Youssra Rebboud
email: youssra.rebboud@eurecom.fr

Pasquale Lisena
email: pasquale.lisena@eurecom.fr

Raphaël Troncy
email: raphael.troncy@eurecom.fr

Prompt-based Data Augmentation for Semantically-Precise Event Relation Classification

Keywords: Event Relation, Information Extraction, Knowledge Graphs, Machine Learning

The process of recognizing and classifying the relationships between events mentioned in the text is a crucial task in natural language processing (NLP) known as event relation extraction. If temporal relations and causality are largely studied in the literature, other types of relations have found less interest. Our study specifically concentrates on four types of event relations: causality, enabling, prevention, and intention. Our main contribution consists of the use of a state-of-the-art language model (GPT-3) to extend an existing small dataset with synthetic examples to address the challenge of insufficient training data. We evaluate the quality of these generated samples by training an event relations extraction system, showing improved performances in classifying event relations.

Introduction

Relation extraction (RE) -the identification and classification of relationships between two named entities in raw text -is a classic natural language processing (NLP) task which is receiving attention from the scientific community [START_REF] Hendrickx | SemEval-2010 Task 8: Multi-Way Classification of Semantic Relations between Pairs of Nominals[END_REF][START_REF] Nasar | Named Entity Recognition and Relation Extraction: State-of-the-Art[END_REF]. A more specific branch of RE aims to automatically detect the relations between events (Event Relation Extraction, ERE). While, in RE, the entity type is crucial for inferring the correct relation 1 , in ERE the relations involve homogeneous entities (namely, pairs of events), requiring specialised methods and approaches. Among event relation, the literature mostly focused on temporal relations [START_REF] Vo | Extracting Temporal Event Relations Based on Event Networks[END_REF], causality [START_REF] Khetan | Causal bert: Language models for causality detection between events expressed in text[END_REF], and coreference of the same event in different textual resources [START_REF] Liu | Extracting events and their relations from texts: A survey on recent research progress and challenges[END_REF].

Apart from causal and temporal structures, event relations may include different concepts such as prevention, intention, enabling, etc. Extracting this variety of relations may serve various downstream applications, including semantic timelines, question answering, and fact checking, supporting the decision making and improving information and entertainment. Apart from a demonstrative proof-of-concept [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF], the automatic extraction of different kind of relations have not been deeply investigated. The first step toward achieving this objective is to develop a dedicated dataset. Although [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF] attempts to construct such an initial dataset for multiple event relations, the result is particularly small in size and exhibited significant imbalances.

The recent advent of generative models has marked a paradigm shift in Natural Language Processing with their capabilities to generate human-like complex text relying only on the provided prompt. In this work, we aim to use prompt-based solutions to extend The Event Relations Dataset [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF] with synthetic sentences including references to event relations, with a particular focus on causality, prevention, intention, and enabling. This dataset will then be used to further evaluate state-of-art techniques based on machine learning, to check if they can be successfully applied in the prediction of event relation types.

This work has two research objectives:

• To investigate if prompt-based generative models are suitable for generating synthetic data for the purpose of populating a dataset of event relation, particularly for such kind of rare and very specific event relation types. • Evaluate the performance of methods based on language models in predicting event relations when trained on synthetic data.

The remainder of this paper is structured as follows. First, we review the existing event relations datasets and event relations extraction in Section 2. We describe our approach for constructing a synthetic event relations dataset by proving GPT-3 with prompts to generate sentences that hold prevention, intention, and enabling relations as well as their constructs in Section 3. Section 4 details a method for extracting event relations from the generated dataset, whose results are discussed in Section 5. Finally, we conclude and outline some future work in Section 6.

Related Work

Events and Event Relationships Datasets

Within the field of events and events relationships, several datasets have been developed with the main objective of capturing events, event coreferences, causal and temporal relations. For instance, ACE 20052 for event extraction, and TimeBank [START_REF] Uzzaman | SemEval-2013 task 1: TempEval-3: Evaluating time expressions, events, and temporal relations[END_REF] and CausalTimeBank [START_REF] Mirza | Annotating causality in the TempEval-3 corpus[END_REF] respectively for temporal and causal events relationship extraction.

In [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF], the FARO ontology for representing event relations has been introduced, as a harmonisation of different models from the literature, including definitions for all relations. In addition, a first Event Relation dataset covering four event relation types: caus lity, prevention, enabli ng, and intention is presented in [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF]. With the exception of causality, these relation types are absent in other annotated datasets. Due to the small size of the dataset and its imbalance, training a model on top of it presented a significant challenge. 3To address this gap, our work aims to increase the size of this dataset using data augmentation techniques. In prior research, numerous techniques were elaborated for data augmentation within the field of events and events relationships extraction [START_REF] Liu | Extracting events and their relations from texts: A survey on recent research progress and challenges[END_REF] such as distant supervision [START_REF] Feng | Effective deep memory networks for distant supervised relation extraction[END_REF] and translation [START_REF] Liu | Event Detection via Gated Multilingual Attention Mechanism[END_REF]. In this work, we intend to leverage capabilities of generative models such as GPT-3 [START_REF] Brown | Language Models are Few-Shot Learners[END_REF]. It is worth to mention that, at the time of writing, no official API for ChatGPT is available.

Events and Events Relationship Extraction

Numerous techniques were adopted to tackle the event relation extraction problem from a general point of view regardless of the entity type -event, person, location, etc. -, including supervised, unsupervised, semi-supervised, and distant supervision approaches [START_REF] Wang | Deep Neural Network Based Relation Extraction: An Overview[END_REF]. Each approach has advantages and limitations: supervised approaches heavily rely on large training datasets; on the other hand, unsupervised approaches fall short in labeling the identified clusters -introducing a barrier for human understanding -and in finding unified evaluation metrics; distant supervision approaches are based on entity alignment between a corpus and and a knowledge base, but demonstrated low accuracy scores due to a bad precision.

Particularly, the Cross-Modal Attention Network [START_REF] Zhao | Modeling dense cross-modal interactions for joint entity-relation extraction[END_REF] achieved state-of-the-art performances by simultaneously learning two tasks: entity recognition and relation classification. The approach involves injecting the token-level information into entity tags, rather than concatenating token and label representations.

In the literature, the extraction of events and their relationships studies mostly a subset of possible relations, namely causality, temporality and coreference [START_REF] Liu | Extracting events and their relations from texts: A survey on recent research progress and challenges[END_REF]. Previous studies demonstrated that models based on the combination of CNN, LSTM and attention mechanism are able to capture causal dependencies, even when the cause and effect are separated by a significant distance within the sentence [START_REF] Yang | A survey on extraction of causal relations from natural language text[END_REF].

Event extraction has been made possible using pretrained language models such as BERT [START_REF] Devlin | BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding[END_REF], as shown in [START_REF] Yang | Exploring pre-trained language models for event extraction and generation[END_REF]. SpanBERT [START_REF] Joshi | SpanBERT: Improving pre-training by representing and predicting spans[END_REF] -an improved version of BERT that excels in predicting text spans instead of single words -has also been employed for event extraction, resulting in notable performance gains [START_REF] Portelli | Improving adverse drug event extraction with spanbert on different text typologies[END_REF].

Building a Synthetic Event Relations Dataset with GPT-3

The Event Relation dataset (Section 2.1) is the only available dataset including multiple event relation types. In this section, we describe our efforts for overcoming the two most important limitations of this dataset: its size and the large unbalance between relation types.

Our data augmentation strategy for expanding the dataset is based on the automatic generation of sentences using a prompt-based model. Using the right prompt as input, the model would provide new synthetic sentences for enriching the dataset.

We use the GPT-3 language model [START_REF] Brown | Language Models are Few-Shot Learners[END_REF], and more precisely the GPT-3.5 text-davinci-003 variant as described in the OpenAI documentation. 4 We are interested in generating sentences that involve events and relationships between them, particularly those related to prevention, intention, and enabling.

Starting Point: The Event Relations Dataset

The Event Relations Dataset [START_REF] Rebboud | Beyond Causality: Representing Event Relations in Knowledge Graphs[END_REF] -later named in this paper the Original Dataset -describes some of the FARO event relation types. It represents the first events and events relations dataset that encapsulates different event relations, ranging from temporal to causal, and extending beyond causality to include intention, prevention, enabling, and the explicit negation of causality -that we will not cover in this work, because it would require a separate discussion. The construction of the dataset was done by manually re-annotating two existing datasets, TimeBank [START_REF] Uzzaman | SemEval-2013 task 1: TempEval-3: Evaluating time expressions, events, and temporal relations[END_REF], and [START_REF] Ning | Joint Reasoning for Temporal and Causal Relations[END_REF], which previously only included temporal and causal relations. The dataset was afterwards extended with more samples for prevention and enabling relation using the same manual validation technique.

The annotation of the aforementioned new event relations types involved also the annotation of the constructs of each relation, which we refer to them in the following as event triggers. It is worth mentioning that these event triggers belong to a bigger class in FARO called Relata, which is an abstraction encompassing two sub classes: Event (immanent) and Condition (transcendent). In this paper, we consider a subset of events that acts as triggers for preventing, causing or intending to cause other events, and a condition that enables the happening of another event.

Example 1. "The move boosts Intelogic Chairman Asher Edelman's stake to 20% from 16.2% and may help prevent Martin Ackerman from making a run at the computer-services concern. "

𝑚𝑜𝑣𝑒 prevents -------⟶ 𝑟𝑢𝑛
In Example 1., there exists an event relationship of type prevention, and the two event triggers that participate in the relation are move and run of type Event.

Example 2. "The government of Prime Minister Brian Mulroney has been under pressure to reduce the deficit, which is expected to reach C$30 billion this year. "

𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 enables ------⟶ 𝑟𝑒𝑑𝑢𝑐𝑒
In Example 2., the event relationship involves the type of enabling, with the two corresponding event triggers being pressure and reduce, in which pressure is an event trigger of type Condition and reduce is of type Event.

Table 1 summarizes the number of event relations per relation type in the Original Dataset after extending it with news agency samples.

Prompt-based Sample Generation of Sentences

When designing the prompt utilized to generate synthetic examples for a specific relation type, we include:

1. the definition that the FARO ontology assigns to that relation type; 2. a subset of relevant examples from the dataset.

We consider a sequence of words Xi = [x 1 , x t1 , …, t2 , x n ], representing an event relationship occurring between two Relata, of a specific relation type ER x . The words x t1 and x t2 respectively represents in the text the two Relata which are the subject and the object of the relations. The definition of the relation type definition(ER x ) is taken from the FARO ontology.

The selection of the prompt is done after a series of attempts. For sentences generation, we started by leveraging only the task description in the prompt. Therefore, the generated sentences where too short and basic, while we need realistic and longer sentences, similarly to those in the Original Dataset.

Table 2 demonstrates an effort to prompt the model to produce sentences that showcase connection between events with the desired relation type, but the resulting answer falls short of meeting our intended expectations.

The prompt text to generate sample sentences including relations of type ERx is written as the following:

Prompt(ERx) = definition(Event) + definition(ER x ) + request(ER) + examples(ERx)
This prompt definition concerns prevention and intention relations. In the context of enabling relation, we include the definition of a condition as follows:

Prompt(ER enable ) = definition(Event) + definition(Condition) + definition(ER enable ) + request(ER) + examples(ERx)
where request(ER) refers to the task description that is given to the language model along with the definitions and examples(ERx) are randomly-selected examples from the existing dataset which will be used to iteratively expand and reformulate the dataset.

Example: Prompt used to generate sentences with event relation of type Enabling Note that the original dataset was re-annotated based on Timebank [START_REF] Uzzaman | SemEval-2013 Task 1: TempEval-3: Evaluating Time Expressions, Events, and Temporal Relations[END_REF] and Event Causality dataset [START_REF] Ning | Joint Reasoning for Temporal and Causal Relations[END_REF], both of which are derived from news articles. This makes the majority of the sentences falling within the political domain. Therefore, introducing the word political in the prompt is to ensure that the generated sentences were coherent and consistent with the original dataset domain.

Prompt-based Event Trigger Annotation

Similarly to sentence generation, we leverage definitions of events to the prompt, adding few examples illustrating the right position for event triggers for each relation types. The prompts have been chosen to acquire the most similar sample pattern to facilitate parsing.

For an event relationship ERx including prevention or intention, the prompt for selecting their event trigger words is designed as follow:

Table 2 Example of prompting attempts that fell short of producing the desired results

Prompt

Answer Limitation Give me an event that enables the happening of other event.

