms of Wolff potentials. The next results are proved in [START_REF] Trudinger | Hessian measures II[END_REF][START_REF] Labutin | Potential estimates for a class of fully nonlinear elliptic equations[END_REF][START_REF] Phuc | Quasilinear and Hessian equations of Lane-Emden type[END_REF]. Theorem 5.3 Let Ω ⊂ R N be a bounded C 2 , uniformly (k-1)-convex domain. Let µ be a nonnegative Radon measure in Ω which can be decomposed under the form

µ = µ 1 + f,
where µ 1 is a measure with compact support in Ω and f ∈ L q (Ω) for some q >

N 2k if k ≤ N 2 , or p = 1 if k > N 2 .
Then there exists a nonnegative function u in Ω, continuous near ∂Ω, such that -u ∈ Φ k (Ω) and u is a solution of the problem

F k [-u] = µ in Ω, u = 0 on ∂Ω.
Furthermore, any nonnegative function u such that -u ∈ Φ k (Ω) which is continuous near ∂Ω and is a solution of above equation, satisfies

1 c 88 W d(x,∂Ω) 8 2k k+1 ,k+1 [µ] ≤ u(x) ≤ c 88 W 2diam Ω 2k k+1 ,k+1 [µ](x), (5.1)
where c 88 is a positive constant independent of x, u and Ω.

Theorem 5.4 Let µ be a measure in M + (R N ) and 2k < N . Suppose that W 2k k+1 ,k+1 [µ] < ∞ a.e. Then there exists u,

-u ∈ Φ k (R N ) such that inf R N u = 0 and F k [-u] = µ in R N and 1 c 88 W 2k k+1 ,k+1 [µ](x) ≤ u(x) ≤ c 88 W 2k k+1 ,k+1 [µ](x), (5.2)
for all x in R N . Furthermore, if u is a nonnegative function such that inf R N u = 0 and -u ∈ Φ k (R N ), then (5.2) holds with µ = F k [-u].

Proof of Theorem E. The condition is necessary. Assume that (1.21) admits a nonnegative solution (u, v), continuous near ∂Ω, such that -u, -v ∈ Φ k (Ω) and u s2 , v s1 ∈ L 1 (Ω). Then by Theorem 5.3 we have

u(x) ≥ 1 c 88 W d(x,∂Ω) 8 2k k+1 ,k+1 [v s1 + µ](x) v(x) ≥ 1 c 88 W d(x,∂Ω) 8 2k k+1 ,k+1 [u s2 + η](x)
for almost all x ∈ Ω.

Using the part 2 of Proposition 2.9, we conclude that (1.22) holds. The condition is sufficient. We define a sequence of nonnegative functions u m , v m , continuous near ∂Ω and such that -u m , -v m ∈ Φ k (Ω), by the following iterative scheme for m ≥ 0,

F k [-u m+1 ] = v s1 m + µ in Ω, F k [-v m+1 ] = u s2 m + η in Ω, u m+1 = v m+1 = 0 on ∂Ω.
(5.3)

Clearly, we can assume that {u m } is nondecreasing as in [START_REF] Phuc | Singular quasilinear and Hessian equation and inequalities[END_REF]. By Theorem 5.3 we have

u m+1 ≤ c 88 W R 2k k+1 ,k+1 [v s1 m + µ] , v m+1 ≤ c 88 W R 2k k+1 ,k+1 [u s2 m + µ] in Ω, (5.4)
where R = 2 diam (Ω). Then, by Proposition 2.8, there exists a constant M * > 0 depending only on N, p, q 1 , q 2 , R such that if

ω(K) ≤ M * Cap G 2k(s 1 +k) s 1 , s 1 s 2 s 1 s 2 -k 2 (K) for any compact set K ⊂ R N with dω = W R 2k k+1 ,k+1 µ] s2 dx + dη, then there holds, v m ≤ c 73 W R 2k k+1 ,k+1 [ω], u m ≤ c 74 W R 2k k+1 ,k+1 [ W R 2k k+1 ,k+1 [ω] s1 ] + c 72 W R 2k k+1 ,k+1 [µ]
in Ω, for all m ∈ N, for some positive constants c 72 , c 73 and c 74 depending only on N, k, s 1 , s 2 , R. Note that we can write

v s1 m + µ = µ 1 + χ Ω δ v s1 m + (1 -χ Ω δ )v s1 m + f , and
u s2 m + η = η 1 + χ Ω δ u s2 m + (1 -χ Ω δ )u s2 m + g ,
where Ω δ = {x ∈ Ω : d(x, ∂Ω) > δ} and δ > 0 is small enough and since u m is continuous near ∂Ω, then v s1 m + µ, u s2 m + η satisfy the assumptions of the data in Theorem 5.3. Therefore the sequence {u m } is well defined and nondecreasing. Thus, {u m } converges a.e in Ω to some function u which satisfies (1.24) in Ω. Furthermore, by the monotone convergence theorem there holds v s1 m → v, u s2 m → u in L 1 (Ω). Finally, by Proposition 5.2, we infer that (1.21) admits a nonnegative solutions u, v, continuous near ∂Ω, with -u, -v ∈ Φ k (Ω) satisfying (1.24).

