Keywords: long run economic growth, interaction between human capital and trade openness, panel data estimations croissance de long terme, complémentarité capital humain / ouverture économique, économétrie des données de panel JEL classi…cation: C23, N10, O11, O40, O50

. The results strongly support the existence of an interaction between trade and education which appears to promote catch-up between countries. Moreover, using a structural change test, I also show that such an interaction is not constant but hinges on the level of GDP per capita already attained by the economy.

Keywords: growth regressions

Introduction

According to the economic literature, human capital should be an important variable in explaining the di¤erences in growth rates among countries. Lucas [START_REF] Lucas | On the mechanisms of economic development[END_REF], Becker, Murphy and Tamura [START_REF] Becker | Human capital, fertility and economic growth[END_REF] and Azariadis and Drazen [START_REF] Azariadis | Threshold externalities in economic development[END_REF], among others, have provided persuasive theoretical models in which growth in GDP is driven by human capital accumulation. Yet, while cross-section analysis seems to con…rm this relation 1 , in studies using panel data, human capital appears either insigni…cantly, or even negatively related to growth 2 . These results suggest that if human capital plays the important role implied by theory, then its in ‡uence on growth may come through more indirect channels. This paper considers one such a mechanism by examining whether human capital a¤ects growth through trade channel.

The relation between trade and growth has long been debated in the literature of economic growth 3 . The often cited bene…cial e¤ects of trade openness include the international transmission of knowledge 4 , e¢cient sectorial reallocation 4 , increasing returns to scale 5 , and incentives to reduce distortions 6 . However, while theory points to several bene…ts of trade openness, the empirical tests are not precise enough to discriminate between the sources of gains from trade. The cross-country studies -from Balassa [START_REF] Balassa | Exports and economic growth: further evidence[END_REF] to Edwards [START_REF] Edwards | Openness, productivity and growth: what do we really know ?[END_REF], Sachs and Warner [START_REF] Sachs | Fundamental sources of long-run growth[END_REF] and Vamvakidis [START_REF] Vamvakidis | Regional trade agreements versus broad liberalisation: which path leads to faster growth ? IMF Working Paper[END_REF] -generally yield evidence of a positive impact of openness on growth. However, few empirical studies have attempted to determine the relative importance of these channels. In one such study, Wacziarg [START_REF] Wacziarg | Measuring the dynamic gains from trade[END_REF] …nds that trade openness mainly a¤ects growth …rst by raising the share of investment in GDP, and secondly by aiding technological transmissions.

This last result is most interesting in the context of this paper since it implies an active role for human capital, as underlined by Pissarides [START_REF] Pissarides | Learning by trading and the returns to human capital in developing countries[END_REF]. If, indeed, trade openness allows countries to gain access to technologies available abroad, it does not guarantee that they are capable to properly implement them for domestic use. Economies may need a certain stock of human capital in order to enhance their ability to use the technologies they acquire 1 See, for instance, Barro [START_REF] Barro | Economic growth in a cross-section of countries[END_REF] 2 Knight, Loayza and Villanueva [START_REF] Knight | Testing the neoclassical theory of economic growth: a panel data approach[END_REF], Islam [START_REF] Islam | Growth empirics: a panel data approach[END_REF], Caselli, Esquivel and Lefort [START_REF] Caselli | Reopening the convergence debate: a new look at cross-country growth empirics[END_REF] 3 Sachs and Warner [START_REF] Sachs | Fundamental sources of long-run growth[END_REF], Edwards [START_REF] Edwards | Openness, trade liberalisation, and growth in developing countries[END_REF] 4 Grossman and Helpman[19] 5 Romer [START_REF] Romer | Endogenous technological change[END_REF] 6 Berthelemy, Pissarides and Varoudakis [START_REF] Berthelemy | Rent seeking, the return to human capital and economic development[END_REF] through trade. In other words, imitation requires not only an object to imitate (provided through trade), but skills as well. Thus, human capital would a¤ect growth through its interaction with trade openness. However, this channel is not the only one to justify a complementary e¤ect between education and trade. As already underlined earlier, openness could a¤ect growth by reducing distortions. Murphy, Shleifer and Vishny [START_REF] Murphy | The allocation of talent: implications for growth[END_REF] and Berthelemy, Pissarides and Varoudakis [START_REF] Berthelemy | Rent seeking, the return to human capital and economic development[END_REF] assume that policy distortions may induce the development of human capital intensive rent-seeking activities. Therefore, trade openness, by reducing these distortions, may generate a reallocation of human capital from rent-seeking activities towards growth enhancing sectors. Consequently, a high level of education combined with openness can lead to faster growth rates.

These two potential e¤ects arising from the interaction of trade and education will have di¤erent impacts. The …rst involves a process of imitation which implies the closing of the technological gap between countries. Technology will circulate through trade leading to a catch-up e¤ect between less developed and more technologically advanced economies. The convergence involved here is di¤erent from the one derived from the Solow [START_REF] Solow | A contribution to the theory of economic growth[END_REF] model on which the Barro[5] regressions are based. Catch-up is not due to factor accumulation but to the circulation of technologies and can be durably hindered by the lack of trade openness or skills. The second e¤ect does not refer to a convergence process since, in this case, the interaction under focus alters permanently the growth rates.

Several empirical studies have already taken into account a possible interaction between education and trade in the growth equation (see for instance Gould and Ru¢n [START_REF] Gould | Human capital, trade and economic growth[END_REF], Berthelemy, Dessus and Varoudakis [START_REF] Berthelemy | Capital humain, ouverture exterieure et croissance: estimation sur donnees de panel d'un modele a coe¢cients variables[END_REF], Levin and Raut [START_REF] Levin | Complementarities between exports and human capital in economic growth: evidence from the semi-industrialized countries[END_REF]). These studies provide robust evidence of a high degree of interaction between trade policies and education expenditures which stimulate economic growth. While the …rst two papers focus on the increase in human capital productivity generated by openness using a Mankiw, Romer and Weil [START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF] speci…cation, the last speci…es human capital as a total factor productivity determinant as in the Feder model [START_REF] Feder | On exports and economic growth[END_REF]. Nevertheless, these papers only account for the direct e¤ect on growth such an interaction between education and trade may generate. They do not consider that such e¤ects could help lagging countries to catch-up with the more developed ones.

This paper tries to test for both e¤ects through an augmented Benhabib and Spiegel speci…cation [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF]. Such a speci…cation allows to model the catch-up process, which is not possible when human capital is only considered as an in-put in the production function. The basic idea behind Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] is to allow the technological term in the production function to depend on the gap in GDP per capita between countries which proxies the catch-up e¤ect due to the di¤usion of knowledge, as well as other variables which characterize each country individuality. The speci…cation is then tested using a long term panel data base covering all the period from 1880 to 1980 for ten OECD countries. The length of the database is an additional contribution in this paper as previous studies on that subject are usually undertaken either on cross-section or over relatively short periods of time.

The results show strong evidence of an interaction between education and trade though mainly explaining the convergence e¤ect among the countries. In addition, using a break in coe¢cient test -built to account both for the temporal and the cross-sectionnal dimensions of the dataset -, I …nd that such an e¤ect is not constant but varies according to the level of GDP per capita.

The paper is organized as follows. The next section discusses the theoretical studies that justify the interaction between trade and human capital. Section 3 develops the empirical speci…cation used to test the impact of education on growth when related to trade. The fourth section deals with the data and provides some descriptive statistics. The …fth section presents the results, and the last one tests the possibility that the estimated coe¢cients may depend on the level of GDP. Finally, section 7 summarizes the main conclusions.

Theoretical background

There are at least two main types of models which support the joint importance of education and trade, in the growth process. The …rst type consists of imitation models and emphasizes the joint necessity for a country to accumulate education and to adopt outward-oriented policies in order to experience a catch-up e¤ect (section 2:1). The second one refers to rent-seeking models de…ning openness as the economic environment most likely to allow for optimal allocation of human capital (section 2:2).

Trade allows import and export of know-how

As underlined by Pissarides [START_REF] Pissarides | Learning by trading and the returns to human capital in developing countries[END_REF], in a world where countries from the South tend to imitate those from the North, trade appears as a route for transferring technology. Through trade, indeed, countries can access new technology coming from innovating economies. However, they are only able to apply this new technology, and hence experience higher growth rates, if their work force has the necessary skills. In other words, more open economies have a greater ability to copy more technologically advanced countries and, in this way, make advantage of their skilled labour force. Symmetrically, economies with a higher skilled labour force can more easily absorb foreign innovations. If we assume that growth rates are driven by technological accumulation, then simultaneously promoting education and strengthening trade openness should boost economic performance, at least among imitating countries. Since new technology tends to come from the more industrial economies to enhance the less advanced countries growth rates, this channel leads to a convergence e¤ect which has been described by imitation models such as Grossman and Helpman [START_REF] Grossman | Innovation and growth in the global economy[END_REF]. However, this catch-up e¤ect is a dynamic one, meaning that when some lagging countries imitate, others generate new technologies. Consequently, the potential growth gains generated by the technological catch-up are never really exhausted.

Trade reduces distortions

The starting point of the model in Berthelemy, Pissarides and Varoudakis [START_REF] Berthelemy | Rent seeking, the return to human capital and economic development[END_REF] is that human capital is not always used in productive activity. In closed countries, for instance, where distortions seem to be particularly developed human capital tends to be mainly used to escape policy -induced constraints. Skilled people are diverted from the productive sector to devote their time to …nd ways of reducing taxes, avoiding restrictions (as lawyers, accountants or traders for example...). These activities are socially useful because they reduce the e¤ects of the distortions on the productive sectors. They, nevertheless, constitute a diversion of human capital from the growth enhancing activities. Though di¢cult to identify which institutional environment allows for an optimal allocation of human capital, trade openness, being a sign that trade barriers are lowered and certainly inducing a reduction in every kind of distortion, appears to provide incentives to drive human capital towards the most productive activities. Thus, trade openness seems essential to boost growth by allowing the human capital productivity to be higher. Such a reduction in misallocation of human capital, as underlined by Berthelemy, Pissarides and Varoudakis [START_REF] Berthelemy | Rent seeking, the return to human capital and economic development[END_REF], can justify the existence of di¤erent growth patterns associated with di¤erent degrees of openness.