One event that enables the happening of another event is a person's decision to take an action. For example, a person's decision to get up and walk across a room enables the person to arrive at their destination.

• Prompt had no examples from the existing dataset. • Answer was too simple. • Did not describe a real-world scenario. • Explanation was given instead of an actual expression.

Describe a situation where an event is an intention to cause another event, for example: Companies such as Microsoft or a combined worldcom MCI are trying to monopolize Internet access.

An example of an event intended to cause another event is when companies such as Microsoft or a combined worldcom MCI attempt to monopolize Internet access. By controlling the majority of the market, these companies can dictate the terms and prices of access, potentially limiting consumer choice and driving up costs...

• Explanation was given instead of an actual expression. definition(Event)

An event is a possible or actual event, which can possibly be defined by precise time and space coordinates. definition(Condition) A condition is the fact of having certain qualities, which may trigger events. definition(ER enable )

The enables relationship connects a condition or an event (trig-ger1), with an other event (trigger 2) it is contributing to realize as an enabling factor. where the last element is the description of the task of retrieving event triggers from the text. This request takes the following shape:

If in this sentence <TEXT OF THE SENTENCE> is present an expression with a <RELATION TYPE> relationship between <𝑥 𝑡1 > (trigger1) and <𝑥 𝑡1 > (trigger2), what would be the trigger1 and trigger2 in these sentences? Give me only one single word for each trigger an only two triggers per sentence. Put each pair between parentheses in a separate line.

For event relations of type enable, the definition of the condition is modified in the following way: The prevent relationship connects an event (trigger1) with the event (trigger 2) for which is the cause of not happening. request(ET)

Prompt
If in this sentence "Subcontractors will be offered a settlement and a swift transition to new management is expected to avert an exodus of skilled workers from Waertsilae Marine's two big shipyards, government officials said." is present an expression with a prevention relationship between settlement (trigger1) and exodus (trigger2), what would be the trigger1 and trigger2 in these sentences? Give me only one single word for each trigger an only two triggers per sentence, put each pair between parentheses in a separate line.

Manual Validation

We use these methods and we generate 600 sentences with each of the relations.

To guarantee the accuracy of the generated set of samples and their appropriate event triggers, we manually validate each synthetic sentence, ensuring its adherence to the given definition. Overall, 90.77% of all generated sentences were correctly representing an event relation of the requested type. After removing the wrong samples from the dataset, we proceed checking the correctness of their extracted event triggers for the remaining correct sentences.

The generated events triggers were not consistent in term of their patterns from one generation to another. For this reason, an additional parsing step was needed. For doing that, we identified the different textual patterns, processed and categorized these patterns by removing irrelevant words such as '(trigger 1)', and retaining only the precise word or sequence of words that represent the essential part of the event. We were able to identify roughly 12 different patterns. Some examples are reported in Table 3.

After this processing, we validated the correctness of the two trigger words, measuring an accuracy of 75.15% for trigger-1 and 66.82% for trigger-2. Sentences with wrong triggers were not eliminated from the dataset, but instead manually fixed. Table 4 shows the detailed accuracy scores for each relation type.

We merged the synthetic data with our original dataset, resulting into a larger and more diverse dataset. We managed to acquire and validate 1507 new sentences -with relative event triggers -making a total 2289 sentences. The statistics of this new dataset -latter in the text named the Augmented Dataset -are reported in Table 5. Based on the above reported results, it is evident that GPT-3, with a clear definition of concepts, particularly definition of relations and their constructs types, along with a limited number of examples (5 for each iteration), is able to reach a considerable accuracy. For event triggers generation, we just included a single sentence example, along with its event triggers. Observing the first generated annotations and realising that, despite the limited amount of training data, the results obtained were reasonably good, we decided to continue without adding further sentences. We considered the accuracy quite high, considering the difficulty of the task.

Events and Event Relation Extraction

In order to fulfil our goal of testing the effectiveness of GPT-3 based data augmentation technique on events and events relationships extraction, and at the same time, to evaluate the performances of existing models, we conducted the following experiment. We fine-tune two instances of the BERT model [START_REF] Devlin | BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding[END_REF]. The first one, named BERTee, is for token classification which we apply on event extraction. The second one, named BERTer, is for sequence classification which we apply on events relation classification. We additionally fine-tune a variant of BERT for event extraction, SpanBERT [START_REF] Joshi | SpanBERT: Improving pre-training by representing and predicting spans[END_REF] taking into consideration that some of our event triggers are represented by more than one word, i.e. a span of words, and SpanBERT is specifically designed to handle this type of representation.

Xi = [x 1 , x t1 , …,x t2 , x n ] is a sentence of n tokens, which is part of the studied dataset. BERTee (Figure 1a) is trained for predicting a tag for each token in the input sequence. The tags are chosen among TAGi=[O,x1 type ,…,x2 type,O ], where x1 type =[Trigger1] and x2 type =[Trig-ger2] are the subject and the object of the event relation, and 'O' is refereed to the rest of tokens in each sentence. The models consists of 12 transformer blocks receiving in input the sequence of tokens Xi and returning the relative contextualized representation H i = [h1, h2,…, hn]. On top of the transformers, a classification layer -consisting of a fully connected layer followed by a softmax activation -maps each contextualized representation ℎ 𝑖 to a probability distribution over the possible labels for that token, i.e., P hi = [P(Trigger1|hi), P(Trigger2|hi), P(O|hi)]. Finally, for x i ∈Xi, we select the most probable tag TAG xi = max(P hi ).

Similarly, BERTer (Figure 1b) takes as input the same sequence of tokens Xi, and uses transformers to compute the contextualized representation H i . This representation feeds a softmax classification layer that maps the hidden states to the event relation types outputting the probability distribution for the label Li ∈L, given L =[causality, enabling, prevention, intention, No-Relation]. We select similarly the most probable label from the outputted labels probabilities. Some of the event mentions present in our work are denoted as a span of words, i.e a sequence of words. Although they present the minority, we wanted to test a variant of BERT called SpanBERT [START_REF] Joshi | SpanBERT: Improving pre-training by representing and predicting spans[END_REF], which is trained mainly to predict a sequence of words rather than a single word. During the training, the model is masking spans of words rather than random single words, pushing the neural network to predict them. The model is similar to the overall architecture of the BERTee in term of transformers blocks: base model with 12 transformer block. In addition to the classification layer, SpanBERT also includes a span classification layer that is designed to predict the label for a span of tokens. For example, given the sentence: "The United Nations passed a resolution to impose an arms embargo on Syria in an effort to pressure the government to end its civil war". The span "arms embargo" is aimed to be tagged by SpanBERT as a single class label, namely as "Trigger1 Trigger1" more robustly.

Results

Events and Events Relationship Extraction

We trained BERTee, BERTer, and SpanBERT models on both the Original and Augmented datasets. However, it is important to note that the test set used for evaluation was extracted The outcomes of the experiment with BERTer are shown in Table 6. We observe that the performance varied across different classes in the Original Dataset. The model showed relatively higher F1 score values for the 'cause' and 'enable' classes, while it struggled to perform well on 'intend' and 'prevent' classes. This is probably due to the low support of these latter classes, with intention relation being less represented in the dataset with only 44 example sentences. Data augmentation leads to significant improvements in the metrics, with an increment of the F1-score for all classes. The highest improvement is observed in the 'prevent' and 'enable' classes, and in particular for the 'intend' class, which received a 115.91% increase in the F1-score with respect to the 44% of the Original Dataset. Despite the absence of added synthetic data for the "cause" relation, we still notice an improvement with a modest margin. Future work will involve data augmentation also for that class.

The performance of BERTee and SpanBERT in events classification is less good in both the Original and Augmented datasets (Table 7). Despite testing different parameters, the results indicated that further improvements are still needed to enhance the performance of this task. Therefore, the current outcomes should be viewed as preliminary, and further investigations will be adopted as future work.

Modeling the Extracted Event Relations in a Knowledge Graph

Event Knowledge Graphs are shown to be an effective data representation way to ease navigation through event flows and their relations and to flexibly retrieve information about these events from the stored knowledge [START_REF] Guan | What is event knowledge graph: A survey[END_REF]. This can serve many applications such as link prediction and fact-checking, and their efficiency tends to be considerable when they are richer in terms of aspects and relationships between them. For this sake, we aim to generate a Knowledge Graph (KG) of events and relations between them from the Augmented dataset. In other words, this KG will be an RDF version of the Augmented dataset.

The KG that we constructed contains events and relations between them. The elements in our KG are classified according to the FARO ontology, which distinguishes between two major types of Relata: Condition and Event (see Section 3.1).

More precisely, the events are typed according to the relation between them. 'Enables' relation, was used to connect two entities in which the subject represents the (Condition) that is necessary for the object (Event) to occur. We also identified the other relations that we focus on in the previous parts between events in our KG, such as "causes", "prevents", and "intends", which relate two entities of type Event. These relations capture different types of causal and temporal dependencies between events, and they allow to reason about complex chains of events and their potential consequences.

To ensure the traceability of our KG, we linked each event in our KG to the sentence it was extracted from and each sentence to its provenance corpus, using the Provenance ontology (PROV-O) [START_REF] Belhajjame | PROV-O: The PROV Ontology[END_REF]. This provenance information allows to track the origin of each piece of information in our KG, and to verify its accuracy and relevance. When the provenance involves scientific datasets or software, it is referenced in the graph using the FaBiO ontology [START_REF] Peroni | FaBiO and CiTO: Ontologies for describing bibliographic resources and citations[END_REF], detailing information about paper title, author and year of publication.

Additionally, in order to have a more complete KG in terms of event relations (temporal relations, comparative relations, etc.), we have incorporated in the same aforementioned way events and their temporal relations from TimeBank [START_REF] Uzzaman | SemEval-2013 task 1: TempEval-3: Evaluating time expressions, events, and temporal relations[END_REF] corpus, We also leverage events which have temporal, comparative and contingent relations from the [START_REF] Hong | Building a Crossdocument Event-Event Relation Corpus[END_REF] dataset, we call it in the rest of the paper Hong dataset.

TimeBank consists of 24k events with 3.4k temporal links, extracted from 183 News article. On the other hand, Hong consists of the annotation of ACE2005 news-wire documents and other news documents about Malaysian Airline 17, resulting in 862 events with 25610 relations between them.

The integration of the earlier stated datasets was made after examining the overlap between some of their event relation definitions with FARO ontology. The mapping of this relations to FARO is shown in 8 and 9.

The resulting graph -containing over 68,000 statements, with 11,917 event relation linkshas been loaded in a triplestore, available for query at http://kflow.eurecom.fr/.

Conclusion and Future Work

In this work, we made a first attempt towards the automatic extraction of event relations with precise semantics from raw text, focusing on a subset of them. We applied GPT-3 to generate synthetic data for the aforementioned event relation types, obtaining good accuracy. The result of this effort is a dataset consisting of 2289 sentences -of which 1507 were synthetic -annotated with the event mentions in the text. The data augmentation method described in this paper can be used to extract even more event relations by properly replacing the definition and the examples to match the required relation types. Furthermore, we used BERT for performing two related tasks: event relation classification and event mentions classification. We utilized also SpanBERT to evaluate its ability to classify events that are expressed as a sequence of words into a single class, even though such events are With the recent release of , new experiments can be performed for improving the synthetic data generation, in particular in the extraction of relevant triggers.

Furthermore, we would like to investigate the interaction between event relation types and the event mentions by jointly extract them from text in a sense of enhancing the event trigger classification by leveraging event relation information. In particular, We plan to test the effectiveness of the [START_REF] Zhao | Modeling dense cross-modal interactions for joint entity-relation extraction[END_REF]model on our own dataset and assess its performance under similar conditions.

In future work, we intend to combine the automatic classification of event relation to classic event identification techniques, in order to automatically annotate news and encyclopedic entries, with the final goal of realizing a KG of interconnected events with precise semantics.

