Laïla Benraïss-Noailles
email: laila.benraiss-noailles@u-bordeaux.fr

Olivier Herrbach
email: olivier.herrbach@u-bordeaux.fr

Catherine Viot
email: catherine.viot@univ-lyon1.fr

The impact of CSR perceptions on employer attractiveness: an empirical study

Keywords: Perception of CSR commitment, Organizational attractiveness, Recent graduates Perception de l'engagement RSE, Attractivité organisationnelle, Etudiants en Masters 1 et 2

This study examines the impact of the perception of a company's commitment to CSR on its attractiveness as an employer, as perceived by management students. More specifically, it aims at checking whether or not future business graduates are more likely to be attracted to an employer based on their perceptions of that company's commitment to CSR.

A questionnaire-based survey was carried out with 295 first-and second-year master in management students in French and Moroccan universities.

Overall, the results show that the perception of CSR has no significant impact on the attractiveness of a potential employer. A country-specific analysis confirmed this result, as no difference between French and Moroccan students was found. The paper's findings show that future employees may not be as sensitive to CSR aspects as could have been expected. It suggests that graduates are likely to be more attracted by their own expected working conditions as opposed to general CSR practices.

of business leaders into question and global warming have made it important for those companies that are conscious about their public image to demonstrate that they uphold the highest ethical standards. This general trend occurs at the worldwide level, albeit at different paces and with country-specific determinants and features.

In France, there has been increased interest in CSR practices since the beginning of the century.

As of 1999, French companies are required by law to devote a portion of their annual report to their commitment towards social responsibility and sustainable development. This requirement now applies to all firms with over 500 employees. In a short time, these practices have become institutionalized, to such an extent that most listed companies now have a CSR or sustainable development manager in charge of implementing the company's social responsibility strategy.

As a result of current legislation, CSR in France includes a growing number of social issues, such as diversity, disability and equal opportunities. Other issues related to a company's reputation are also reflected in CSR: change management and responsible restructuring, the promotion of social dialogue, quality of work life, etc. Therefore, CSR -as originally and currently conceived -extends beyond abiding by legal standards to defining a broader management philosophy and is one way to increase attractiveness and social branding as evidence of fair working conditions.

In Morocco, the royal speech delivered during the 3 rd Intégrales de l'Investissement (investment conference, in 2005) encouraged companies to establish CSR indicators and tools: "Social responsibility requires more than charitable compassion since it is an essential condition for the sustainability, long-term profitability and social acceptability of investments and economic growth. Keeping this in mind, we are closely monitoring the development of a flow of financial investments and placements combining legitimate goals of cost-efficiency and profits with the equally legitimate universal goals of social responsibility and sustainable human development.

Morocco, in light of its legislation and political choices, is an able and willing partner and destination for socially responsible investors and upholds the highest social, environmental, and governance standards". In accordance with this initiative, the CGEM (Confédération Générale des Entreprises du Maroc -General Confederation of Moroccan Companies) introduced in 2005 the first prize designed to reward socially-responsible companies for their commitment to CSR and organized a national conference to promote CSR in Morocco. This confederation has since adopted a social responsibility charter and introduced its own social label [START_REF] Benraïss-Noailles | Etat des lieux de la RSE au Maroc[END_REF].

The impact of Morocco's opening up to the international market has also contributed to companies paying greater attention to their social responsibility practices. The presence of subsidiaries of multinational firms that are submitted to the requirements of their parent companies in terms of CSR has driven companies to adopt responsible practices. A study comparing the CSR practices of Moroccan companies listed on the stock exchange confirmed this trend [START_REF] Benraïss-Noailles | La RSE en vitrine: l'empreinte des maisons mères[END_REF]. Based on the sustainable development reports available on the institutional websites of the companies under study, the survey concluded that the subsidiaries of international French groups make their practices visible and adopt a communications strategy that makes it easier to understand CSR commitments in comparison to domestic companies.

The present research aimed at assessing whether the alleged beneficial effect of CSR initiatives in terms of organizational attractiveness is empirically supported. This was done using an online survey of two samples of future business graduates in France and Morocco. In so doing, we first contribute to the CSR literature by subjecting one of its important assumptions to empirical enquiry. Secondly, we also relate this literature to the literature on organizational image and attractiveness.

Organizational attractiveness

Organizational attractiveness refers to the "envisioned benefits that a potential employee sees in working for a specific organization" (Berthon, Ewing & Hah, 2005: 156). It affects a company's workforce in several ways. For instance, it may influence their compensation systems: potential employees are willing to make salary concessions if an employer has a good reputation [START_REF] Cable | The value of organizational image in the recruitment context: a brand equity perspective[END_REF]. Organizational attractiveness also has an impact on a number of employee attitudes, including motivation [START_REF] Berthon | Captivating company: dimensions of attractiveness in employer branding[END_REF][START_REF] Lievens | The relation of instrumental and symbolic attributes to a company's attractiveness as an employer[END_REF][START_REF] Lievens | Organizational identity and employer image: towards a unifying framework[END_REF], job satisfaction [START_REF] Barakat | The influence of corporate social responsibility on employee satisfaction[END_REF][START_REF] Bodderas | The influence of the employer brand on employee attitudes relevant for service branding: an empirical investigation[END_REF][START_REF] Davies | Employer branding and its influence on managers[END_REF], commitment (Kunerth & Moseley, 2011;[START_REF] Priyadarshi | Employer brand image as predictor of employee satisfaction, affective commitment and turnover[END_REF][START_REF] Vaijayanthi | Employer branding as an antecedent to organization commitment: an empirical study[END_REF][START_REF] Love | Workplace branding: leveraging human resources management practices for competitive advantage through 'best employer' surveys[END_REF], loyalty [START_REF] Priyadarshi | Employer brand image as predictor of employee satisfaction, affective commitment and turnover[END_REF][START_REF] Mignonac | A matter of prestige? Examining the role of perceived organizational image in turnover intentions of key people[END_REF] and retention [START_REF] Tulasi | Employer brand in India: a strategic HR tool for competitive advantage[END_REF][START_REF] Love | Workplace branding: leveraging human resources management practices for competitive advantage through 'best employer' surveys[END_REF].

Several studies have demonstrated the impact of attractiveness on the intention to apply to and join a company [START_REF] Agrawal | Effect of employer brand image on application intentions of B-school undergraduates[END_REF][START_REF] Collins | The relationship between early recruitment-related activities and the application decisions of new labor-market entrants: a brand equity approach to recruitment[END_REF], the probability of accepting a job offer, the applicant-to-hire conversion rate, the time required to fill a position and the number and quality of applicants [START_REF] Highhouse | Assessing company employment image: an example in the fast food industry[END_REF][START_REF] Cable | The value of organizational image in the recruitment context: a brand equity perspective[END_REF][START_REF] Knox | Measuring and managing employer brand image in the service industry[END_REF][START_REF] Agrawal | Effect of employer brand image on application intentions of B-school undergraduates[END_REF][START_REF] Lemmink | The role of corporate image and company employment image in explaining application intentions[END_REF].

Potential applicants use a wide range of information sources to learn about employers, including those that are not voluntarily provided by companies [START_REF] Cable | The value of organizational image in the recruitment context: a brand equity perspective[END_REF]. Such sources include word-of-mouth (Van Hoye, 2012) and social media [START_REF] Sivertzen | Employer branding: employer attractiveness and the use of social media[END_REF]. Empirical research has shown that candidates are most attracted to companies with a strong employer brand equity [START_REF] Highhouse | Assessing company employment image: an example in the fast food industry[END_REF][START_REF] Cable | The value of organizational image in the recruitment context: a brand equity perspective[END_REF][START_REF] Knox | Measuring and managing employer brand image in the service industry[END_REF][START_REF] Lemmink | The role of corporate image and company employment image in explaining application intentions[END_REF]. Previous research also confirmed that adequate employer branding increases a company's attractiveness for future graduates [START_REF] Shahzad | Relationship between perceived employer branding and intention to apply: evidence from Pakistan[END_REF][START_REF] Agrawal | Effect of employer brand image on application intentions of B-school undergraduates[END_REF].

CSR and corporate attractiveness

Companies seek to set themselves apart from their competitors in order to attract the best candidates [START_REF] Rynes | Recruitment, job choice, and post-hire consequences: A call for new research directions[END_REF]. The characteristics of job position (e.g., pay) and company (e.g. location) influence job choice [START_REF] Chapman | Applicant attraction to organizations and job choice: a meta-analytic review of the correlates of recruiting outcomes[END_REF]. Creating a positive impression in the minds of current and potential employees is an ongoing process that cannot be achieved only through generous compensation programs; many other factors contribute to attractiveness, such as work culture, HR policies, job satisfaction, the company's social image, and many others (Kashikar-Rao, 2014).

However, these factors do not effectively differentiate between companies seeking candidates with the same profiles. Certain CSR practices, on the other hand, may be strong distinguishing factors that allow a responsible company to stand out from other potential employers [START_REF] Jones | Why are job seekers attracted by corporate social performance? Experimental and field tests of three signal-based mechanisms[END_REF][START_REF] Peretti | Ressources Humaines[END_REF]. CSR is a broad and evolving construction which makes its definition quite difficult [START_REF] Zweers | CSR in recruitment advertising: Does it really pay to be green?[END_REF]. [START_REF] Elkington | Cannibals with Forks: The Triple Bottom Line of 21st Century Business[END_REF], most definitions of CSR use a generic term that includes economic (basic need for financial success), social (linked to problems of poverty, health and income inequality) and environmental (business activities impact on the environment and natural resources).

For [START_REF] Kashikar-Rao | Role of CSR in employer branding: Emerging paradigm[END_REF], there is a link between organizational reputation, organizational perceived CSR and employer brand. Now, companies have realized that CSR is inextricably linked to their reputation among external and internal stakeholders. As a result, CSR performance may enhance a company's attractiveness as an employer [START_REF] Backhaus | Exploring relationship between corporate social responsibility and employer attractiveness[END_REF], leading [START_REF] Ewing | Employment branding in the knowledge economy[END_REF] to consider CSR commitments, when upheld, to be assets in terms of perception of a company and its potential attractiveness. For [START_REF] Grigore | The role of corporate social responsibility in building employer's brand[END_REF], socially engaged companies have an advantage over other companies in terms of attractiveness.

They propose a model in which values, CSR strategy and benefits for employees as a result of CSR practices play a mediating role between employees' perceptions of CSR and the employer brand (in terms of being a responsible employer). Indeed, "by implementing a CSR strategy and sharing information about this subject publicly, organizations send signals to the job market and enhance their attractiveness" (Capelli, Guillot-Soulez & Sabadie, 2015: 6).