Proof of Theorem F The condition is necessary. Assume that (1.21) admits nonnegative solution (u, v), such that -u, -v ∈ Φ k (R N ) and u s2 , v s1 ∈ L 1 loc (R N ). Then by Theorem 5.3 we have

u(x) ≥ 1 c 88 W 2k k+1 ,k+1 [v s1 + µ](x) v(x) ≥ 1 c 88 W 2k k+1 ,k+1 [u s2 + η](x) for almost all x ∈ R N .
Using Proposition 2.9-(ii), we conclude that (1.22) holds.

The condition is sufficient. We defined a sequence of nonnegative functions u m , v m , continuous near ∂Ω and such that -u m , -v m ∈ Φ k (Ω), by the following iterative scheme for m ≥ 0,

F k [-u m+1 ] = v s1 m + µ in R N , F k [-v m+1 ] = u s2 m + η in R N , inf R N u m+1 = inf R N v m+1 = 0.
As in the previous proofs {u m } is nondecreasing. By Theorem 5.3 we have

u m+1 ≤ c 88 W 2k k+1 ,k+1 [v s1 m + µ] v m+1 ≤ c 88 W 2k k+1 ,k+1 [u s2 m + µ] a.e. in R N .
Then, by Proposition 2.7, there exists a constant M * > 0 depending only on N, p, q 1 , q 2 , R such that if

ω(K) ≤ M * Cap I 2k(s 1 +k) s 1 , s 1 s 2 s 1 s 2 -k 2 (K) for any compact set K ⊂ R N with dω = W 2k k+1 ,k+1 µ] s2 dx + dη, then v m ≤ c 69 W 2k k+1 ,k+1 [ω], u m ≤ c 70 W 2k k+1 ,k+1 [ W 2k k+1 ,k+1 [ω] s1 ] + c 68 W 2k k+1 ,k+1 [µ]
in Ω, for all m ∈ N, where c 68 , c 69 and c 70 depend on N, k, s 1 , s 2 , R. Therefore the sequence {u m } is well defined and nondecreasing. Thus, {u m } converges a.e in Ω to some function u for which (1.27) is satisfied in R N . Furthermore, by the monotone convergence theorem we have v s1 m → v, u s2 m → u in L 1 loc (R N ). Finally, by Proposition 5.2, we obtain that (1.21) admits a nonnegative solutions u, v with -u, -v ∈ Φ k (R N ) satisfying (1.27). Remark 6.3 The condition 0 < q 1 < N N -1 is a necessary and sufficient condition in order

(I 1 [ω]) q1 be locally integrable in R N for any ω ∈ M + b (R N ). Theorem 6.4 Let q 1 ≥ 1, q 1 q 2 > 1 and ω ∈ M b (R N + ). If ω(K) ≤ c 90 Cap ρ q 1 +2 q 1 , q 1 q 2 q 1 q 2 -1 (K) ∀ K ⊂ R N + , K Borel,
for some c 90 > 0, then

I 1 W q 1 +2 q 1 +1 , q 1 +1 q 1 [ω] q2 ρ χ R N + ≤ c 91 I 1 [ω] a.e. in R N + . (6.4) Proof. Step 1. For any compact K ⊂ x ∈ R N + : I q 1 +2 q 1 [f ρ χ R N + ](x) > λ , we have ω(K) ≤ c 90 Cap ρ q 1 +2 q 1 , q 1 q 2 q 1 q 2 -1 (K) ≤ c 90 λ -q 1 q 2 q 1 q 2 -1 ˆRN + f q 1 q 2 q 1 q 2 -1 ρ dx
by assumption and the definition of the capacity. Hence,

λ q 1 q 2 q 1 q 2 -1 ω I q 1 +2 q 1 [f ρ χ R N + ] > λ ≤ c 90 ˆRN + f q 1 q 2 q 1 q 2 -1 ρ dx ∀ λ > 0.
This implies an estimate in Lorentz space,

||I q 1 +2 q 1 [f ρ χ R N + ]|| L q 1 q 2 q 1 q 2 -1 ,∞ (R N ,dω) ≤ ||f || L q 1 q 2 q 1 q 2 -1 (R N ,χ R N + ρ dx)
∀ f ≥ 0. (6.5)

Step 2. Since, for any g ∈ C c (R N + ), ˆRN + I q 1 +2 q 1

[gω]f ρ dx = ˆRN I q 1 +2 q 1

[f ρ χ

R N +

]gdω, we infer, using duality between L p,1 and L p ,∞ , Holder's inequality therein and (6.5), that ˆRN + I q 1 +2 q 1

[gω]f ρ dx ≤ ||I q 1 +2 q 1

[f ρ χ

R N + ]|| L q 1 q 2 q 1 q 2 -1 ,∞ (R N ,dω)
||g|| L q 1 q 2 ,1 (R N ,dω)

≤ ||f || L q 1 q 2 q 1 q 2 -1 (R N ,χ R N + ρ dx)
||g|| L q 1 q 2 ,1 (R N ,dω) ∀ f, g ≥ 0.

