Guillermo Moncecchi

Jean-Luc Minel

Dina Wonsever

The Influence of Syntactic Information on Hedge Scope Detection

In this paper we elaborate over the use of sequential supervised learning methods on the task of hedge cue scope detection. We address the task using a learning methodology that proposes the use of an iterative, error-based approach to improve classification performance. We analyze how the incorporation of syntactic constituent information to the learning and post-processing steps produces a performance improvement of almost twelve points in terms of F-score over previously unseen data.

Introduction

The use of speculative language poses an interesting problem for Natural Language Processing (NLP), since it potentially reflects the subjective position of the writer towards the truth value of certain facts; when the fact is extracted or used for inference, this certainty information should not be lost. Hedging, a term first introduced by [START_REF] Lakoff | Hedges: A study in meaning criteria and the logic of fuzzy concepts[END_REF] to describe the use of 'words whose job is to make things fuzzier or less fuzzy' is 'the expression of tentativeness and possibility in language use' [START_REF] Hyland | The author in the text: Hedging scientific writing[END_REF], and is extensively used in scientific writing. Several authors have studied the phenomenon of speculation from philosophical, logical and linguistic points of view, generating a rich and heterogeneous theory about speculative language. All this body of work should aid in the construction of an NLP system for speculative sentence detection.

The most common approach in the NLP literature (see, for example, [START_REF] Farkas | The CoNLL-2010 shared task: Learning to detect hedges and their scope in natural language text[END_REF]) to speculative language detection within scientific writing is through hedge detection: the presence of lexical tentativeness or possibility marks ('hedge cues') within a sentence is considered an indicator of speculation. The linguistic devices used to express hedging are diverse: besides modal verbs, epistemic lexical verbs (such as 'suggest' or 'indicate that'), adjectives ('likely'), adverbs ('probably'), and even nouns ('possibility') are frequently used to express uncertainty. For example, in the following sentence, the modal verb 'may' is epistemically used to avoid bald assertion:

Example 1. Since clinical remission has been observed in a significant fraction of DLCL cases, these markers may serve as critical tools for sensitive monitoring of minimal residual disease and early diagnosis of relapse.

The sentence expresses speculation, and the hedge 'may' serves to show the possibility of a certain procedure. But there is another fact within the sentence (that clinical remission have been observed in many cases) which is not hedged, so marking the whole sentence as speculative could lead to the wrong assumption that this fact is merely a possibility. The notion of hedge scope [START_REF] Morante | Learning the scope of hedge cues in biomedical texts[END_REF][START_REF] Vincze | The bioscope corpus: biomedical texts annotated for uncertainty, negation and their scopes[END_REF] captures the idea that it is possible that only a fragment of a sentence is hedged. In the previous example, we could consider that the scope of the 'may' hedge cue could be the verb phrase that starts with the very word3 : Example 2. Since clinical remission has been observed in a significant fraction of DLCL cases, these markers {may serve as critical tools for sensitive monitoring of minimal residual disease and early diagnosis of relapse}.

Hedges, when lexically marked, can be considered linguistic operators, therefore inducing a scope. However, the very notion of hedge scope is introduced for the Bioscope corpus annotation [START_REF] Vincze | The bioscope corpus: biomedical texts annotated for uncertainty, negation and their scopes[END_REF], and is never formally defined, presenting instead a series of criteria to identify them, based mainly on syntax. We have previously suggested [START_REF] Moncecchi | Recognizing Speculative Language in Research Texts[END_REF] defining hedge scope as the part of meaning in the sentences that is hedged, i.e. where the tentativeness or possibility holds. From this perspective, scope is a semantic notion, that can be, however, strongly related with syntax.

In this paper we show how we applied supervised sequential learning methods to the task of scope detection. In particular, we want to analyze how the incorporation of different linguistic information (including lexical and syntactic information) to the learning and post-processing steps allowed to improve prediction performance4 . To achieve this improvement we propose an iterative methodology, where errors committed on a held out corpus are inspected and general rules (mostly based on expert knowledge on syntax) are proposed to solve them. These rules are incorporated as features for the learning process.

In the next section we review previous work on hedge cue identification and scope detection, with special emphasis on this second task. Section 3 presents the learning methodology. In section 4 we analyze performance improvement after incorporating diverse learning attributes, mainly from syntactic scope information, and discuss when these improvements hold on previously unseen data. Finally, we analyze the presented approach, suggesting future lines of research.

Related Work

The first approaches to speculation detection aimed to classify each sentence as speculative or not speculative. [START_REF] Medlock | Weakly supervised learning for hedge classification in scientific literature[END_REF], for example, pro-posed to use a semi-supervised learning approach to solve this binary classification task. [START_REF] Morante | Learning the scope of hedge cues in biomedical texts[END_REF], adapting previous work on negation detection, reduced speculation detection to hedge cue identification: a sentence would be considered speculative if it included one or more hedge cues.

For the scope detection task, [START_REF] Morante | Learning the scope of hedge cues in biomedical texts[END_REF] proposed to address it as a case of sequential classification: given a sentence and a hedge cue within it, classify each sentence token, indicating if it is part of the hedge scope, using a so-called FOL marking: assign class F to the first token of the predicted scope, and L the last one, marking every other token with class O. The classifier input included both the sentence and the hedge cue since a sentence could include two or more cues. For this task, only those sentences where a hedge cue had been found were used as learning instances.

Several works have been presented using this same approach, specially during the CoNLL-2010 Shared Task [START_REF] Farkas | The CoNLL-2010 shared task: Learning to detect hedges and their scope in natural language text[END_REF], ranging from pure rule-based studies, based on sentence syntactic structure (Ozgur et al., 2009;Ovrelid et al., 2010;[START_REF] Velldal | Resolving speculation: Maxent cue classification and dependency-based scope rules[END_REF], to machine learning and hybrid systems [START_REF] Morante | Memory-based resolution of in-sentence scopes of hedge cues[END_REF], incorporating lexical and syntactical information. However, the task of scope resolution is still far from resolved: the best reported result for the CoNLL evaluation corpus is a F-measure of 0.696 [START_REF] Velldal | Speculation and negation: Rules, rankers, and the role of syntax[END_REF], using gold-standard hedge cues.

Methodology

To address the sequential learning task of scope detection, we followed a learning methodology that starts with an initial guess of attributes for supervised learning and a learning method, and builds a classifier; this classifier is in turn evaluated on a held out corpus, and errors are handled to an expert, who proposed rules for solving, based on present or proposed lexical, positional or syntactic attributes. In this section we briefly explain the process. For a detailed explanation see [START_REF] Moncecchi | Enriching the bioscope corpus with lexical and syntactic information[END_REF][START_REF] Moncecchi | Improving speculative language detection using linguistic knowledge[END_REF][START_REF] Moncecchi | Recognizing Speculative Language in Research Texts[END_REF].

We used the abstracts section of the Bioscope corpus [START_REF] Vincze | The bioscope corpus: biomedical texts annotated for uncertainty, negation and their scopes[END_REF] as a learning source and for evaluation purposes. The Bioscope corpus is a freely available corpus of medical free texts, biological full papers and biological abstracts, annotated at a token level with negative and speculative keywords, and at sentence level with their linguistic scope. We first divided the corpus into a 80% training corpus and a 20% evaluation corpus; since we were using an iterative methodology, we further separated a 10% held out corpus from the training set, and used it for tuning.

Following previous studies, we addressed scope detection as a sequential learning task, using FOL classes for each token. Table 1 shows a scope learning instance for the following sentence:

Example 3. This finding {suggests that the BZLF1 promoter may be regulated by the degree of squamous differentiation}.

The input format is the standard learning format used in the CoNLL Shared Tasks [START_REF] Buchholz | Conll-x shared task on multilingual dependency parsing[END_REF]. For each hedge cue in a sentence, a learning instance is generated. In the previous example, the hedge cue is the word 'suggests' (you can see it marked with a B value for the 'Hedge cue' attribute in Tables 123), while its scope is the text span between tokens 3 and 16. There is another, different, learning instance coming from the same sentence, originated by the hedge cue 'may' spanning from tokens 5 to 16 (according to the Bioscope corpus annotation guidelines). Each learning instance could include an arbitrary number of additional attributes, resulting, for example, from lexical or syntactical analysis.