While the …rst channel refers to a convergence e¤ect, the second allows the economies to experience di¤erent growth paths. The forces described above operate simultaneously. Therefore, testing them amounts to …nding a model which includes the catch up e¤ect and the endogenous growth one.

The econometric speci…cation

The Benhabib and Spiegel model [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] can incorporate the two scenarios listed above. According to Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF], the accumulation of technology depends on several variables including a catch-up e¤ect and the educational attainment of the labour force. To this structure, I add a variable which interacts trade with human capital that a¤ects the technological term both directly and through the catch-up term.

Benhabib and Spiegel technological term

Assuming a Cobb-Douglas technology, the production function can be expressed as:

Y t;i = A t;i K ® t;i L t;i (1)
where Y t;i is the level of production, K t;i the physical capital, L t;i the labour and A t;i the total factor productivity.

Taking log di¤erences in (1), the change in GDP can be speci…ed as a linear function of the change in the production factors:

¢ ln Y t;i = ¢ ln A t;i + ®¢ ln K t;i + ¯¢ ln L t;i (2)
Then, following Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF], the change in technology is assumed to depend on two factors:

ln A t+1;i ¡ ln A t;i = c + dH t;i + eH t;i ¤ Ã y t;max ¡ y t;i y t;i ! (3)
where y t;i is the level of production per capita and y t;max the maximum level of production per capita attained within the set of countries of interest at year t.

The …rst factor (H t;i ) is the level of human capital which proxies the domestic endogenous innovation process. The second one (H t;i ¤ ³ yt;max¡y t;i

y t;i
´)is a term interacting the level of human capital with a variable representing the technological gap between the country i and the most developed country. This last term, …rst proposed by Nelson and Phelps [START_REF] Nelson | Investment in humans, technological di¤usion, and economic growth[END_REF] to estimate the role of human capital in the adoption of new technologies, represents the catch-up e¤ect due to the di¤usion of technology.

One objection to this speci…cation is that Benhabib and Spiegel did not model the channel through which technological catch-up can take place. They assumed perfect and unbounded di¤usion of knowledge, uncorrelated with the degree of openness of the countries. As it is more likely that imitation requires at once commercial interaction between countries and skills within the imitating economies, in what follows an estimator of trade openness is incorporated in the Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] speci…cation.

The augmented Benhabib and Spiegel speci…cation

To account for the two channels through which trade might interact with education, the Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] speci…cation is altered in two ways. First, an interactive variable involving education and trade is inserted directly as a factor of the technological term. This variable accounts for the direct e¤ect I referred to earlier. Then, the catch-up e¤ect is speci…ed to depend not only on the level of human capital, but also on the degree of openness to the outside world. While the …rst interactive variable represents the improvement in technological accumulation coming from a simultaneous increase in education and trade, the second variable proxies the convergence e¤ect. Through the …rst term I capture the Berthelemy, Pissarides and Varoudakis [START_REF] Berthelemy | Rent seeking, the return to human capital and economic development[END_REF] productive reallocation of human capital e¤ect which is due to trade. Through the second one, I capture the Grossman and Helpman [START_REF] Grossman | Innovation and growth in the global economy[END_REF] technological catch-up e¤ect. Both e¤ects are expected to enhance growth.

ln A t+1;i ¡ln A t;i = c+dH t;i +eH t;i ¤T r t;i +f H t;i ¤ à y t;max ¡ y t;i y t;i ! ¤© (T r t;i ) (4)
where © is a linear function of trade (T r t;i ) which preserves the Benhabib and Spiegel term but also allows for the catch up e¤ect due to the interaction between trade and education.

© (T r t;i ) = g + hT r t;i (5)
Going back to (2) and replacing the technological term with the equations speci…ed above yields:

¢ ln Y t;i = c + ®¢ ln K t;i + ¯¢ ln L t;i + dH t;i + eH t;i ¤ T r t;i +f H t;i ¤ à y t;max ¡ y t;i y t;i ! (g + hT r t;i ) + u i;t (6)
Expanding ( 6) results in the following testable speci…cation:

¢ ln Y t;i = c + ®¢ ln K t;i + ¯¢ ln L t;i + (d ¡ f g)H t;i + (e ¡ f h)H t;i ¤ T r t;i +f gH t;i ¤ à y t;max y t;i ! + f hH t;i ¤ T r t;i ¤ à y t;max y t;i ! + u i;t (7)
The sign of the coe¢cient on both human capital and the pure interactive e¤ect between trade and education is ambiguous: depending on whether the catch-up phenomenon dominates the direct e¤ect on technological accumulation the overall impact can be positive or negative. [START_REF] Banks | Banks'cross national time-series data archive user's manual[END_REF] The long term evidences

The data

The interactive e¤ect between trade and education is analysed by using a data set derived from three di¤erent databases provided by Banks [START_REF] Banks | Banks'cross national time-series data archive user's manual[END_REF], Maddison [START_REF] Maddison | l'economie mondiale 1820-1992[END_REF] and Jones and Obstfeld [START_REF] Jones | Saving, investment and gold: a reassessment of historical current account data[END_REF] and including exports, GDP, population growth, enrollment rates and investment share in GDP. The resulting data base is a long term annual panel from 1880 to 1980, including 10 OECD countries 7 . For the purpose of the article, the war period observations are excluded from the dataset.

The estimators

The investment variable is proxied by the share of capital formation in GDP, the variation in labour by the population growth, human capital by the enrollment rates and the indicator of trade by the share of exports in GDP corrected for the size of countries. Since trade and education are the variables of interest, the following two sections are devoted to discussing the relevance of the indicators used.

The estimator of human capital

When selecting a measure of human capital, one must choose between building a stock estimator and using an education rate. The latter option seems problematic since, as underlined by Pritchett [START_REF] Pritchett | Where has all education gone ? IBRD[END_REF], the enrollment rates simply represent ‡ow investment rates. They, then, do not account appropriately for human capital accumulation. However, testing for the correlation between the education rates and the corresponding human capital stocks on the recent period casts a serious doubt over the ‡ow nature of the enrollment rates. It appears, indeed, that education rates are highly correlated to the existing human capital stocks 8 for most of the countries under consideration, suggesting that their changes over time could appropriately proxy the evolution of the stock variables. In this case trading the stock estimators for the education rates would not be a too rough approximation.

Another reason is driving such a choice: stock estimators are not available for the length of the dataset used. Consequently, building stock estimators from education rates would require either to extrapolate values from a already existing stock (but available data do not correspond exactly to the de…nition of the education rates such as those listed in the Banks [START_REF] Banks | Banks'cross national time-series data archive user's manual[END_REF] and Maddison[26] 8 Comparing the Nehru, Swanson and Dubey [START_REF] Nehru | A new database on human capital stocks in developing and industrial countries: sources, methodology and results[END_REF] average number of years in secondary school* and the Banks [START_REF] Banks | Banks'cross national time-series data archive user's manual[END_REF] * Such a choice makes sense since on 1960-1979 the changes in the overall education rate (primary + secondary) are certainly driven by the variance in the secondary education rate. Indeed, for the countries of interest, the primary education rate had, at this time, already reached a ceiling. Nevertheless, the education rate is used in the regressions as a lagged variable. There is no reason, indeed, to think that current education has a direct in ‡uence on economic performance. The process takes time, for instance, the time for a student to exit school and …nd a job. Therefore, the regressions are performed using the education variable lagged by ten years.

A proxy of trade

The measure of trade used is the share of exports in GDP. Several economists have pointed out the limits of such an indicator, stressing in particular the endogeneity between GDP and exports as a severe drawback (among them Bradford and Chakwin [START_REF] Bradford | Alternative explanation of the tradeoutput correlation in the east asian economies[END_REF], Rodrik [START_REF] Rodrik | Getting interventions right: how South Korea and Taiwan grew rich[END_REF] and Cyrus, Frankel and Romer [START_REF] Cyrus | Trade and growth in east asian countries: cause and e¤ect ?[END_REF]). Nevertheless, the share of exports in GDP is a convenient estimator for essentially two reasons. First, as underlined by Alesina and Wacziarg [START_REF] Alesina | Openness, country size and the government[END_REF], such a variable appears to be a good measure of the fraction of the economy which actually interacts with the rest of the world. Secondly, it has the advantage of measuring the openness both in cross-country and time series data (Vamvakidis[37]). Imports would certainly do a better proxy to account for the imitation process since the products entering the country incorporate the technologies used abroad. Yet, since data on imports are not available on a long period for the range of countries of interest, and providing that for the available values of imports both series are highly correlated 9 , the regressions are performed using the share of exports in GDP.

The trade estimator is also corrected for the size of the countries. As underlined by Alesina and Wacziarg [START_REF] Alesina | Openness, country size and the government[END_REF], small countries, because they cannot bene…t from access to a large market, tend to be more open to trade whereas bigger countries have "naturally" less incentive to trade with the rest of the world. This hypothesis points toward a negative relationship between country size and the degree of trade openness that should be controlled for. I choose not to proxy the size of countries by their population, as usually done in the literature, to avoid the endogeneity problem between this last variable and the growth rate. Rather, I turn to the Lee [START_REF] Lee | International trade, distorsions and long-run economic growth[END_REF] measure 10 of free 9 According to the NBER macrohistory database the correlation coe¢cient between exports and imports is 0.93 for France (data from 1878 to 1938), 0.97 for US (data from 1867 to 1969), 0.89 for UK (data from 1856 to 1950). 10 Regressing the import share on natural resource endowments (proxied by the land size 11

Cahiers de la MSE -2000.22 trade openness 11 .

The trade variable eventually derived has the following speci…cation:

T r t;i = X t;i GDP t;i

LeeV ar i

Where X GDP represents the share of exports in GDP and LeeV ar the measure calculated by Lee [START_REF] Lee | International trade, distorsions and long-run economic growth[END_REF]. It proxies the degree to which a country interacts with the rest of the world through trade, controlling for the fact that bigger countries naturally need to trade less.

Some empirical evidence

This section is devoted to a short historical review of the evolution of GDP per capita (section 4.3.1), export share in GDP (section 4.3.2) and enrollment rate (section 4.3.3) over the twentieth century. The graphs are inserted at the end of the paper. For the sake of clarity, the number of countries which behaviour is analysed is reduced to four. Therefore, the selected exemples belong to di¤erent geographical areas so as to make them representative of the sample.