Several other studies establish the importance of CSR and its effects on organizational attractiveness [START_REF] Albinger | Corporate social performance and attractiveness as an employer to different job seeking populations[END_REF][START_REF] Bauer | Green career choices: The influence of ecological stance on recruiting[END_REF][START_REF] Gomes | Employer branding constrains applicants' job seeking behaviour?[END_REF][START_REF] Greening | Corporate social performance as a competitive advantage in attracting a quality workforce[END_REF][START_REF] Lievens | The relation of instrumental and symbolic attributes to a company's attractiveness as an employer[END_REF]. [START_REF] Aiman-Smith | Are you attracted? Do you intend to pursue? A recruiting policy-capturing study[END_REF] evaluated two dimensions of CSR, ecological practices and lay off policy. They found that the CSR's ecological dimension was a better predictor of organizational attractiveness. [START_REF] Behrend | Effects of pro-environmental recruiting messages: The role of organizational reputation[END_REF] confirmed this result and showed that a pro-environmental message on the recruitment site improves students' perception of the organization's reputation.

In another study based on a sample of business administration students, Turban and Greening (1997b) examined the effects of CSR as a multidimensional approach. The results show that an organization's rating on employee relations, product quality and the environment were the strongest predictors of organizational attractiveness. In another later research, [START_REF] Greening | Corporate social performance as a competitive advantage in attracting a quality workforce[END_REF] conducted an experiment in which CSR was manipulated in terms of relations with unions (weak and strong). The authors found that job seekers were more attracted to organizations with a positive reputation (strong union relations).

The perception of a company's CSR activities is therefore a key factor in building an employer brand image and in the choices of future applicants [START_REF] Grigore | The role of corporate social responsibility in building employer's brand[END_REF][START_REF] Bhattacharya | Using corporate social responsibility to win the war for talent[END_REF][START_REF] Backhaus | Exploring relationship between corporate social responsibility and employer attractiveness[END_REF][START_REF] Albinger | Corporate social performance and attractiveness as an employer to different job seeking populations[END_REF]Turban & Greening, 1997a). [START_REF] Jones | Why are job seekers attracted by corporate social performance? Experimental and field tests of three signal-based mechanisms[END_REF] identified the main CSR signals that affect organizational attractiveness. They included potential candidates' anticipated pride in being associated with a prestigious organization commended for its CSR efforts, as well as the expected work conditions resulting from the company's efforts to increase employee well-being through responsible practices on their behalf. After reviewing this literature, we propose the following hypothesis:

The perception of CSR affects a company's attractiveness as an employer.

Methods

To test our hypothesis, a quantitative empirical study was conducted using two samples of students in two different countries. An online questionnaire was used to collect data. We used measurement scales that demonstrated strong psychometric properties. This methodological approach is consistent with the positivist epistemological position adopted for this research and has been a characteristic of most empirical investigations in the field of employer branding to date.

Sample

The Sphinx software was used to design the questionnaire and make it available online in order to collect data. The questionnaire was disseminated in Moroccan (sample 1) and French (sample 2) universities using a link sent by mail to students directly and shared on social media sites (Linkedin, Viadeo and Facebook) between the months of September and December 2016. The current study is based on convenience sample. Interviewing future graduating students allows to target a population of potential employees and thus respond to the issue of the present study, namely the attractiveness of potential employees and the determinants of their choice of a future employer. Moreover, targeted individuals were students who were about to complete their studies. These upcoming graduates will be looking for employment in the coming months.

Using this type of sample is a widespread practice in studies exploring the impact of CSR on organizational attractiveness (e.g. Turban & Greening, 1997b;[START_REF] Behrend | Effects of pro-environmental recruiting messages: The role of organizational reputation[END_REF].

Screening questions were used in order to include only first-and second-year professional master's students in management sciences as respondents for this study. The average age of respondents for the first sample (n=154) was 23 years and 11 months. Men represented 65% of the respondents and second-year master students represented 54.5% of the sample. The respondents in the second sample (n=141) had an average age of 23 years and 6 months. Men represented 56% of the sample and second-year master students represented 62.4% of the sample. The average age of respondents was 23 years and 11 months for sample 1 (n=154) and 23 years and 6 months in sample 2 (n = 141). Men represented respectively 65% and 56% in sample 1 and sample 2. 54.5% of respondents were second-year master students in sample 1 (62.44%, in sample 2). Therefore, both samples were taken from generation Y, as popularized by [START_REF] Coupland | Generation X: Tales for an accelerated culture[END_REF], referring to individuals born between 1977 and 1995.

Measures

The survey instrument was administered in French. Except for the demographic variables, the response categories for all variables ranged from 1 ("strongly disagree") to 7 ("strongly agree").

The respondents were first asked to think of the specific company they would like to work for when they start their career. Then, they were told to answer all items included in the questionnaire based on their assessment of this company.

The scale proposed by [START_REF] Jones | Why are job seekers attracted by corporate social performance? Experimental and field tests of three signal-based mechanisms[END_REF] was used to measure the perception of the potential future employer's CSR activities. The items in this scale measure two facets of CSR perceptions. The first dimension, CSR-Community, is measured by four items: "This company gives back to its community (locally, nationally, and/or internationally)", "This company takes part in voluntary or charitable activities", "This company is active in helping its community"

and "This company tries to have a positive influence on its community". Four additional items reflected the second facet, the perception of CSR-Environment: "This company has good environmental policies", "This company is concerned about environmental sustainability", "This company tries to reduce its impact on the environment" and "This company is an environmentally friendly company".

The organizational attractiveness of the employer of choice is measured by an overall scale of organizational attractiveness [START_REF] Rampl | Employer brand trust and affect: linking brand personality to employer brand attractiveness[END_REF], including three items: "I would accept a job offer from X", "I would attend to apply for a position at X" and "I would very much like to work for X".

Results

Descriptive analysis -Moroccan sample

Future graduates were asked to answer an initial question about the company they intended to apply to after graduation. A descriptive analysis of the responses shows that for the Moroccan respondents, private Moroccan companies topped the list (49%). These companies included primarily the OCP1 (13%), the banking sector (14%) and telecommunications (14%). The other responses in this category were divided between many other companies (mainly three large groups Al Omrane, Managem and Menara Holding). Next, came multinational companies (30%) -led by a three-way tie between Renault, Procter & Gamble and Unilever, followed by Google and L'Oréal2 -and the public sector (21%). If we take OCP's past into consideration (public company until 2008, but still associated with the public sector in respondents' minds), a third of the respondents hoped to work in this sector.

In our sample, respondents are prepared to relocate in order to find their first job (92% will look for a job in another city, primarily in Casablanca) and a large majority (64%) had no previous experience in the desired company (64%). The absence of a first experience (e.g.: internships) in the desired company emphasizes the important role that a company's reputation could play in respondents' choices, since the choice is not based on the desired company knowledge.

Descriptive analysis -French sample

For the French respondents, two sectors topped the list: agri-food and ICT (17%) with Danone and Google being cited the most often in their respective sectors. The third most attractive sector was cosmetics with L'Oréal (12%). The aeronautics sector came in fourth place (9% of respondents), with Dassault as the top choice among this industry. More than a third of respondents (36%) indicated that they did not plan to move to a new location for their first job, while more than 20% indicated that they plan to start their career abroad. Out of the respondents who planned to move to relocate, Paris and the surrounding region was the leading destination for 34% of the respondents.

Results of the exploratory factor analysis

The exploratory factor analysis of the scale used to measure perception of CSR showed that it explains 50% of the total variance (Table 2). The resulting structure is one-dimensional, as seven out of the eight items in the scale loaded on one dimension. The second dimension was deleted because it was mono-item. Table 1 shows the factor structure of the scale. The exploratory analysis of the measurement scale for organizational attractiveness showed the expected one-dimensional structure and explained 81% of the variance (Table 3).

Results of the confirmatory analysis and discussion

The AMOS module of the SPSS software was used to assess the overall fit of the model to the data. The figure 1 shows the results. These results show that the perception of CSR commitment has no significant impact on a company's attractiveness as an employer. The respondents did not appear to be attracted to companies because of their strong commitment to CSR activities. Therefore, our hypothesis was not supported. A differentiated analysis of the two samples allowed us to confirm this impact. Figure 2 presents the correlations of the model for the two samples separately. It shows that the impact of the perception of a company's level of CSR commitment remained insignificant whether the respondents were from Moroccan (0.03) or French (0.11) universities. Therefore, the hypothesis was not supported for these two samples.

Figure 1: Research model

Chi-2=57 for 34 df (p< 0,001). RMSEA=0,048

Figure 2. Country effect

Chi-2=76 for 34 df (p< 0,001). RMSEA=0,090.

Chi-2=48 for 34 df (p< 0,001). RMSEA=0,055.

Discussion

Based on two samples in different countries, perceptions of CSR did not appear to be related to organizational attractiveness. There are a number of possible explanations for these results. The first stems from the fact that CSR has become a widespread practice and no longer represents a means for firms to differentiate themselves. Indeed, since companies (and their subsidiaries) are required to communicate about their CSR activities and initiatives, all companies now CSR Perception inform the public about their responsible practices for employees, society and the environment.

It takes an expert eye to differentiate between more and less responsible companies, and an indepth understanding of sustainable development reports, which is not often accessible to the general public. Such information concerning CSR -except when negative and reported in the media, therefore tarnishing a company's reputation -does not impact choice and unaided awareness of companies perceived as having a positive reputation.

Another explanation could stem from a fear of being manipulated -applicants may pay particular attention to the credibility of companies' messages about their CSR actions. CSR is often considered to be a marketing tool as it is increasingly present in institutional or advertising communications [START_REF] Gauthier | Marketing et développement durable[END_REF]. Due to repeated exposure, a neutralizing effect may appear in relation to these messages. From a managerial perspective, this could raise questions about the importance of this type of message, at least in relation to recent graduates as a target. In any event, the credibility of corporate messages must be closely considered, in order to avoid the potentially negative effects caused by messages perceived as lacking credibility.

Next, the findings for the sample taken from French universities could result from the fact that students learn about CSR in management programs at these universities. This may have increased the respondents' awareness of this dimension in a potential employer. It is possible that this awareness (at least for certain respondents) produced positive effects, as has already been observed for students in the USA [START_REF] Jones | Why are job seekers attracted by corporate social performance? Experimental and field tests of three signal-based mechanisms[END_REF], a country in which even greater emphasis is placed on CSR in corporate message strategies. Another explanation for this result comes from the rather feminine characteristic of French society (Masculinity score: 43)3 which means that the dominant values in society are concern caring for others, equality of the genders, solidarity and quality of life. While the score obtained by Morocco is intermediate (53), it is not conclusive. This result confirms the conclusions of [START_REF] Katz | Culture based expectations of corporate citizenship: A propositional framework and comparison of four cultures[END_REF] who found that countries with a female tendency were more willing to support the environment.