To evaluate classifier performance, we took a perfect-match approach: we considered an evaluation instance as correctly identified only if every token in the sentence were correctly classified. This means that, for the case of scope detection, the scope was considered correctly detected if both the first and last token of the scope were correctly marked. Classification performance was measured in terms of the traditional figures of precision, recall, and F-score. For the scope detection task, these three numbers coincided, since every False Positive (instances with incorrectly classified scope), implied a False Negative (instances where the correct scope was not identified).

The sequential learning process acts as expected: from the learning instances, a model is constructed using a learning method. We used linear-chain Conditional Random Fields, a statistical method, since it had produced state-of-theart result in many sequential learning tasks [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting and labeling sequence data[END_REF]. Besides the usual lexical or syntactical attributes related to each word (e.g. POS tags, lemmas, syntax tree parent), we introduced a special type of attributes, which we called knowledge rules. Their name comes from the fact that they try to model a prediction on the target classes, given an instance, product of an expert rule. For example, a rule that states 'the scope of the keyword 'may' is the same as the scope of the parent node of the hedge cue in the syntax tree', would be represented by an attribute (with exactly the same possible values as the target class), whose F and L tokens will be, respectively, the first and last token of the syntax scope of the parent node of 'may'. Note that these knowledge rules are not post-processing rules, but just learning attributes. The learning method will decide if they are relevant for learning, just as any other attribute. We will see some examples later in this article.

In the next section we show how we defined a base classifier and incrementally improved it. Since our procedure is based on analyzing errors, we used a held out corpus to evaluate improvements; classifier measurement on the evaluation corpus was only done after finishing the iterative process. At the end of the section, we will show how each classifier performed on the evaluation corpus, so as to measure if the improvements on the held out corpus hold.

Classification Improvement

Baseline Classifier

What we considered as learning attributes for our baseline classifier were the same lexical attributes (word, lemma, and part-of-speech class5 ) we used for hedge cue identification, adding the hedge cue attribute learned in the previous step; all four attributes were considered in a window of size 5, centered in the current token. Figure 1 shows the learning instance for this classifier, for our previous example. This classifier also includes a set of post-processing rules that are fired when one (or both) scope limits could not be identified: when we transform the task of scope detection into a sequential classification one and use a FOL format, it is possible that not exactly one F and one L class are predicted for an instance. In this case, we know for sure that the correct scope could not be identified, and should try to back off to a simpler guess. The set of rules we used to modify the classifier results on evaluation data to correct those cases were a slightly modified form of the rules presented by [START_REF] Morante | Learning the scope of hedge cues in biomedical texts[END_REF].

This baseline classifier, including the post-processing rules, obtained an Fscore of 0.66 on the held out corpus.

Iteration 1: Adding Syntax Information

We observed that most scopes were associated with the sentence syntactic constituents, particularly those that included the hedge cue (this was reasonable, since the corpus annotation guidelines were mostly based on syntax). For example, the scope of a lexical verb such as 'suggest' matched the parent component of the hedge cue in the parse tree, as the parse tree in Figure 1 shows: To improve classification, we included as a learning attribute a knowledge rule stating that the scope of the hedge cue was the syntactic scope of the parent of the hedge cue (i.e. the parent in the parse tree of its first word), modulo final periods. Or, in terms of attribute values: in hc parent scope = F if the token is the first word of the parent of the HC = L if the token is the last word of the parent of the HC = O otherwise Since this criterion does not hold for every part-of-speech (and not even for every use of the verb, as, for example, passive voice construction or raising verbs cases show), we also included as a learning attribute the part-of-speech of the hedge cue parent. Tables 2 and3 show the classifier attributes for the two hedge cues in sentence 3. We can see that in the case of 'suggests', the in hc parent scope rule matches the cue scope, while that does not happen in the case of the hedge cue 'may'. We expected that the classifier could confirm or discard a correlation between the knowledge rule and the scope, based on training data, as it does with every other attribute.

After a grid search for the best attributes and fine-tuning learning parameters (w.r.t performance on the held-out corpus), we obtained a classifier with an Fmeasure of 0.705 on the held out corpus, improving by four points with respect to the base classifier.

Iteration 2: Adding Ancestors in the Syntax Tree

After the previous iteration, we elaborated a list of the 116 errors the classifier had committed in the held-out corpus and tried to understand why it had been wrong. As we have already said, this method could be applied because attributes for learning corresponded quite directly with a few observable properties related to the linguistic phenomenon we aimed to study. Analyzing these errors, we found that assuming that only parent scopes were related with hedge scopes was not enough. In some cases, such as those of passive voice use, hedge scopes matched the syntactic scope of ancestors in the tree other than the parent. To cope with this, we added the same syntactic information for hedge cue grandparents and great-grandparents in the syntax tree, and their part-of-speech tags, exactly the same way we did in the previous section. We fed the classifier with all of them in the form of knowledge rules, and let it select which scope to use, depending on the remaining attributes. Performance, again, improve clearly, yielding a 0.740 F-score, eight points above the baseline, and four points above the previous iteration.

Iteration 3: Adjusting Ancestor Scopes

Until this iteration we had assumed that syntactic scopes matched hedge scopes, i.e. that we could find an ancestor of the hedge cue in the syntax tree whose scope matched the scope of the hedge cue. Studying classification errors, we found that this assumption was not always true. In this iteration we aimed to see whether this was a general problem, and whether we could adjust constituent scopes to achieve concordance.

We first studied, for every hedge cue in the training corpus, if its hedge scope matched the syntactic scope of one of its ancestors, resulting from the sentence parsing. We found that for about 80% of the hedge cues this was actually true. To improve this matching, we studied the cases of misalignment where they accounted for a greater proportion and greater number of instances. We found two main causes for misalignments: the scope of the hedge cue included several non-nested scopes, not allowing a characterization of the hedge scope in terms of syntactic scopes, or the hedge scope matched just a portion of an ancestor scope, excluding some subconstituents. In both cases, scopes should be modified (adding or eliminating subcomponents) to match hedge scopes. Consider, for example, the following sentence:

Example 4. Our results lend further support to the {hypothesis that inflammatory and immune responses of monocytes/macrophages may be modulated at the molecular level by signals originating from tissue structural cells such as fibroblasts}.

Its syntactic analysis tree (partially depicted in Figure 2), shows that the hedge cue scope (shown underlined) includes the parent noun phrase of the hedge cue (excluding the initial determiner) and the clause to the right of the grandparent constituent. We can observe that this mismatch was probably originated by an incorrect parsing decision. The clause starting with 'that inflammatory. . . ' could be considered within the syntactic scope of the NP clause headed by the noun hedge cue 'hypothesis'. Several other, similar errors appeared in the held out corpus.

Recall that to obtain sentence constituents we used an external parser: therefore, it was not possible (in our working scenario) to correct parsing errors such as the one shown. However, analyzing the relation between hedge scopes and syntactic constituents in the training corpus it was possible for us to derive a series of rules to 'correct' the extracted features and adjust the syntactic scopes based mainly on lexical information. This procedure did not aim to derive a correct parse, but only to produce better features for the task, in certain clearly identified cases. Similar comments could be made to those cases where the syntax constituent was correctly derived, but, due to annotation idiosyncrasies, certain parts should not be included in the hedge scope. After looking at several of these mismatches, we modified our definition of 'constituent scope', incorporating several rules to cope with specific, 'pathological' cases. Once again, performance improved on the held out corpus, but this time only by 1.5 points, giving an F-measure of 0.756.

Iteration 4: Handling Misclassified examples

Another source of errors beside wrong scope selection and mismatches between hedge and syntactic scopes, were the cases where the classifier failed to classify one sentence token as the first element of the scope and one token as the last one. In these cases, we could be sure that this evaluation instances would be misclassified, because they were 'badly-formed', since they did not meet the very definition of scope. After the last iteration, more than half of classification errors corresponded to these cases.