  (a) BERTee for Event Extraction (b) BERTer for Event Relation Classification

Figure 1 :

 1 Figure 1: BERTee and BERTer architectures

Table 1

 1 Total number of relations in the Original Dataset.

	Relation type	Cause Intend Prevent Enable Not-Cause
	Number of relations	283	44	89	124	3

  Event Triggers ER enable = definition(Event) + definition(Condition) + definition(ER enable ) + request trig (ER enable , sentence, x t1 , x t2 ) Example: Prompt used to generate event triggers with event relation of type Prevention

	definition(Event)	An event is a possible or actual event, which can possibly be
		defined by precise time and space coordinates.
	definition(Condition) A condition is the fact of having certain qualities, which may
		trigger events.
	definition(ER enable )

Table 3

 3 Three of the different textual patterns which GPT-3 was returning in output for the Event Triggers selection.

	Pattern Number		Event Triggers
	0			Entitles, Buy
	1	Approval (trigger1), Acquire (trigger2)
	2	"Trigger1 (military): success Trigger2 (diplomatic): risks"
	Table 4
	Percentage of Correct Sentences and Event Trigger Words with GPT-3
	Relation types	Intention Prevention Enabling Total
	Correct Generated Sentences(%)	93.82	97	81.5	90.77
	Correct ET1 (%)	75.13	81.83	68.5	75.15
	Correct ET2 (%)	73.47	77	50	66.82
	Number of Checked Examples	600	600	600	1800

Table 5

 5

	Augmented Dataset Statistics
	Relation Type Original Dataset Augmented Dataset
	Prevent	93	646
	Enable	118	573
	Intend	44	615
	Cause	283	283
	No-Relation	72	172
	total	610	2289

Table 6

 6 Event relationship extraction results on the test set with BERTer on both the original and the Augmented dataset

	Dataset	Class	Precision Recall	F1-Score
		cause	0.64	0.83	0.72
		enable	0.77	0.67	0.71
	Original Dataset	intend	0.67	0.33	0.44
		prevent	0.62	0.67	0.64
		cause	0.75	0.75	0.75 (+4.16%)
		enable	0.93	0.96	0.95 (+33.80%)
	Augmented Dataset	intend	0.98	0.92	0.95 (+115.91%)
		prevent	0.96	0.93	0.94 (+46.86%)

Table 7

 7 Event extraction results on the test set after finetuning BERTee and SpanBERT for token Classification on both the original Dataset and the augmented dataset. (*): The reported metrics are the macro average of Precision, Recall, and F1-score. The macro average F1 score computes the F1 score independently for each class and then takes the unweighted average of the scores. This means that each class is given equal weight, regardless of the number of samples it contains. For this reason, the F1 score values may seem to not reflect the overall precision and recall metrics

	Model	Dataset	Precision(*) Recall(*) F1-score(*)
	SpanBERT	Original Dataset Augmented Dataset 0.34 0.34	0.36 0.32	0.12 0.18
	BERTee	Original Dataset Augmented Dataset 0.33 0.33	0.31 0.34	0.12 0.23

Table 8

 8 Mapping of Event Relation Dataset (Hong) relations to FARO

		FARO		Hong
	Category	Super-type	Type	Type	Super-type
			Immediately before	Before	Before
			Meets	Before	Meet
			Starts	Starts	Starts
	Temporal	Temporally related to	Ends	Finish	Finish
		Contains(the subproperty)	During	During
			Overlaps	Overlaps	Overlap
			Simulations to	Equality	Equality
	Causality Enabling	Contingently related to	Causes Enables	Causality Condition	Contingency
			Opposite to	Opposite
	Comparison Comparatively related to	Alternative to	Negation	Comparison
		Contrasting version of	Competition

Table 9

 9 Mapping of TimeBank relations to FARO The reported results show that the augmented dataset -and in general synthetic data -improve the ability of the model to generalize and correctly classify sequences, even for classes with a limited number of training examples in the original dataset. However, for event mention classification the performance is still relatively low and needs further improvements.All code used for the experiments reported in this paper, as well as the resulting dataset is available at https://github.com/ANR-kFLOW/event-relation-classification.

	Types	FARO	TimeBank
		Simultanious to	SIMULTANIOUS
		Before	BEFORE
		Imediately before	IBEFORE
		Inverse of (Imediately before)	IAFTER
		Contains(the subproperty)	During
	Temporal	Contains	INCLUDES
		Inverse of (contains)	IS INCLUDED
		Starts	BEGINS
		Inverse of (starts)	BEGUN BY
		Ends	ENDS
		Inverse of (ends)	ENDED BY
	less frequent.

https://catalog.ldc.upenn.edu/LDC2006T06

More details are provided in Section 3.1

https://platform.openai.com/docs/guides/completion

Acknowledgments

This work has been partially supported by the French National Research Agency (ANR) within the kFLOW project (Grant n°ANR-21-CE23-0028).


Mabel Cuesta

Rosa Pardo

POSITIVE SOLUTIONS FOR SLIGHTY SUBCRITICAL ELLIPTIC PROBLEMS VIA ORLICZ SPACES

Keywords: Positive solutions, subcritical nonlinearity, changing sign weight. [2020]58E07, 35J20, 35B32, 35J25, 35J61

This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sens. Using an Orlicz-Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais-Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions.

Introduction

In this paper we study the classical positive solutions to the Dirichlet problem for a class of semilinear elliptic equations whose nonlinear term is of subcritical nature in a generalized sens and involves indefinite nonlinearities. More precisely, given Ω ⊂ R N , N > 2, a bounded, connected open subset, with C 2 boundary ∂Ω, we look for positive solutions to:

(1.1)

-∆u = λu + a(x)f (u), in Ω, u = 0, on ∂Ω, where λ ∈ R is a real parameter, a ∈ C 1 ( Ω) changes sign in Ω,

(1.2) f (s) := g(s) + h(s), with h(s) := |s| 2 * -2 s [ln(e + |s|)] α , 2 * = 2N
N -2 is the critical Sobolev exponent, α > 0 is a fixed exponent, and g ∈ C 1 (R) satisfies

(H)      (H) 0 lim s→0 f (s) |s| p-2 s = L 1 ,
for some L 1 > 0, and p ∈ 2, 2N N -2

(H) ∞ lim s→∞ g(s)

|s| q-2 s = L 2 , for some L 2 ≥ 0, and q ∈ 2, 2N N -2

(H) g |g (s)| ≤ C(1 + |s| q-2 ), for s ∈ R.

The second author is supported by grants PID2019-103860GB-I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo 920894.

We will say that g (or even f ) satisfies hypothesis (H) whenever (H) 0 , (H) ∞ , and (H) g are satisfied. Since we are interested in positive solutions, we (1.3) redefine f to be zero on (-∞, 0],

note that f (0) = 0 and that (1.4)

lim s→0 + f (s) s -L 1 |s| p-2 = 0.
When λ = 0, a(x) ≡ 1 and g(s) ≡ 0, this kind of nonlinearity has been studied in [START_REF] Castro | A priori bounds for Positive Solutions of Subcritical Elliptic Equations[END_REF][START_REF] Castro | A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions Discrete and Continuous Dynamical[END_REF][START_REF] Castro | Equivalence between uniform L 2 * (Ω) a-priori bounds and uniform L ∞ (Ω) a-priori bounds for subcritical elliptic equations[END_REF][START_REF] Mavinga | A priori bounds and existence of positive solutions for semilinear elliptic systems[END_REF], and in [START_REF] Damascelli | A priori estimates for some elliptic equations involving the p-Laplacian Nonlinear Analysis[END_REF] for the case of the p-laplacian operator, with α > p N -p . It is known the existence of uniform L ∞ a priori bounds for any positive classical solution, and as a consequence, the existence of positive solutions. When α → 0, there is a positive solution blowing up at a non-degenerate point of the Robin function as α → 0, see [START_REF] Clapp | Alberto A solution to a slightly subcritical elliptic problem with non-power nonlinearity[END_REF] for details.

From [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF] it is known that (λ 1 , 0) is a bifurcation point of positive solutions (λ, u λ ) to the equation (1.1) .

For f behaving like |u| p-2 u at zero with 2 ≤ p ≤ 2 * , the influence of the negative part of the weight a is displayed under the sign of Ω a(x)ϕ 1 (x) p dx, where ϕ 1 is the first positive eigenfunction for -∆ in H 1 0 (Ω). Specifically, whenever

Ω a(x)ϕ 1 (x) p dx < 0 the bifurcation of positive solutions from the trivial solution set is 'on the right' of the first eigenvalue, in other words for values of λ > λ 1 . And whenever Ω a(x)ϕ 1 (x) p dx > 0 the bifurcation from the trivial solution set is 'on the left' of the first eigenvalue, in other words for values of λ < λ 1 .

Inspired by the work of Alama and Tarantello in [START_REF] Alama | On semilinear elliptic problems with indefinite nonlinearities Calculus Var[END_REF], we will focus our attention to the case a(x) changing sign and (1.5) is satisfied, and, among other things, we will prove the existence of a turning point for a value of the parameter Λ > λ 1 , and in particular the existence of solutions when λ = λ 1 . We will use local bifurcation and variational techniques.

All throughout the paper, for v : Ω → R, v = v + -v -where v + (x) := max{v(x), 0} and v -(x) := max{-v(x), 0}.

Let us also define Ω ± := {x ∈ Ω : ±a(x) > 0}, Ω 0 := {x ∈ Ω : a(x) = 0}, and assume that both Ω + , Ω -are non empty sets.

For this nonlinearity the Palais-Smale condition of the energy functional becomes a delicate issue, needing Orlicz spaces and a Orlicz-Sobolev embedding theorem.

In order to prove (PS) condition, Alama and Tarantello ( [START_REF] Alama | On semilinear elliptic problems with indefinite nonlinearities Calculus Var[END_REF]) assume that the zero set Ω 0 has a non empty interior. This is also a common hypothesis for other authors when dealing with changing sign superlinear nonlinearities [START_REF] Chang | Mei-Yue Dirichlet problem with indefinite nonlinearities[END_REF][START_REF] Ramos | Christophe Superlinear indefinite elliptic problems and Pohožaev type identities[END_REF][START_REF] Tehrani | Infinitely many solutions for an indefinite semilinear elliptic problem in R N[END_REF]. But this is a technical hypothesis. (PS)-condition will be proved in Proposition 3.1 without assuming that hypothesis. We neither use Ambrosetti-Rabinowitz.

Let us now denote

(1.6) C 0 = inf{C ≥ 0 : f (s) + C ≥ 0 for all s ≥ 0},
and remark that hypothesis (H) implies that C 0 < +∞. Observe also that

(1.7) f (s) + C 0 s ≥ 0, for all s ≥ 0; f (s)s + C 0 s 2 ≥ 0, for all s ∈ R.
Let u be a weak solution to (1.1). By a regularity result, see Lemma 2.1, u ∈ C 2 (Ω)∩C 1,µ (Ω). So by a solution, we mean a classical solution.

Assume that u is a non-negative nontrivial solution. It is easy to see that the solution is strictly positive. Indeed, adding ±C 0 a(x)u to the r.h.s. of the equation, splitting a = a + -a -, taking into account (1.4), and letting in each side the nonnegative terms, we can write

-∆ + a -(x) f (u) u + C 0 + C 0 a(x) + u (1.8) = λu + a(x) + f (u) + C 0 u + C 0 a(x) -u, in Ω.
Now, the strong Maximum Principle implies that u > 0 in Ω, and ∂u ∂ν < 0 on ∂Ω.

Our main result is the following theorem.

Theorem 1.1. Assume that g ∈ C 1 (R) satisfies hypothesis (H). Let C 0 > 0 be defined by (1.6). If a changes sign in Ω, and (1.5) holds, then there exists a Λ ∈ R,

λ 1 < Λ < min λ 1 int (Ω 0 ) , λ 1 int Ω + ∪ Ω 0 + C 0 sup a +
and such that (1.1) has a classical positive solution if and only if λ ≤ Λ.

Moreover, there exists a continuum (a closed and connected set) C of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point (λ, u) = (λ 1 , 0) which is unbounded. The paper is organized in the following way. Section 2 contains a regularity result and a non existence result. (PS)-condition and an existence of solutions result for λ < λ 1 based in the Mountain Pass Theorem will be proved in Section 3. A bifurcation result for λ > λ 1 is developped in Section 4. The main result is proved in Section 5. Appendix A contains some useful estimates. Orlicz spaces, a Orlicz-Sobolev embeddings theorems, and variational techniques, also including a (PS) condition in Orlicz-Sobolev spaces setting and the Mountain Pass Theorem, will be treated in Appendix B.

A regularity result and a non existence result

Next, we recall a regularity Lemma stating that any weak solution is in fact a classical solution.