Knowledge about the dominant cultural dimensions may also provide additional insight. The collectivist culture which is common in Morocco could lead to a company's CSR dimension being seen as self-evident, since CSR actions and commitments are to be expected. According to [START_REF] Labelle | La conjugaison des facteurs contextuels, organisationnels et individuels comme déterminant de la sensibilité des PME au sujet du développement durable[END_REF], the society defines what is expected from the company. Indeed, a collectivist society (like Morocco) will rather promote solid relations where each one assumes responsibility for the other members of his/her group. Employer -employee relationships are perceived as a family bond 4 . Therefore, potential applicants would give little consideration to these actions in differentiating companies in their job search, and CSR messages would not be as necessary as in other settings. Previous studies have shown that culture influences the impact of perceptions of CSR [START_REF] Kang | The impact of corporate social responsibility activities on organisational attractiveness across different cultures[END_REF].

Finally, it could also be that CSR perceptions may not be as important in terms of determining future employee choices as other aspects, such as the quality of the specific job position that people expect to have. More specifically, CSR commitments are not a determining factor for candidates seeking their first job. Their concerns are probably elsewhere (e.g. a good salary).

The role of CSR could be more decisive for more experienced potential employees. This will be the subject of future research. Duarte, Gomes and Gonçalves das Neves (2014) have indeed shown that applicants' responses are impacted mostly by engagement in practices toward employees and in the economic domain compared to engagement in practices that benefit community and environment.

Conclusion

This research showed that the perception of a potential employer's CSR commitment does not have a significant impact on student intentions, whether from French or Moroccan universities. These findings could be further examined through future research in order to understand expectations regarding CSR in the two contexts.

Future research could examine the influence of CSR communication on a company's attractiveness as an employer in differentiating the aspects related to the environment from those related to the company's involvement in the community. This could be conducted by studying the reaction of potential applicants to different company profiles determined in advance using the scenario method. An important dimension in that respect could be attributions made about CSR motives by organizations. [START_REF] Joo | A moderated mediation model of CSR and organizational attractiveness among job applicants: roles of perceived overall justice and attributed motives[END_REF] have indeed shown that CSR communications are more effective when future employees perceive a company as being "other-centered" as opposed to "self-centered" in terms of the reasons for their CSR policies.

It should be noted that there are limitations to this research. The first is inherent to the sample.

First, since the respondents are first and second-year professional master's students and are therefore not actually seeking employment, they may idealize their potential employer. It would be worthwhile to question them again once they have graduated and are actively seeking employment, in order to assess how their perceptions of potential employers' CSR commitment 4 According to the same website. have changed. Second, since the students were not in an authentic job search situation, they did not have the necessary time to look for information about their chosen companies. Respondents spontaneously cited companies (unaided awareness). Depending on the information that they could have found about their chosen company's CSR commitment, they may have been inclined to modify their choice.

Another limitation is methodological in nature. the measure of Attractiveness by an overall scale may bias the results. To rule out this bias, it would be more suitable to use a multidimensional scale, such as that of [START_REF] Berthon | Captivating company: dimensions of attractiveness in employer branding[END_REF]. Using the scale for employer attractiveness by [START_REF] Berthon | Captivating company: dimensions of attractiveness in employer branding[END_REF] The last limitation of this work comes from the deepening of the impact of culture on the perception of CSR practices of companies, only a global approach of national culture was used for the two significant dimensions (masculinity vs feminity and 'individualim vs collectivism).

The literature shows that culture influences both the design and implementation of CSR [START_REF] Matten | Implicit'' and ''Explicit'' CSR: A conceptual framework for a comparative understanding of corporate social responsibility[END_REF][START_REF] Labelle | La conjugaison des facteurs contextuels, organisationnels et individuels comme déterminant de la sensibilité des PME au sujet du développement durable[END_REF][START_REF] Gautier | Les enjeux culturels de la labellisation RSE des PME[END_REF], and the perception of corporate CSR activities by stakeholders [START_REF] Albinger | Corporate social performance and attractiveness as an employer to different job seeking populations[END_REF][START_REF] Aiman-Smith | Are you attracted? Do you intend to pursue? A recruiting policy-capturing study[END_REF][START_REF] Bauer | Green career choices: The influence of ecological stance on recruiting[END_REF][START_REF] Behrend | Effects of pro-environmental recruiting messages: The role of organizational reputation[END_REF][START_REF] Gomes | Employer branding constrains applicants' job seeking behaviour?[END_REF][START_REF] Greening | Corporate social performance as a competitive advantage in attracting a quality workforce[END_REF][START_REF] Lievens | The relation of instrumental and symbolic attributes to a company's attractiveness as an employer[END_REF]. However, little research has been conducted to date on the perceptions of CSR initiatives by potential employees [START_REF] Mueller | The effects of corporate social responsibility on employees' affective commitment: a cross-cultural investigation[END_REF][START_REF] Zweers | CSR in recruitment advertising: Does it really pay to be green?[END_REF]. Future comparative research could take into account the culture's dimensions in order to identify and measure the impact of each dimension on the perception of CSR initiatives by current and / or potential employees, on the basis of Hofstede's VSM 2013 (Value Survey Module, 2013).

  could have produced differing results. It comprises five dimensions of attractiveness (Interest value, Social value, Economic value, Development value, and Application value) and the results could have shown which dimensions are the most affected by the perception of an employer's CSR commitments.

Office Chérifien des Phostphates (Cherifien Phosphates Office)

These companies also appear in the 2018 Universum ranking (hhttps://www.challenges.fr/entreprise/voici-leclassement-des-entreprises-qui-font-rever-les-etudiants-en-2018_576332). These companies play right into the aspirations of generation Y. For students, they convey a likeable and innovative image while offering a multicultural environment or the opportunity to work abroad.

https://www.hofstede-insights.com/country-comparison/france,morocco/ Hofstede Insights is a consulting firm (Itim International until 2017) for more than 35 years, founded with the support of Geert Hofstede.


EA Jean-Jacques Chardin

John Webster's The Duchess of Malfi (1614): the aesthetics of instability

My contention in this paper is that Webster's The Duchess of Malfi conforms to no predefined aesthetic code. The complexity of the technique of characterization, the fascination for dismembered bodies, and the use of exuberant and at times very stilted imagery, bespeak the fluidity of identity, a deep interrogation on the value of death and the co-presence of fixed and evanescent meaning. The play's structural pivot is uncertainty as to the value of all moral and aesthetic categorizations.

Critical assessments of The Duchess of Malfi have long discussed the merits and demerits of the play judged by some as a great poetic achievement only second to Shakespeare's, and by others as a complete dramatic failure for its gross structural inconsistencies and lack of verisimilitude. Building upon Rowland Wymer's and Inga-Stina Ewbank's insightful contributions 1 , I would like to suggest 1 Rowland. Wymer, "The Duchess of Malfi.

that the play should be understood in terms of its patently complex aesthetics and that its pivotal structural device is fluctuation, not only in the way Webster constructs his characters, but also in the presentation of the giddy world which Duke Ferdinand's court epitomizes, and in Webster's use of visual, unstable rhetoric.

1.Character construction: mirrors and foils It has been claimed that The Duchess of Malfi repeats many of the themes and situations of The White Devil, first performed in 1612 (Wymer,52). The Italian setting, the corruption of the Catholic church and the attempt to fashion one's individual life in a world of destruction are common to both plays. Yet, as contended by Rowland Wymer, one of the main differences between the two texts lies in the way Webster constructs his characters. More so than The White Devil, The Duchess of Malfi suggests emotional and psychological complexity in the strategy of characterization. The play presents two symmetrical trios of male and female characters, the Cardinal, Ferdinand and Bosola on the one hand (with Antonio acting as a foil for them), and the Duchess, Julia and Cariola on the other hand (with the Old Lady of II, and 2 as an allegory of the chaos of the world). But within each of these groups, the similarities are strongly balanced by telling differences.

The Cardinal and Ferdinand are first introduced by Antonio who spies on them as intriguers and poisonous predators, as suggested by the animal imagery that defines them. According to Antonio, "the spring in [the Cardinal's] face is nothing but the engendering of toads" (1, 1, 152-3)2 , and Ferdinand uses the law to his benefit as if it were "a foul cobweb to a spider" (1,1,171). The two brothers' characterization is amplified by Antonio who reveals that they are both affected by an excess of melancholy, one of the four humours defined by classical and Renaissance medicine as cold and dry like the earth. But the symptoms of the disease are not the same in the two brothers. In the case of the Cardinal, melancholy manifests itself in the form of jealousy and solitariness. Bosola's incidental comment on travellers who return as "arrant knaves […] because they [carry] themselves always along with them (1, 1, 42-3) is obviously meant to apply to the Cardinal who is entirely focused upon his secluded self. When Bosola comes to kill him at the end of the play, and nobody comes to his rescue, the Cardinal aptly compares himself to a young hare, an animal which Renaissance mythographers viewed as symbolic of loneliness3 : "Shall I die like a leveret / Without any resistance? Help, help, help! / I am slain" (5,5,. In The Anatomy of Melancholy Burton diagnoses solitariness as one of the signs of a diseased mind and reports Hippocrates' description of the melancholy man as unable to "endure the light, or to sit in lightsome places […] and daring not to come in company for fear he should be misused" (Burton,387). Burton completes his description by adding that melancholy men "will diet themselves, feed, and live alone" (Burton,396).

Ferdinand's melancholy, on the other hand, is of an altogether different nature. The fit of lycanthropia he suffers from transforms him into a hybrid creature roaming about Saint Mark's Church at night in a frenzied rage, and attempting to catch his own shadow on the ground, to the bemused eyes of Malateste and the Marquis of Pescara (5, 2, 37). Burton identifies lycanthropia not as melancholy but as raving madness: "Aetius and Paulus call it a kind of melancholy; but I should rather refer it to madness as most do" (Burton,141). As for Bosola, he too is portrayed as a melancholy figure, and again there are many differences with the Cardinal's and Ferdinand's affections. An embittered reviler portrayed by Antonio as "the only court gall" (1,1,23), Bosola is fond of dark secluded places where he can work in close designs, as implied by the image of the moth eating a piece of cloth ("want of action/ Breeds all malcontents: and their close rearing,/ Like moths in cloth, do hurt for want of wearing," 1, 1, 78). Bosola is the prototype of the Jacobean malcontent, dressed in black like Hamlet, a garb that Antonio considers inappropriate because of his new position as the Duchess's provisor of the horse: I do understand your inside.