As we have previously mentioned, these cases were handled using postprocessing rules, based mainly in positional and lexical information. Since we had available the sentence syntax structure, we decided to modify these rules and base them on syntax, probably reflecting better the corpus annotation guidelines. We ended using only three post-processing rules:

-If the hedge cue is a conjugated verb (except when in passive voice or in the case of raising verbs such as 'seem'), use the next verb phrase up in the syntax tree that includes it. -If the hedge cue is 'or', 'neither' or 'either', use the first noun phrase that includes it. -In every other case, use the first clause that includes the hedge cue.

After modifying post-processing rules, performance dramatically improved by more than ten percentage points, obtaining a 0.860 F-measure value.

Iteration 5: Post-processing Rules

After we had studied the three types of errors identified in section 4.3 and modified the classifier attributes and post-processing rules, improving classification, we had only 49 errors left. We studied these errors, and identified several patterns where the classifier did not manage to predict the correct scopes. We found that in most cases these errors probably corresponded to situations where, despite having enough attributes, the absence of enough training data prevented the classifier to infer them. To try to solve this problem, we took a rule-based approach: we ignored the classifier predictions and deterministically assigned the scope limits. We had to be careful about being very precise in the determination of the situations these rules fired, the avoid introducing false positives. After applying these rules, and fine tuning parameters, performance improved to its best value on the held out corpus with 0.885 in terms of F-measure.

Results on the Evaluation Corpus

To estimate the performance of our classifier, we rebuilt the classifiers in each one of the methods iteration, this time training on the whole training corpus (also including sentences in the held out corpus). We do not only wish to estimate the prediction ability of our best classifier on future data, but also to evaluate if the application of the methodology actually produced a performance improvement after each iteration, when faced with unseen data. This is of particular importance for our method: since improvement was based on errors on a held out corpus, we risked overfitting our built classifiers to this corpus. It can be observed that improvements in the held out corpus hold for the evaluation corpus, except for iteration #3 and #5. In both cases, overfitting occurred. It seems that specific rules used to adjust scopes or even final results were too tailored to solve errors in the held out corpus, and actually harmful for the overall performance on unseen data. On the other hand, general decisions, such as the addition of new syntax attributes or a general modification of postprocessing rules clearly improved performance. At the end of the day, the best classifier improved performance by more than twelve points in term of F-measure, compared to the baseline classifier.

Final Remarks

In this article we presented a series of experiment on the detection of hedge scopes, and showed how we improved classifier performance incorporating attributes resulting from syntactic analysis. All the results presented suggest that the methodology proposed successfully identified syntax information as an important predictor of hedge scopes, clearly improving baseline results for scope detection.

However, there were still several instances that could not be correctly classified, especially those that did not correspond with the scope of any ancestor of the hedge cue. We found that the classification method used was highly successful in finding the correlation between syntactic scopes and hedge scopes, when that was possible. In almost every case the correct ancestor in the tree was selected by the classifier to be used as the hedge scope, independently of the hedge cue part-of-speech tag. Adjusting the original syntactic scope of the hedge cue ancestor to exclude or add certain spans (such as certain prepositional phrases or clauses at the end or at the beginning of the scope) seemed useful to improve classification, but produced no improvement on the evaluation data. We think that the absence of clear annotation rules for the selected corpus, indicating when they should be included or excluded acted as a source of ambiguity, and we found several corpus instances where the reasons for pruning or not pruning the hedge scope were not clear. The rules for scope pruning that we developed were based purely on data observation, causing an overfit to the training corpus.

Another source of hedge scope detection errors was syntax parsing. Several of the remaining errors on the held out corpus corresponded to cases such as the one presented in section 4.4, where syntax constituents were incorrectly identified by the parser, yielding wrong syntactic scopes and consequently wrong hedge scopes. Improving parsing results would almost surely imply improving hedge scope identification.

In future research, we plan to apply this methodology to similar phenomena involving cues and scope, such as negation, event recognition, factuality or modality identification. Previous work on those topics suggests that they present several similarities and relation with syntax, with respect to the here addressed phenomenon of hedging.

Fig. 1 .

 1 Fig. 1. Parsing tree for sentence 3. The scope of the hedge cue is shown in bold

Fig. 2 .

 2 Fig. 2. Hedge scope for sentence 4

Table 1 .

 1 Base classifier attributes

	Token Word	Lemma	POS Hedge cue	Scope
	1	This	This	DT O	O
	2	finding	finding	NN O	O
	3	suggests	suggest	VBZ B	F
	4	that	that	IN	O	O
	5	the	the	DT O	O
	6	BZLF1	BZLF1	NN O	O
	7	promoter	promoter	NN O	O
	8	may	may	MD O	O
	9	be	be	VB O	O
	10	regulated	regulate	VBN O	O
	11	by	by	IN	O	O
	12	the	the	DT O	O
	13	degree	degree	NN O	O
	14	of	of	IN	O	O
	15	squamous	squamous	JJ	O	O
	16	differentiation differentiation NN O	L
	17	.	.	.	O	O

Table 2 .

 2 Learning Instance after Iteration 1 for sentence 3, hedge cue 'suggests'

	Token Word	Lemma	POS HC	PPOS	in-PScope Scope
	1	This	This	DT O	VP	O	O
	2	finding	finding	NN O	VP	O	O
	3	suggests	suggest	VBZ B	VP	F	F
	4	that	that	IN	O	VP	O	O
	5	the	the	DT O	VP	O	O
	6	BZLF1	BZLF1	NN O	VP	O	O
	7	promoter	promoter	NN O	VP	O	O
	8	may	may	MD O	VP	O	O
	9	be	be	VB O	VP	O	O
	10	regulated	regulate	VBN O	VP	O	O
	11	by	by	IN	O	VP	O	O
	12	the	the	DT O	VP	O	O
	13	degree	degree	NN O	VP	O	O
	14	of	of	IN	O	VP	O	O
	15	squamous	squamous	JJ	O	VP	O	O
	16	differentiation differentiation NN O	VP	L	L
	17	.	.	.	O	VP	O	O
	Token Word	Lemma	POS HC	PPOS	in-PScope Scope
	1	This	This	DT O	VP	O	O
	2	finding	finding	NN O	VP	O	O
	3	suggests	suggest	VBZ O	VP	O	O
	4	that	that	IN	O	VP	O	O
	5	the	the	DT O	VP	O	F
	6	BZLF1	BZLF1	NN O	VP	O	O
	7	promoter	promoter	NN O	VP	O	O
	8	may	may	MD B	VP	F	O
	9	be	be	VB O	VP	O	O
	10	regulated	regulate	VBN O	VP	O	O
	11	by	by	IN	O	VP	O	O
	12	the	the	DT O	VP	O	O
	13	degree	degree	NN O	VP	O	O
	14	of	of	IN	O	VP	O	O
	15	squamous	squamous	JJ	O	VP	O	O
	16	differentiation differentiation NN O	VP	L	L
	17	.	.	.	O	VP	O	O

Table 3 .

 3 Learning Instance after Iteration 1 for sentence 3, hedge cue 'may'

Table 4 .

 4 Table 4 summarizes the results 6 . Scope detection: results on evaluation data. Number in parenthesis show results on the held out corpus.

	Classifier	F-score
	Baseline	0.737 (0.664)
	It. #1 (Adding syntax info)	0.749 (0.705)
	It. #2 (Adding ancestors)	0.802 (0.740)
	It. #3 (Adjusting ancestors scopes)	0.800 (0.756)
	It. #4 (Handling misclassified examples) 0.852 (0.860)
	It. #5 (Post-processing rules)	0.831 (0.885)

In this work, hedge cues will be underlined and their scope marked with brackets, annotated with the name of the hedge cue in case there are multiply nested scopes

Obviously, for scope to be detected we have first to identify sentence hedge cues; in this paper (for space reasons), we assume that this step is completed, and use gold-standard hedge cues, concentrating on the problem of scope detection

We applied the GENIA tagger[START_REF] Tsuruoka | Developing a robust Part-of-Speech tagger for biomedical text[END_REF] to the original corpus to get the lexical information for each word, and the Stanford Parser[START_REF] Klein | Accurate unlexicalized parsing[END_REF] to get each sentence syntax tree

Note that these results are not comparable with state-of-the art numbers, since most of them are calculated on the evaluation corpus of the CoNLL 2010 Shared Task, which, for example, includes full articles. Actually, we evaluated our methods against the CoNLL corpus, obtaining competitive results. See[START_REF] Moncecchi | Recognizing Speculative Language in Research Texts[END_REF] for details.