Income, growth and convergence

Figure 1 presents evidence of the catch-up e¤ect towards US. This phenomenon seems to be mainly relevant after the second world war though.

The trade estimator

Figure 2 shows the measure of trade openness of four countries: France, Japan, UK and US, from 1870 to 1994. These indexes can not be read of each country) and on trade distortion (among them distance to major world exporters as a proxy for transportation costs), and replacing the coe¢cients on trade distortions with zeros, Lee [START_REF] Lee | International trade, distorsions and long-run economic growth[END_REF] de…nes an alternative measure of free trade openness for each country. Such a measure should be relevant to correct the trade variable even on a long term setting since it is calculated from variables which are unlikely to change over time. 11 However, the estimations are robust to these corrections. Correcting the trade estimator either by the population of countries, or by the Lee variable or even not correcting for it yield the same results. This robustness con…rms Vamvakidis [START_REF] Vamvakidis | Regional trade agreements versus broad liberalisation: which path leads to faster growth ? IMF Working Paper[END_REF] result according to which correction for the size of countries was mainly relevant for cross-country analysis.

directly as the share of exports in GDP since they are corrected for the "size" of the countries. Nevertheless, they give a good idea of the ranking of countries and of the evolution of those countries trade orientation over the 20th century. At the end of the 19th century, the degree of openness is quite heterogenous: while UK is rather outward oriented, Japan is totally closed. Then, the two World Wars appear to create big breaks in the French, British and Japanese openness trends: before World War I, the di¤erent trends are quite stable and slightly increasing, after that, the picture gets blurred. The countries tend to close and converge towards a low level of openness. This tendency, however, reverses after World War II when every country clearly tends to interact more and more with the rest of the world. This e¤ect has a catch-up connotation as, by the end of the period, every country's openness level lies in the interval 0.77 -0.85.

The education rate

In …gure 3,the cumulative enrollment rate in primary and secondary school 12is presented. While Japan has increased its education rate all through the period, France and UK appear to have su¤ered from the …rst and second World Wars. The US experience the highest education rates throughout the 20th century, but lose the supremacy by the end of the period. By contrast with the previous graph, catch-up here is mainly relevant before the …rst world war. After that, the picture gets less clear: while France and the UK experience a collapse in their trends, from a 60% enrolment rate to less than 40% of 5-19 year olds in primary and secondary schools, the US jump to a higher path and Japan continues to increase its education rate smoothly. After the second world war, all countries seem to reach a ceiling between 80% and 90%.

The results of the augmented Benhabib and Spiegel speci…cation

Results are reported in table 1. A Fisher test on the regressions shows that the …xed e¤ects estimators perform better. The time dummies are included in the regressions though not reported. The robustness of the speci…cation is also checked by adding the share of export in GDP and the initial GDP per capita to the initial speci…cation. The Benhabib and Spiegel interactive term appears to be positive, but only weakly signi…cant. By contrast, the interactive term ampli…ed by the level of openness is positive and strongly signi…cant. Education is negative and signi…cant, suggesting, as found by Benhabib and Spiegel, that catch-up is certainly a more important channel through which human capital a¤ects growth than the endogenous country-speci…c technological progress. The size of the coe¢cient even indicates that once the convergence e¤ect taken into account (f g in equation 7), H remains negatively correlated to growth suggesting that education is a cost for society. The pure interactive variable -share of export in GDP times education rate -is signi…cant and negative, which indicates that trade interacts with education but mainly in the catchup process. A Student test provides some evidence of equality (in absolute value) between the coe¢cient of Inter and Complbs implying, if we go back to the initial speci…cation, that the only channel through which the interactive variable a¤ects growth is technological catch-up 13 .

The results are robust to the inclusion of new variables. Adding the share of exports in GDP and the initial GDP per capita does not change the signi…cance or the magnitude of the coe¢cients. It suggests that Catchupbs and Complbs are not driven by a traditional convergence e¤ect.

Finally, from the regression presented above, the Benhabib and Spiegel term seems to have little in ‡uence on growth, and the catch-up e¤ect appears to be better captured when weighted by trade and education. In other words, improving only the level of human capital as argued by Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] is not enough to experience a catch-up e¤ect. In order to catch up faster a country should simultaneously develop both its education and its trade.

There is also weak evidence that the interaction between trade and education a¤ects growth directly. While a complementary e¤ect between education and trade could justify an increase in the speed of convergence, it is not a relevant variable in explaining permanent di¤erences in growth rates. However, the Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] speci…cation may not be the most appropriate way to test the e¤ect of openness on human capital reallocation between sectors. As underlined by Berthélémy, Dessus and Varoudakis [START_REF] Berthelemy | Capital humain, ouverture exterieure et croissance: estimation sur donnees de panel d'un modele a coe¢cients variables[END_REF], such a reallocative e¤ect and its consequences on human capital productivity should be better captured by a variable coe¢cients model which would allow the structural coe¢cient on education to vary according to other factors 14 .

A structural change test

The next issue the paper analyses is whether the interaction between trade and education always a¤ects the convergence in the same way, or if it varies according to the stage of development attained by the countries. The idea behind this is that the complementary e¤ect could hinge on the level of income already accumulated by the country. Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] have shown that the e¢ciency of technology adoption may change with the relative position of the country on the GDP per capita scale. By separating their sample into three equal sub-samples on the basis of initial per capita income, they …nd that the catch-up e¤ect is mainly relevant for the poorest countries whereas the innovation term is positive and signi…cant for the richest ones. This paper proposes to test this conjecture in a more systematic way by reviewing every sub-sample that can be generated from the initial sample instead of assuming a priori the critical level of GDP per capita.

The test

In order to test whether H t;i ¤ T r t;i ¤ ³ yt;max y t;i ´always a¤ects growth by the same amount whatever the level of GDP per capita, the test should allow the coe¢cient of H t;i ¤ T r t;i ¤ ³ yt;max y t;i ´to depend on GDP per capita. The Chow test is not appropriate since it studies the variation in coe¢cient according to time and not to the level of another variable. The Durlauf and Johnson [START_REF] Durlauf | Local versus global convergence across national economies[END_REF] test and its extensions 15 are only applicable to cross-sections. Therefore, they are not suitable for the purpose of this paper. The test presented below is an attempt to reconcile both bunches of methods by taking into account the panel aspect of the dataset as well as the cross-sectional one.

Recalling the initial speci…cation: 14 Testing a Mankiw, Romer and Weil [START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF] speci…cation over a panel of 83 countries on 1960-1990, Berthélémy, Dessus and Varoudakis [START_REF] Berthelemy | Capital humain, ouverture exterieure et croissance: estimation sur donnees de panel d'un modele a coe¢cients variables[END_REF] con…rm the impact of trade openness on human capital productivity. However a similar regression performed on the Banks [START_REF] Banks | Banks'cross national time-series data archive user's manual[END_REF] and Maddison[26] database does not provide conclusive results. 15 Berthelemy and Varoudakis [START_REF] Berthelemy | Economic growth, convergence clubs and the role of …nancial development[END_REF] The next step of the test is to endogenise the threshold. Since, I do not wish to impose a certain level of GDP per capita as a threshold, I choose to estimate each possible threshold in the database by performing the regressions for every available value of y. Each value of y used as a potential threshold (y¤) provides a certain t-statistic associated with the coe¢cient of Complbs inf t;i that I then plot. In what follows, the existence of a cuto¤ point in the growth path will be infered whenever the graph shows a t-statistic signi…cant at the 5% level.

Evidence of thresholds

When H t;i ¤ T r t;i ¤ ³ yt;max y t;i
´depends on the level of GDP per capita, the distribution of T ¡ stat has the shape presented in …gure 4 . This graph suggests there should be a threshold in the database at lny = 7:97. From the graphs, such a level of log GDP per capita was reached by Canada and the European countries around the …rst World War and Japan after the second World War. The US is the only country which has always been above the threshold.

I then estimate the coe¢cients which result from a model with the threshold at 7.97 and report them in table 2. In this speci…cation, countries with log of GDP per capita lower than 7.97 are represented by Complbs inf and the others by Complbs sup .

The new speci…cation restores the signi…cance of the Benhabib and Spiegel catch-up term. One can explain this result by a better speci…ed equation. However, even if signi…cant, the Benhabib and Spiegel catch-up term is weak whereas the catch-up term ampli…ed by trade and education is stronger. Such a result implies that education enhances the speed of adoption of technology from abroad, but this e¤ect is much bigger when education and trade interact.

Another signi…cant outcome from this regression is that the interactive variable has a stronger impact for a country lying above the threshold point of GDP per capita = 7:97. This result suggests that at a relatively low level of income countries experience a relatively "weak" catch-up e¤ect while at a higher one this e¤ect becomes stronger. Consequently, and contrary to Benhabib and Spiegel [START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF] , the catch-up e¤ect does not decrease when income increases. It works more like a learning by doing process: the more the country develops, the more it is able to take advantages of the new technologies it absorbs from outside. Symmetrically, opening up and simultaneously de-veloping the level of education allows economies to experience higher growth rates. .000

This paper uses panel data to empirically evaluate whether education and trade a¤ect growth when interacting together. The regressions provide robust evidence that such a complementary e¤ect has accelerated technological catch-up between OECD countries all through the twentieth century.

Going back to the initial purpose of this article, this result also means that education can a¤ect growth positively. While education, taken as a production factor, appears unrelated to growth, it becomes important when interacting with trade. Consequently, this paper underlines the importance for a country of coordinating economic policies concerning trade and education.

However, as shown in the paper, simultaneously promoting investment in human capital as well as in education does not a¤ect growth constantly with equal intensity. Since the interactive e¤ect is mainly relevant in explaining catch-up, such a policy should essentially bene…t lagging countries. And, moreover, since the catch-up e¤ect itself also exhibits nonlinearity as it increases with GDP per capita, the more countries develop, the easier it should get for them to catch-up.

10

  Cahiers de laMSE -2000.22 database) or to sacrify 40 years to determine the starting values, possibility that would compromise the aim of the paper, i.e. studying long run growth.