[…] Because you would not seem to appear to th'world Puffed up with your preferment, you continue This out-of-fashion melancholy. Leave it, leave it. (2,1, The metaphor suggests that Bosola's melancholy is perhaps just affectation, like Hamlet's or Jaques's in As You Like It, but simulation serves him as a front to observe the world around him and pass sardonic judgements on human behaviour. Bosola is an eye debunking courtly life as a breeder of hypocrisy, deformity and vice.

Bosola also shares some of the traits of the Machiavel who, as a spy to Duke Ferdinand, unscrupulously accepts to do anything to gain his ends: Oh, this base quality Of intelligencer! Why every quality i'th'world Prefers but gain or commendation. Now, for this act I am certain to be raised. And men that paint weeds to the life are praised. (3, 2, 314-7) But Bosola's cynicism is heavily balanced by his moral evolution in the course of the play. At the moment he strangles the Duchess in a horrific display of cruelty, he weeps and appears as a penitent murderer poignantly describing his victim's body lying on turtles' feathers:

Oh, she's gone again! There the cords of life broke. Oh, sacred innocence, that sweetly sleeps On turtles' feathers, whilst a guilty conscience Is a black register wherein is writ All our good deeds and bad. (4,2,.

The elegiac tone and the feeling of remorse sit rather awkwardly with his function as murderer and Bosola's moral development in the final acts of the play shows him to be what Rowland Wymer calls "a mobile signifier" (247). So the characterization of the male figures is expressive of a world of artifice, disease and delusion, but each of them appears as a series of variations on similar motifs and it would be difficult to conclude with Catherine Belsey that "they [resolve] into figures in a pattern (Belsey,117).

The trio of female characters (the Duchess, Julia and Cariola) is built on the same technique of perspectives. Although they belong to three different social classes (the Duchess is an aristocrat, Julia is a courtesan and Cariola is a lady-in-waiting), the three of them are submitted to male authority and are killed by men. But Cariola is an entirely submissive figure who cannot accept to live without her mistresss whereas the Duchess and Julia attempt to lead a sexual life of their own, outside the normative codes of the patriarchal society, albeit unsuccessfully.

Julia is a lustful pleasure seeker who changes affections several times in the course of the play. She is the Cardinal's mistress (she claims to have been Delio's previously, 2,4,46), and she falls in love with Bosola in a somewhat melodramatic scene where she first threatens him with a pistol: Bosola: Come; come, I'll disarm you, And arm you thus. [Takes her pistol and embraces her] Yet this is wondrous strange. Julia: Compare thy form and my eyes together, You'll find my love no such great miracle. Now you'll say I am a wanton. The love exchange between the two reverses the conventions of courtly love as it is the woman who woos the man, and the pistol, an overtly sexual symbol, testifies to a reversal of gender roles.

Like Julia the Duchess is "not [a] figure cut in alabaster" (1,1,442), and she too proposes to her suitor. But she is forced to do so as court conventions do not permit that Antonio, her steward, should speak first. Although she is dominated by sexual desire, she is also a loving wife and a caring mother, far from the "lascivious whore" Lisa Jardine considers her to be after her marriage with Antonio (Jardine,72). And as has often been observed, her fortitude and calm compliance with death turn her into a paragon of stoicism completely detached from the contingencies of the world:

Tell my brothers, That I perceive death, now I am well awake, Best gift is they can give or I can take. I would fain put off my last woman's fault: I'd not be tedious to you (4,2, The iteration of the first person pronoun five times within four lines, together with the strong stresses on "they" and "I" in a very regular iambic pentameter(211), shows that the Duchess adopts the composed voice of the determined, self-assured victim who freely accepts to die at Bosola's hands. Her dignified death appears as the moment when she can eventually assert her autonomous self outside all the constraints of the world.

Contrary to Catherine Belsey's view that the figures of the play are patterned on "antitheses whose function is to identify and define" (Belsey,125), I would like to argue that character construction in The Duchess of Malfi relies on an elaborate system similar to that of a mirror cabinet producing contrasted effects of reflection and distortion. Each character is made of a succession of images that the spectator has to take in without being able to reconstruct a complete picture and the technique commands that he constantly accommodate his gaze. Identities are unstable, they are full of "wild turnings" (2,4,12), in the words of the Cardinal addressed to Julia, just like the world they inhabit.

2.The giddiness of the world

The Duchess's remarriage is perceived by Ferdinand as a threat to the stability of the dukedom. Ferdinand realizes that the Duchess is not going to conform to the role he and his brother have planned for her. Brian Chalk has brilliantly shown that Ferdinand and the Cardinal attempt to monumentalize the Duchess into the figure of the obedient widow only to realize that she is not prepared to play the game. They discover that her marriage to Antonio "shatters the illusion of fixity that [their] sister provides for [them]" (Chalk,390). As a result Ferdinand is intent to set the world right again, and his obsession finds expression in his perverted desire to control the Duchess's face with a compass: "I would then have a mathematical instrument made for her face that she might not laugh out of compass" (1, 1, 131-2). But the greater threat to the stability of Calabria is the two brothers' evil designs.

After the Duchess has given birth to a third child, Ferdinand has behaved suspiciously and Antonio is worried that things might soon turn worse, because "those houses that are haunted are most still/ Till the devil be up" (3, 1, 23-4). Ferdinand's court is hellish and the devil lurks close by, possibly in every one of the palace's inhabitants, as suggested by Bosola's retort to Ferdinand's proposal that he spy on the Duchess: Bosola: It seems you would create me One of your familiars. Ferdinand: Familiar? What's that? Bosola: Why, a very quaint invisible devil-in flesh, An intelligencer.

[…] Take your devils, Which hell calls angels! These cursed gifts would make You a corrupter, me an impudent traitor; And should I take these, they'd take me to hell. (1, 1, 251-9) Bosola feigns to refuse the Duke's money which he considers a devilish payment for a hellish job. In medieval iconography, the devil is often represented as holding a mirror, as in Hieronymus Bosch's Garden of Earthly Delights (c. 1550) where a mirror is placed on a devil's bottom and a lady sitting just in front sees frightening images reflected in it. The devil's mirror deceives by creating illusions and collapsing all ordinary distinctions. In this respect it opposes the Christus speculum topos in which, according to Calvin's commentaries on Paul's epistles, we may without deception contemplate our own election. In the medieval European imagination, the devil is a protean figure who embodies metamorphosis, counter nature and the world upside down.

In the passage quoted above, the money offered is presented as a gift whereas Ferdinand in just bribing Bosola. A gift is a special commodity given without an explicit agreement for reward. In Nicomachean Ethics (book VIII, 13), Aristotle considers the gift as the most virtuous expression of what he defines as "friendship of pleasure," because "it is noble to do well by another without a view to repayment" (143). But in Bosola's view, the bribe signifies treachery and turns Ferdinand into a corrupter. In the barter economy set up by the Duke, the gift demands a counter-gift (spying), that is a form of repayment which contradicts the value of pure gratuity. Bosola explicitly compares the Italian court to hell where devils are called angels, and debunks a situation in which doing evil is considered a sign of gratitude while generosity leads him to become a villain: The Italian court is a place where good and evil are no longer defined by antithesis, rather they encroach upon one another, which generates indistinctness in the system of values. There definitely is something rotten in the dukedom of Calabria and the numerous references to corpses, skeletons and dismembered bodies, together with the masque of the madmen (4, 2) are signs that the world of The Duchess of Malfi is disordered, depraved, "rotten and rotting others" has Bosola has it (4, 2, 304). In this respect, the Old Lady's pockmarked, painted face (2, 1; 2, 2) sets off the Duchess's purity, but it also reduces humanity to the level of putrefying flesh and allegorizes a nightmarish world inhabited by dereliction, destruction and death.

I
In The Duchess of Malfi death is spectacularly dramatized in a succession of striking visual tableaux. The spectator is shown the Duchess and Cariola being strangled by Bosola (4, 2) and Julia kissing the poisoned bible offered by the Cardinal (5, 2). The concluding scene of the play stages the death of Antonio stabbed by Bosola, and the Cardinal, Ferdinand and Bosola killing one another in an ostentatious display of murders somehow reminiscent of the final scene of Hamlet.

Unsurprisingly death has been looming from the very beginning of the play. In the wooing scene between the Duchess and Antonio (1, 1), the will the Duchess means to write is the occasion for Webster to establish a connexion between death and love: a will gives instructions for what should happen after the person's death but the Duchess reveals her desire (will) and her sexual drive. Right from the beginning of the play the language of love is inflected with references to death. When the Duchess is delivered of her son (2, 3), Bosola hears "a woman shriek" (1), which he interprets as the sound of an owl:

List again! It may be 'twas the melancholy bird, Best friend of silence and of solitariness, The owl, that screamed so. (7-9)

In many classical and medieval bestiaries, the owl is a bird of night portending death. Later in the scene, Bosola picks up the horoscope ill-fatedly dropped by Antonio and reads that the child's life will be short: "Mars being in a human sign, joined to the tail of the Dragon, in the eighth house, doth threaten a violent death. Caetera non scrutantur" (2, 3, 61-3). These references show that the Duchess's married life is fated to end in disaster. Finally, the connexion between love, life and death is symbolically amplified by the madmen in the dirge they sing to torture the Duchess in her prison:

Oh, let us howl some heavy note Some deadly, dogged howl, Sounding as from the threat'ning throat Of beasts and fatal fowl. As ravens, screech-owls, bulls, and bears! We'll bill and bawl our parts, Till irksome noise have cloyed your ears And corrosived your hearts. At last, whenas our choir wants breath, Our bodies being blest, We'll sing like swans, to welcome death, And die in love and rest. (4, 2, 60-71)

The madmen's song, sung "to a dismal kind of music," is supposed to offer a prefiguration of the Duchess's oncoming death, but it fails to move the Duchess. As shown by Catherine Belsey, the masque reveals "the transformation not of the Duchess but of Ferdinand" himself (131) and functions as a reflection of the world of hell and death he has created around him.