Franc ¸oise Demengel

Jean-Michel Coron

GENERALIZED EIGENVALUES FOR FULLY NONLINEAR SINGULAR OR DEGENERATE OPERATORS IN THE RADIAL CASE

Keywords: AMS Subject Classifications: 35J25, 35J60, 35P15, 35P30. 1

In this paper we extend some existence results concerning generalized eigenvalues for fully nonlinear operators, singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the p-Laplace operator by Del Pino and Manasevich.

Introduction

The extension of the concept of eigenvalue for fully nonlinear operators has seen a remarkable development in these last years, let us mention the works of Quaas, Sirakov [START_REF] Quaas | On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators[END_REF], Ishii, Yoshimura [START_REF] Ishii | Demi-eigen values for uniformly elliptic Isaacs op erators[END_REF], Juutinen [START_REF] Juutinen | On the principal eigenvalue of a very badly degenerate equation[END_REF], Patrizi [START_REF] Patrizi | The Neumann problem for singular fully nonlinear operators[END_REF], Armstong [START_REF] Armstrong | Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations[END_REF], and previous papers of the author with Isabeau Birindelli [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF][START_REF] Birindelli | Eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF] which all deal with the existence of eigenvalues and corresponding eigenfunctions for different fully-nonlinear operators in bounded domains.

In [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF] we defined the concept of first eigenvalue on the model of [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] and we proved some existence results for Dirichlet problems, and for the eigenvalue problem.

The simplicity of the first eigenvalue which is known in the case of the p-Laplacian, for Pucci's operators, and for operators related but homogeneous of degree 1, remains an open problem for general operators fully nonlinear singular or degenerate homogeneous of degree 1 + α with α > -1. However, in [START_REF] Birindelli | Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case, To appear in ZAA[END_REF] we proved some uniqueness results in the case where the domain is a ball or an annulus and when the operator is radial.

Concerning the "other eigenvalues," little is known about them, except for the Pucci's operators and, for the p-Laplacian, in the radial case.

More precisely in [START_REF] Pino | Global bifurcation from the eigenvalues of the p-Laplacian[END_REF] the authors prove that in the radial case for the p-Laplace operator, there exists an infinite denumerable set of eigenvalues, which are simple and isolated; in [START_REF] Busca | Nonlinear eigenvalues and bifurcation problems for Pucci's operator[END_REF] the authors prove the same result for Pucci's operators. Moreover, in each of these papers, the authors establish some bifurcation results of positive (respectively negative) solutions for some related partial differential equations.

Here we consider also the radial case for the model operator

F (Du, D 2 u) = |∇u| α M a,A (D 2 u)
where a and A are two positive numbers, a ≤ A, α > -1 and M a,A is the Pucci's operator M a,A (M ) = Atr(M + )atr(M -).

We prove the existence of a denumerable set of eigenvalues (µ k ) k which are simple and isolated, and some continuity results for the eigenvalues with respect to the parameters α, a, A.

Assumptions, notation and previous results in the general case

We begin with some generalities about the operators that we consider.

Let Ω be some bounded domain in R N . Let S be the space of symmetric matrices on R N . For α > -1, F α satisfies:

(H1) F α : Ω×R N \{0}×S → R, is continuous and for all t ∈ R ⋆ , µ ≥ 0, for any x ∈ Ω, p ∈ R N \{0}, X ∈ S, F α (x, tp, µX) = |t| α µF α (x, p, X); (H2) there exist 0 ≤ a ≤ A, such that for any x ∈ Ω, p ∈ R N \{0}, M ∈ S, N ∈ S, N ≥ 0 a|p| α tr(N ) ≤ F (x, p, M + N ) -F (x, p, M ) ≤ A|p| α tr(N ); Let us now recall the definition of viscosity solutions.

Definition 2.1. Let Ω be a bounded domain in R N and suppose that f is continuous on Ω × R; then v, continuous in Ω, is called a viscosity super solution (respectively sub-solution) of F (x, ∇u, D 2 u) = f (x, u) if for all x 0 ∈ Ω:

-Either there exists an open ball B(x 0 , δ), δ > 0 in Ω on which v = cte = c and 0 ≤ f (x, c), for all x ∈ B(x 0 , δ) (respectively 0 ≥ f (x, c)).

-Or for all ϕ ∈ C 2 (Ω), such that vϕ has a local minimum on x 0 (respectively a local maximum) and ∇ϕ(x 0 ) = 0, one has F (x 0 , ∇ϕ(x 0 ), D 2 ϕ(x 0 )) ≤ f (x 0 , v(x 0 )) (respectively F (x 0 , ∇ϕ(x 0 ), D 2 ϕ(x 0 )) ≥ f (x 0 , v(x 0 ))).

One can also extend the definition of viscosity solution to upper semicontinuous sub-solutions and lower semicontinuous super solutions, as is done in the paper of Ishii [START_REF] Ishii | Viscosity solutions of non-linear partial differential equations[END_REF].

We shall consider in the sequel radial solutions, which will be solutions of differential equations of order two. These solutions will be C 1 everywhere and C 2 on each point where their gradient is not zero, so it is easy to see that these solutions are viscosity solutions.

We now recall the definition of the first eigenvalue and first eigenfunction adapted to this context, on the model of [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF].

We define

λ + (Ω) = sup{λ : ∃ ϕ > 0, F (x, ∇ϕ, D 2 ϕ) + λϕ 1+α ≤ 0 in Ω}, λ -(Ω) = sup{λ : ∃ ϕ < 0, F (x, ∇ϕ, D 2 ϕ) + λ|ϕ| α ϕ ≥ 0 in Ω}.
Remark 2.2. Let us observe that in this definition, for λ + (respectively λ -), the supremum can be taken over either continuous and bounded functions, or lower semicontinuous and bounded functions (respectively continuous and bounded functions, or upper semicontinuous and bounded).

We proved in [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF] the following existence result of "eigenfunctions."

Theorem 2.3. Suppose that Ω is a bounded regular domain. There exists ϕ ≥ 0 such that

F (x, ∇ϕ, D 2 ϕ) + λ + (Ω)ϕ 1+α = 0 in Ω ϕ = 0 on ∂Ω.
Moreover, ϕ > 0 inside Ω is bounded and continuous. Symmetrically there exists ϕ ≤ 0 such that

F (x, ∇ϕ, D 2 ϕ) + λ -(Ω)|ϕ| α ϕ = 0 in Ω ϕ = 0 on ∂Ω.
Moreover, ϕ < 0 inside Ω is bounded and continuous.

These eigenvalues have the following properties, called maximum and minimum principles.

Theorem 2.4. Suppose that Ω is a bounded regular domain. If λ < λ + , every upper semicontinuous and bounded sub-solution of

F (x, ∇u, D 2 u) + λ|u| α u ≥ 0 which is ≤ 0 on the boundary is ≤ 0 inside Ω. If λ < λ -,
every lower semicontinuous and bounded super-solution of

F (x, ∇u, D 2 u) + λ|u| α u ≤ 0 which is ≥ 0 on the boundary is ≥ 0 inside Ω.
The maximum and minimum principle and an iterative process permits us to prove the existence of solutions for the Dirichlet problem

F (x, ∇u, D 2 u) + λ|u| α u = f in Ω u = 0 on ∂Ω,
where f is supposed to be continuous and bounded, and λ < inf(λ + , λ -). Moreover, if f ≤ 0 and λ < λ + , (respectively f ≥ 0 and λ < λ -), there exists a nonnegative (respectively nonpositive) solution.

We now give an increasing property of the eigenvalues λ ± with respect to the domain.

Proposition 2.5. Suppose that Ω and Ω ′ are some regular bounded domains such that Ω ′ ⊂⊂ Ω. Then λ ± (Ω ′ ) > λ ± (Ω).