Lemma 2.1. If u ∈ H 1 0 (Ω) weakly solves (1.1) with a continuous function f with polynomial critical growth

|f (x, s)| ≤ C(1 + |s| 2 * -1 ), then, u ∈ C 2 (Ω) ∩ C 1,µ (Ω) and u C 1,µ (Ω) ≤ C 1 + u 2 * -1 L (2 * -1)r (Ω) , for any r > N and µ = 1 -N/r. Moreover, if ∂Ω ∈ C 2,µ , then u ∈ C 2,µ (Ω).
Proof. Due to an estimate of Brézis-Kato [START_REF] Brézis | Remarks on the Schrödinger operator with singular complex potentials[END_REF], based on Moser's iteration technique [START_REF] Moser | A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations[END_REF], u ∈ L r (Ω) for any r > 1; and by elliptic regularity u ∈ W 2,r (Ω), for any r > 1 (see [START_REF] Struwe | Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems[END_REF]Lemma B.3] and comments below).

Moreover, by Sobolev embeddings for r > N and interior elliptic regularity

u ∈ C 1,α (Ω) ∩ C 2 (Ω). Furthermore, if ∂Ω ∈ C 2,α , then u ∈ C 2,α (Ω).
Proposition 2.2. Let f satisfy hypothesis (H) and let C 0 be defined in (1.6). Assume that a changes sign in Ω.

(1) Problem (1.1) does not admit a positive solution u ∈ H 1 0 (Ω) for any λ ≥ λ 1 int Ω + ∪ Ω 0 + C 0 sup a + .

(

) If int (Ω 0 ) = ∅, then λ 1 int (Ω 0 ) < +∞ and (1.1) 2
does not admit a positive solution for any

λ ≥ λ 1 int (Ω 0 ) . Proof. 1. Let λ ≥ λ 1 int Ω + ∪Ω 0 +C 0 sup a +
, and assume by contradiction that there exists a non-negative non-trivial solution u ∈ H 1 0 (Ω) to (1.1) for the parameter λ. Since the Maximum Principle u > 0 in Ω, see (1.8).

Let φ be the positive eigenfunction of -∆, H 1 0 (int Ω + ∪ Ω 0 of L 2 -norm equal to 1. For simplicity we will denote also by φ the extension by 0 of φ in all Ω. By Hopf's maximum principle we have

∂ φ ∂ν < 0 on ∂ int Ω + ∪ Ω 0
, where ν is the outward normal.

Again if we multiply the equation (1.1) by φ and integrate along int Ω + ∪ Ω 0 we find, after integrating by parts,

0 > ∂(int(Ω + ∪Ω 0 )) u ∂ φ ∂ν dσ + int(Ω + ∪Ω 0 ) λ 1 int (Ω + ∪ Ω 0 ) -λ + C 0 a + (x) u φ dx = Ω + a + (x) f (u) + C 0 u φ dx > 0, a contradiction. 2. Let λ ≥ λ 1 int (Ω 0
) and assume by contradiction that there exists a positive solution u ∈ H 1 0 (Ω) of problem (1.1) for the parameter λ. Let φ be a positive eigenfunction associated to λ 1 int (Ω 0 ) < +∞. For simplicity we will also denote by φ the extension by 0 in all Ω. If we multiply equation (1.1) by φ and integrate along Ω 0 we find, after integrating by parts,

int (Ω 0 ) ∇u • ∇ φ dx = λ int (Ω 0 ) u φ dx.
On the other hand

int (Ω 0 ) ∇u • ∇ φ dx = λ 1 (int (Ω 0 )) int (Ω 0 ) φu dx + ∂(int (Ω 0 )) u ∂ φ ∂ν dσ. Hence 0 > ∂(int (Ω 0 )) u ∂ φ ∂ν dσ = λ -λ 1 int (Ω 0 ) int (Ω 0 ) u φ dx ≥ 0, a contradiction.
3. An existence result for λ < λ 1

In this section we prove the existence of a nontrivial solution to equation (1.1) for λ < λ 1 , through the Mountain Pass Theorem.

3.1. On Palais-Smale sequences. In this subsection, we define the framework for the functional J λ associated to the problem (1.1) λ . Hereafter we denote by • the usual norm of H 1 0 (Ω):

u = Ω |∇u| 2 dx 1/2 . Given f (s) = h(s) + g(s) defined
F (s) + 1 2 C 0 s 2 ≥ 0, for all s ≥ 0.
Consider the functional J λ : H 1 0 (Ω) → R given by

J λ [v] := 1 2 Ω |∇v| 2 dx - λ 2 Ω (v + ) 2 dx - Ω a(x)F (v + ) dx.
Observe that for all v ∈ H 1 0 (Ω),

J λ v + ≤ J λ [v].
The functional J λ is well defined and belongs to the class C 1 with

J λ [v] ψ = Ω ∇v∇ψ dx -λ Ω v + ψ dx - Ω a(x)f (v + )ψ dx,
for all ψ ∈ H 1 0 (Ω). Consequently, non-negative critical points of the functional J λ correspond to non-negative weak solutions to (1.1).

The next Proposition proves that Palais-Smale sequences are bounded whenever λ < λ 1 (int Ω 0 ), where λ 1 (int Ω 0 ) may be infinite. Proposition 3.1. Assume that g ∈ C 1 (R) satisfies hypothesis (H) and assume also that λ < λ 1 (int Ω 0 ) ≤ +∞.

Then any (PS) sequence, that is, a sequence satisfying

(J 1 ) J λ [u n ] ≤ C, (J 2 ) J λ [u n ] ψ ≤ ε n ψ , where ε n → 0 as n → +∞ is a bounded sequence.
Proof. 1. Let {u n } n∈N be a (PS) sequence in H 1 0 (Ω) and assume by contradiction that u n → +∞. Let us first prove the following claim:

Claim. Let v ∈ H 1 0 (Ω) be the weak limit of v n = un un and assume that v n → v, strongly in L 2 * -1 (Ω) and a.e. Then v = 0 a.e. in Ω.

Assume that v ≡ 0 and denote

γ n = u n . Let ω n := {x ∈ Ω : v + n (x) > 1}, then for any ψ ∈ C 1 0 (Ω), ln(e + γ n ) α γ 2 * -1 n (u + n (x)) 2 * -1 [ln(e + γ n v + n (x))] α |ψ| ≤ |v + n (x)| 2 * -1 ψ ∞ , ∀x ∈ ω n .
Let x ∈ Ω \ ω n , using the estimates (A.1),

ln(e + γ n ) α γ 2 * -1 n (u + n ) 2 * -1 [ln(e + γ n v + n )] α |ψ| ≤ |v + n | 2 * -2 ψ ∞ ≤ ψ ∞
Besides, by the reverse of the Lebesgue dominated convergence theorem, see for instance [2, Theorem 4.9, p. 94] , there exists h i ∈ L 1 (Ω), 1 ≤ i ≤ 3 such that, up to a subsequence,

|v + n | 2 * -1 ≤ h 1 , |v + n | p-1 ≤ h 2 |v + n | 2 * -2 ≤ h 3 , , a.e.
x ∈ Ω, for all n ∈ N, and therefore

ln(e + γ n ) α γ 2 * -1 n f (u + n )ψ ≤ C (h 1 + h 2 + h 3 + 1)) ψ ∞ ∈ L 1 (Ω).
By Lebesgue dominated convergent theorem we have

ln(e + γ n ) α γ 2 * -1 n a(•)f (u + n )ψ → a(•)(v + ) 2 * -1 ψ strongly in L 1 (Ω).
We have used here that if v + (x) = 0, then

lim n→+∞ ln(e + γ n ) ln(e + γ n v + n (x)) = 1,
and if v + (x) = 0, then lim n→+∞ ln(e + γ n ) ln(e + γ n v + n (x)) α |v + n (x)| 2 * -1 ≤ lim n→+∞ |v + n (x)| 2 * -2 = 0.
On the other hand

ln(e + γ n ) α γ 2 * -1 n Ω ∇u n • ∇ψ dx → 0.
Hence, using (J 2 ) for an arbitrary test function ψ, multiplying by ln(e+γn) α γ 2 * -1 n and passing to the limit we find

Ω a(x)(v + ) 2 * -1 ψ dx = 0 ∀ψ ∈ C 1 0 (Ω).
In particular v + = 0 a.e. in Ω \ Ω 0 .

Assume that int Ω 0 = ∅, and that λ < λ 1 (int Ω 0 ). Thus, for any

ψ ∈ C 1 0 (int Ω 0 ) we have from (J 2 ) int Ω 0 ∇u n • ∇ψ dx -λ int Ω 0 u + n ψ dx = o(1).
Dividing by u n and passing to the limit we have

int Ω 0 ∇v • ∇ψ dx = λ int Ω 0 v + ψ dx. From the Maximum Principle, v ≥ 0 in int Ω 0 . Since λ < λ 1 (int Ω 0 ) then it must be v + ≡ 0 in int Ω 0 . Hence v + ≡ 0 in Ω.
On the other hand, taking u - n as a test function in the condition (J 2 ),

- Ω |∇u - n | 2 dx - Ω a(x)f (u + n )u - n dx = Ω |∇u - n | 2 dx ≤ n u -
n so u - n → 0 and then v -≡ 0, and we conclude the proof of the claim.

2. In order to achieve a contradiction, we use a Hölder inequality, and properties on convergence into an Orlicz space, cf. Appendix B.

To this end, the analysis of Lemma A.2 give us the existence of

α * > 0 such that the function s → s 2 * -1 [ln(e+s)] α is increasing along [0, +∞[ if α ≤ α * .
In this case we will denote

(3.2) m(s) = s 2 * -1 [ln(e + s)] α If α > α * the function s → s 2 * -1 [ln(e+s)] α possesses a local maximum s 1 in [0, +∞[. Let us denote by s 1 the unique solution s > s 1 such that s 2 * -1 1 [ln(e + s 1 )] α = s 2 * -1 [ln(e + s)] α
and define the non-decreasing function Since

(3.3) m(s) :=    s 2 * -1 [ln(e+s)] α if s ∈ [s 1 , s 1 ], s 2 * -1 1 [ln(e+s 1 )] α if s ∈ [s 1 , s 1 ]. It follows that (3.4) s → M (s) = s 0 m(t) dt is a N -function in [0, +∞[.
v n 0 in H 1 0 (Ω) and strongly in L 2 (Ω), it follows from (J 2 ) applied to ψ = u n that (3.5) lim n→∞ Ω a(x) f (u + n )u n u n 2 dx = lim n→∞ Ω a(x) f (u + n ) u n v + n dx = 1.
Since the Hölder inequality into Orlicz spaces, see Proposition B.11.(ii),

(3.6) Ω a(x) f (u + n ) u n v + n dx ≤ a ∞ u n f (u + n ) M * v + n M
By Theorem B.3 and Theorem B.12 we have

(3.7) v n -v M → 0.
Moreover, since there exists C > 0 such that m(s) ≤ Cs 2 * -1 and M (s) ≤ Cs 2 * for all s ≥ 0, and the sequence {u n } n∈N ⊂ H 1 0 (Ω), then, for each n ∈ N, there exists a C n such that

|u n m(u n )| ≤ C n , |M (|u n |)| ≤ C n .
By using definition B.8 of M * and identities of Proposition B.9 we have

M * |m(u n )| = |u n m(u n )| -M (|u n |) then, for each n ∈ N, Ω M * |m(u n )| dx ≤ 2C n . Observe that f (s) ≤ C(1 + m(s)), that M * f (s) ≤ M * C(1 + m(s)) , see Proposition B.11.(iii)
, and by convexity of M * , that

f (u + n ) M * ≤ Ω M * C(1 + m(u + n ) dx + 1 ≤ C n ,
see Proposition B.11.(i), and the r.h.s. is bounded for each n. Consequently, a(x [START_REF] Krasnoselskiȋ | Rutickiȋ Convex functions and Orlicz Spaces[END_REF], Theorem 14.2).

) f (u + n ) un ∈ L M * (Ω), which is the dual of L M (Ω) (see
On the other hand, from

J 2 , for all ψ ∈ C ∞ c (Ω), (3.8) Ω ∇v n ∇ψ dx -λ n Ω v n ψ dx - Ω a(x) f (u + n ) u n ψ dx ≤ ε n u n ψ .
Taking the limit, and since [START_REF] Donaldson | Trudinger Orlicz-Sobolev Ppaces and Imbedding Theorems[END_REF]),

C ∞ c (Ω) is dense in L M (Ω) (see
(3.9) lim n→∞ Ω a(x) f (u + n ) u n ψ dx = 0, for all ψ ∈ L M (Ω). Moreover, since (3.7), v n → v = 0 in L M (Ω).
Hence [2, Proposition 3.13 (iv)], and (3.9) imply

lim n→∞ Ω a(x) f (u + n ) u n v n dx = 0,
which contradicts (3.5), ending the proof.