20

  Cahiers de la MSE -2000.[START_REF] Knight | Testing the neoclassical theory of economic growth: a panel data approach[END_REF]

Figure 1 :Figure 2 :

 12 Figure 1: Log of GDP per capita: Canada, France, Japan, US

Figure 3 :Figure 4 :

 34 Figure 3: Education rate: France, Japan, UK, US

  According to this table 7 countries over 10, essentially the european countries, are characterised by a high correlation between the stock variable and the education rate.

			education rates (in primary and secondary school) on 1960-1979
	yields the following correlation coe¢cients: Canada Denm. France Germ. Italy Japan Norway UK US	Sweden
	.327	.964	.934	.704	.987 .396	.946	.832 -.742 .958

Table 1 :

 1 Estimation of a Benhabib and Spiegel augmented speci…cation ln Y t+1;i ¡ ln Y t;i . T-stat are in parentheses.

			Speci…cation (7) robustness robustness
				to Tr	to Lny t
		I/Y	0.095	0.095	0.091
			(2.56)	(2.56)	(2.39)
		n	.375	.375	.366
			(1.93 )	(1.92 )	(1.87 )
		H	-0.070	-0.069	-0.079
			(-2.22)	(-2.05)	(-1.98)
		Inter	-0.145	-0.146	-0.149
	(H t;i ¤ T r t;i )	(-2.73)	(-2.00)	(-2.76)
	Catchupbs (H t;i ¤ ³ Yt;max Y t;i ´)	0.018 (1.38)	0.018 (1.30)	0.022 (1.29)
	(H t;i ¤ T r t;i ¤	Complbs ³ Y t;max Y t;i ´)	0.150 (4.11)	0.150 (3.72)	0.150 (4.12)
		Lny t			.0007
					(.39 )
		Tr		.0003
				(.006 )
		Tests
		Obs	710	710	710
		R 2	0.29	0.29	0.30
	Fixed e¤ects test (p-value)	.000	.000	.000
	Hausman test (p-value)	.010	.040	.030
	Dependant variable:

Table 2 :

 2 Model with threshold Dep. Var.: ln Y t+1;i ¡ ln Y t;i

		I=Y	.079
				(2.14)
			n	.358
				(1.86)
		H	-.060
				(-1.93)
	H t;i ¤ T r t;i	-0.205 (-3.70)
	H t;i ¤	³ yt;max y t;i	´0.026 (2.02)
	Complbs inf	0.135 (3.72)
	Complbs sup	0.179 (4.84)
		tests
		Obs	710
		R 2	0.31
	Fixed e¤ects test (p-value)	.000
	Hausman test (p-value)

Cahiers de la MSE -2000.[START_REF] Knight | Testing the neoclassical theory of economic growth: a panel data approach[END_REF]

Cahiers de la MSE -2000.[START_REF] Knight | Testing the neoclassical theory of economic growth: a panel data approach[END_REF]

Canada, Denmark, Germany, France, Italy, Japan, Norway, Sweden, UK, USA

The data for the war periods are not available

Cahiers de la MSE -2000.[START_REF] Knight | Testing the neoclassical theory of economic growth: a panel data approach[END_REF]

If e ¡ fh = ¡fh in equation[START_REF] Benhabib | The role of human capital in economic development: evidence from aggregate cross-country data[END_REF], then e = 0.

´, the variable which is assumed to be the one a¤ected by the level of development: ´is included in the regression, in which case the existence of a threshold is infered. Formally, the test is the following: ´.


Denis Bonheure

AND Colette De Coster

Ann Derlet
email: aderlet@univ-tlse1.fr

INFINITELY MANY RADIAL SOLUTIONS OF A MEAN CURVATURE EQUATION IN LORENTZ-MINKOWSKI SPACE

In this paper, we show that the quasilinear equation

has a positive smooth radial solution at least for any α > 2 = 2N/(N -2), N ≥ 3. Our approach is based on the study of the optimizers for the best constant in the inequality

We also prove that the best constant is not achieved for α = 2 . As a byproduct, our arguments combined with Lusternik-Schnirelmann category theory allow to construct a sequence of radial solutions.

Introduction

It is well known [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] that the Lane-Emden equation [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF] -∆u = |u| α-2 u in R N , admits no nontrivial nonnegative solution for 2 < α < 2 , N ≥ 3, while, for α = 2 , any positive solution can be written in the form

u δ,a (x) = β N δ δ 2 + |x -a| 2 N -2 2N
, as proved by Caffarelli, Gidas and Spruck [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF]. For α > 2 , the set of all positive radial solutions is a one-parameter family {u a (r) = au 1 (a (α-2)/2 r) : a > 0}, where u 1 is strictly decreasing in r (see for instance [START_REF] Gui | On the stability and instability of positive steady states of a semilinear heat equation in R n[END_REF]). Non radial singular solutions have been constructed by Dancer, Guo and Wei [START_REF] Dancer | Non-radial singular solutions of Lane-Emden equation in R N[END_REF]. We mention that it is still open whether all smooth positive solutions are radially symmetric around some point or not.

The prescribed mean curvature equation in Euclidian space -div ∇u

1 + |∇u| 2 = |u| α-2 u in R N ,
has also been the object of many studies. It has been considered, among others, by Ni and Serrin [START_REF] Ni | Existence and non-existence theorems for ground states for quasilinear partial differential equations[END_REF] and del Pino and Guerra [START_REF] Del Pino | Ground states of a prescribed mean curvature equation[END_REF]. It is known that this problem has infinitely many radial positive solution if α ≥ 2 and no smooth positive solutions if α ≤ (2N -2)/(N -2). In contrast with the non-existence result for the Lane-Emden equation in the subcritical range, del Pino and Guerra proved the existence of many positive solutions when α = 2 -, for sufficiently small > 0.

In this work, we aim to study the following prescribed mean curvature equation in the Lorentz-Minkowski space ( 2)

Q(u) = |u| α-2 u in R N , where (3)
Q(u) = -div ∇u 1 -|∇u| 2 .
The quasilinear operator Q is a classical object in Riemannian geometry. The Lorentz-Minkowski space L N +1 = {(x, t) ∈ R N ×R}, with the flat metric N j=1 (dx j ) 2 -(dt) 2 is the natural framework of classical relativity. If M is an N -dimensional hypersurface of L N +1 that is the graph of a smooth function u ∈ C 1 (Ω) with ∇u L ∞ < 1, the local mean curvature of M is given by Q(u), see for instance [START_REF] Bartnik | Spacelike hypersurfaces with prescribed boundary values and mean curvature[END_REF][START_REF] Cheng | Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces[END_REF]. The determination of maximal or constant mean curvature hypersurfaces is an important issue in classical relativity. The volume integral Ω 1 -|∇u| 2 gives the area integral in L N +1 and surfaces of maximal area (or simply maximal surfaces) solve the equation Q(u) = 0 in Ω.

For functions defined on the whole of R N , the operator Q is relevant in Maxwell-Born-Infeld field theory, see for instance [START_REF] Born | Foundation of the new field theory[END_REF][START_REF] Born | Foundation of the new field theory[END_REF][START_REF] Kiessling | Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequence[END_REF][START_REF] Kiessling | On the quasi-linear elliptic PDE -∇•(∇u/ 1 -|∇u| 2 ) = 4π k a k δs k in physics and geometry[END_REF]. Basically, in this theory, which is fully relativistic, it is assumed that there is a maximal field strength. This lead Born and Infeld to consider the following Lagrangian density, expressed in Lorentz-Minkowski space,

L BI = b 2   1 -1 - | E| 2 -| B| 2 b 2 - ( E. B) 2 b 4   ,
where E is the electric field, B is the magnetic field and b is the maximal admissible value of the electric field. Up to our knowledge, the equation (2) has never been considered in the literature, at least in R N . We refer to [START_REF] Bereanu | Variational methods for nonlinear perturbations of singular φ-laplacians[END_REF][START_REF] Bereanu | Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space[END_REF][START_REF] Brezis | Periodic solutions of the forced relativistic pendulum[END_REF][START_REF] Coelho | Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball[END_REF] for recent results on the existence of radial solutions for BVPs involving Q in the ball with either Dirichlet or Neumann conditions.

Supercritical problems are usually difficult to tackle through variational methods. For instance, concerning the Lane-Emden equation, Farina [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF] has obtained a Liouville-type result for C 2 solutions of (1) with finite Morse index. Basically, if the dimension is small (N ≤ 10), the only finite Morse index solution is 0 except at the critical exponent where the above-mentioned positive solutions arise as constrained minimizers on a manifold of codimension 1.

In contrast, we show here that the quasilinear equation ( 2) has a smooth positive radial solution for any α > 2 , N ≥ 3 by using simple arguments from Critical Point Theory and the Calculus of Variations. In fact, when α > 2 , we have enough compactness to deal with the problem in a standard way. Indeed, we minimize the volume integral (4)

R N (1 -1 -|∇u| 2 ),
truncated in a convenient way, constrained to the unit sphere of L α (R N ). Then we prove a gradient estimate which is uniform with respect to the truncation parameter.

Our first main result is the following. We restrict here our attention to the existence of radially symmetric solutions. On the one hand, we expect that all positive smooth solutions are indeed radially symmetric, though this is an open question. On the other hand, our solution arises as a constrained minimizer and its Schwarz symmetric rearrangement yields a radially symmetric minimizer (and therefore a radially symmetric solution). Surprisingly, our approach to establish the existence of a solution of (2) fails in the critical case α = 2 . Indeed, as stated in Theorem 1.2 below, the solution of Theorem 1.1 realizes the best constant in an inequality between the volume integral (4) and the L α -norm. This inequality still holds for α = 2 but the best constant is not achieved. We emphasize that this contrasts with the Sobolev inequality.

In the sequel, we denote by X the functional space

X := u ∈ D 1;2 (R N ) : ∇u ∈ L ∞ (R N ) and ∇u L ∞ ≤ 1 , endowed with the norm u D 1;2 (R N ) := R N |∇u| 2 1/2 .
We establish the following Sobolev-type inequality.

Theorem 1.2. There exists C > 0 such that

(5) R N (1 -1 -|∇u| 2 ) ≥ C R N |u| α N α+N
for every u ∈ X if and only if α ≥ 2 . Moreover, the best constant

inf u∈X \{0} R N (1 -1 -|∇u| 2 ) R N |u| α N N +α
is achieved by a radial solution of (2) for α > 2 while it is not achieved for α = 2 .