For the convenience of the reader we give a short proof here; we do it for λ + . Let ϕ be an eigenfunction for λ + (Ω). Then by the strict maximum principle there exists ǫ > 0 such that ϕ ≥ 2ǫ on Ω ′ . Define

λ ′ = λ + (Ω) inf Ω ′ ϕ 1+α (ϕ-ǫ) 1+α > λ + (Ω). Then the function ϕ -ǫ is some positive function which satisfies in Ω ′ F (x, ∇(ϕ -ǫ), ∇∇(ϕ -ǫ)) + λ ′ (ϕ -ǫ) 1+α ≤ 0 which implies by the definition of λ + (Ω ′ ) that λ + (Ω) < λ ′ ≤ λ + (Ω ′ ).
The following property of eigenvalues will be needed in Section 4.

Proposition 2.6. Suppose that there exists µ ∈ R, and u continuous and bounded such that

F (x, ∇u, D 2 u) + µ|u| α u = 0 , u ≥ 0, u ≡ 0 in Ω u = 0 on ∂Ω.
Then µ = λ + . Symmetrically suppose that there exists µ ∈ R, and u continuous and bounded such that

F (x, ∇u, D 2 u) + µ|u| α u = 0, u ≤ 0, u ≡ 0 in Ω u = 0 on ∂Ω. Then µ = λ -.
Proof of Proposition 2.6. We consider only the first case; the other can be treated in the same manner. By the strict maximum principle, [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF], [START_REF] Birindelli | Strict maximum principle[END_REF], u > 0 in Ω. By the definition of the first eigenvalue, µ ≤ λ + . If µ < λ + , then the minimum principle would imply that u ≤ 0 in Ω, a contradiction.

We now recall some regularity and compactness results which will be used in the last section (see [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF]).

Proposition 2.7. Suppose that Ω is a bounded regular domain. Suppose that F satisfies the previous assumptions. Let f be a continuous and bounded function in Ω. Let u be a continuous and bounded viscosity solution of

F (x, ∇u, D 2 u) = f in Ω u = 0 on ∂Ω. (2.2)
Then for any γ < 1 there exists some constant C, which depends only on |f | ∞ , γ, a, A, and N , such that for any (x, y) ∈ Ω2

|u(x) -u(y)| ≤ C|x -y| γ .
Corollary 2.8. Suppose that Ω is a bounded regular domain. Suppose that F satisfies the previous assumptions. Suppose that (f n ) is a sequence of continuous and uniformly bounded functions, and (u n ) is a sequence of continuous and bounded viscosity solutions of

F (x, ∇u n , D 2 u n ) = f n in Ω u n = 0 on ∂Ω.
Then the sequence (u n ) is relatively compact in C(Ω). Moreover, if f n converges, even simply, to some continuous and bounded function f , and if for a subsequence σ(n), u σ(n) → u, then u is a solution of the equation with the right-hand side f .

(See [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF]).

Remark 2.9. Under some additional assumptions on the regularity of F , one has some Lipschitz regularity of the solutions. This assumption is satisfied in the case of the operator considered in the following sections.

We end this section by giving some properties of the first demi-eigenvalues for some particular operators related to Pucci's operators: Let 0 < a < A and the Pucci's operator be

M a,A (D 2 u) = Atr((D 2 u) + ) -atr((D 2 u) -),
where (D 2 u) ± denote the positive and negative part of the symmetric matrix D 2 u. For α > -1 the operator

F (∇u, D 2 u) = |∇u| α M a,A (D 2 u)
satisfies the assumption (H1), (H2). We denote by λ ± a,A,α its corresponding first eigenvalues. Then we have the following.

Proposition 2.10. If a < A, one has λ + a,A,α (Ω) < λ - a,A,α (Ω). Moreover, if λ eq is the first eigenvalue for the operator |∇u| α ∆u, λ + a,A,α ≤ aλ eq < Aλ eq ≤ λ - a,A,α . Proof of Proposition 2.10. Let φ > 0 be some eigenfunction for the eigenvalue λ + a,A,α (Ω). We observe that a∆φ ≤ Atr(

D 2 φ) + -atr(D 2 φ) -= M a,A (D 2 φ).
This implies that a∆φ|∇φ| α + λ + a,A,α |φ| α φ ≤ 0 and then, by the definition of λ eq , aλ eq ≥ λ + a,A,α . In the same manner let φ < 0 be such that ∆φ|∇φ| α = -λ eq |φ| α φ, then

|∇φ| α Atr((D 2 φ) + ) -atr((D 2 φ) -) ≥ |∇φ| α A∆φ = -Aλ eq |φ| α φ
and by the definition of λ - a,A,α this implies that Aλ eq ≤ λ - a,A,α . The question of the simplicity of the first eigenvalues for general operators satisfying (H1),.. (H3), is an open problem. The difficulty resides in the fact that one cannot establish some strict comparison principle. More precisely we should need the following result:

If u ≥ v and F (x, ∇u, D 2 u) = f ≤ F (x, ∇v, D 2 v) = g, then either u > v everywhere, or u ≡ v.
The difficulty when one wants to prove this result resides on the points where test functions have their gradient equal to zero.

However, we proved in [START_REF] Birindelli | Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case, To appear in ZAA[END_REF] the simplicity result in the radial case. It will be precised in the forthcoming section; this will be an argument for the existence and the properties of the other eigenvalues in the case of the operator |∇u| α M a,A (D 2 u).

The radial case

Let Ω be a ball B(0, 1) or an annulus B(0, 1) \ B(0, ρ) for some ρ ∈ (0, 1). We suppose that there exists F such that, for any radial function u(x) = g(|x|), F (x, ∇u, D 2 u) = F (r, g ′ , g ′′ ). In that case the conditions on F imply that

|g ′ | α γ 1 g ′′ + γ 2 (N -1) |x| g ′ ≤ F (x, ∇φ, D 2 φ) ≤ |g ′ | α Γ 1 g ′′ + Γ 2 (N -1) |x| g ′ ,
where

γ 1 = a if g ′′ > 0 A if g ′′ < 0 , γ 2 = a if g ′ > 0 A if g ′ < 0, Γ 1 = A if g ′′ > 0 a if g ′′ < 0 , Γ 2 = A if g ′ > 0 a if g ′ < 0.
In this situation one can define the first radial eigenvalues λ ± rad (Ω) λ + rad (Ω) = sup{λ : ∃ϕ > 0, radial, F (r, ϕ ′ , ϕ ′′ ) + λϕ 1+α ≤ 0 in Ω}, λ - rad (Ω) = sup{λ : ∃ϕ < 0, radial, F (r, ϕ ′ , ϕ ′′ ) + λ|ϕ| α ϕ ≥ 0 in Ω}. Acting as in the general case, one can prove the existence of eigenfunctions for each of these eigenvalues, and using the maximum and minimum principle one derives that λ ± rad (Ω) = λ ± (Ω) in the sense given in theorem 2.3 for the operator F (x, ∇u, D 2 u).

Remark 3.1. In the case of the ball, for any constant sign viscosity solution of F (r, u ′ , u ′′ ) + λ ± |u| α u = 0 in B(0, 1) u = 0 on {r = 1}, u is decreasing from r = 0 for λ + and increasing from r = 0 for λ -. In particular if u is C 1 , 0 is the unique point where u ′ is zero.

In the case of an annulus B(0, 1) \ B(0, ρ), if u is a positive (respectively negative) viscosity solution of

F (r, u ′ , u ′′ ) + λ ± |u| α u = 0 in B(0, 1) \ B(0, ρ) u = 0 on {r = 1} and {r = ρ}, (3.1)
then there exists a unique point r = r u such that u is increasing (respectively decreasing) on [ρ, r u ], and decreasing (respectively increasing) on [r u , 1]. In particular if u is C 1 , r u is the unique point where u ′ is zero.

The uniqueness result obtained in [START_REF] Birindelli | Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case, To appear in ZAA[END_REF] is the following.

Proposition 3.2. Suppose that Ω is a ball or an annulus. Suppose that ϕ and ψ are two positive radial eigenfunctions in the viscosity sense, for the eigenvalue λ + , which are zero on the boundary; then there exists some positive constant c such that ϕ = cψ.

Remark 3.3. Of course the same result holds for the negative eigenfunctions corresponding to λ -.

From now on we shall denote by an abuse of notation by M a,A (r, g ′ , g") the operator g → Γ 1 g ′′ + Γ 2 (N -1) r g ′ and F will be

F (r, g ′ , g ′′ ) = |g ′ | α Γ 1 g ′′ + Γ 2 (N -1) r g ′ , (3.2)
where Γ 1 and Γ 2 are the multivalued functions defined at the beginning of Section 3.