Therefore, ||I q 1 +2 q 1

[gω]|| L q 1 q 2 (R N ,χ R N + ρ dx) ≤ ||g|| L q 1 q 2 ,1 (R N ,dω) . (6.6)

Step 3. Taking g = χ B t (x) and since for q 1 ≥ 1

W q 1 +2
q 1 +1 , q 1 +1 q 1

[ν](x) = ˆ∞ 0 ν(B ρ (x))

ρ N -q 1 +2 q 1 q1 dx ≤ c 89 ˆ∞ 0 ν(B ρ (x)) ρ N -q 1 +2 q 1 dx q1 = c 89 I q 1 +2
q 1

[ν](x)

q1 ∀ν ∈ M + b (R N ), ∀x ∈ R N ,
We have ˆt 0 max{x N , s}ds t max{x N , t}, ˆ∞ t ω(B 2r (x))

r N -q 1 +2 q 1 q1 dr r q2-1

≤ c 94   ˆ∞ t r -q 1 +2 q 1 (q 1 q 2 -1) (max{x N , r})

1 q 1 q 2 -1 q1 dr r   q-1
t -(q 1 +2)(q 2 -1) q 1 q 2 -1 (max{x N , t}) -q 1 (q 1 -1) q 1 q 2 -1 , by (6.7) and ω(B 2t (x))

t N -q 1 +2 q 1
q1-1 t 2 q 1 ≤ c 95 t -q 1 +2 q 1 (q 1 q 2 -1) (max{x N , t})

1 q 1 q 2 -1 q2-1 t 2 q 1
= c 95 t -(q 1 +2)(q 1 -1) q 1 (q 1 q 2 -1) + 2 q 1 (max{x N , t}) -q 1 -1 q 1 q 2 -1 .

Thus, ˆt 0 max{x N , s}ds ˆ∞ t ω(B 2r (x))

r N -q 1 +2 q 1 q1 dr r q2-1 ω(B 2t (x)) t N -q 1 +2 q 1
q1-1 t 2/q1 ≤ c 96 , and we obtain (6.9). Combining (6.12), (6.13) and (6.14) we obtain (6.10). Moreover, we deduce (6.11) from (6.10) and [1, Theorem 2. where ω(x) = ρ(P[σ 1 ]) q2 + σ 2 in R N . Therefore, we infer that if

I 1 (I 2 [(I 1 [ω]) q1 ]) q2 χ R N + ρ ≤ c 103 I 1 [ω] in R N + (6.18)
for some c 103 > 0 small enough, then (6.1) admits a positive solution (u, v). On the other hand, we deduce (6.18) from Lemma 6.2 and Theorem 6.4. The proof is complete.

Remark 6. [START_REF] Bidaut-Véron | Removable singularities and existence for a quasilinear equation with absorption or source term and measure data[END_REF] The system

-∆u = v q1 + 1 µ in Ω -∆v = u q2 + 2 η in Ω u = 3 σ 1 , v = 4 σ 2 in ∂Ω, (6.19)
where d(.)µ, d(.)λ belong to M + b (Ω), σ 1 , σ 2 to M + (∂Ω) and the j are positive numbers, is analyzed in [START_REF] Bidaut-Véron | Semilinear elliptic equations and systems with measure data: existence and a priori estimates[END_REF]Th 4.6]. Therein it is proved that if ˆΩ (G[µ] + P[λ]) max{q1,q2} d(x)dx < ∞, (6.20)

which is equivalent to a capacitary estimate, and min q 2 q 1 + 1 q 2 + 1 , q 1 q 2 + 1

q 1 + 1 < N + 1 N -1 , (6.21)
and if the j are small enough, then (6.19) admits a positive solution. Now condition (6.21) is a subcriticality assumption (for at least one of the two exponents q j ) since there is no condition on the boundary measures.