The fact that inequality [START_REF] Bonheure | On the multiplicity of nodal solutions of a prescribed mean curvature problem[END_REF] does not hold below the critical exponent is rather clear since the volume integral (4) is bounded from above by the Dirichlet energy. This does not mean that (2) has no non trivial nonnegative solutions for α < 2 though we conjecture that this is indeed the case. One can for instance exclude the existence of fast decaying solution but we are not able to prove a complete non-existence result for α < 2 . Also the existence of a positive solution of (2) in the critical case α = 2 remains an interesting open question.

At last, as a natural extension of our existence result, we combine our previous approach with Lusternik-Schnirelmann category theory to obtain a sequence of solutions whose volume integral diverge. Namely we prove the following multiplicity result.

Theorem 1.3. For any α > 2 , equation (2) has a sequence of radial solutions

(u k ) k∈N such that R N (1 -1 -|∇u k | 2 ) → +∞ as k → ∞.
Again, we first consider an auxiliary problem and conclude by a sharp uniform estimate on the gradient of our solutions. Note that we do not provide sign information on solutions though one could probably argue as in [START_REF] Ramos | A Bahri-Lions theorem revisited[END_REF][START_REF] Bonheure | On the multiplicity of nodal solutions of a prescribed mean curvature problem[END_REF] to obtain a sequence of sign changing solutions. We leave this, as well as the existence of infinitely many positive solutions, as open questions.

The paper is organized as follows. Section 2 contains some preliminary results on the functional spaces we will work with. In Section 3, we establish the existence of at least one classical solution of (2) (see Theorem 1.1 above). Section 4 is devoted to the proof of the inequality in Theorem 1.2 and especially to the existence of optimizers for the best constant in this inequality. Finally, in Section 5, we obtain infinitely many solutions of (2) as stated in Theorem 1.3.

With some abuse of notation, we will sometimes consider radial functions as functions of one variable, thus writing u(|x|) or u(x) or u(r). For any set A of functions, A rad is defined as the set of all radially symmetric functions of A. Throughout the paper, C denotes a positive constant that can change from line to line.

Functional framework and preliminary results

Let us set a 0 (s) = (1 -s) -1/2 for all s < 1. Equation ( 2) can be written as

-div a 0 (|∇u| 2 )∇u = |u| α-2 u in R N .
We introduce the energy functional

I 0 (u) := 1 2 R N A 0 (|∇u| 2 ),
where A 0 (t) = t 0 a 0 (s) ds for all t ≤ 1. This functional is well defined on X = {u ∈ D 1;2 (R N ) : ∇u ∈ L ∞ (R N ) and ∇u L ∞ ≤ 1}, because we have

1 2 |∇u| 2 ≤ 1 -1 -|∇u| 2 = |∇u| 2 1 + 1 -|∇u| 2 ≤ |∇u| 2 .
Lemma 2.1. Let u ∈ X . Then |∇u| ∈ L q (R N ) for every q ≥ 2, and u ∈ L s (R N ) for every s ≥ 2 . Moreover, u can be assumed to be continuous and such that

lim |x|→∞ u(x) = 0.
Proof. Since |∇u| ≤ 1 and |∇u| ∈ L 2 (R N ), we infer that |∇u| ∈ L q (R N ) for every q ≥ 2. It then follows that u ∈ L qN/(N -q) (R N ) for every q ≥ 2, and, by interpolation, u ∈ L s (R N ) for every s ≥ 2 . Observe also that since u ∈ W 1,r (R N ) for some r > N , it can be assumed to be continuous and moreover lim |x|→∞ u(x) = 0.

Working with the functional I 0 in X requires some care. Since I 0 is weakly lower semi-continuous, a natural way to obtain a solution of (2) consists in minimizing I 0 constrained to the manifold

M 0 := u ∈ X : R N |u| α = 1 .
However, it is not clear that minimizers solve an associated Euler-Lagrange equation. Indeed, the functional I 0 is C 1 only at points u ∈ X with Lipschitz constant Lip(u) strictly less than 1. Without this condition, minimizers solely solve a variational inequality.

To overcome this lack of differentiability on the boundary of X , we will work with an auxiliary functional. This type of truncation argument has already been used in [START_REF] Coelho | Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation[END_REF][START_REF] Coelho | Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball[END_REF] to deal with Dirichlet boundary condition in an interval or a ball. Here, one of the novelties is that an a priori L ∞ bound on minimizers cannot be derived from the solely boundedness of the gradient. Therefore, we truncate the volume integral in a different way than in [START_REF] Coelho | Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation[END_REF][START_REF] Coelho | Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball[END_REF] and we deal with a different functional framework.

We now define our auxiliary functional. For

θ ∈ ]0, 1[, define a θ : R → R + by (6) a θ (s) = a 0 (s) for 0 ≤ s ≤ 1 -θ and a θ (s) = γs p + δ for s > 1 -θ,
where γ and δ are chosen in such a way that a θ is C 1 . The exponent p will be chosen later according to the value of α in (2).

In the sequel, we will work with the spaces D 1;r rad (R N ) and D 1;(2,q) rad (R N ), defined respectively as the closure of the smooth compactly supported radially symmetric functions for the norms

u D 1;r := R N |∇u| r 1 r and u D 1;(2,q) := R N |∇u| 2 1 2 + R N |∇u| q 1 q , with 1 < q, r < ∞. Consider the manifold M := u ∈ D 1;(2,q) rad (R N ) : R N |u| α = 1 .
We will look for critical points of I θ constrained to M where

I θ : D 1;(2,q) rad (R N ) → R + is defined by I θ (u) := 1 2 R N A θ (|∇u| 2 ),
and A θ (t) = t 0 a θ (s) ds. We next recall some elementary facts. We quote them in separate lemmas for further references in the text. We do not provide the details for Lemma 2.2 which follows from standard arguments. We refer for instance to [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF][START_REF] Lions | Symétrie et compacité dans les espaces de Sobolev[END_REF][START_REF] Bonheure | Positive solutions of a second-order singular ordinary differential equation[END_REF] for Lemma 2.3, whereas Lemma 2.4 can easily be deduced from [START_REF] Lions | Symétrie et compacité dans les espaces de Sobolev[END_REF]. Below, q := qN/(N -q) for q < N .

Lemma 2.2. Let u ∈ D 1;(2,q) rad (R N ). Then u ∈ D 1;r rad (R N ) for every r ∈ [2, q]. If q < N then u ∈ L s (R N ) for every s ∈ [2 , q ]; if q = N then u ∈ L s (R N ) for every s ∈ [2 , +∞[; if q > N then u ∈ L s (R N ) for every s ∈ [2 , +∞]. Moreover, the embeddings are continuous. Lemma 2.3. Let r ∈ [2, q] if q < N , and r ∈ [2, N [ if q ≥ N .
Then there exists C > 0 (depending only on N and r) such that for all u ∈ D 1;(2,q) rad (R N ), there holds

|u(x)| ≤ C|x| -N -r r ∇u L r , for almost all x ∈ R N \ {0}. Lemma 2.4. Let (u n ) n ⊂ D 1;
(2,q) rad (R N ) be a bounded sequence. If q < N then for any s ∈ ]2 , q [, there exists a subsequence which converges weakly in D 1;(2,q) rad (R N ) and strongly in L s (R N ). If q > N , the same result holds for any s ∈ ]2 , +∞[. We close this section by a uniform estimate on the regularization schema. Observe that for θ 1 := 1/(2p + 1), the function a θ defined in ( 6) is given by

a θ1 (s) = 1/ √ 1 -s if 0 ≤ s ≤ 1 -θ 1 and a θ1 (s) = γ p s p if s > 1 -θ 1 ,
where

γ p = √ 2p + 1 ((2p + 1)/2p) p .
Therefore, for all θ ∈ ]0, θ 1 ] and s ∈ R + we have [START_REF] Born | Foundation of the new field theory[END_REF] γ p p + 1

s p+1 ≤ A θ1 (s) ≤ A θ (s), and (8)
A θ (s) ≥ A θ1 (s) ≥ s, if s ≤ 2p 2p+1 , ≥ γp p+1 2p 2p+1 p s, if s > 2p 2p+1 .
Inequalities ( 7) and ( 8) lead to uniform estimates (with respect to θ) in

D 1;(2,2p+2) rad (R N
). They will be important keys in the sequel to obtain a priori bounds independent of the truncation parameter θ. As for an upper bound on I θ , we observe that for all

u ∈ D 1;(2,2p+2) (R N ), (9) A θ (|∇u| 2 ) ≤ C (|∇u| 2p+2 + |∇u| 2 ).
for some constant C depending on θ. The functional I θ is then well defined in D 1;(2,q) rad (R N ) with q := 2p+2 and it is straightforward that

I θ is C 1 on D 1;
(2,q) rad (R N ). The preceding lemmas suggest to choose p in the definition of a θ such that q = 2p + 2 satisfies q > α. Indeed, a lower bound in D 1;(2,2p+2) (R N ) will follow from [START_REF] Born | Foundation of the new field theory[END_REF] and [START_REF] Born | Foundation of the new field theory[END_REF] whereas M is weakly closed as soon as q > α.

Existence of a positive solution for supercritical exponents

In this section, we prove that equation ( 2) has at least one positive solution.

3.1. The auxiliary problem. We will first look for a solution of the modified problem

-div a θ (|∇u θ | 2 )∇u θ = λ θ α|u θ | α-2 u θ in R N ,
where a θ is defined in [START_REF] Bonheure | Positive solutions of a second-order singular ordinary differential equation[END_REF]. It will turn out that if the parameter θ is small enough, this solution also solves the original equation [START_REF] Bartnik | Spacelike hypersurfaces with prescribed boundary values and mean curvature[END_REF]. From now on, we assume θ ∈ ]0, θ 1 ]. Recall also that q > α (which can be written as q > N α/(N + α)), θ 1 = 1/(2p + 1) and q = 2p + 2.

Proposition 3.1. Let α > 2 and q > N α N +α . Then there exists

u θ ∈ D 1;(2,q) rad (R N ) such that (10) c 1 θ := min v∈M I θ (v) = I θ (u θ ) > 0.
For any minimizer u θ of (10), there exists λ θ ∈ R + such that u θ is a weak solution of the equation

(11) -r N -1 a θ (|u θ | 2 )u θ = λ θ αr N -1 |u θ | α-2 u θ , i.e. +∞ 0 r N -1 a θ (|u θ | 2 )u θ v = λ θ α +∞ 0 r N -1 |u θ | α-2 u θ v, for every v ∈ D 1;(2,q) rad (R N ). Moreover, for every s ∈ [2 , q ], s ∈ [2 , +∞[ or s ∈ [2 , +∞] if q < N , q = N and q > N respectively, there exist C 1 , M 1 > 0 independent of θ ∈ ]0, θ 1 ] such that (12) max{ u θ D 1;(2,q) , u θ L s } ≤ C 1 and c 1 θ ≤ M 1 .
Proof. We proceed in several steps.