Death is one of the great structural motifs of the play to which all human experiences are finally reduced. Significantly, life and death are both described by echoing topographical images and references. When the Duchess enters her married life at the beginning of the play, she confesses to Cariola that she is stepping into an unchartered territory with no hand to guide her: "I am going into a wilderness/ Where I shall find nor path nor friendly clew/ To be my guide" (1, 1, 349-51). The wilderness is a waste land, uninhabited and uncultivated. As an echo to the Duchess's remark, Julia gently embraces death in the final act of the play, visiting like Mortimer in Marlowe's Edward II, a yet unknown country she cannot even name: "I go, I know not wither" (5, 2, 289). 4A morbid anxiety about dismantled bodies and putrefying flesh hangs over the world of The Duchess of Malfi. After Ferdinand has given the Duchess a dead man's hand to kiss (4, 1, 43), Bosola presents her with a fearful image of the final destruction and absolute depravity of the flesh: Thou art a box of wormseed, at best but a salvatory of green mummy. What's this flesh? A little crudded milk, fantastical puff paste. Our bodies are weaker that those paper prisons boys used to keep flies in-more contemptible, since ours is to preserve earthworms. (4,2, Man is no more than a bundle of perishable matter and life and death are vain, as suggested by Antonio just before he dies: "In all our quest of greatness/ Like wanton boys, whose pastime is their care,/ We follow after bubbles blown in th 'air" (5, 4, 62-4). Antonio seems to rephrase Gloucester's tragic outcry in King Lear, "As flies to wanton boys are we to the Gods" (4, 1, 36), and the Homo Bulla motif, typical of vanity scenes and emblems of death where Thanatos putti are seen to blow soap bubbles, testifies to the brevity and insignificance of human life. Delio's moralizing comments upon the huge waste of the end add to this sombre picture by hinting at oblivion and obliteration:

These wretched eminent things Leave no more fame behind'em than should one Fall in a frost and leave his print in snow-As soon as the sun shines, it ever melts Both form and matter. (5,5, Even though the Duchess's child will be established in her mother's right, the snow of death melts everything away. The world of The Duchess of Malfi is dominated by vacuity and meaninglessness but the vanitas motif is not balanced by its aeternitas pendant so common in baroque visualisations of death. Life and death are mere tales told by madmen, full of sound and fury, signifying nothing, and the only way to express this frightening reality is by using a language whose meaning is evanescent.

3.The value of signs

The Duchess of Malfi is a play concerned with language, or more precisely with the make-up and proper ends of language. The first encounter between the Duchess and Antonio is a good introduction to the complex linguistic games and semantic slips abounding in the play. After the ceremonies held the previous night to celebrate her brothers' visit, the Duchess thinks it "fit, like thrifty husbands [they] enquire/ What's laid up for tomorrow" (1,1,. She means by this that she should consider the way to write her will, but Antonio understands that she intends to do her accounting: Antonio: I'll fetch your grace The particulars of your revenue and expense. Duchess: Oh, you are an upright treasurer; but you mistook, For when I said I meant to make inquiry What's laid up for tomorrow, I did mean What's laid up yonder for me. Antonio: Where? Duchess: In heaven. (1,1, In the course of the conversation meaning becomes abstruse because the matter of the speech is unclear and the phrasing is sophisticated, as when the Duchess urges Antonio to be less humble and more adventurous:

Duchess: This goodly roof of yours is too low built; I cannot stand upright in't, nor discourse, Without I raise it higher. Raise yourself, Or, if you please, my hand to help you so. Antonio: Ambition, madam, is a great man's madness, That is not kept in chains and close-pent rooms, But in fair, lightsome lodgings, and is girt With the wild noise of prattling visitants, Which makes it lunatic beyond all cure. (1,1, The image of the roof of humility paraphrases Jospeh Hall's description of the humble man as "a True Temple of God built with a low roof"5 (Hall,94). The textual embedding is meant to parody the moral implications of the original text, which obviously Antonio does not notice, and his reply to the Duchess is a piling up of long-drawn, convoluted metaphors obfuscating meaning. To claim that ambition is measured by chattering visitors living in beautiful lodgings is awkward and the phrasing is roundabout. Such linguistic excess is a barrier to social communication and the Duchess does not even care to answer Antonio, she simply states that she makes him lord of her properties: "You may discover what a wealthy mine / I make you lord of" .

The misunderstanding between the Duchess and Antonio is amplified as she proposes to him, playfully deploying double entendre and what Barbara Correll defines as "discursive maneuvering" (365). When she offers her ring to cure his eye, the imagery is purposefully contorted: "One of your eyes is bloodshot-use my ring to't./ They say 'tis very sovereign" (1, 1, 395-6). The bloodshot eye is a metaphor for his misunderstanding (he cannot see what she is driving at), but it can also be an image of male erection and the ring she offers as a remedy is of course a bawdy pun for the female sex. Gratiano uses the same image in the final couplet of The Merchant of Venice, fearing "no other thing/ So sore as to keep safe Nerissa's ring." The whole passage is replete with gross sexual innuendo which Antonio can barely construe at first. Right from the outset, the language of the play operates on equivocation, as the Duchess finally admits after she has made her proposal:

We are forced to woo, because none dare woo us; And as a tyrant doubles with his words, And fearfully equivocates, so we Are forced to express our violent passions In riddles and in dreams. (1,1, Ferdinand also uses the language of equivocation whose design is to deceive his enemies: he pretends he wants Antonio's advice when he plans to have him killed ("Send Antonio to me; I want his head in a business," 3, 5, 27). Catherine Belsey is right to draw attention to the fact that the Duchess's equivocation is meant to "establish relationship" with Antonio whereas Ferdinand's purpose is just "to entangle his victims" ( 129), but the fact remains that in The Duchess of Malfi, communication is often suspended, if not entirely blocked, and the Duchess's double entendres are self-reflective comments on the language of the play as a whole.

Communication depends on shared use of language but the characters appear to be confined into a language of their own. Antonio and the Duchess lie to secure their position as husband and wife, Ferdinand deliberately withholds information, as when he refuses to explain why he does not want his sister to remarry ("Do not you ask the reason, but be satisfied/ I say I would not," 1, 1 250-1), and Bosola and the Cardinal both comment on their own private designs in numerous asides not intended to be shared by others.

Moreover, the play incorporates several dumb shows which the characters watch in silence, like the madmen's caperings (4, 2). Visual tableaux are effective and need no words of explanation but when verbal comments are provided, they merely describe what has been shown. During the dumb show of the Cardinal's arming (3, 4), the Duchess, Antonio and their children are banished from the Loreto shrine and a long ditty is sung by some churchmen. As in a conventional emblem, a verbal comment is adduced to the visual element but it simply puts into words what is being staged in the show. In this case the function of words is to repeat, and not to create meaning. Later in the play, the Echo scene (5, 3) provides evidence of a language reduced to meaningless stammering. The Duchess has become a lost voice and all she can do is to repeat some of Antonio's words. Of course the echo serves to reveal Antonio's oncoming fate, but the scraps of sentences she utters are like a broken syntax that makes no sense outside the pattern of iteration and exemplifies a language deprived of its creative energy.

Catherine Belsey has rightfully observed that the stereotyped rhetoric of emblems informs the language of the play (Belsey,, and in many instances, the characters use aphorisms and maxims reminiscent of emblematic discourse. When the Cardinal dies at the hand of his brother he dramatically exclaims that "I suffer now for what hath former been;/ Sorrow is held the eldest child of sin" (5, 5, 53-4), and Ferdinand meets his own death echoing his brother with a couplet which strangely contradicts his attitude all along the play: "Whether we fall by ambition, blood, or lust,/ Like diamond we are cut to our own dust" (5, 5, 70-1). These sententiae certainly express abstract moral truths and suggest that in the case of the two brothers punishment is justified, but they ring false in the derelict world of the end of the play where moral concepts have no place. In the same way Delio's closing lines calling for a life of integrity that none of the characters has been able to construct accentuate even more blatantly the discrepancy between language and reality: "Integrity of life is fame's best friend,/ Which nobly, beyond death, shall crown the end" (5,5,.

Approached from a historical perspective, the issue of language in The Duchess of Malfi should be referred to the early modern semantic conceptions and understandings of the relationship between signs and their meanings. The play questions the Neo-Platonic-Hermetic tradition that names are the essence of what is named. As rightly pointed out by Robert Ornstein in a classic of Jacobean criticism, Webster "presents in art the sceptical, pragmatic nominalism of the late Renaissance" (Ornstein,134). By exploring the energy and the limits of language, the play suggests that the connection between signs and signifieds is not motivated naturally, it also intuits that the reality which it is the function of language to capture is too complex, too shady and too unstable to ever be fixed in and by a rigid linguistic system.

To wrap up, The Duchess of Malfi should be understood in terms of its aesthetic instability. The technique of characterization rejects rigid codifications and allegorical figurations. The Duchess, Antonio, Bosola, the Cardinal and Ferdinand all grapple with inner conflicts, they are aggregates of contrasting facets in a huge catoptric construction where images are superimposed over, and displaced by, other images. The play is an exploration of evil, but none of the characters can be reduced to the totality of what evil means. When "rightly gaz'd upon," character construction shows "nothing but confusion," but "ey'd awry," it reveals the fluidity of identity and the paradoxes of human nature.

The world of Calabria is inhabited by corruption and decay and it rests on no valid ethical system since the distinction between good and evil is continually blurred. Just as Bosola's guilty conscience is "a black register wherein is writ/ All our good deeds and bad--a perspective/ That shows us hell" (4, 2, 343-5), reality is ungraspable because made of shifting perspectives whose final image is the effacement consequent upon death.

The language of the play oscillates between over elaborate rhetoric and a form of sterile vocality which bespeaks a degree of skepticism in the power of words to express reality. As stated by Inga-Stina Ewbank, the language of the play serves to define "the nature of the whole world in which the Duchess in particular and man in general live" (271).

Fascination with death and decay, together with the direct presentation of horrors onstage, align the play with some of the main features of baroque aesthetics. The Duchess of Malfi conforms to the baroque impulse which José Antonio Maravall defines in Culture of the Baroque as "the pursuit of the extreme" (Maravall,87). But the tendency to fragmentation in the presentation of human identities expresses more mannerist concerns, while some of the language of the play is shared between the use of maxims and sententiae, recalling the classical genre of the emblem, and a more pregnant questioning of the power of language. Classical, mannerist, and baroque, Webster's play is a brilliant, hybrid composition which evades ready-made aesthetic categorizations

  would have you curse yourself now, that your bounty, Which makes men truly noble, e'er should make Me a villain. Oh, that to avoid ingratitude For the good deed you have done me, I must do All the ill man can invent! Thus the devil Candies all sins o'er, and what heaven terms vile That names he complimental. (1, 1, 264-70)

References are to the Norton text, ed. Michael Neill (New York and London : Norton and Co, 2015).

See for instance JoannisPierii Valeriani, Hieroglyphica [1556], Liber XIII, « De Lepore ».