Remark 3.4. We shall most of the time use more correctly the definition which is valid when g is Lipshitz, and when Γ 1 and Γ 2 are determined:

F (r, g ′ , g ′′ ) = Γ 1 d dr ( |g ′ | α g ′ 1 + α ) + Γ 2 (N -1) r |g ′ | α g ′ , the derivative d dr ( |g ′ | α g ′ ) 1+α
) being taken in the distributional sense. We end this section by giving one consequence of the Hopf principle in the case of the operator F . Remark 3.5. Suppose that u is a nonnegative solution in the viscosity sense of F (r, u ′ , u ′′ ) = f on [0, R) for some R ≤ ∞, with f continuous and nonpositive; then either u > 0 everywhere, or u ≡ 0. In particular if u satisfies F (r, u ′ , u ′′ ) = -λ|u| α u with λ > 0, and if u(r o ) = 0 then u must change sign on r o .

4. The functions w + and w -.

In this section we prove the existence and uniqueness of some radial solutions of

|w ′ | α M a,A (r, w ′ , w ′′ ) = -|w| α w in R + , w(0) = 1, w ′ (0) = 0.
This will permit us as in [START_REF] Busca | Nonlinear eigenvalues and bifurcation problems for Pucci's operator[END_REF], [START_REF] Pino | Global bifurcation from the eigenvalues of the p-Laplacian[END_REF] to prove the existence of an infinite numerable set of radial eigenvalues for the operator |∇w| α M a,A (D 2 w) in the ball. Proposition 4.1. There exists a unique C 1 solution of the equation

|w ′ | α (M a,A (r, w ′ , w ′′ )) = -|w| α w in R + , w(0) = 1, w ′ (0) = 0. (4.1)
Moreover, w is C 2 around each point where w ′ = 0. This proposition will be a consequence of the following three results.

Proposition 4.2. For all r o ≥ 0, and for all k o = 0 there exists some δ > 0 such that there is existence and uniqueness of a solution to

a |k ′ | α k ′ N -1 r + d dr ( |k ′ | α k ′ 1 + α ) = -|k| α k for r ∈ (r o , r o + δ), or r ∈ (r o -δ, r o ) ∩ R + , k(r o ) = k o , k ′ (r o ) = 0, (4.2)
A|k ′ | α k ′ N -1 r + a d dr ( |k ′ | α k ′ 1 + α ) = -|k| α k for r ∈ (r o , r o + δ), or r ∈ (r o -δ, r o ) ∩ R + , k(r o ) = k o , k ′ (r o ) = 0, (4.3) A |k ′ | α k ′ N -1 r + d dr ( |k ′ | α k ′ 1 + α ) = -|k| α k for r ∈ (r o , r o + δ), or r ∈ (r o -δ, r o ) ∩ R + , k(r o ) = k o , k ′ (r o ) = 0, (4.4) a|k ′ | α k ′ N -1 r + A d dr |k ′ | α k ′ 1 + α = -|k| α k for r ∈ (r o , r o + δ), or r ∈ (r o -δ, r o ) ∩ R + k(r o ) = k o , k ′ (r o ) = 0. (4.5)
Moreover, k is C 2 around each point where k ′ = 0.

In a second step we shall prove the following existence and uniqueness result.

M a,A (r, w ′ , w ′′ ) = - |w| α w |w ′ | α , (w(r o ), w ′ (r o )) = (w o , w ′ o ).
Moreover, if on (r 1 , r 2 ) ⊂ (0, ∞) w is a maximal solution,

lim r→r i ,r∈(r 1 ,r 2 ) w ′ (r) = 0, w is C 2 on (r 1 , r 2 ), d dr (|w ′ | α w ′ (r)) exists everywhere on (r 1 , r 2 ) and d dr (|w ′ | α w ′ )(r + 1 )w(r + 1 ) < 0, and
d dr (|w ′ | α w ′ )(r - 2 )w(r - 2 ) < 0. Proposition 4.4. Let δ be such that, on C([0, δ]), k in (4.2) with r o = 0 and k o = 1 is well defined and |k -1| C([0,δ]) < 1 2 .
Then there exists some constant c 1 depending on a, A, N such that |k ′ | ≤ c 1 . Moreover, there exists r 1 > 0 which depends only on a, A, and N such that k ′ and k ′′ are < 0 on (0, r 1 ).

Remark 4.5. The analogous result holds for the situations in (4.3), (4.4), (4.5).

We postpone the proof of these three propositions, and we conclude the local existence and uniqueness result, arguing as follows.

Let r o = 0, k be the solution of (4.2) with k o = 1, and, according to Proposition 4.4, let r 1 be such that on (0, r 1 ], k ′ and k ′′ are negative. Let w be the solution given by Proposition 4.3 of

M a,A (r, w ′ , w ′′ ) = - |w| α w |w ′ | α in R + , w(r 1 ) = k(r 1 ), w ′ (r 1 ) = k ′ (r 1 ) = 0 (4.6)
on some neighborhood (r 1 -δ 1 , r 1 ). By the equation one must have w ′′ (r 1 ) < 0. Then by uniqueness w = k on (r 1δ 1 , r 1 ). We can continue replacing r 1 by r 1δ 1 and finally obtain that w = k on the left of r 1 as long as w ′ = 0, i.e., until 0. So we have obtained the existence and uniqueness of a solution on a neighborhood on the right of zero.

We can extend the solution on the right of r 1 . If w ′ (r) = 0 for all r ≥ r 1 , the result is given by Proposition 4. The equation can also be written as

     d dr (r (N -1)(1+α) |k ′ | α k ′ )(r) = - (α + 1)r (N -1)(1+α) |k| α k(r) a in R + k(0) = 1, k ′ (0) = 0, (4.7)
or equivalently, defining φ p ′ (u) = |u| p ′ -2 u and p ′ = α+2 α+1 , as

k(r) = 1 - r 0 φ p ′ α + 1 as (N -1)(1+α) s 0 t (N -1)(1+α) |k| α k(t) dt ds. (4.7)
We use the properties of the operator T defined by

T (k)(r) = 1 - r 0 φ p ′ α + 1 as (N -1)(1+α) s 0 t (N -1)(1+α) |k| α k(t) dt ds (4.8) which satisfies on [0, δ] ||T (k) -1|| ∞ ≤ δ φ p ′ (α + 1)δ||k|| α+1 ∞ a((N -1)(1 + α) + 1) ≤ c 1 δ p ′ ||k|| ∞ ≤ c 1 δ p ′ (||k -1|| ∞ + 1),
where

c 1 = (α+1) a((N -1)(1+α)+1) p ′ -1 . If δ < 1 3 |α|+1 c 1 1 p ′ , T sends the ball {u ∈ C([0, δ]) : ||u -1|| C([0,δ]) ≤ 1
2 } into itself. We now prove that it is contracting. We observe that for k with values in [

1 2 , 3 2 ] (α + 1) a((N -1)(1 + α) + 1) 1 2 α+1 s ≤ α + 1 as (N -1)(1+α) s 0 t (N -1)(1+α) |k| α k(t) dt ≤ (α + 1) a((N -1)(1 + α) + 1) 3 2 α+1 s,
and then by the mean value theorem for (u, v) ∈ B C([0,δ]) (1, 1 2 )

φ ′ p ( α + 1 as (N -1)(1+α) s 0 t (N -1)(1+α) u 1+α (t)dt -φ p ′ ( α + 1 as (N -1)(1+α) s 0 t (N -1)(1+α) v 1+α (t)dt ≤ c 1 s p ′ -1 |u α+1 -v α+1 | L ∞ ([0,s]) sup 3 2 -α , 1 2 -α ≤ c 1 s p ′ -1 |u -v| L ∞ ([0,s]) sup 3 2 -α , 1 2
-α sup 3 2 α , 1 2 α ≤ c 1 s p ′ -1 |u -v| L ∞ ([0,s]) 3 |α| .
This implies that

|T (u) -T (v)| ≤ c 1 δ p ′ p ′ |u -v| L ∞ ([0,δ]) 3 |α| ≤ 1 3 |u -v| L ∞ ([0,δ]) .
Then the fixed-point theorem implies that there exists a unique fixed point in C([0, δ]).