Theorem 3.2. Assume the hypothesis of Proposition 3.1 and let {u n } n∈N be a (PS) sequence in H 1 0 (Ω). Then, there exists a subsequence, denoted by {u n } n∈N , such that u n → u in H 1 0 (Ω). Proof. From Proposition 3.1 we know that the sequence is bounded. Consequently, there exists a subsequence, denoted by {u n } n∈N , and some u ∈ H 1 0 (Ω) such that u n u weakly in

H 1 0 (Ω), (3.10) Ω a(x)g(u n )|u n -u| dx → 0, (3.11) u n → u a.e. (3.12)
By testing (J 2 ) against ψ = u n -u and using (3.10), and (3.11) we get

u n -u 2 = Ω ∇u n • ∇(u n -u) dx + o(1) ≤ a ∞ Ω |u n | 2 * -1 [ln(e + |u n |)] α |u n -u| dx + o(1). Claim. Ω |u n | 2 * -1 [ln(e + |u n |)] α |u n -u|dx = o(1),
In order to prove this claim, we use as in the above proposition, a Hölder inequality and a compact embedding into an Orlicz space, c.f. Appendix B.

By Theorem B.3 and Theorem B.12 we have

(3.13) u n -u M → 0,
where m, and M are defined by (3.2)-(3.4), as in the above proposition.

On the other hand, since there exists C > 0 such that m(s) ≤ Cs 2 * -1 and M (s) ≤ Cs 2 * for all s ≥ 0, and the sequence

{u n } n∈N is bounded in H 1 0 (Ω), then |u n m(u n )| ≤ C, |M (|u n |)| ≤ C for all n ∈ N
By using definition B.8 of M * and identities of Proposition B.9 we have Now, using Holder's inequality (B.6) and that s 2 * -1 [ln(e+s)] α ≤ m(s) for all s ≥ 0, we get

M * |m(u n )| = |u n m(u n )| -M (|u n |) then Ω M * |m(u n )| dx ≤ C for all n ∈ N.
Ω |u n | 2 * -1 [ln(e + |u n |)] α |u n -u|dx ≤ u n -u M m(u n ) M * ≤ (C+1) u n -u M
and it follows from (3.13) that u n -u → 0. Proof. We verify the hypothesis of the Mountain Pass Theorem, see [13, Theorem 2, §8.5]. Observe that the derivative of the functional J λ :

H 1 0 (Ω) → H 1 0 (Ω)
is Lipschitz continuous on bounded sets of H 1 0 (Ω); also the (PS) condition is satisfied, see Proposition 3.1. Clearly J λ [0] = 0.

1. Let now u ∈ H 1 0 (Ω) with u = r, for r > 0 to be selected below. Then,

(3.14) J λ [u] = r 2 2 - λ 2 Ω (u + ) 2 dx - Ω a(x)F (u + ) dx.
From hypothesis (H) we have

Ω a(x)G(u + ) dx ≤ C Ω |u| p + |u| q dx ≤ C (r p + r q ) .
where

G(s) := s 0 g(t) dt. Now, definition (1.2) implies that Ω a(x)F (u + ) dx ≤ C r p + r q + r 2 * .
In view of (3.14), and thanks to the Poincaré inequality we get

J λ [u] ≥ 1 2 1 - |λ| λ 1 r 2 -C r p + r q + r 2 * ≥ C 1 r 2 ,
taking |λ| < λ 1 , r > 0 small enough, and using that p, q, 2 * > 2.

2. Now, fix some element 0 ≤ u 0 ∈ H 1 0 (Ω), u 0 > 0 in Ω + , u 0 ≡ 0 in Ω -. Let v = tu 0 for a certain t = t 0 > 0 to be selected a posteriori. Since (3.15) f

(tu 0 ) = |t| 2 * -2 t f (u 0 ) ln(e + |u 0 |) ln(e + |tu 0 |) α + g(tu 0 ), then f (tu 0 )/t → +∞ as t → +∞ in Ω + .
From definition, and integrating by parts,

F (s) = s 0 t 2 * -1 ln(e + t) α + g(t) dt = 1 2 * sh(s) + G(s) + α 2 * s 0 1 ln(e + t) α+1 t 2 * e + t dt.
It can be easily seen that lim s→+∞ G(s) sf (s) = 0. Therefore, using l'Hôpital's rule we can write

lim s→+∞ F (s) sf (s) = 1 2 * ∈ 0, 1 2 , (3.16) hence lim t→+∞ F (tu 0 ) tu 0 f (tu 0 ) = 1 2 * ∈ 0, 1 2 in Ω + . (3.17)
Let C 0 ≥ 0 be such that F (s) + 1 2 C 0 s 2 ≥ 0 for all s ≥ 0 (see (1.7)), and let (3.18)

Ω + δ := {x ∈ Ω + : a(x) = a + (x) > δ}.
By definition, u 0 ≡ 0 in Ω -, so, introducing ± 1 2 C 0 (tu 0 ) 2 , splitting the integral, and using (3.17)-(3.18) we obtain

- Ω a(x)F (tu 0 ) dx = - Ω + a + (x)F (tu 0 ) dx ≤ C 0 t 2 2 Ω + a + (x)u 2 0 dx - Ω + δ a + (x) 1 2 C 0 (tu 0 ) 2 + F (tu 0 ) dx ≤ C + C 0 t 2 2 Ω + a + (x)u 2 0 dx - δt 2 2 Ω + δ C 0 u 2 0 + u 0 f (tu 0 ) 2 * t dx.
Hence, there exists a positive constant C > 0 such that

J λ [tu 0 ] = t 2 2 u 0 2 -t 2 λ 2 u 0 2 L 2 (Ω) - Ω + a + (x)F (tu 0 ) ≤ C(1 + t 2 ) - δ t 2 2 Ω + δ C 0 (u 0 ) 2 + u 0 f (tu 0 ) 2 * t dx < 0
for t = t 0 > 0 big enough.

Step 3. We have at last checked that all the hypothesis of the Mountain Pass Theorem are accomplished. Let

Γ := {g ∈ C [0, 1]; H 1 0 (Ω) : g(0) = 0, g(1) = t 0 u 0 }, then, there exists c ≥ C 1 r 2 > 0 such that c := inf g∈Γ max 0≤t≤1 J λ [g(t)] is a critical value of J λ , that is, the set K c := {v ∈ H 1 0 (Ω) : J λ [v] = c, J λ [v] = 0} = ∅. Thus there exists u ∈ H 1 0 (Ω), u ≥ 0, u = 0 such that for each ψ ∈ H 1 0 (Ω), we have (3.19) Ω ∇u • ∇ψ dx = Ω λu + + a(x)f (u + ) ψ dx.
and thereby u is a nontrivial weak solution to (3.19). By Lemma 2.1, u is a classical solution, and by (1.8), u > 0 in Ω.

A bifurcation result for λ > λ 1

Next Proposition uses Crandall -Rabinowitz's local bifurcation theory, see [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF], and Rabinowitz's global bifurcation theory, see [START_REF] Rabinowitz | Some global results for nonlinear eigenvalue problems[END_REF].

λ 1 < Λ < min λ 1 int (Ω 0 ) , λ 1 int Ω + ∪ Ω 0 + C 0 sup a + where C 0 > 0 is such that f (s) + C 0 s ≥ 0 for all s ≥ 0, (see definition (1.6)).
Moreover, there exists an unbounded continuum (a closed and connected set) C of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point (λ, u) = (λ 1 , 0).

Proof. Proposition 2.2 proves the upper bounds for Λ. Next we concentrate our attention in proving that Λ > λ 1 . Choosing λ as the bifurcation parameter, we check that the conditions of Crandall -Rabinowitz's Theorem [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF] are satisfied. For r > N , we define the set W 2,r + := {u ∈ W 2,r (Ω) : u > 0 in Ω}, and consider W 2,r + (Ω) ∩ W 1,r 0 (Ω) endowed with the topology of W 2,r (Ω). If r > N , we have that

W 2,r + (Ω) ∩ W 1,r 0 (Ω) → C 1,µ 0 (Ω) for µ = 1 -N r ∈ (0, 1)
. Moreover, from Hopf's lemma, we know that if ũ is a positive solution to (1.1) then ũ lies in the interior of W 2,r + (Ω) ∩ W 1,r 0 (Ω). We consider the map F :

R×W 2,r + (Ω)∩W 1,r 0 (Ω) → L r (Ω) for r > N , F : (λ, u) → -∆u -λu -a(x)f (u)
The map F is a continuously differentiable map. Since hypothesis (i), g(0) = 0, and so a(x)F (0) = 0, F (λ, 0) = 0 for all λ ∈ R, and since F u (x, 0) = 0,

D u F (λ 1 , 0)w := -∆w -λ 1 w, D λ,u F (λ 1 , 0)w := -w. Observe that N D u F (λ 1 , 0) = span[ϕ 1 ], codim R D u F (λ 1 , 0) = 1, D λ,u F (λ 1 , 0)ϕ 1 = -ϕ 1 ∈ R D u F (λ 1 , 0) ,
where N (•) is the kernel, and R(•) denotes the range of a linear operator. Hence, the hypotheses of Crandall-Rabinowitz theorem are satisfied and (λ 1 , 0) is a bifurcation point. Thus, decomposing

C 1,µ 0 (Ω) = span[ϕ 1 ] ⊕ Z, where Z = span[ϕ 1 ] ⊥ ,
there exists a neighbourhood U of (λ 1 , 0) in R× C 1,µ 0 (Ω), and continuous functions λ(s), w(s), s ∈ (-ε, ε), λ : (-ε, ε) → R, w : (-ε, ε) → Z such that λ(0) = λ 1 , w(0) = 0, with Ω wϕ 1 dx = 0, and the only nontrivial solutions to (1.1) in U , are (4.1) λ(s), sϕ 1 + s w(s) : s ∈ (-ε, ε) .

Set u = u(s) = sϕ 1 + s w(s). Note that by continuity w(s) → 0 as s → 0, which guarantees that u(s) > 0 in Ω for all s ∈ (0, ε) small enough.

Next, we show that λ(s) > λ 1 for all s small enough. Since (3.15), and hypothesis (H) 0 on f , note that a(x)f (su) s p-1 u p-1 → L 1 a(x) as s → 0. In fact, as w(s) → 0 uniformly as s → 0, hypothesis (H) 0 yields a(x)f sϕ 1 + s w(s)

s p-1 ϕ 1 + w(s) p-1 -→ L 1 a(x) uniformly in Ω as s → 0.
Hence, multiplying and dividing by ϕ 1 + w(s) p-1 , we deduce

1 s p-1 Ω a(x)f u(s) ϕ 1 → s→0 L 1 Ω a(x)ϕ p 1 .
Now we prove that λ(s) > λ 1 arguing by contradiction. Assume that there is a sequence (λ n , u n ) = λ(s n ), u(s n ) of bifurcated solutions to (1.1) in U , with λ(s n ) ≤ λ 1 . Multiplying (1.1) λn by ϕ 1 and integrating by parts

0 ≤ λ 1 -λ(s n ) s p-1 n Ω u(s n )ϕ 1 = 1 s p-1 n Ω a(x)f u(s n ) ϕ 1 → L 1 Ω a(x)ϕ p 1 < 0
which yields a contradiction, and consequently, Λ > λ 1 . Finally, Rabinowitz's global bifurcation Theorem [START_REF] Rabinowitz | Some global results for nonlinear eigenvalue problems[END_REF] states that in fact, the set C of positive solutions to (1.1) emanating from (λ 1 , 0) is a continuum (a closed and connected set) which is either unbounded, or contains another bifurcation point, or contains a pair of points (λ, u), (λ, -u) whith u = 0. Since (1.8), any non-negative non-trivial solution is strictly positive, moreover (λ 1 , 0) is the only bifurcation point to positive solutions, so C can not reach another bifurcation point. Since (1.3), neither C contains a pair of points (λ, u), (λ, -u) whith u = 0, which state that C is unbounded, ending the proof.

Proof of Theorem 1.1

First we prove an auxiliary result.