In Marlowe's Edward II, Mortimer Junior reports to Queen Isabella that Mortimer « scorns the world, and as a traveller/ Goes to discover countries yet unknown »(5, 6,. Edward II was published in 1594 and Webster may have used it as a source of inspiration.

Hall's Characters of Vertues and Vices was issued in 1608.


Ibrahim Soufi
email: soufimath@gmail.com

Existence of solutions for a p-Laplacian boundary value problem on the half-line

Keywords: p-Laplacian, critical point, variational method. 2010 Mathematics subject classification: 35B38, 74G35

In this paper we consider p-Laplacian boundary value problem on a half line. Using critical point theory and Mountain Pass Theorem, we prove the existence of nontrivials solutions.

Introduction

This paper is concerned with the following p-Laplacian boundary value problem on the half-line

                
ρ(t)Φ p (u (t)) + s(t)Φ p (u(t)) = f (t, u(t)), t = t j , a.e. t > 0, ∆ ρ(t j )Φ p (u (t j )) = I j (u(t j ))h(t j ), j = 1, 2, 3, ..., u (0 + ) = u(0) = 0, u (+∞) = u(+∞) = 0 (1.1) where Φ p u = |u| p-2 u, p > 1, s ∈ L ∞ [0, +∞) with ess inf [0,+∞) s(t) > 0, ρ be a continuously differentiable and bounded function such that max ρ L 1 , ρ L q < +∞,

1 p + 1 q = 1, inf [0,+∞)
ρ(t) > 0, Here t 0 = 0 < t 1 < t 2 < . . . < t j < . . . < t m → +∞, as m → ∞, are the impulse points, while the impulsive functions I j : R -→ R are assumed continuous. Finally ∆ ρ(t j )Φ p (u (t j )) = ρ(t + j )Φ p (u (t + j ))ρ(t - j )Φ p (u (t - j )) j = 1, 2, 3, ..., where u (t + j ) = lim t→t + j u (t) and u (t - j ) = lim t→t - j u (t) stand for the right and the left limits of u at t j , respectively. As for h : [0, +∞) -→ [0, +∞), it is a continuous function that satisfies ∑ +∞ j=1 h(t j ) < +∞.

and f : [0, +∞) × R -→ R is a Carathéodory function, i.e. (i) f (., u) is measurable, for each u ∈ R, (ii) f (t, .) is continuous, for a.e. t ∈ [0, +∞). Some of the ideas used in this paper are motivated by the work in [START_REF] Shi | Infinitely many solutions for a p-Laplacian boundary value problem with impulsive effects[END_REF][START_REF] Shi | Multiple solutions for p-Laplacian boundary value problems with impulsive effects[END_REF], and also we mention [START_REF] Heidarkhani | Some multiplicity results to the existence of three solutions for a Dirichlet boundary value problem involving the p-Laplacian[END_REF][START_REF] Huang | Solutions to Periodic BVP for Second Order Impulsive Differential Equation with a p-Laplacian Operator[END_REF][START_REF] Lin | Infinitely many homoclinic orbits of second-order p-Laplacian systems[END_REF][START_REF] Lu | Existence of periodic solutions of a p-Laplacian neutral functional differential equation[END_REF][START_REF] Ma | Existence of infinitely many periodic solutions for ordinary p-Laplacian systems[END_REF][START_REF] Tian | Periodic solutions of non-autonomous second-order systems with a p-Laplacian[END_REF][START_REF] Xu | Some existence results on periodic solutions of ordinary p-Laplacian systems[END_REF][START_REF] Xu | Existence of weak solutions for p-Laplacian probem with impulsive effects[END_REF], was studied the existence of solutions in a bounded interval [a, b]. This paper is devted to the generalization of problem in [START_REF] Shi | Infinitely many solutions for a p-Laplacian boundary value problem with impulsive effects[END_REF][START_REF] Shi | Multiple solutions for p-Laplacian boundary value problems with impulsive effects[END_REF], when we consider the problem in the half-line.

Preliminaries

The functional framework

Symbol L p (0, +∞) for p > 1 means the space of such measurable real valued functions defined on [0, +∞) that ∞ 0 |u (t)| p dt < +∞. Solutions to (1.1) will be considered in the space W 1,p 0,ρ,s (0, +∞) which is defined as follows. We say that u ∈ W 1,p 0,ρ,s (0, +∞) if u ∈ L p (0, +∞) and if there exists a function g ∈ L p (0, +∞), called a weak derivative, and such that

+∞ 0 u(t)ϕ (t)dt = - +∞ 0 g(t)ϕ(t)dt for all ϕ ∈ C ∞ c (0, +∞), where C ∞ c (0, +∞) is the space of compactly supported func- tions from C ∞ ([0, +∞)), R). We denote g := u . We endow the space W 1,p 0,ρ,s (0, +∞) with its norm u = +∞ 0 ρ(t)|u (t)| p + s(t)|u(t)| p dt 1 p ,
or the equivalent norm

u p = p √ ρu L p + p √ su L p .
Let the spaces C l,ρ [0, +∞) be defined by

C l,ρ [0, +∞) = u ∈ C([0, +∞), R) : lim t→+∞ ρ(t)u(t) exists endowed with the norm u ∞,ρ = sup t∈[0,+∞) ρ(t)|u(t)|.
Moreover the space W 1,p 0,ρ,s (0, +∞) is reflexive. We have:

Lemma 2.1. (a) The operator T : W 1,p 0,ρ,s (0, +∞) -→ T(W 1,p 0,ρ,s (0, +∞)) ⊂ L p (0, +∞) × L p (0, +∞) : = L P 2 (0, +∞) u -→ T(u) = p √ su, p √ ρu is an isometric isomorphism. (b) W 1,p 0,ρ,s (0, +∞) is a reflexive space.
Proof. (a) It is clear that T is a linear operator and that T conserves norms, i.e.,

∀ u ∈ W 1,p 0,ρ,s (0, +∞), Tu L p 2 = u p . Indeed Tu L p 2 = p √ su, p √ ρu L p 2 = p √ ρu L p + p √ su L p = u p .
(b) Since L p (0, +∞)) is a reflexive Banach space, the cartesian product L 2 p (0, +∞)) is also a reflexive Banach space with respect to the norm

u L p 2 = u 1 L p + u 2 L p , where u = (u 1 , u 2 ) ∈ L p 2 (0, +∞).
From part (a), T(W 1,p 0,ρ,s (0, +∞)) is a closed subspace of L p 2 (0, +∞); then by [5, Theorem 4.10.5], the space T(W 1,p 0,ρ,s (0, +∞)) is reflexive. Consequently W 1,p 0,ρ,s (0, +∞) is also reflexive. Lemma 2.2. [START_REF] Corduneanu | Intergral Equations and Stability of Feedback Systems[END_REF] Let D ⊂ C l,ρ ([0, +∞), R) be a bounded set. Then D is relatively compact if the following conditions hold: (a) D is equicontinuous on any compact sub-interval of R + , i.e.

∀ J ⊂ [0, +∞) compact, ∀ ε > 0, ∃ δ > 0, ∀ t 1 , t 2 ∈ J : |t 1 -t 2 | < δ =⇒ |ρ(t 1 )u(t 1 ) -ρ(t 2 )u(t 2 )| ≤ ε, ∀ u ∈ D, (b) D is equiconvergent at +∞ i.e., ∀ ε > 0, ∃ T = T(ε) > 0 such that ∀ t : t ≥ T(ε) =⇒ |ρ(t)u(t) -ρ(+∞)u(+∞)| ≤ ε, ∀ u ∈ D.

Critical point theory

Now we recall some essential facts from critical point theory (see [START_REF] Badiale | Semilinear Elliptic Equations for Beginners[END_REF][START_REF] Prodi | A Primer of Nonlinear Analysis[END_REF]). Definition 2.3. Let (X, • ) be a Banach space, Ω ⊂ X an open subset, and J : Ω -→ R a functional. We say that J is Fréchet differentiable at u ∈ Ω if there exists an operator A ∈ X such that lim

v∈Ω, v →0 J(u + v) -J(u) -Av v = 0.
The operator A, which is unique, is called the Fréchet differential of J at u and is denoted by A = J F (u) or A = J (u).

Definition 2.4. Let X be a Banach space, Ω ⊂ X an open subset, and J : Ω -→ R a functional. We say that J is Gâteaux differentiable at u ∈ Ω if there exists A ∈ X such that lim t→0 J(u + tv) -J(u) t = Av, for all v ∈ X. The operator A, which is unique, is denoted by A = J G (u) or merely J when there is no confusion.

The mapping which sends to every u ∈ Ω the mapping J G (u) is called the Gâteaux differential of J and is denoted by J G . Proposition 2.5. [START_REF] Prodi | A Primer of Nonlinear Analysis[END_REF] Let X be a Banach space, Ω ⊂ X an open subset, and J : Ω -→ R a Gâteaux differentiable functional at some point u ∈ Ω. If J G is continuous at u, then J is Fréchet differentiable at u and J F (u) = J G (u).

We say that J ∈ C 1 if J G is continuous at every u ∈ Ω. Definition 2.6. Let X be a Banach space, Ω ⊂ X an open subset, and

J : Ω -→ R a Gâteaux differentiable functional. A point u ∈ Ω is called a critical point of J if J (u) = 0, i.e., J (u)v = 0, for every v ∈ X. If further J(u) = c, we say that u is a critical point of J at level c.
Clearly, every point of a local minimum of a Gâteaux differentiable functional J is a critical point J.

Definition 2.7. Let X be a Banach space. A functional J : X -→ R is called coercive if, for every sequence (u k ) k∈N ⊂ X, u k → +∞ =⇒ J(u k ) → +∞.
Definition 2.8. Let X be a Banach space. A functional J : X -→ (-∞, +∞] is said to be sequentially weakly lower semi-continuous (swlsc for short) if

J(u) ≤ lim inf n→+∞ J(u n ), as u n u in X, when n → ∞.
Then we have Lemma 2.9. [START_REF] Brezis | Analyse Fonctionnelle. Th Ì Ąeorie et Applications[END_REF] [Minimization Principal] Let X be a reflexive Banach space and J a functional defined on X such that

(1) lim u →+∞ J(u) = +∞ (coercivity condition), (2)
J is sequentially weakly lower semi-continuous.

Then J is lower bounded on X and achieves its lower bound at some point u 0 .