  2k k+1 ,k+1 [µ B (x)] q dx ≤ c 4 µ(B) for all ball B s.t. B ∩ supp µ = ∅,(1.6)

s

  a.e. in Ω, by inequality (2.10) in Lemma 2.3. Let M ω denote the centered Hardy-Littlewood maximal function which is defined for any f ∈ L 1 loc (R N , dω) by

  and u has the property that for any k > 0 there exist λ + k and λ - k belonging to M + b ∩ M 0 (Ω), respectively concentrated on the sets u = k and u = -k, with the property that µ + k µ + s , µ - k λ - s in the narrow topology of measures and such that ˆ{|u|<k} |∇u| p-2 ∇u.∇ϕdx = ˆ{|u|<k} ϕdµ 0

Lemma 6 . 5

 65 Let α > 0, s > 1 such that α + 2 s < N -1 where s = s s-1 . For all η ∈ M + (R N -1 ), there holdsˆRN (I α [η ⊗ δ {x N =0} ]) s x N dx ˆRN-1 ˆ∞ 0 η(B t (x ))where I β is the Riesz potential of order β in R N -1 . As a consequence, we haveCap ρ α,s (E × {x N = 0}) Cap I α+2/s -1 ,s (E) ∀E ⊂ R N -1 , E Borel.(6.11)Proof. We haveˆRN (I α [η ⊗ δ {x N =0} ]) s x n dx ≥ ˆRN ˆ4x N 2x N (η ⊗ δ {x N =0} )(B r (x)) [η ⊗ δ {x N =0} ]) s x n dx ≤ ˆRN ˆ∞ x N η(B r (x )) r N -α dr r s dx N dx ≤ c 99 ˆRN-1 ˆ∞ 0 η(B t (x )) t N -1-α-2 s s dt t dx .

(6. 13 )

 13 On the other hand, by [20, Proposition 5.1], there holds

5 . 1 ]

 51 , which ends the proof.Proof of Theorem 6.1 The following estimates are cclassicalG(x, y) x N y N |x -y| N -2 max{|x -y|, x N , y N } 2 ≤ c 100 y N |x -y| N -1 ,(6.15)P(x, z) = c 101 x N |x -z| N ≤ c 101 1 |x -z| N -1 . (6.16) Thus, G [(P[σ 1 ]) q2 ] + P[σ 2 ] ≤ c 102 I 1 [ω],(6.17)

26 )

 26 Conversely,, if µ and η are bounded, there exists positive constant c 31 depending on N, k, s 1 , s 2 such that, if 0 < s 1 < N k N -2k and (1.26) holds with c 31 instead of c 30 , then (1.25) admits a nonnegative solution (u, v) with -u and -

  . Statement (i): We assume that (2.19) holds. Put ω = (W α,β [µ]) s + η and apply (2.19) to K = B 2ρ (x). Since by homogeneity

	Cap I αβ(q+β-1) q	,	qs qs-(β-1) 2

diam (Ω) 1,p [µ](x)a.e. in Ω.(3.2)

Further results

The method exposed in the previous sections, can be applied to types of problems. We give below an example for a semilinear system in R N + = {x = (x , x N ), x ∈ R N -1 , x N > 0}.

where we have identified ∂R N + and R N -1 . We denote by P (resp. G) the Poisson kernel in R N + (resp the Green kernel in R N ). The Poisson potential and the Green potential, P[.] and G[.], associated to -∆ are defined respectively by

see [START_REF] Marcus | Nonlinear second order elliptic equations involving measures[END_REF]. We set ρ(x) = x N and define the capacity Cap ρ α,s by

If there exists a constant c > 0 such that if

for all Borel sets K ⊂ R N + and G ⊂ R N -1 , then the problem (6.1) admits a solution.

All solutions in above theorem are understood in the usual very weak sense:

a.e. in R N + .

To prove Theorem 6.1 we need the following basic estimate, Lemma 6.2 Assume that 0 < q 1 < N N -1 . Then for any ω ∈ M + b (R N ),

where c 89 > 0 depends on q 1 , q 2 and N .

Proof. The proof of Lemma 6.2 is similar to the one of Lemma 2.2 and details are omitted.

we deduce that for almost all x ∈ R N + , ˆRN

ρ dy ≤ c 90 ω(B t (x)), from (6.6), which implies

ρ dy r N max{x N , r} for any x ∈ R N + , r > 0 where the symbol is defined by

from which follows

Therefore, if the following inequality holds

it will imply (6.4).

Step 4. We claim that (6.9) holds. Since B r (y) ∈ B 2r (x), y ∈ B t (x), r ≥ t,

By integration by part,

= q 2 ˆ∞ 0 ˆt 0 max{x N , s}ds ˆ∞ t ω(B 2r (x))

r N -q 1 +2 q 1 q1 dr r q2-1 ω(B 2t (x))

t N -q 1 +2 q 1 q1 dt t = q 2 ˆ∞ 0 ˆt 0 max{x N , s}ds ˆ∞ t ω(B 2r (x))

r N -q 1 +2 q 1 q1 dr r q2-1 ω(B 2t (x))