Step 1: Lower bounds on c 1 θ . The inequalities ( 7) and ( 8) imply the existence of a positive constant C depending only on p such that, for all θ ∈ ]0, θ 1 ] and all u ∈ D 1;(2,q) rad (R N ), ( 13)

I θ (u) ≥ C R N |∇u| 2 and I θ (u) ≥ C R N |∇u| q .
As α > 2 , we have 2 < N α N +α < q and we deduce by interpolation and Sobolev inequality that for all u ∈ M,

(14) I θ (u) ≥ C R N |∇u| N α N +α ≥ C R N |u| α N N +α = C > 0,
for some C > 0 which depends only on p, α and N . This implies that inf

v∈M I θ (v) > 0.
Step 2: Existence of a minimizer. Let (u n ) n ⊂ M be a minimizing sequence, i.e.

I θ (u n ) → inf v∈M I θ (v)
as n → ∞. Choosing ū ∈ M a smooth function such that |∇ū(x)| < 1 -θ 1 for all x ∈ R N , we can assume w.l.g. that (15)

I θ (u n ) ≤ R N (1 -1 -|∇ū| 2 ) =: M 1
for any n ∈ N and any θ ∈ ]0, θ 1 ]. It then follows from ( 13) and ( 15) that (u

n ) n is bounded in D 1;(2,q) rad (R N ). Since α > 2 and q > N α N +α , Lemma 2.4 implies that, up to a subsequence, (u n ) n con- verges weakly to u θ in D 1;(2,q) rad (R N ) and strongly in L α (R N ) as n → ∞. Obviously, R N |u θ | α = 1 and u θ ∈ M.
Moreover, I θ being convex and continuous, I θ is weakly lower semi-continuous and

I θ (u θ ) ≤ lim inf n→∞ I θ (u n ) = inf v∈M I θ (v).
Since u θ ∈ M, we conclude that I θ (u θ ) = inf v∈M I θ (v).

Step 3: A priori bounds on the family {u θ : θ ∈ ]0, θ 1 ]}. From (15), we infer that ( 16) 13) and ( 16), u θ is bounded in D 1; (2,q) uniformly in θ. The a priori bound in L s follows from Lemma 2.2 according to whether q < N , q = N or q > N .

c 1 θ = I θ (u θ ) ≤ M 1 . By (
Step 4: The Euler-Lagrange equation. By the Lagrange multiplier rule, there exists

λ θ ∈ R such that for all ϕ ∈ D 1;(2,q) rad (R N ), I θ (u θ )(ϕ) = λ θ α R N |u θ | α-2 u θ ϕ.

This means that

-div a θ (|∇u θ | 2 )∇u θ = λ θ α|u θ | α-2 u θ in R N ,
in the weak sense. As u θ is radial, [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] follows.

Observe that it is standard to prove that u θ is a classical solution of ( 11) on ]0, +∞[. If q > N then the solution is bounded and we can apply the regularity theory of Lieberman [START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF] to deduce that the weak solution u θ is also C 1,α for some 0 < α < 1 in a neighborhood of the origin. We can deduce the regularity at the origin from even simpler arguments if q < N . Observe that for α > 2 , we have N -N/α > N α/(N + α). In particular, Proposition 3.

1 holds if q > N -N/α. Lemma 3.2. Let α > 2 and N -N α < q < N . If u θ is a minimizer of (10), it is bounded in C 1 (R N ) and either u θ > 0 or u θ < 0 on R N .
Proof. As u θ is a solution of ( 11) on ]0, +∞[, it is standard to check that, for r > 0, u θ is regular. On the other hand, one observes that r N -1 a θ (|u θ | 2 )u θ satisfies the Cauchy condition at the origin so that it has a finite limit as r → 0. This limit must be zero otherwise we have

r N -1 a θ (|u θ | 2 )|u θ | 2 ≥ Cr -N -1 q-1
near 0, which is not integrable because q < N . This contradicts the fact that, as u θ is a weak solution of (11), we have

+∞ 0 r N -1 a θ (|u θ | 2 )|u θ | 2 = λ θ α.
We now claim that u θ is bounded. Integrating the equation, we get

a θ (|u θ (r)| 2 )u θ (r) = λ θ α r N -1 r 0 s N -1 |u θ (s)| α-1 ds,
for all r ∈ [0, ∞[. Using the estimate from Proposition 3.1, it follows that

a θ (|u θ (r)| 2 ) |u θ (r)| ≤ Cλ θ αr N (q -α+1) q -N +1 u θ α-1
L q , with C > 0. Moreover, we have N (q -α + 1)/q -N + 1 > 0 since we assume N -N α < q, and therefore u θ (0) = 0 and, for r ≤ 1, we conclude that

a θ (|u θ (r)| 2 ) |u θ (r)| ≤ Cλ θ α u θ α-1
L q . We next deduce from Lemma 2.3 and Proposition 3.1 that for all r > 1,

a θ (|u θ (r)| 2 )u θ (r) = λ θ α r N -1 1 0 s N -1 |u θ (s)| α-1 ds + r 1 s N -1 |u θ (s)| α-1 ds ≤ Cλ θ α u θ α-1 L q + λ θ α r N -1 u θ α-1 L 2 r 1 s N -1 s -(N -2)(α-1) 2 ds ≤ C 1 + r 1-(N -2)(α-1)
and since α > 2 , we have

1 - N -2 2 (α -1) < - N 2
so that the claim follows.

As u θ (0) = 0 on proves by standard arguments that u θ is a classical solution.

To show that any minimizer satisfies either u θ > 0 or u θ < 0, we argue by contradiction. Indeed, if u θ changes sign, then |u θ | ∈ M and I θ (|u θ |) = I θ (u θ ). In other words, v = |u θ | is also a minimizer, and vanishes at some point r 0 ∈ [0, ∞[. Since v is a solution of [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] with min [0,+∞[ v = v(r 0 ) = 0 and the solutions of [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] are regular, we also have v (r 0 ) = 0, which contradicts the local uniqueness of the solution of the Cauchy problem. This concludes the proof.

Back to the original equation (2)

. We now prove that the solution obtained in Proposition 3.1 is a solution of our original problem (2) provided the parameter θ is small enough.

In the sequel, (u θ , λ θ ) is the solution of ( 17)

-div a θ (|∇u θ | 2 )∇u θ = λ θ α|u θ | α-2 u θ in R N ,
obtained in Proposition 3.1. We first estimate the Lagrange multiplier through an argument of the Calculus of Variations.

Lemma 3.3. For all θ ∈ ]0, θ 1 ], we have 0 < λ θ = N N +α c 1 θ .
Proof. Multiplying (17) by u θ and integrating, we obtain (18)

R N a θ (|∇u θ | 2 )|∇u θ | 2 = λ θ α R N |u θ | α = λ θ α.
Next, we prove that

(19) R N a θ (|∇u θ | 2 )|∇u θ | 2 = N α 2N + 2α R N A θ (|∇u θ | 2 ).
To this end, consider the function f : R + → R defined by f (t) := I θ t N α u θ (tx) . For all t ∈ R + , t N/α u θ (tx) ∈ M, and f achieves its minimum at t = 1. A change of variable yields

f (t) = 1 2 R N A θ t 2N α +2 |∇u θ (tx)| 2 dx = 1 2t N R N A θ t 2N α +2 |∇u θ (y)| 2 dy.
From the last equality and Lebesgue's dominated convergence theorem, it is easy to see that f is differentiable. Hence, as f (1) is a minimum, we have

f (1) = 1 2 2N α + 2 R N a θ (|∇u θ | 2 )|∇u θ | 2 -N R N A θ (|∇u θ | 2 ) = 0,
which proves [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF].

Combining [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] with [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF], we conclude that

λ θ = N 2N + 2α R N A θ (|∇u θ | 2 ) = N N + α c 1 θ > 0.
An important consequence of this lemma is that the uniform estimate on the levels c 1 θ from Proposition 3.1 yields a uniform estimate on the Lagrange multiplier. This estimate allows to deduce that, for θ small, our regularization leads to a solution of an unperturbed equation (with Lagrange multiplier). Proposition 3.4. Assume N -N/α < q < N . For α > 2 and θ small enough, the function u θ obtained in Proposition 3.1 is a radial solution of (20)

-div ∇u

1 -|∇u| 2 = λ|u| α-2 u in R N , with λ = R N |∇u| 2 1 -|∇u| 2 > 0.
Moreover either u θ > 0 or u θ < 0 on R N .

Proof. Consider the solution (u θ , λ θ ) of

-div a θ (|∇u θ | 2 )∇u θ = λ θ α|u θ | α-2 u θ in R N ,
obtained in Proposition 3.1. Let us prove the existence of a constant E > 0 such that for all θ ∈ ]0, θ 1 ] and all r > 0,

a θ (|u θ (r)| 2 )u θ (r) ≤ E. (21)
We argue as in Lemma 3.2 to deduce uniform estimates. First, using the uniform estimates from Proposition 3.1 and Lemma 3.3, it follows that for all r < 1 and all θ ∈ ]0, θ 1 ],

a θ (|u θ (r)| 2 ) |u θ (r)| ≤ Cλ θ α u θ α-1 L q ≤ C, with C > 0 independent of θ ∈ ]0, θ 1 ]
. Moreover, by Lemma 2.3, Proposition 3.1 and Lemma 3.3, we have for all r > 1 and θ ∈ ]0, θ 1 ],

a θ (|u θ (r)| 2 )u θ (r) ≤ Cλ θ α u θ α-1 L q + λ θ α r N -1 ∇u θ α-1 L 2 r 1 s N -1 s -(N -2)(α-1) 2 ds ≤ C 1 + r 1-(N -2)(α-1) 2 ,
where C > 0 is still independent of θ. As α > 2 , we have

1 - N -2 2 (α -1) < - N 2 .
This proves [START_REF] Kawohl | Rearrangements and convexity of level sets in PDE[END_REF].