In the case of equation ( 4.3) one is led to consider

T (k)(r) = k o - r ro φ p ′ α + 1 as N + s ro t N + |k| α k(t) dt ds
with N + = (N -1)(1+α)A a . For equation (4.5) we shall consider

T (k)(r) = k o - r ro φ p ′ α + 1 As N - s ro t N -|k| α k(t) dt ds with N -= (N -1)(1+α)a A
. Finally for equation (4.4)

T (k)(r) = k o - r ro φ p ′ α + 1 As (N -1)(1+α) s ro t (N -1)(1+α) |k| α k(t) dt ds.
Proof of Proposition 4.3. We prove the local existence by proving that, for each (w o , w ′ o ) with w ′ o = 0 and for all r o > 0, there exists a neighborhood around r o and a solution to the equation which satisfies the condition (w(r o ), w ′ (r o )) = (w o , w ′ o ). We suppose that w ′ o = 0 and we introduce the function

f 2 (r, y 1 , y 2 ) = M - m(y 2 )(N -1) r - |y 1 | α y 1 |y 2 | α , (4.9)
where M and m are respectively the functions

M (x) = x A if x > 0 x a if x < 0 (4.10)
and

m(x) = Ax if x > 0 ax if x < 0. ( 4.11)
The functions M and m are Lipschitzian, hence f 2 is Lipschitzian with respect to y = (y 1 , y 2 ) around (w o , w ′ o ) when w ′ o = 0. Let f 1 (r, y 1 , y 2 ) = y 2 , and f (y 1 , y 2 ) = (f 1 (y 1 , y 2 ), f 2 (y 1 , y 2 )); then the standard theory of ordinary differential equations implies that

(y ′ 1 , y ′ 2 ) = f (y 1 , y 2 ), (y 1 , y 2 )(r o ) = (w o , w ′ o ) has a unique solution around (w o , w ′ o ) when w ′ o = 0. Then w = y 1 is a local solution of w ′′ = M - m(w ′ )(N -1) r - |w| α w |w ′ | α (4.12)
with the initial condition w

(r o ) = w o , w ′ (r o ) = w ′ o .
If w is a solution on (r 1 , r 2 ) and lim r→r 2 ,r<r 2 w ′ (r) exists and is = 0, w ′′ has also a finite limit from the equation, thus lim r→r 2 ,r<r 2 (y ′ 1 , y ′ 2 )(r) exists and is finite and one can continue, replacing r o by r 2 and (w o , w ′ o ) by (w(r 2 ), lim r→r 2 ,r<r 2 w ′ (r)). If lim r→r 2 ,r<r 2 w ′ (r)) is zero, one gets at least when α > 0, lim r→r 2 w ′′ (r) = ±∞, so that one cannot get a continuation, since the solutions of (y ′ 1 , y ′ 2 ) = f (y 1 , y 2 ) must be C 1 . We prove the last facts concerning d dr (|w ′ | α w ′ ). Suppose that w(r 2 ) > 0 and assume by contradiction that lim r→r 2 d dr (|w ′ | α w ′ (r)) ≥ 0. Then the equation on the left of r 2 is, since it is clear from the equation that w ′ cannot be nonnegative,

A d dr |w ′ | α w ′ 1 + α + a(N -1) r |w ′ | α w ′ = -|w| α w
which yields a contradiction when r → r 2 . Suppose now that w(r 2 ) < 0 and lim r→r 2 d dr (|w ′ | α w ′ (r)) ≤ 0; then from the equation w ′ cannot be ≥ 0 on the left of r 2 , and one is led to solve on the left of r 2 :

a d dr |w ′ | α w ′ 1 + α + A(N -1) r |w ′ | α w ′ = -|w| α w.
This is absurd by passing to the limit when r → r 2 . Suppose that w(r 1 ) > 0 and assume by contradiction that lim r→r 1 ,r>r 1 d dr (|w ′ | α w ′ )(r) ≥ 0; by the equation w ′ cannot be ≥ 0 thus this equation is on the right of r 1 :

A d dr |w ′ | α w ′ 1 + α + a(N -1) r |w ′ | α w ′ = -|w| α w.
This is absurd by passing to the limit when r → r 1 . Suppose that w(r 1 ) < 0 and that lim r→r 1 ,r>r

1 d dr (|w ′ | α w ′ )(r) ≤ 0; then the equation on the right of r 1 is a d dr |w ′ | α w ′ 1 + α + A(N -1) r |w ′ | α w ′ = -|w| α w.
This is absurd letting r go to r 1 .

Proof of Proposition 4.4. We can observe that |k ′ | α k ′ is differentiable for r > 0 and has a limit < 0 for r → 0. Moreover we shall give some constant δ 1 which depends only on a, A, α, N such that k ′ = 0 and d dr (|k ′ | α k ′ ) remains < 0 on (0, δ 1 ).

We begin to prove that d dr (|k ′ | α k ′ ) < 0 around zero. One has for r > 0

(|k ′ | α k ′ )(r) = - 1 + α ar No r 0 (|k| α k)(s)s No ds,
where N o = (N -1)(1 + α), and thus (|k ′ | α k ′ ) is continuously differentiable for r = 0, as the primitive of some continuous function, and

d dr (|k ′ | α k ′ )(r) = N o (1 + α) ar No+1 r 0 (|k| α k)(s)s No ds - 1 + α a (|k| α k)(r).
For the point 0, one has

lim r→0 |k ′ | α k ′ (r) r = -lim r→0 1 + α ar No+1 r 0 |k| α k(s)s No ds = - 1 + α a(N o + 1) < 0.
Using the fact that k tends to 1 when r goes to zero we get that

lim r→0 d dr (|k ′ | α k ′ )(r) = 1 + α a ( N o N o + 1 -1) = - (1 + α) A(N o + 1) < 0 and then |k ′ | α k ′ is C 1 on 0.
Moreover we prove that there exists a neighborhood on the right of zero which depends only on the data, such that d dr (|k ′ | α k ′ ) < 0 on it. For that aim we begin to establish some Lipschitz estimate on the solution with some constant which depends only on the data.

We have chosen δ (which depends only on a, A, α, and N ) such that, for r ∈ [0, δ], k(r) ∈ [ 1 2 , 3 2 ]. We now observe that k ′ is then bounded by

|k ′ | α+1 (r) ≤ 1 + α a(N o + 1) 3 2 α+1 r.
We have obtained the fact that there exists some constant c 2 which depends only on the constant a, N, A such that |k ′ | ≤ c 2 on (0, δ). We derive from this that on [0, δ] |k(r) -1| ≤ c 2 r, and also that

|(|k| α k)(r) -1| ≤ (1 + α) sup 3 2 α , 1 2 α c 2 r = c 3 r,
and thus d dr

(|k ′ | α k ′ )(r) + 1 + α a(N o + 1) ≤ (1 + α)N o ar No+1 r 0 ||k| α k -1|(s)s No ds + 1 + α a |(|k| α k)(r) -1| ≤ c 3 (1 + α)r a N o N o + 2 + 1 .
We have obtained the fact that as long as r < No+2 2(No+1) 2 c 3 ≡ r 1 , d dr (|k ′ | α k ′ ) remains negative (and thus so does k ′ ). This ends the proof of Proposition 4.4.

To finish the proof of Proposition 4.1, i.e., to prove global existence result, suppose that w is a solution on [0, r 1 ). If w ′ (r 1 ) = 0 we use Proposition 4.3, if w ′ (r 1 ) = 0, using lim r→r 1 , r>r 1 d dr (|w ′ | α w ′ )(r)w(r 1 ) < 0 we consider on the right of r 1 , equation ( 4.2) if w(r 1 ) > 0, and equation ( 4.4) if w(r 1 ) < 0. We have obtained a solution on R + .

We now prove that the solution w is oscillatory.

Proposition 4.6. The solution of (4.1) is oscillatory; i.e., for all r > 0 there exists τ > r such that w(τ ) = 0.

Proof of Proposition 4.6. First step. We suppose that a = A. We follow the arguments in [START_REF] Pino | Global bifurcation from the eigenvalues of the p-Laplacian[END_REF], but this process is classical in ODE oscillation theory, [START_REF] Hartmann | Ordinary Differential Equations[END_REF].