Definition 2.10. Let X be a real Banach space, J ∈ C 1 (X, R). If any sequence (u n ) ⊂ X for which (J(u n )) is bounded in R and J (u n ) -→ 0 as n → +∞ in X possesses a convergent subsequence, then we say that J satisfies the Palais-Smale condition, (PS) condition for brevity. Let X be a Banach space and let J ∈ C 1 (X, R) satisfy J(0) = 0. Assume that J satisfies (PS) and there exist positive numbers ρ and α such that

(1) J(u) ≥ α if u = ρ,
(2) there exists u 0 ∈ X such that u 0 > ρ and J(u 0 ) < α.

Then there exists a critical point. Furthermore it is characterized by

J (u) = 0, J(u) = inf γ∈Γ max t∈[0,1] J(γ(t)),
where

Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = u 0 }.

Some embedding results

Lemma 2.12. W 1,p 0,ρ,s (0, +∞) embeds continuously in C l,ρ [0, +∞).

Proof. For u ∈ W 1,p 0,ρ,s (0, +∞) , we have

|ρ(x)u(x)| = |ρ(x)u(x) -(pu)(0)| = x 0 (ρu) (t)dt ≤ x 0 ρ (t)u(t)dt + x 0 ρ(t)u (t)dt = x 0 ρ (t) p s(t) p s(t)u(t)dt + x 0 q ρ(t) p ρ(t)u (t)dt ≤ 1 inf [0,+∞) p s(t)   +∞ 0 ρ (t) q dt   1 q   +∞ 0 s(t)|u| p (t)dt   1 p +   +∞ 0 ρ(t)dt   1 q   +∞ 0 ρ(t)|u | p (t)dt   1 p ≤ M 1   +∞ 0 s(t)|u| p (t)dt   1 p + M 2   +∞ 0 ρ(t)|u | p (t)dt   1 p ≤ max M 1 , M 2 u p ≤ 2 max M 1 , M 2 u . Therefore, we obtain u ∞,ρ ≤ k u , where k = 2 max M 1 , M 2 , M 1 = 1 inf [0,+∞) p s(t) ρ L q (0, +∞) , M 2 = ρ L 1 (0, +∞) Lemma 2.13. W 1,p 0,ρ,s (0, +∞) embeds continuously in L p [0, +∞). Proof. For u ∈ W 1,p 0,ρ,s (0, +∞) , we have u p L p (0,+∞) = +∞ 0 |u(t)| p dt ≤ 1 inf [0,+∞) s(t) +∞ 0 s(t)|u(t)| p dt ≤ 1 inf [0,+∞) s(t) +∞ 0 s(t)|u(t)| p dt + 1 inf [0,+∞) s(t) +∞ 0 ρ(t)|u (t)| p dt = 1 inf [0,+∞) s(t) u p Lemma 2.14. The embedding W 1,p 0,ρ,s (0, +∞) → C l,ρ [0, +∞) is compact. Proof. Let D ⊂ W 1,p
0,ρ,s (0, +∞) be a bounded set. Then Lemma 2.12 establishes the boundedness of D in C l,ρ [0, +∞). Let R > 0 be such that for all u ∈ D, u ≤ R. We will check the conditions in Lemma 2.2. (i) D is equicontinuous on every compact sub-interval of [0, +∞).

Let u ∈ D and x 1 , x 2 ∈ J ⊂ [0, +∞) where J is a compact sub-interval, we have

|ρ(x 2 )u(x 2 ) -ρ(x 1 )u(x 1 )| = x 2 x 1 (ρu) (t)dt = x 2 x 1 ρ (t)u(t) + ρ(t)u (t) dt ≤ x 2 x 1 ρ (t)u(t)dt + x 2 x 1 ρ(t)u (t)dt = x 2 x 1 ρ (t) p s(t) p s(t)u(t)dt + x 2
x 1 q ρ(t) p ρ(t)u (t)dt by using the Hölder's inequality, we obtain

|ρ(x 2 )u(x 2 ) -ρ(x 1 )u(x 1 )| ≤ 1 inf [0,+∞) p s(t)   x 2 x 1 ρ (t) q dt   1 q   x 2 x 1 s(t)|u| p (t)dt   1 p +   x 2 x 1 ρ(t)dt   1 q   x 2 x 1 ρ(t)|u | p (t)dt   1 p ≤ max     1 inf [0,+∞) p s(t)   x 2 x 1 ρ (t) q dt   1 q ,   x 2 x 1 ρ(t)dt   1 q     u p ≤ 2R max     1 inf [0,+∞) p s(t)   x 2 x 1 ρ (t) q dt   1 q ,   x 2 x 1 ρ(t)dt   1 q     → 0, as |x 2 -x 1 | → 0, ∀u ∈ D.
(ii) D is equiconvergent at +∞.

|ρ(x)u(x) -ρ(+∞)u(+∞)| ≤ 1 inf [0,+∞) p s(t)   +∞ x ρ (t) q dt   1 q   +∞ x s(t)|u| p (t)dt   1 p +   +∞ x ρ(t)dt   1 q   +∞ x ρ(t)|u | p (t)dt   1 p ≤ max     1 inf [0,+∞) p s(t)   +∞ x ρ (t) q dt   1 q ,   +∞ x ρ(t)dt   1 q     u p ≤ 2R max     1 inf [0,+∞) p s(t)   +∞ x ρ (t) q dt   1 q ,   +∞ x ρ(t)dt   1 q     → 0,
when x tends to +∞, ∀u ∈ D.

Existence of weak solutions

Take v ∈ W 1,p 0,ρ,s (0, +∞), multiply the equation in problem (1.1) by v, and then integrate over (0, +∞), we get

- +∞ 0 ρ(t)Φ p (u (t)) v(t)dt + +∞ 0 s(t)Φ p (u(t)) dt = +∞ 0 f (t, u(t))v(t)dt. The left-hand term is - +∞ 0 ρ(t)Φ p (u (t)) v(t)dt = - +∞ ∑ j=0 t j+1 t j ρ(t)Φ p (u (t)) v(t)dt = +∞ 0 ρ(t)Φ p (u (t))v (t)dt + +∞ ∑ j=1 I j(u(t j ))v(t j ) + ρ(0)ϕ p (u (0))v(0) + ρ(+∞)Φ p (u (+∞))v(+∞). I. Soufi Hence +∞ 0 ρ(t)Φ p (u (t))v (t) + s(t)Φ p (u(t))v(t) dt + +∞ ∑ j=1 I j (u(t j ))h(t j )v(t j ) = +∞ 0 f (t, u(t))v(t)dt.
This leads to the natural concept of weak solution for problem (1.1). Definition 3.1. We say that a function u ∈ W 1,p 0,ρ,s (0, +∞) is a weak solution of problem (1.1) if

+∞ 0 ρ(t)Φ p (u (t))v (t) + s(t)Φ p (u(t))v(t) dt + +∞ ∑ j=1 I j (u(t j ))h(t j )v(t j ) = +∞ 0 f (t, u(t))v(t)dt.
for all v ∈ W 1,p 0,ρ,s (0, +∞).

In order to study problem (1.1), we consider the functional J : W 1,p 0,ρ,s (0, +∞) -→ R defined by

J(u) = 1 p u p + +∞ ∑ j=1 h(t j ) u(t j ) 0 I j (t)dt - +∞ 0 F(t, u(t))dt, (3.1)
where

F(t, u) = u 0 f (t, s)ds.

The sublinear case

Theorem 3.2. Suppose that the following conditions hold: (H 1 ) There exist a constant µ ∈ [0, p -1) and positive functions a (H 2 ) There exist constants k > 0 and γ ∈ [0, p -1) such that

1 , b 1 ∈ L 1 [0, +∞) such that | f (t, x)| ≤ a 1 (t)|x| µ + b 1 (t),
|I j (t)| ≤ d|t| γ , ∀ t ∈ R, ∀ j ∈ {1, 2, . . .}.
Then problem (1.1) has at least one weak solution.

Proof. Claim 1 The functional J is well defined. Given u ∈ W 1,p 0,ρ,s (0, +∞), Assumptions (H 1 ) and (H 2 ) guarantee that

|F(t, u(t))| ≤ a 1 (t) µ + 1 |u(t)| µ+1 + b 1 (t)|u(t)|. Hence +∞ 0 F(t, u(t))dt ≤ 1 µ + 1 +∞ 0 a 1 (t) (ρ(t)) µ+1 |ρu| µ+1 dt + +∞ 0 b 1 (t) ρ(t) |ρ(t)u(t)|dt ≤ 1 (µ + 1) inf [0,+∞) ρ(t) µ+1 u µ+1 ∞,ρ +∞ 0 a 1 (t)dt + 1 inf [0,+∞) ρ(t) u ∞,ρ +∞ 0 b 1 (t)dt ≤ k µ+1 (µ + 1) inf [0,+∞) ρ(t) µ+1 u µ+1 +∞ 0 a 1 (t)dt + k inf [0,+∞) ρ(t) u +∞ 0 b 1 (t)dt ≤ k µ+1 (µ + 1) inf [0,+∞) ρ(t) µ+1 u µ+1 a 1 L 1 + k inf [0,+∞) ρ(t) u b 1 L 1 and +∞ ∑ j=1 u(t j ) 0 h(t j )I j (τ)dτ ≤ dk γ+1 (γ + 1) inf [0,+∞) ρ(t) γ+1 u γ+1 +∞ ∑ j=1 h(t j ). Then |J(u)| ≤ 1 p u p + dk γ+1 (γ + 1) inf [0,+∞) ρ(t) γ+1 u γ+1 +∞ ∑ j=1 h(t j ) + k µ+1 (µ + 1) inf [0,+∞) ρ(t) µ+1 u µ+1 a 1 L 1 + k inf [0,+∞) ρ(t) u b 1 L 1 < ∞.
Claim 2 J is sequentially weakly lower semi-continuous.

Let (u n ) ⊂ W 1,p 0,ρ,s (0, +∞) be a sequence such that u n u in W 1,p 0,ρ,s (0, +∞), as n → ∞. Then (u n ) converges uniformly to u on [0, +∞) and lim inf n→+∞ u n ≥ u .
The continuity of functions f and I j , j ∈ {1, 2, . . .} together with Lebesgue Dominated Convergence Theorem yield

lim inf n→+∞ J(u n ) = lim inf n→+∞ 1 p u n p + +∞ ∑ j=1 u n (t j ) 0 h(t j )I j (τ)dτ - +∞ 0 F(t, u n (t))dt ≥ 1 p u p + +∞ ∑ j=1 u(t j ) 0 h(t j )I j (τ)dτ - +∞ 0 F(t, u(t))dt = J(u).
Therefore, J is sequentially weakly lower semi continuous. Claim 3 J is coercive. In view of (H 1 ) and (H 2 ), Lemma 2.12 guarantees that

J(u) ≥ 1 p u p - dk γ+1 (γ + 1) inf [0,+∞) ρ(t) γ+1 u γ+1 +∞ ∑ j=1 h(t j ) - k µ+1 (µ + 1) inf [0,+∞) ρ(t) µ+1 u µ+1 a 1 L 1 - k inf [0,+∞) ρ(t) u b 1 L 1 . (3.2) Since µ < p -1 and γ < p -1, (3.2) implies lim u -→+∞ J(u) = +∞.
Lemma 2.9 guarantees that J has a local minimum which is a critical point of J.