Finally, by construction of a θ , [START_REF] Kawohl | Rearrangements and convexity of level sets in PDE[END_REF] implies that |u θ (r)| ≤ 1 -for some > 0, and hence u θ solves (20) for θ small enough. More precisely, we have for all r ≥ 0 and all θ < min{θ 1 , 1/(1 + E 2 )},

|u θ (r)| ≤ E √ 1 + E 2 ,
and the result follows for θ < min{θ 1 , 1/(1+E 2 )}. The fact that λ := λ θ is bounded away from zero follows from Lemma 3.3.

Proof of Theorem 1.1. By Proposition 3.4, we know that for θ small enough, u θ is a radial solution of (20) i.e. u θ is a solution of

-r N -1 v 1 -|v | 2 = λr N -1 |v| α-2 v in ]0, +∞[.
Observe that w t defined by w t (r) = tu θ (r/t) solves

-r N -1 w 1 -|w | 2 = λ 1 t α r N -1 |w| α-2 w in ]0, +∞[.
Then w t is a solution of the original equation ( 2) if t = λ 1/α . Remark 3.5. Note that, for θ ∈ ]0, θ 1 ], w t satisfies in fact

I θ (|∇w t | 2 ) = min I θ (|∇v| 2 ) : v ∈ D 1;(2,q) rad (R N ), R N |v| α = λ α+N α
,

where λ = c 1 θ N/(N + α) > 0.

Optimizers in the inequality involving the volume integral

This section deals with the proof of Theorem 1.2 stated in the introduction. This theorem will follow from Proposition 4.1, Proposition 4.2, Proposition 4.4 and Proposition 4.5 below. Proposition 4.1. Assume α ≥ 2 . Then there exists a constant C > 0, depending only on α and N , such that for all u ∈ X ,

(22) R N (1 -1 -|∇u| 2 ) ≥ C R N |u| α N α+N .
Proof. If α ≥ 2 then 2 ≤ N α/(N + α). Hence, using the fact that ∇u L ∞ ≤ 1 and Sobolev inequality, we have for all u ∈ X ,

R N (1 -1 -|∇u| 2 ) ≥ 1 2 R N |∇u| 2 ≥ 1 2 R N |∇u| N α N +α ≥ C R N |u| α N N +α
, where C > 0 depends only on α and N .

Observe that the exponent N α/(α + N ) in the L α -norm naturally arises in the proof when using Sobolev inequality. The presence of this exponent can also be explained from the invariance of the inequality [START_REF] Kiessling | Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequence[END_REF] under the homeomorphisms φ t : u(•) → tu(•/t) for t > 0.

We next show that the inequality [START_REF] Kiessling | Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequence[END_REF] does not hold whatever C > 0 when α < 2 .

Proposition 4.2. If α < 2 then inf u∈X \{0} R N (1 -1 -|∇u| 2 ) R N |u| α N N +α = 0. Proof. It is straightforward to construct a sequence (u n ) n ⊂ D 1;2 (R N ) such that ∇u n L ∞ ≤ 1, u n L α = 1, and R N |∇u n | 2 → 0 as n → ∞. Then we have for all n ∈ N, R N (1 -1 -|∇u n | 2 ) = R N |∇u n | 2 1 + 1 -|∇u n | 2 ≤ R N |∇u n | 2 ,
and the conclusion follows.

We now focus on the best constant for which [START_REF] Kiessling | Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequence[END_REF] holds when α > 2 . We will use the following lemma. For the definition and basic properties of the symmetric rearrangement, the reader is referred to [START_REF] Kawohl | Rearrangements and convexity of level sets in PDE[END_REF][START_REF] Talenti | Inequalities in rearrangement invariant function spaces[END_REF] (among many others). Since we adapt a rather classical lemma from [START_REF] Talenti | Inequalities in rearrangement invariant function spaces[END_REF], we keep the notations therein. In particular, the symmetric rearrangement u of u is the function whose graph is the Schwarz symmetrization of |u|, see for instance [29, Definition 1.C].

Lemma 4.3. For all u ∈ X , we have the inequality

(23) R N 1 -1 -|∇u| 2 ≥ R N 1 -1 -|∇u | 2 ,
where u : R → R denotes the symmetric rearrangement of u.

Proof. First we observe that u is well defined if u ∈ X because u is Lipschitz continuous and all the level sets {x ∈ R N : u(x) > t} (t ∈ R) have finite measure. In addition, by the Pólya-Szegö inequality (see for instance [10, Theorem 4.7]), we have

∇u L ∞ ≤ ∇u L ∞ ≤ 1 and R N |∇u | 2 ≤ R N |∇u| 2 .
Therefore the right-hand side of (23) makes sense and both sides of the inequality are finite because ∇u is square integrable for u ∈ X .

It is proven in [29, Theorem 1.C] (see also [START_REF] Kawohl | Rearrangements and convexity of level sets in PDE[END_REF]) that the inequality

R N Φ(|∇u|) ≥ R N Φ(|∇u |)
holds for any Lipschitz-continuous u which decays at infinity and any convex, in-

creasing function Φ : [0, ∞[→ [0, ∞[ satisfying Φ(0) = 0.
For all n ∈ N, let us consider the functions

H n , G n : [0, ∞[→ [0, ∞[ defined by H n (s) = 1 -(1 -s) 1/2 , for s < 1 -1/n 2 , = 1 - 1 n + n 2 (s -1 + 1 n 2 ), for s ≥ 1 -1/n 2 ,
and G n (s) = H n (s 2 ). Observe that G n is convex, increasing and satisfies G n (0) = 0. Hence, by [29, Theorem 1.C], we know that (24)

R N G n (|∇u|) ≥ R N G n (|∇u |).
As u ∈ D 1;2 (R N ), the measure of the set A := {x ∈ R N : |∇u| ≥ 1/2} is finite and the fact that u ∈ X implies that, for all n ≥ 2,

|G n (|∇u(x)| 2 )| ≤ h(x) with h ∈ L 1 (R N ) defined by h(x) = 1, for x ∈ A, = |∇u| 2 , for x ∈ A.
Hence, we can apply Lebesgue's dominated convergence theorem to prove that (25)

R N G n (|∇u|) → R N 1 -1 -|∇u| 2 .
as n goes to infinity. We can argue in the same way to prove that (26)

R N G n (|∇u |) → R N 1 -1 -|∇u | 2 .
We then conclude by ( 24), ( 25) and [START_REF] Ni | Existence and non-existence theorems for ground states for quasilinear partial differential equations[END_REF].

With this lemma at hand we can prove the following proposition.

Proposition 4.4. If α > 2 , the infimum C(α) := inf u∈X \{0} R N (1 -1 -|∇u| 2 ) R N |u| α N N +α
is achieved by a radial solution of (2).

Proof. By Lemma 4.3 and the L α -norm-preserving property of the symmetric rearrangement, we may restrict our attention to a minimizing sequence (u n ) n ⊂ X of radial functions. Since the quotient is invariant under the family of homeomorphisms φ t : v(•) → tv(•/t) (t > 0), we may assume that

R N |u n | α = 1.
It is easily seen that (u n ) n is a priori bounded in X . Lemma 2.1 then provides a bound in D 1;q (R N ) for every q ≥ 2. From Lemma 2.4, we deduce the required compactness to conclude that (u n ) n weakly converges in D 1;2 (R N ) to a function u ∈ X with R N |u| α = 1. The fact that u realizes the infimum C(α) follows from the weak lower semi-continuity (with respect to the weak convergence in D 1;2 (R N )) of the volume integral.

To show that w t = t u(•/t) solves (2) for some t > 0, we first prove that |∇u| is bounded away from 1. Denoting by u θ a minimizer of I θ over M (see Proposition 3.1), we have for all θ > 0, [START_REF] Ramos | A Bahri-Lions theorem revisited[END_REF]

I θ (u θ ) ≤ I θ (u) ≤ I 0 (u),
where the second inequality follows from the ordering property of the family I θ . Moreover, we have established in Section 3 that I θ (u θ ) = I 0 (u θ ) for θ small enough. As u is a minimizer of I 0 this implies that the inequalities in [START_REF] Ramos | A Bahri-Lions theorem revisited[END_REF] are in fact equalities. In particular, I θ (u) = I θ (u θ ), and u is a minimizer of I θ over M too. The arguments of Proposition 3.4 now apply so that

|u (r)| ≤ E √ 1 + E 2 < 1,
for some E > 0 and we conclude as in the proof of Theorem 1.1.

We now turn to the case of the critical exponent α = 2 .

Proposition 4.5. The infimum

C(2 ) = inf u∈X \{0} R N (1 -1 -|∇u| 2 ) R N |u| 2 N N +2
is not achieved.

Proof. Assume by contradiction that C( 2) is achieved by some u ∈ X . As above, we may suppose that u is radial. Let us prove that (28)

∞ 0 r N -1 1 + N α u 2 1 -|u | 2 -N (1 -1 -|u | 2 ) ≤ 0.
Define for all t ∈ [0, 1],

f (t) := 1 2 R N A 0 t 2N α +2 |∇u(tx)| 2 dx = 1 2t N R N A 0 t 2N α +2 |∇u(y)| 2 dy.
Let t ∈ (0, 1) be fixed. As 1 is a minimum of f , the mean value theorem yields the existence of t ∈ (t, 1) such that [START_REF] Talenti | Inequalities in rearrangement invariant function spaces[END_REF] f ( t) ≤ 0.