Proposition 5.1. For each λ ∈ (λ 1 , Λ), the following holds: (i) Problem (1.1) λ admits a positive solution

u λ = inf u(x) : u > 0 solving (1.1) λ ,
in other words u λ is minimal. (ii) Moreover, the map λ → u λ is strictly monotone increasing, that is, if λ < µ < Λ, then u λ (x) < u µ (x) for all x ∈ Ω, and ∂u λ ∂ν (x) > ∂uµ ∂ν (x) for all x ∈ ∂Ω. (iii) Furthermore, u λ is a local minimum of the functional J λ .

Proof. (i.a) Step 1. Existence of positive solutions for any λ ∈ (λ 1 , Λ).

Let λ ∈ (λ 1 , Λ) be fixed. By definition of Λ, there exists a λ 0 ∈ (λ, Λ) such that the problem (1.1) λ 0 admits a positive solution u 0 . It is easy to verify that u 0 > 0 is a supersolution to (1.1) λ . Indeed, for any

ψ ∈ H 1 0 (Ω) with ψ ≥ 0 in Ω Ω ∇u 0 ∇ψ dx-λ Ω u 0 ψ dx- Ω a(x)f (u 0 )ψ dx = (λ 0 -λ) Ω u 0 ψ dx ≥ 0.
Moreover, for every δ > 0 satisfying (5.1)

0 < δ < λ -λ 1 2L 1 a -∞ 1 p-2 1 ϕ 1 ∞
the function u = δϕ 1 is a subsolution for (1.1) λ whenever λ > λ 1 . Let δ > 0 satisfying (5.1) and such that g(s) ≥ 0 for any s ∈ [0, δ ϕ 1 L ∞ (Ω) ]. For any ψ ∈ H 1 0 (Ω), ψ > 0 with in Ω we deduce

δ Ω ∇ϕ 1 ∇ψ dx -λδ Ω ϕ 1 ψ dx - Ω a(x)f (δϕ 1 )ψ dx = -(λ -λ 1 )δ Ω ϕ 1 ψ dx - Ω a(x)f (δϕ 1 )ψ dx = -(λ -λ 1 )δ Ω ϕ 1 ψ dx - Ω a(x) (δϕ 1 ) 2 * -1 [ln(e + δϕ 1 )] α + g(δϕ 1 ) ψ dx ≤ -(λ -λ 1 )δ Ω ϕ 1 ψ dx + a - ∞ Ω h(δϕ 1 ) + g(δϕ 1 ) ψ dx < 0.
This allow us to take u = δϕ 1 as a subsolution for (1.1) λ with u < u 0 . The sub-and supersolution method now guarantees a positive solution u to (1.1) λ , with u ≤ u ≤ u 0 .

(i.b) Step 2. Existence of a minimal positive solution u λ for any λ ∈ (λ 1 , Λ).

To show that there is in fact a minimal solution, for each x ∈ Ω we define u λ (x) := inf u(x) : u > 0 solving (1.1) λ .

Firstly, we claim that u λ ≥ 0, u λ ≡ 0. Assume by contradiction that u λ ≡ 0. This would yield a sequence u n of positive solutions to (1.1) λ such that ||u n || C(Ω) → 0 as n → ∞, or in other words, (λ, 0) is a bifurcation point from the trivial solution set to positive solutions. Set

v n := un ||un|| C(Ω)
. Observe that v n is a weak solution to the problem

(5.2) -∆v n = λv n + a(x)f (u n )/||u n || C(Ω) in Ω ; v n = 0 on ∂Ω .
It follows from (H) 0 that a(x)f (un)

||un|| C(Ω)
→ 0 in C(Ω) as n → ∞. Therefore, the right-hand side of (5.2) is bounded in C(Ω). Hence, by the elliptic regularity, v n ∈ W 2,r (Ω) for any r > 1, in particular for r > N . Then, the Sobolev embedding theorem implies that

||v n || C 1,α (Ω) is bounded by a constant C that is independent of n. Then, the compact embedding of C 1,µ (Ω) into C 1,β (Ω) for 0 < β < µ yields, up to a subsequence, v n → Φ ≥ 0 in C 1,β (Ω). Since ||v n || C(Ω) = 1, we have that ||Φ|| C(Ω) = 1. Hence, Φ ≥ 0, Φ ≡ 0.
Using the weak formulation of equation (5.2), passing to the limit, and taking into account that λ is fixed and v n → Φ, we obtain that Φ ≥ 0, Φ ≡ 0, is a weak solution to the equation

-∆Φ = λΦ in Ω , Φ = 0 on ∂Ω.
Then, by the maximum principle it follows that Φ = ϕ 1 > 0, the first eigenfunction, and λ = λ 1 is its corresponding eigenvalue, which contradicts that λ > λ 1 .

Secondly, we show that u λ solves (1.1) λ . We argue on the contrary. Observe that the minimum of any two positive solutions to (1.1) λ furnishes a supersolution to (1.1) λ . Assume that there are a finite number of solutions to (1.1) λ , then u λ (x) := min u(x) : u > 0 solves (1.1) λ and u λ is a supersolution. Choosing ε 0 small enough so that ε 0 ϕ 1 < u λ , the sub-supersolution method provides a solution ε 0 ϕ 1 ≤ v ≤ u λ . Since v is a solution and u λ is not, then v u λ , contradicting the definition of u λ , and achieving this part of the proof.

Assume now that there is a sequence u n of positive solutions to (1.1) λ such that, for each x ∈ Ω, inf u n (x) = u λ (x) ≥ 0, u λ ≡ 0. Let u 1 := min{u 1 , u 2 }. Choosing ε 1 small enough so that ε 1 ϕ 1 < u 1 , the subsupersolution method provides a solution ε 1 ϕ 1 ≤ v 1 ≤ u 1 . We reason by induction.

Let u n := min{v n-1 , u n+1 }. Choosing ε n small enough so that ε n ϕ 1 < u n , the sub-supersolution method provides a solution ε n ϕ 1 ≤ v n ≤ u n ≤ v n-1 . With this induction procedure, we build a monotone sequence of solutions v n , such that

(5.3) 0 < v n ≤ u n ≤ v n-1 ≤ u n-1 ≤ • • • ≤ v 1 .
Since monotonicity and Lemma 2.1,

v n C(Ω) ≤ v 1 C(Ω)
, by elliptic regularity, v n C 1,µ (Ω) ≤ C for any µ < 1, and by compact embedding

v n → v in C 1,β (Ω) for any β < α.
Using the weak formulation of equation (1.1) λ , passing to the limit, and taking into account that λ is fixed, we obtain that v is a weak solution to the equation (1.1) λ . Hence

v(x) ≥ u λ > 0. Moreover, since (5.3), v n (x) ↓ v(x) pointwise for x ∈ Ω, so inf v n (x) = v(x)
. Also, and due to (5.3), u n (x) ↓ v(x) pointwise for x ∈ Ω, and inf u n (x) = v(x).

On the other hand, by construction u n ≤ u n+1 , so, for each

x ∈ Ω, v(x) = inf u n (x) ≤ inf u n (x) = u λ (x)
. Therefore, and by definition of u λ , necessarily v = u λ , proving that u λ solves (1.1) λ , and achieving the proof of step 2.

(ii) The monotonicity of the minimal solutions is concluded from a subsupersolution method. Reasoning as in step 1, u µ is a strict supersolution to (1.1) λ , so w := u µ (x) -u λ (x) ≥ 0, w ≡ 0. Moreover, w = 0 on ∂Ω, and we can always choose c 0 := C 0 a ∞ > 0 where C 0 is defined by (1.6), so that a -(x)f (s) + c 0 ≥ 0 and a + (x)f (s) + c 0 ≥ 0 for all s ≥ 0, then

-∆ + a -(x)f θu µ + (1 -θ)u λ + c 0 w = (µ -λ)u µ + λw + a + (x)f θu µ + (1 -θ)u λ + c 0 w > 0 in Ω,
finally, the Maximum Principle implies that w > 0 in Ω, and ∂w ∂ν < 0 on ∂Ω, ending the proof of step 3. Theorem 2] if there exists an ordered pair of L ∞ bounded sub and super-solution u ≤ u to (1.1) λ , and neither u nor u is a solution to (1.1) λ , then there exist a solution u < u < u to (1.1) λ such that u is a local minimum of J λ at H 1 0 (Ω). Reasoning as in (i), u := u µ with µ > λ is a strict super-solution to (1.1) λ , and u := δϕ 1 is a strict sub-solution for δ > 0 small enough, such that u(x) < u(x) for each x ∈ Ω. This achieves the proof.

(iii) Since [4,
Proof of Theorem 1.1. Theorem 3.3 provides the existence of positive solutions for λ < λ 1 , and Proposition 5.1 provide the existence of minimal positive solutions for λ ∈ (λ 1 , Λ).

(a) Step 1. Existence of a second positive solution for λ ∈ (λ 1 , Λ).

Fix an arbitrary λ ∈ (λ 1 , Λ), and let u λ be the minimal solution to (1.1) λ given by Proposition 5.1, minimizing J λ . A second solution follows seeking a solution through variational arguments [14, Theorem 5.10] and the Mountain Pass procedure shown below.

First, reasoning as in Proposition 5.1(iii), we get a local minimum ũλ > 0 of J λ . If ũλ = u λ , then ũλ is the second positive solution, ending the proof. Assume that ũλ = u λ . Now we reason as in [14, Theorem 5.10] on the nature of local minima. Thus, either

(i) there exists ε 0 > 0, such that inf J λ (u) : u -ũλ = ε 0 > J λ (ũ λ ), in other words, ũλ is a strict local minimum, or (ii) for each ε > 0, there exists u ε ∈ H 1 0 (Ω) such that J λ has a local minimum at a point u ε with u ε -ũλ = ε and J λ (u ε ) = J λ (ũ λ ).
Let us assume that (i) holds, since otherwise case (ii) implies the existence of a second solution.

Consider now the functional

I λ : H 1 0 (Ω) → R given by I λ [v] = J λ [u λ + v] -J λ [u λ ], more specifically I λ [v] := 1 2 Ω |∇v| 2 dx - λ 2 Ω (v + ) 2 dx - Ω Gλ (x, v + ) dx.
where

Gλ (x, s) := a(x) F (u λ (x) + s) -F (u λ (x)) -f (u λ (x))s = a(x) 1 2 f (u λ (x))s 2 + o(s 2 )
.

Obviously I λ [v + ] ≤ I λ [v],
and observe that

I λ [v] = 0 ⇐⇒ J λ [u λ + v] = 0. Fix now some element 0 ≤ v 0 ∈ H 1 0 (Ω)∩L ∞ (Ω), v 0 > 0 in Ω + , v 0 ≡ 0 in Ω -. Let v =
tv 0 for a certain t = t 0 > 0 to be selected a posteriori, and evaluate

I λ [tv 0 ] = 1 2 t 2 ∇v 0 2 L 2 (Ω) -λ v 0 2 L 2 (Ω) - Ω Gλ (x, tv 0 ) dx.
Reasoning as in the proof of Theorem 3.3 for large positive t, since F (t)/t 2 → ∞ as t → ∞, and using also (3.1) we obtain that

I λ [tv 0 ] ≤ C(1 + t + t 2 ) - Ω + a + (x) F (u λ + tv 0 ) + 1 2 C 0 (u λ + tv 0 ) 2 ≤ C(1 + t + t 2 ) -δ Ω + δ F (u λ + tv 0 ) + 1 2 C 0 (u λ + tv 0 ) 2 dx, so I λ [tv 0 ] < 0
for t = t 0 big enough, and where Ω + δ is defined by (3.18). Thus, the Mountain Pass Theorem implies that if

Γ := {g ∈ C [0, 1]; H 1 0 (Ω) : g(0) = 0, I λ [g(1)] < 0}, then, there exists c > 0 such that c := inf g∈Γ max 0≤t≤1 I λ [g(t)]
is a critical value of I λ , and thereby

K c := {v ∈ H 1 0 (Ω) : I λ [v] = c, I λ [v] = 0} is non empty.
Since for any g ∈ Γ we have

I λ [g + (t)] ≤ I λ [g(t)
] for all t ∈ [0, 1], it follows that g + ∈ Γ, and we derive the existence of a sequence v n such that

I λ [v n ] → c, I λ [v n ] → 0, v n ≥ 0.
On the other hand, w n := u λ + v n is a (PS) sequence for the original functional

J λ . Since Theorem 3.2, if λ < λ 1 (int Ω 0 ), v n → v λ en H 1 0 (Ω), so I λ [v] = 0 and I λ [v] = c > 0, hence v λ ≥ 0 is a nontrivial critical point of I λ . Consequently, w λ := u λ + v λ is a positive critical point of J λ , such that, for each ψ ∈ H 1 0 (Ω), we have Ω ∇w λ • ∇ψ dx = Ω λw λ + a(x)f (w λ ) ψ dx,
and thereby w λ := u λ + v λ ≥ u λ , w λ = u λ is a second positive solution to (1.1) λ .