Finally, it is easy to check that under (H 1 ), the functional J is Gâteaux differentiable and the Gâteaux derivative at a point u ∈ X is

J (u), v = +∞ 0 ρ(t)Φ p (u (t))v (t) + s(t)Φ p (u(t))v(t) dt + +∞ ∑ j=1 I j (u(t j ))h(t j )v(t j ) - +∞ 0 f (t, u(t))v(t)dt, (3.3)
for all v ∈ W 1,p 0,ρ,s (0, +∞). Therefore u is a weak solution of problem (1.1).

Example 3.3.

                   -e -t u (t) + u(t) = ln(|x| + 1) 1 + t 2 a. e , t ≥ 0, t = t j , u(0) = u (0 + ) = 0, u(+∞) = u (+∞) = 0, ∆(e -j u (j)) = 1 2 j u(j), j ∈ {1, 2, 3, ...}. (3.4)
It can be easily checked that all condition of Theorem 3.2 are satisfied with

f (t, x) = ln(|x| + 1) 1 + t 2 , p = 2, ρ(t) = e -t , s(t) = 1, a 1 (t) = 1 1 + t 2 , b 1 (t) = 2 1 + t 2 , µ = 1, d = 2, γ = 1, h(t) = 1 2 t , +∞ ∑ j=1 h(j) = 1, I j (s) = √ s.
Therefor problem (3.4) has at least one nontrivial solution.

ρ(t) p u p +∞ 0 a 2 (t)dt + k inf [0,+∞) ρ(t) u +∞ 0 b 2 (t)dt ≤ k p p inf [0,+∞) ρ(t) p u p a 2 L 1 + k inf [0,+∞) ρ(t) u b 2 L 1 .
Then

J(u) ≥ 1 p     1 - k p inf [0,+∞) ρ(t) p a 2 L 1     u p - k inf [0,+∞) ρ(t) u b 2 L 1 - dk γ+1 (γ + 1) inf [0,+∞) ρ(t) γ+1 u γ+1 +∞ ∑ j=1 h(t j ). (3.5) Since a 2 L 1 < inf [0,+∞) ρ(t) p k p and γ < p -1, (3.5) implies that lim u -→+∞ J(u) = +∞.
Then Lemma 2.9 guarantees that problem (1.1) has at least one weak solution. x p = 0, uniformly in t ≥ 0. (H 6 ) There exist positive functions c 1 , c 2 ∈ L 1 ((0, +∞), [0, +∞)), and σ > p such that (a) F(t, x) ≥ c 1 (t)|x| σc 2 (t), for a.e. t ≥ 0 and all x ∈ R, (b) σF(t, x) ≤ x f (t, x), for a.e. t ≥ 0 and all x ∈ R \ {0}.

Nontrivial weak solution

(H 7 ) There exists 0 < γ ≤ p such that

γ x 0 I j (t)dt ≥ xI j (x) > 0, ∀ x ∈ R \ {0}, ∀ j ∈ {1, 2, . . .}.
Then problem (1.1) has at least one nontrivial weak solution.

Proof. Claim 1 Let 0 < ε < inf [0,+∞) s(t) p . From (H 5 ), there exists δ > 0 such that |x| ≤ δ =⇒ |F(t, x)| ≤ ε|x| p .
Using Lemma 2.13, we deduce that

+∞ 0 |F(t, u(t))dt| ≤ ε u p L p ≤ ε inf [0,+∞) s(t) u p , for a.e. t ≥ 0, whenever u ∞ ≤ δ. Let 0 < ρ ≤ δ k and α = 1 p - ε inf [0,+∞) s(t) ρ p . Then for u = ρ,
we have

J(u) = 1 p u p + +∞ ∑ j=1 u(t j ) 0 h(t j )I j (τ)dτ - +∞ 0 F(t, u(t))dt ≥ 1 p u p - +∞ 0 F(t, u(t))dt ≥ 1 p - ε inf [0,+∞) s(t) u p = α.
Assumption (1) in Lemma 2.11 is then satisfied. Claim 2 From (H 7 ), there exists c 3 > 0 such that

x 0 I j (t)dt ≤ c 3 |x| γ , for every x ∈ R.
Now (H 6 )(a) and Lemma 2.12 guarantee that for some

v 0 ∈ W 1,p 0,ρ,s (0, +∞), v 0 = 0, J(ξv 0 ) = 1 p ξ p v 0 p + +∞ ∑ j=1 ξv 0 (t j ) 0 h(t j )I j (τ)dτ - +∞ 0 F(t, ξv 0 (t))dt ≤ 1 p ξ p v 0 p + c 3 ξ γ k γ u 0 γ inf [0,+∞) ρ(t) γ +∞ ∑ j=1 h(t j ) -|ξ| σ +∞ 0 c 1 (t)|v 0 (t)| σ dt + +∞ 0 c 2 (t)dt.
Since σ > p ≥ γ, then for u 0 = ξv 0 , J(u 0 ) ≤ 0, as ξ → +∞. Thus Assumption (2) in Lemma 2.11 is satisfied.

Claim 3 J satisfies the (PS) condition. Notice first that by (H 3 ), J ∈ C 1 (W 1,p 0,ρ,s (0, +∞), R). Now, let (u n ) be a sequence in W 1,p 0,ρ,s (0, +∞) such that (J(u n )) is bounded and lim n→+∞ J (u n ) = 0. We shall prove that the sequence (u n ) is bounded. Using (H 6 )(b) and (H 7 ), there exists some K > 0 such that (3.9) If p ≥ 2, by the results of [START_REF] Shi | Multiple solutions for p-Laplacian boundary value problems with impulsive effects[END_REF], there exists c p > 0 such that |x| p-2 x -|y| p-2 y (xy) ≥ c p |x -y| p , p ≥ 2.

K ≥ σJ(u n ) -J (u n ), u n = σ p - 1
Therefore, we obtain J (u n ) -J (u 0 ) u nu 0 ≥ c p u nu 0 p + +∞ ∑ j=1 h(t j ) I j (u n (t j )) -I j (u(t j )) u n (t j )u(t j ) -+∞ 0 f (t, u n )f (t, u 0 ) u nu 0 dt.

Lemma 2 . 11 .

 211 (see, e.g., [12, Theorem 2.2 ] or [15, Theorem 3.1 ]) [Mountain Pass Lemma]

  for a.e. t ∈ [0, +∞) and all x ∈ R.

3.2 The limit case

  µ = p -1 Assume that (H 2 ) holds both with (H 3 ) There exist positive functions a 2 , b 2 ∈ L 1 (0, +∞) with a 2 L 1 < ≤ a 2 (t)|x| p-1 + b 2 (t), for a.e. t ∈ [0, +∞) and all x ∈ R. Arguing as in the proof of Theorem 3.2, we can prove that J is sequentially weakly lower semi-continuous. In addition, under (H 2 ) and (H 3 ), we have the estimates:

	[0,+∞) Theorem 3.4. inf	ρ(t)	p
	and							k p
	| f (t, x)| Then problem (1.1) has at least one weak solution.
	Proof. |F(t, u(t))| ≤	a 2 (t) p	|u(t)| p + b 2 (t)|u(t)|,
	0	+∞	F(t, u(t))dt ≤	0	+∞	a 2 (t) p	|u(t)| p + b 2 (t)|u(t)| dt
			≤			k p
				p inf [0,+∞)

  Suppose that the following conditions hold: (H 4 ) There exist positive functions ϕ, g such that ϕ ∈ L 1 ((0, +∞), R) and g ∈ C(R, R) with | f (t, x)| ≤ ϕ(t)g(x), for a.e. t ∈ [0, +∞) and all x ∈ R.

	x→0 (H 5 ) lim	F(t,x)

Theorem 3.5.

  (t)dt -I j (u n (t j ))u n (t j )≥ σ p -1 u n p .Since σ > p, then the sequence (u n ) is bounded in W (0, +∞). Next, we prove that (u n ) converges strongly to some u in W (u n ) is bounded in the reflexive Banach space W 1,p 0,ρ,s (0, +∞), there exists a subsequence of (u n ) still denoted (u n ) such that (u n ) converges weakly to some u in W 1,p 0,ρ,s (0, +∞). Then (u n ) converges uniformly to u on [0, +∞) by Lemma 2.14. Thus )I j (u n (t j )) -I j (u(t j )) u n (t j )u(t j ) = 0 (3.6) Φ p (u 0 (t)) u n (t)u 0 (t) dt ) I j (u n (t j )) -I j (u(t j )) u n (t j )u(t j ) (t, u n )f (t, u 0 ) u nu 0 dt.

	+	+∞ ∑	u n h(t j ) σ p -u n (t j ) +∞ 0	σF(t, u n (t)) -f (t, u n (t))u n (t) dt
		j=1
					1,p
					0,ρ,s 1,p 0,ρ,s (0, +∞).
		+∞
	lim n→+∞ h(t j and ∑ j=1 +∞
	lim n→+∞	0
			+∞
			+ h(t j -∑ j=1 +∞	f
			0

0 I j Since f (t, u n (t))f (t, u(t)) u n (t)u(t) dt = 0. (

3.7)

Since lim n→+∞ J (u n ) = 0 and (u n ) converges weakly to some u, we see that

lim n→+∞ J (u n ) -J (u), u nu = 0. (3.8)

Calculating in (3.8) directly we see that

J (u n ) -J (u), u nu = +∞ 0 ρ(t) Φ p (u n (t)) -Φ p (u 0 (t)) u n (t)u 0 (t) dt + +∞ 0 s(t) Φ p (u n (t)) -

It follows directly that u n → u 0 in W 1,p 0,ρ,s (0, +∞). If 1 < p < 2, by the results of [START_REF] Bai | Three solutions for a p-Laplacian boundary value problem with impulsive effects[END_REF], there exists d p > 0 such that

It follows directly that u n → u 0 in W 1,p 0,ρ,s (0, +∞). Therefore, J satisfies the (PS) condition. All conditions of Lemma 2.11 are then fulfilled; as a consequence J has a critical point which is a nontrivial weak solution of problem (1.1).

Example 3.6. Consider the boundary value problem