(Note that we cannot conclude as in Lemma 3.3 that f (1) = 0 because f may not be well defined for t > 1.) Here, the mean value theorem applies because f is continuous on [t, 1] and differentiable on (t, 1). In order to prove the differentiability of f in s ∈ (t, 1), observe first that from the strict inequality s(2N/α)+2 |∇u| 2 < 1 a.e. in R N , we deduce the differentiability of the integrand. Moreover the derivative of the integrand satisfies the uniform estimate

2N α + 2 s 2N α +1 |∇u| 2 1 - s 2N α +2 |∇u| 2 ≤ C |∇u| 2 ,
which holds for all s close to s and almost every x ∈ R N . Lebesgue's dominated convergence theorem implies then that f is differentiable on (t, 1), and the inequality ( 29) is equivalent to

(30) -N t-N-1 R N 1 -1 - t 2N α +2 |∇u| 2 + t-N R N 1 2 2N α + 2 t 2N α +1 |∇u| 2 1 - t 2N α +2 |∇u| 2 ≤ 0.
Next, we consider (t k ) ⊂ (0, 1) such that t k → 1 as k → ∞. From what precedes, we infer the existence of a sequence ( tk ) ⊂ (0, 1) still converging to 1 as k goes to ∞, and satisfying (30) with t = tk for all k ∈ N. This implies that, for every k ∈ N,

0 ≤ R N N α + 1 t 2N α +1 k |∇u| 2 1 - t 2N α +2 k |∇u| 2 ≤ N tk R N 1 -1 - t 2N α +2 k |∇u| 2 ≤ N t 2N α +1 k R N |∇u| 2 ≤ N R N |∇u| 2 .
Hence, it follows from Fatou's Lemma, Lebesgue's dominated convergence theorem and (30)

that |∇u| 2 √ 1-|∇u| 2 ∈ L 1 (R N ) and R N |∇u| 2 1 -|∇u| 2 ≤ lim inf k→∞ R N N α + 1 t 2N α +1 k |∇u| 2 1 - t 2N α +2 k |∇u| 2 ≤ lim inf k→∞ N tk R N 1 -1 - t 2N α +2 k |∇u| 2 = N R N 1 -1 -|∇u| 2 .
This implies that (28) holds.

To conclude, we define the function g : [0, 1[→ R by g(s) := (1 + N α -N )s -N √ 1 -s + N and we compute g(0) = 0, g (0) = 1 + N α -N 2 and g (s) = N 4(1-s) 3/2 . Therefore we have g(s) > 0 for s ∈ ]0, 1[ if and only if 1 + N α -N 2 ≥ 0, which is true if and only if α ≤ 2 . Hence, we infer that

0 < ∞ 0 r N -1 g(u 2 ) 1 -|u | 2 = ∞ 0 r N -1 1 + N α u 2 1 -|u | 2 -N (1 -1 -|u | 2 ) ,
which contradicts (28).

A multiplicity result

In this section, we use again the auxiliary functional I θ defined in Section 2.

Since the manifold M is symmetric and I θ is an even functional, Lusternik-Schnirelmann category theory provides a sequence of critical values for I θ constrained to M. More precisely, let A denote the set of closed and symmetric (with respect to the origin) subsets of D 1;(2,q) rad (R N ). We define the usual min-max values

c k θ := inf A∈Γ k max u∈A I θ (u),
where Γ k := {A ⊂ M : A ∈ A, A is compact and γ(A) ≥ k}, and γ(A) is the genus of the set A. We refer e.g. to [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF] for the definition of the genus and for more details on Lusternik-Schnirelmann theory.

We first show that these levels are indeed critical levels of I θ . It is clear that M ⊂ A and γ(M) = +∞. Next we show that I θ satisfies the Palais-Smale condition on M by which we mean that every sequence (u n ) n ⊂ M such that I θ (u n ) is bounded and I θ|M (u n ) → 0 admits a converging subsequence. Here I θ|M denotes the derivative of I θ constrained to M. Denoting by

T u M := {v ∈ D 1;(2,q) rad (R N ) : R N |u| α-2 uv = 0}
the tangent space to M at u, the projection P u :

D 1;(2,q) rad (R N ) → T u M is given by P u (w) = w -u R N |u| α-2 uw.
Then, for any w ∈ D

1;(2,q) rad (R N ) we have v = P u (w) ∈ T u M and I θ|M (u)(v) = I θ|M (u)(P u (w)) = I θ (u)(w) -λI θ (u)(u),
where λ = R N |u| α-2 uw. To prove the Palais-Smale condition, we will use the following convexity inequalities.

Lemma 5.1. There exist γ 2 , γ q > 0 such that for every u, v ∈ D

1;(2,q) rad (R N ), (31) I θ ( u + v 2 ) ≤ 1 2 I θ (u) + 1 2 I θ (v) -γ 2 R N |∇u -∇v| 2

and

(32)

I θ ( u + v 2 ) ≤ 1 2 I θ (u) + 1 2 I θ (v) -γ q R N |∇u -∇v| q .
Proof. Since I θ has a uniformly positive definite second derivative, we can apply [START_REF] Nápoli | Mountain pass solutions to equations of p-Laplacian type[END_REF]Lemma 2.3] to deduce (31). In order to prove (32), we first observe that [16, Lemma 2.1] allows to show that s → A θ (s 2 ) is strongly q-monotone. This yields, for some γ q > 0, the inequality

A θ u (r) + v (r) 2 2 ≤ 1 2 A θ (u (r) 2 ) + 1 2 A θ (v (r) 2 ) -2γ q |u (r) -v (r)| q
where u, v are given functions in D 1;(2,q) rad (R N ) and r > 0. Mutliplying by r N -1 and integrating from 0 to +∞, we deduce (32).

We now turn to the verification of the Palais-Smale condition. Lemma 5.2. For α > 2 , the functional I θ satisfies the Palais-Smale condition on M.

Proof. Let (u n ) n ⊂ M be a Palais-Smale sequence, i.e. I θ (u n ) is bounded and

I θ|M (u n ) → 0. Since I θ is coercive, it is clear that (u n ) n is bounded in D 1;
(2,q) rad (R N ) and therefore, by Lemma 2.4, up to a subsequence, there exists u ∈ M such that u n converges weakly to u in D 1;(2,q) rad (R N ) and strongly in L α (R N ) as n → ∞. Since (u n ) n is a Palais-Smale sequence, we have, as n → ∞,

I θ (u n )(P un (u n -u)) = I θ (u n )(u n -u) -λ n I θ (u n )(u n ) → 0,
where we have written

λ n = R N |u n | α-2 u n (u n -u). Now, using the fact that (u n ) n is bounded in D 1;(2,q) rad (R N ) and u n → u in L α (R N ), we infer λ n → 0 and I θ (u n )(u n ) is bounded. Hence, we deduce that (33) lim sup n→∞ I θ (u n )(u n -u) ≤ 0.
To complete the proof, it remains to show that (u n ) n converges strongly to u, which amounts to prove that

u n -u = R N |∇u n -∇u| 2 1 2 + R N |∇u n -∇u| q 1 q → 0,
as n → ∞. Since I θ is locally bounded, we may assume that I θ (u n ) converges. By weak lower semi-continuity, we infer where for short we have written µ = µ k θ and v = u k θ . Then, using the equation, we compute

F (r) = r N -1 a θ (|v | 2 )|v | 2 + r[ r N -1 a θ (|v | 2 )v v + r N -1 a θ (|v | 2 )v v ] -r N v v a θ (|v | 2 ) - N 2 r N -1 A θ (|v | 2 ) + µN r N -1 |v| α +µαr N |v| α-2 vv + N α v(r N -1 a θ (|v | 2 )v ) + N α r N -1 a θ (|v | 2 )|v | 2 = r N -1 (1 + N α )a θ (|v | 2 )|v | 2 - N 2 A θ (|v | 2 ) .
As v and v are bounded, we have F (0) = 0. To estimate F at +∞, we integrate the equation and we obtain This completes the proof of (35).

We are now able to complete the proof of Theorem 1.3.

Proposition 5.5. For α > 2 and θ small enough, the function t k u k θ (r/t k ) where u k θ is given by Proposition 5.3, and

t k =   R N |∇u k θ | 2 1 -|∇u k θ | 2   1/α
, is a solution of (2).

Proof. Fix k ≥ 1. The proof follows from arguments that were used in Section 3. Indeed, since we have an estimate of the Lagrange multiplier µ k θ and on u k θ in D 1;(2,q) rad (R N ) as well as in L q (R N ) which are independent of θ, we infer, as in the proof of Proposition 3.4, that

|∇u k θ | ≤ E √ 1 + E 2 ,
for some E > 0. The result then follows for θ < min{θ 1 , 1/(1 + E 2 )} as in the proof of Theorem 1.1.

Theorem 1 . 1 .

 11 If α > 2 , equation (2) has a positive radial classical solution.

I

  θ (u) ≤ lim inf n→∞ I θ (u n ),whereas the convexity of I θ and (33) implies lim supn→∞ I θ (u n ) ≤ I θ (u) + lim sup n→∞ I θ (u n )(u n -u) ≤ I θ (u).Hence I θ (u n ) converges to I θ (u). Using again the lower semi-continuity of I θ , (31) and (32), we conclude thatI θ (u) ≤ lim inf n→∞ I θ ( u n + u 2 ) ≤ I θ (u) -γ 2 lim sup n→∞ R N |∇u n -∇u| 2 and I θ (u) ≤ lim inf n→∞ I θ ( u n + u 2 ) ≤ I θ (u) -γ q lim sup n→∞ R N bounded in C 1 (R N ) and satisfying R N |∇u k θ (x)| 2 < ∞.Let us define the functionF (r) = r N a θ (|v | 2 )|v | 2 -1 2 r N A θ (|v | 2 ) + µr N |v| α + N α r N -1 v va(|v | 2 ) = r r N -1 v a θ (|v | 2 ) v -1 2 r N A θ (|v | 2 ) + µr N |v| α + N α v r N -1 a θ (|v | 2 )v ,

r N - 1 0 (s N - 1 µα 1 0(s N - 1 1 ( 2 -N - 2 2N 2 -N - 2 2 2 + 0 FF

 101111222220 a θ (|v | 2 )v = -µα r 0 (s N -1 |v| α-2 (s)v(s)) ds.Using the decay estimate of Lemma 2.3, the a priori bound of Proposition 5.3 and the arguments of Lemma 3.2, we deduce thatr N -1 a θ (|v | 2 )|v | ≤ µα r |v| α-1 (s)) ds. ≤ |v| α-1 (s)) ds + C r α ).Hence, we deduce thata θ (|v | 2 )|v | ≤ C(r 1-N + r α ),and since a θ (|v | 2 ) ≥ 1, the same estimate holds for |v |. This implies, again by Lemma 2.3 and Proposition 5.3, thatF (r) ≤ r N a θ (|v | 2 )|v | 2 + µr N |v| α + N α r N -1 v va θ (|v | 2 ) ≤ C(r 2-N + r 2+N -(N -2)α r 2N -(N -2)α + r N -N -2 2 α + r -(r) ≤ lim sup r→∞ F (r) -lim r→0 (r) = 0.