(b) Step 2. Existence of a classical positive solution for λ = Λ.

We prove the existence of a solution for λ = Λ. For each λ ∈ (λ 1 , Λ), problem (1.1) admits a minimal positive weak solution u λ and λ → u λ is increasing, see Proposition 5.1. Taking the monotone pointwise limit, let us define u Λ (x) := lim λ↑Λ u λ (x).

Since step 1, for any λ ∈ (λ 1 , Λ) there exists a second positive solution to (1.1) λ . Let's denote it by ũλ = u λ . Now, define the pointwise limit (5.7) ũ λ 1 (x) := lim sup

λ→λ 1 ũ λ (x).
Reasoning as in step 2, ũ λ 1 < +∞ and ũ

λ 1 ∈ C 2 (Ω) ∩ C 1,µ (Ω) is a classical solution to (1.1) λ 1 .
Moreover, ũ λ 1 > 0. Assume on the contrary that ũ λ 1 = 0. By the Crandall-Rabinowitz's Theorem [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF], the only nontrivial solutions to (1.1) in a neighborhood of the bifurcation point (λ 1 , 0) are given by (4.1)). Since Proposition 5.1, those are the minimal solutions u λ , and due to ũλ = u λ , ũλ are not in a neighbourhood of (λ 1 , 0), contradicting the definition of ũ λ 1 (x), (5.7)

Hence, ũ λ 1 ≥ 0, and reasoning as in (1.8), the Maximum Principle implies that ũ λ 1 > 0.

Appendix A. Some estimates First, we prove an useful estimate of ln(e+s) ln(e+as) . Lemma A.1. Let 0 < a ≤ 1 be fixed. Then for all s ≥ 0, (A. [START_REF] Alama | On semilinear elliptic problems with indefinite nonlinearities Calculus Var[END_REF] ln(e + s) ln(e + as) ≤ ln e a ≤ 1 a .

Proof. Denote (s) = ln(e+s) ln(e+as) for all s ≥ 0. Then 1 ≤ (s) ≤ (s 0 ) where s 0 > 0 is the unique value where (s) = 0. When computing s 0 we find (s 0 ) = 0 ⇐⇒ (e + as 0 ) ln(e + as 0 ) -a(e + s 0 ) ln(e + s 0 ) = 0 and therefore max = (s 0 ) = ln(e + s 0 ) ln(e + as 0 ) = e + as 0 a(e + s 0 ) .

Notice that we have (s 0 ) ≤ 1 a . In order to find a better upper bound of ln( e+as 0 e+s 0 ) let us denote for all s ≥ 0 θ(s) = (e + as) ln(e + as) -a(e + s) ln(e + s).

Then, there exists χ ∈ (0, s 0 ) such that and the first inequality of (A.1) is achieved. The second one is obvious.

0 -e(1 -a) = θ(s 0 ) -θ(0) = θ (χ)s 0 =⇒ e(1 -a) s 0 = -θ (χ).

Then

Next lemma is about the variations of h(s) = s Let us define for s ≥ 0, θ(s) := ln(e + s) -α 2 * -1 s s + e .

We have:

     θ(0) = 1, θ(s) → +∞ as s → +∞, θ (s) = s+e(1-α 2 * -1 ) (e+s) 2
.

Hence:

(1) If α 2 * -1 ≤ 1 then θ (s) ≥ 0 for all s ≥ 0 and in particular θ(s) ≥ 0 and therefore h (s) ≥ 0 for all s ≥ 0;

(2) if α 2 * -1 > 1 then θ (s 0 ) = 0 for s 0 = e α 2 * -1 -1 .
Let us compute θ(s 0 ):

θ(s 0 ) = ln α 2 * -1 - α 2 * -1 + 2,
and hence:

(i) if θ(s 0 ) ≥ 0 then θ(s) ≥ 0 for all s ≥ 0 and therefore h (s) ≥ 0 for all s ≥ 0;

(ii) if θ(s 0 ) < 0 then there exists s 1 < s 2 such that

θ(s) > 0 ∀s ∈ [0, +∞[ \ ]s 1 , s 2 [ =⇒ h (s) > 0 ∀s ∈ [0, +∞[ \ ]s 1 , s 2 [.
Notice that t → ln t is greater that t → t -2 somewhere between some t 1 < 1 and the value t * =the unique solution > 2 of the equation

ln t * = t * -2.
Finally the statement of the lemma holds for α * = t * (2 * -1).

Appendix B. A compact embedding using Orlicz spaces

For references on Orlicz spaces see [START_REF] Krasnoselskiȋ | Rutickiȋ Convex functions and Orlicz Spaces[END_REF][START_REF] Rao | Theory of Orlicz Spaces[END_REF]. Throughout Ω ⊂ R N is an bounded open set. We will denote L(Ω) = {ϕ : Ω → R : ϕ is Lebesgue measurable}. Assume also that M satisfies the ∆ 2 -condition, that is,

(B.1) ∃K > 0, ∀s ∈ [0, +∞[, M (2s) ≤ KM (s). Let {u n } n∈N in H 1 0 (Ω) be a sequence satisfying (1) sup n∈N u n 2 * < ∞, (2
) there exists u ∈ H 1 0 (Ω) such that lim n→+∞ u n (x) = u(x) a.e. Then there exists a subsequence {u n k } k∈N such that

(B.2) lim k→∞ Ω M |u n k (x) -u(x)| dx = 0.
In order to proof this theorem we need some definitions. Proof. Let fix ε > 0 and let δ > 0 be such that

∀n ∈ N, ∀A ⊂ Ω mesurable , |A| < δ =⇒ A M (|u n |)dx ≤ ε.
Using Fatou's lemma we infer that also

∀A ⊂ Ω mesurable , |A| < δ =⇒ A M (|u|)dx ≤ ε. Let Ω n = {x ∈ Ω : |u n (x) -u(x)| > M -1 (ε)}.
As a consequence of Egoroff's theorem, the sequence (u n ) n∈N converge in measure to u so there exists n 0 ∈ N such that

|Ω n | < δ.
Then, using the convexity of M and (B.1) it comes

Ω M |u n -u| dx = Ωn M |u n -u| dx + Ω\Ωn M |u n -u| dx ≤ 1 2 Ωn (M 2|u n | + M 2|u| dx + |Ω| M M -1 (ε) ≤ K 2 Ωn M (|u n |) + M (|u|) dx + |Ω|ε ≤ (K + |Ω|)ε.
In order to prove that, for the sequence of our theorem, the set Proof. For the Valle Poussin's theorem see [START_REF] Natanson | Theory of functions of a real variable[END_REF] page 159. To prove the second statement remark that the function Φ = Φ•M -1 satisfies (B.3).

Here M -1 stand for the right-hand inverse.

Proof of theorem B.3. Let us take Φ(s) = |s| 2 * . From hypothesis (1) of the theorem, the set K = {u n : n ∈ N} satisfies (B.4) for some D > 0.

Then the conclusion follows from lemma B.5 and Lemma B.6 .

Remark B.7. Whenever (B.2) is satisfied we say that the sequence {u n k } k∈N converges in M -mean to u.

One can formulate Theorem B.3 as a compact embedding of H 1 0 (Ω) in some vector space endowed of the Luxembourg norm associate to M (see [START_REF] Krasnoselskiȋ | Rutickiȋ Convex functions and Orlicz Spaces[END_REF][START_REF] Rao | Theory of Orlicz Spaces[END_REF]). Instead, we are going to use the Orlicz-norm which is more suitable to our purposes. We will will see later in Theorem B.12 that the convergence in M -mean implies the convergence with respect to the Orlicz-norm, provided that the ∆ 2 -condition is satisfied. • M is a norm in the real vector space L M (Ω) = u ∈ L(Ω) : u M < +∞ .

(see [START_REF] Krasnoselskiȋ | Rutickiȋ Convex functions and Orlicz Spaces[END_REF] for the details). Let us prove the following less trivial properties: (ii) The divide the proof in 3 steps.

Step 1: For all v ∈ L(Ω),

Ω uv dx ≤ u M if ρ(v, M * ) ≤ 1 ρ(v, M * ) u M if ρ(v, M * ) > 1

  Furthermore, (a) For every, λ ∈ λ 1 , Λ), (1.1) admits at least two classical ordered positive solutions. (b) For λ = Λ, problem (1.1) admits at least one classical positive solution. (c) For every λ ≤ λ 1 , problem (1.1) admits at least one classical positive solution.

  which implies that there exists K > 0 such that m(2s) ≤ Km(s) for all s ≥ 0 and consequently M satisfies the ∆ 2 -condition (B.1).

  Finally, by inequality (B.5) of Proposition B.12 we get sup m(|u n |) M * , n ∈ N ≤ C + 1.

3. 2 .

 2 An existence result for λ < λ 1 . The next Theorem provides a solution to (1.1) for λ < λ 1 based on the Mountain Pass Theorem. Theorem 3.3. Assume that Ω ⊂ R N is a bounded domain with C 2 boundary. Assume that the nonlinearity f defined by (1.2) satisfies (H), ant that the weight a ∈ C 1 (Ω). Then, the boundary value problem (1.1) λ has at least one classical positive solution for any λ < λ 1 .

Proposition 4 . 1 .

 41 Let us define Λ := sup{λ > 0 : (1.1) admits a positive solution}. If (1.5) holds then,

  -θ (s) = a ln e + s e + as ≤ a ln 1 a) + 1 -a a ln(1/a) + 1 -a ≤ ln(1/a) + 1,

Definition B. 1 .

 1 We will say that a function M : [0, +∞[→ [0, +∞[ is a N -function if and only if (N1) M is convex, increasing and continuous, The proof of the following property is trivial, we just quoted it for the sake of completeness. Proposition B.2. Any N -function M admits a representation of the form M (s) = s 0 m(t)dt where m : [0, +∞[→ [0, +∞[ is a non-decreasing right-continuous function satisfying m(0) = 0 and lim s→+∞ m(s) = +∞. Thus, m is the right-derivative of M .Our first aim is to prove the following result: Theorem B.3. Let M : [0, +∞[→ R be a N -function such that lim s→+∞

Definition B. 4 .

 4 Let K ⊂ L(Ω). We say that K has equi-absolutely continuous integrals if and only if ∀ε > 0 there exists h > 0 such that∀ϕ ∈ K, ∀A ⊂ Ω mesurable , |A| < h =⇒ A |ϕ(x)| dx < ε.Lemma B.5. Let M : [0, +∞[→ R be a N -function satisfying the ∆ 2 condition (B.1). Let {u n } n∈N be a sequence of measurable functions converging a.e. to some function u and such that the set M |u n | : n ∈ N has equi-absolutely continuous integrals. Then (B.2) holds.

MΦthen the family K 1 =

 1 |u n | : n ∈ N has equi-absolutely continuous integrals we are going to use the following lemma : Lemma B.6. Let K ⊂ L(Ω) and let Φ : [0, +∞[→ [0, +∞[ |u| dx ≤ D. Then all the functions u ∈ K are integrable and K has equi-absolutely continuous integrals (Valle Poussin's theorem). Moreover, if M : [0, +∞[→ [0, +∞[ is a continuous increasing function satisfying {M |u| : u ∈ K} has equi-absolutely continuous integrals.

Definition B. 8 .

 8 Let M be a N -function. The complementary of M defined for all s ≥ 0 is the functionM * (s) := max st -M (t) : t ≥ 0 .As before, we give the following trivial result for the sake of completeness:Proposition B.9. If m is the right derivative of M then m * (s) = sup{t : m(t) ≤ s}is the right derivative of M * and M * is a N -function. Furthermore, for all s ≥ 0 we have sm(s) = M (s) + M * (m(s)), sm * (s) = M (m * (s)) + M * (s). Next, let us introduce the Orlicz norm associated to M : Definition B.10. Let M be a N -function and let M * be its complementary. Let us denote for any v ∈ L(Ω) ρ(v, M * ) = Ω M * |v| dx and define the Orlicz norm of any u ∈ L(Ω) by u M := sup Ω uv dx : v ∈ L(Ω), ρ(v, M * ) ≤ 1 .

Proposition B. 11 .

 11 (i) For all u ∈ L(Ω), (B.5) u M ≤ Ω M (|u|) dx + 1.

(

  ii) For any u and v in L(Ω) it holds (B.6) Ω uv dx ≤ u M v M * (Holder's inequality). (iii) For any u and v in L(Ω) we have u M ≤ v M if |u| ≤ |v| a.e. Proof. (i) This follows from the definition of • M and the inequality |uv| ≤ M (|u|) + M * |v| .

Step 2 : 3 :

 23 Indeed, the first case follows directly from the definition. If ρ(v, M * ) > 1 then by convexityM * |v| ρ(v, M * ) ≤ M * |v| ρ(v, M * )and thereforeρ |v| ρ(v, M * ) , M * ≤ 1 ρ(v, M * ) Ω M * |v| dx = 1andΩ u v ρ(v, M * ) dx ≤ u M . If u M ≤ 1 then ρ m(|u| , M * ) ≤ 1. Set u n = uχ {|u|≤n} for all n ∈ N. Since u n is bounded then ρ (m(|u n | , M * < +∞. Assume by contradiction that Ω M * m(|u| dx > 1 and let n 0 ∈ N be such that Ω M * m |u n 0 | dx > 1. We have M * m(|u n 0 | < M |u n 0 | + M * m(|u n 0 |)| = |u n 0 | m |u n 0 | and therefore, by (i), ρ m(|u n 0 |), M * < Ω |u n 0 | m |u n 0 | dx ≤ u n 0 M ρ m(|u n 0 |), M * which contradicts u n 0 M ≤ u M ≤ 1.This is trivial from the definition of u M , step 1 and the fact that |u|m(|u|) = M (|u|) + M * m(|u| .Step If u M ≤ 1 then ρ(u, M ) ≤ u M . Let us remark that for all s ≥ 0 M * (m(s)) + M (s) = sm(s).

  2 * -1 [ln(e+s)] α for s ≥ 0. Lemma A.2. There exists α * > 2(2 * -1) such that h is an increasing function on ]0, +∞[ if and only if α ≤ α * . Moreover, if α > α * there exists s 1 < s 2 such that h is increasing in [0, +∞[ \ ]s 1 , s 2 [.

	Proof. We have
	h (s) =	s 2 * -2 [ln(e + s)] α+1 (2 * -1) ln(e + s) -	αs s + e	,
	so	h (s) ≥ 0 ⇐⇒ ln(e + s) ≥	α 2 * -1	s s + e	.

Acknowledgments

We would like to thank Professors Xavier Cabré, Carlos Mora and Guido Sweers for helpful discussion and references about Orlicz-Sobolev spaces.

This work was started during Pardo's visit to the LMPA, Université du Littoral Côte d'Opale ULCO, whose invitation and hospitality she thanks.

We next see that u Λ < +∞, reasoning on the contrary. Assume that there exists a sequence of solutions u n := u λn such that u λn → +∞ as λ n → Λ. Set v n := u n / u n , then there exists a subsequence, again denoted by v n such that v n v in H 1 0 (Ω), and v n → v in L p (Ω) for any p < 2 * and a.e. Arguing as in the claim of Proposition 3.1, v ≡ 0. Moreover (5.4)

On the other hand, from the weak formulation, for all

)

Taking the limit, and since

Hence [2, Proposition 3.13 (iv)], and (5.6) imply

which contradicts (5.4) and yields u Λ < +∞.

By Sobolev embedding and the Lebesgue dominated convergence theorem,

Now, by substituting ψ = u n in (5.5), using Hölder inequality and Sobolev embeddings we obtain

By compactness, for a subsequence again denoted by u n , u n u * in H 1 0 (Ω), u n → u * in L p (Ω) for any p < 2 * and a.e. By uniqueness of the limit, u Λ = u * . Finally, by taking limits in the weak formulation of u n as λ n → Λ, we get

The existence of a classical positive solution for λ < λ 1 is done in Theorem 3.3. Let's look for a solution when λ = λ 1 .

Set v 0 = m(|u|). From step 2, ρ(v 0 , M * ) ≤ 1 and then

Now we prove Holder's inequality. From step 2 applied to M * and

and Holder's inequality follows.

The proof of (iii) is trivial.

Finally, we give the following compact embedding result:


Abstract. As pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sucient. In this note we x the problem and its minor consequences on other results in the same paper.

Corrected statement of Theorem 2.2, and related results

First recall some notations used in [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]2].

Denition 1. Let P be a set of positive integers, the natural density dens(P) of P is given by

dens(P) = lim X→∞ 1 X k≤X 1 P (k)
if the limit exists, where 1 P is the indicator function of the set P.

For γ 1 , . . . , γ N ∈ R, we denote by γ 1 , . . . , γ N Q the Q-vector space spanned by these elements.

The following theorem corrects the awed hypothesis in [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Th. 2.2].

Theorem 1. Let N ≥ 1 and let γ 1 , . . . , γ N ∈ (0, π) be distinct real numbers

such that π / ∈ γ 1 , . . . , γ N Q . Let D ≥ 1 and x c 1 , . . . , c N ∈ C D . Let F = (F 1 , F 2 , . . . , F D ) : N → R D be the function dened by F (k) = N n=1 c n e ikγn + c n e -ikγn = N n=1 (a n cos(γ n k) + b n sin(γ n k)) .
The image of F is contained in a compact subset of the subspace

V F of R D generated by the vectors a 1 , . . . , a N , b 1 , . . . , b N .
Then, for every subspace H ⊂ R D not containing V F and every vector α ∈ R D , one has dens(F ∈ α + H) = 0.

In particular, if

V F ⊂ D d=1 {x ∈ R D : x d = 0}
, then, for every α 1 , . . . , α D ∈ R, one has dens(F d = α d ) = 0 for all 1 ≤ d ≤ D, and the density

dens(F 1 > α 1 , F 2 > α 2 , . . . , F D > α D )
exists.

Date: September 18, 2020. (1) Note that the dierence with [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Th. 2.2] lies in the hypothesis on the real numbers γ 1 , . . . , γ N : we need that π / ∈ γ 1 , . . . , γ N Q . In particular, the hypothesis is stronger than the one in [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Th. 2.2], but still weaker than the full Linear Independence.

(2) In the case π ∈ γ 1 , . . . , γ N Q , Bailleul observed in [START_REF] Bailleul | Explicit Kronecker-Weyl theorems and applications to prime number races[END_REF]Th. 1.5] that the subtorus generated by γ 1 , . . . , γ N over Z may not be connected, this is the reason of the gap in the proof of [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Th. 2.2]. This is also the cause of the dierence between the continuous case ( [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Th. 1.2]) and the discrete case. Indeed, the subtorus {(yγ 1 , . . . , yγ N ) : y ∈ R}/(2πZ) N is connected unconditionally : it is the continuous image of the connected set R. This last remark corrects [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Rem. 2.3.(i) and Rem. 3.7], note also that one should read R/2πZ in [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Rem. 2.3.(i)] instead of Z/2πZ.

The main application of this result is the analogue of Chebyshev's bias in rings of polynomials with coecients in nite elds. Let us recall some denitions.

Denition 2. Let p α be a prime power, and Q ∈ F p α [t]. Let a 0 , a 1 , . . . , a D mod Q be distinct invertible congruence classes, and let π(k; Q, a i ) denote the number of irreducible polynomials in F p α [t] with degree at most k that are congruent to a i mod Q.

If, for each permutation σ, the set

P Q;a σ(0) ,a σ(1) ,...,a σ(D) := {k ∈ N : π(k; Q, a σ(0) ) > π(k; Q, a σ(1) ) > . . . > π(k; Q, a σ(D)
)} admits a natural density, we say that the irreducible polynomial race is weakly inclusive. Moreover, if every set of the form {k ∈ N : π(k; Q, a i ) = π(k; Q, a j )}, i = j, has natural density equal to zero, we say that the ties have density zero.

The hypothesis in [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Cor. 2.5] has to be changed accordingly.

Corollary 2. Let p α be a prime power, and Q ∈ F p α [t]. Suppose that there exists M ≥ 1 and γ 1 , . . . , γ M ∈ (0, π) such that π / ∈ γ 1 , . . . , γ M Q , and such that for each character χ mod Q, there exists

1 ≤ m ≤ M with L( 1 2 + iγ m , χ) = 0 but L( 1 2 + iγ m , χ ) = 0 for χ = χ.
Then, every irreducible polynomial race in F p α [t] modulo Q is weakly inclusive and the ties have density zero.

Finally, the hypothesis in Theorem 1 is now too strong to deduce [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Cor. 2.6], we should thus consider this statement as not proved. Indeed, one needs to take into account all the zeros of the Dirichlet L-functions of the quadratic characters modulo f (T )(T -u), but we only have information on the zeros of the Dirichlet L-functions of the primitive quadratic character.

Proof of the corrected statement

The key lemma is the following. Lemma 3. Let N ≥ 1 and γ 1 , . . . , γ N ∈ (0, π) be distinct real numbers such that

π / ∈ γ 1 , . . . , γ N Q . Then the sub-torus A = {(kγ 1 , . . . , kγ N ) : k ∈ Z}/(2πZ) N is connected.
Proof. We will show that A = {(yγ 1 , . . . , yγ N ) : y ∈ R}/(2πZ) N , then the conclu- sion follows from the fact that this sub-torus is connected. The rst inclusion (⊂) is immediate.

Let (e 1 , . . . , e d ) be a basis of γ 1 , . . . , γ N Q , such that for all 1 ≤ i ≤ N , one has γ i = d k=1 g i,k e k with g i,k ∈ Z. By hypothesis, the set {2π, e 1 , . . . , e d } is linearly independent over Q, thus, by the discrete version of the KroneckerWeyl Equidistribution Theorem (see e.g. [START_REF] Bailleul | Explicit Kronecker-Weyl theorems and applications to prime number races[END_REF]Th. 1.2]), one has {(ke 1 , . . . , ke d ) : k ∈ Z} = (R/2πZ) d . In particular, for every y ∈ R and > 0, there exists ∈ Z and m

1 , . . . , m d ∈ Z such that max 1≤k≤d | e k -ye k -m k 2π| < / max 1≤i≤N d k=1 |g i,k |.
Thus, for all 1 ≤ i ≤ N , we have

| γ i -yγ i - d k=1 g i,k m k 2π| ≤ d k=1 |g i,k | • | e k -ye k -m k 2π| < .
Using the fact that g i,k ∈ Z, this shows that y(γ 1 , . . . , γ N ) ∈ A, for all y ∈ R, which concludes the proof.

We can now give the proof of Theorem 1.

Proof of Theorem 1. We follow the proof from [De], the only aw is in the proof of [START_REF] Devin | Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes[END_REF]Lem. 3.5].

Let S be the unit sphere of R D and, let f : (V ∩ S) × A be the function dened by

f (u, θ) := N n=1 2 Re( u, c n e iθn ) (1) = N n=1 u, a n cos θ n + u, b n sin θ n ,
this function is analytic in the variable θ.

Since the vectors a 1 , . . . , a N , b 1 , . . . , b N span V , there exists at least one of them, say v j , such that u, v j = 0. So, the function f (u, •) is a linear combination of the 2N characters of A dened by (θ 1 , . . . , θ N ) → e iθn , (θ 1 , . . . , θ N ) → e -iθn , 1 ≤ n ≤ N , with at least one non-zero coecient. Note that these 2N characters are all distinct, and also distinct from the trivial character (θ 1 , . . . , θ N ) → 1. This follows from the fact that the values of the derivative of these functions restricted to {(yγ 1 , . . . , yγ N ) : y ∈ R}/(2πZ) N ⊂ A at y = 0 are respectively γ n , -γ n , 1 ≤ n ≤ N , and 0 which are distinct. Thus by a result of DedekindArtin ( [START_REF] Lang | Algebra[END_REF]VI,Th. 4.1]), those characters are linearly independent and the function f (u, •) is not constant on A. Moreover, Lemma 3 implies that A is connected, and the fact that N > 0 and γ 1 = 0 ensures that it is not restricted to {0}, so that a non-constant analytic function on A is indeed non-locally constant.

The rest of the proof follows similarly to [De].

Acknowledgments

I thank Alexandre Bailleul for his careful reading of [De] and for helpful discussions that led to this correction, as well as the referee for their useful remarks that improved the text.