We assume by contradiction that there exists r o such that w does not vanish on [r o , ∞). Then one can consider the function

y(r) = r (N -1)(1+α) |w ′ | α w ′ (r) |w| α w(r) ,
which satisfies the equation

y ′ (r) = - (α + 1)r (N -1)(1+α) a - (α + 1)|y| α+2 (r) r (N -1)(1+α) 2 .
Integrating between r 0 and t one gets that

y(t) + (α + 1) t r 0 |y| α+2 (r) r (N -1)(1+α) 2 dr = - (α + 1)t (N -1)(1+α)+1 a ((N -1)(1 + α) + 1) + y(r 0 ) + (α + 1)r (N -1)(1+α)+1 o a ((N -1)(1 + α) + 1)
.

In particular we obtain that y(t) ≤ 0 for t large enough.

For the next step it will be useful to remark that if, in place of the equation, we had the inequality

d dr r (N -1)(1+α) |w ′ | α w ′ (r) ≤ -r (N -1)(1+α) |w| α w a ,
the conclusion would be the same.

We obtain that -y(t) = |y(t)| ≥ Ct (N -1)(1+α)+1 for some constant C > 0, as soon as t is large enough. Let

k(t) = t r 0 |y| α+2 (r) r (N -1)(1+α) 2 dr;
then using the previous considerations k(t) ≥ c 1 t N (1+α)+2 for some positive constant c 1 .

Coming back to the equation, always for t large,

(α + 1)k(t) ≤ |y(t)| = (k ′ (t)t (N -1)(1+α) 2 ) 1 α+2
and thus

(1 + α) α+2 k α+2 (t) ≤ k ′ (t)t (N -1)(1+α) 2 .
Integrating between t and s, s > t, we obtain the fact that for some positive constant

c 2 1 k α+1 (t) - 1 k α+1 (s) ≥ c 2 1 t (N -1)(1+α) 2 -1 - 1 s (N -1)(1+α) 2 -1 . Letting s go to infinity, 1 k α+1 (t) ≥ c 2 1 t (N -1)(1+α) 2 -1 .
From this one gets a contradiction with k(t) ≥ c 1 t N (1+α)+2 . This ends the proof of the first step. Second step: a < A. We argue along the model of [START_REF] Busca | Nonlinear eigenvalues and bifurcation problems for Pucci's operator[END_REF]. We suppose as in the first step that there exists r o such that w does not vanish on [r o , ∞). We begin by proving that, if w > 0 for r ≥ r o , then for r ≥ r o

d dr r (N -1)(1+α) |w ′ | α w ′ (r) ≤ -r (N -1)(1+α) |w| α w(r) a ,
and then following the previous arguments in the first step we obtain that if y(r) = r (N -1)(1+α) |w ′ | α w ′ (r) |w| α w(r) , then

y(t) + (α + 1) t ro |y| α+2 (r) r (N -1)(1+α) 2 dr ≤ - t (N -1)(1+α)+1 a ((N -1)(1 + α) + 1) + y(r o ) + r (N -1)(1+α)+1 o a ((N -1)(1 + α) + 1) ,
a contradiction if y > 0 for t large enough.

To prove that

d dr r (N -1)(1+α) |w ′ | α w ′ (r) ≤ -r N -1)(1+α) |w| α w(r) a ,
let us note that in the case w ′ ≤ 0 and d dr ( |w ′ | α w ′ 1+α ) ≤ 0 equality holds in the previous inequality; if w ′ ≥ 0 and d dr ( |w ′ | α w ′ 1+α ) ≥ 0 the equation is impossible. For the other cases, we assume first that w ′ ≤ 0; this implies if

d dr |w ′ | α w ′ 1+α ≥ 0 that a d dr |w ′ | α w ′ 1 + α + a(N -1) r |w ′ | α w ′ ≤ A d dr ( |w ′ | α w ′ 1 + α ) + a(N -1) r |w ′ | α w ′ = -|w| α w, which implies the result. If w ′ ≥ 0 and d dr ( |w ′ | α w ′ 1+α ) ≤ 0 a d dr |w ′ | α w ′ 1 + α + a(N -1) r |w ′ | α w ′ ≤ a d dr ( |w ′ | α w ′ 1 + α ) + A(N -1) r |w ′ | α w ′ = -|w| α w.
This also implies the result. We now assume that w < 0 on [r o , ∞). Then we prove that there exists r ⋆ such that w ′ (r ⋆ ) = 0 and w ′ > 0 on (r ⋆ , ∞). Indeed, by the equation, if

w ′ (r ⋆ ) = 0, by Proposition 4.3 lim r→r ⋆ ,r>r ⋆ d dr (|w ′ | α w ′ )(r) > 0.
This implies that w ′ is increasing on r ⋆ , thus w ′ is > 0 on a neighborhood on the right of r ⋆ . Moreover, if there exists r ′ > r ⋆ such that w ′ (r ′ ) = 0, we argue as before and then w ′ > 0 after r ′ . From these remarks, it is sufficient to discard w ′ < 0 on [r o , ∞). Then in that case necessarily d dr (|w ′ | α w ′ ) > 0 on [r o , ∞) by the equation, and then w satisfies

d dr (|w ′ | α w ′ (r)r N -) = -(1 + α) r N -|w| α w(r) a > 0.
Let g(r) ≡ (|w ′ | α w ′ (r)r N -); g is monotone increasing, and since w ′ < 0, it has a limit c 1 ≤ 0 at +∞. On the other hand, since w ′ < 0 there exists c 2 ∈ [-∞, 0) such that lim r→+∞ w(r) = c 2 , thus from the equation satisfied by w, lim r→+∞ g ′ (r) = +∞, which is a contradiction with lim r→+∞ g(r) = c 1 ≤ 0. Finally, w ′ > 0 after r o . We recall that N + = A(1+α)(N -1) a . Distinguishing the cases d dr (|w ′ | α w ′ ) > 0 and d dr (|w ′ | α w ′ ) < 0 on the right of r o and arguing as we already did before we obtain that w satisfies

d dr (|w ′ | α w ′ (r)r N + ) ≤ - (1 + α)r N + |w| α w(r) a .
Then defining

y(r) = r N + |w ′ | α w ′ (r) |w| α w(r) one has y ′ (t) + (α + 1)|y(r)| α+2 r (N + )(α+1) + (α + 1)r N + a ≤ 0. (4.13)
Hence integrating between r o and t one gets for some constant c 1 > 0

|y(t)| = -y(t) ≥ c 1 t N + +1 . Let k(t) = t ro |y| α+2 (r) r N + (α+1) dr ≥ ct N + +α+3
. From the equation (4.13) integrated between r o and t, using

k ′ (t) = |y| α+2 (t) t N + (α+1) , we get (α + 1) α+2 k α+2 (t) ≤ k ′ (t)t N + (α+1) , hence, for some positive constant c 2 k -(α+1) (t) -k -(α+1) (s) ≥ c 2 (t -N + (α+1)+1 -s -N + (α+1)+1 )
for s > t. Letting s go to infinity and using lim k(t) = +∞, one derives that k -(α+1) (t) ≥ c 2 t -N + (α+1)+1 , which is a contradiction with k(t) ≥ c 1 t N + +α+3 . We have obtained that w is oscillatory. This ends the proof of Proposition 4.6.

For the sake of completeness, we give some properties of the function w inherited from the property of the eigenfunctions in the viscosity sense [START_REF] Birindelli | Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case, To appear in ZAA[END_REF]. Lemma 4.7. Between two successive zeros of w, there exists a unique zero of w ′ .

Proof. Suppose that w is of constant sign on B(0, t) \ B(0, s), s < t and w(s) = w(t) = 0; then w 1 (x) = w(µ

1 2+α x) is an eigenfunction for one of the first demi-eigenvalues µ = λ + (B(0, t) \ B(0, s)) if w > 0, or µ = λ -(B(0, t) \ B(0, s)) if w < 0.
Then by the uniqueness of the first eigenfunction in the radial case, if w > 0, by Remark 3.1, w is increasing on [s, r w ] and decreasing on [r w , t] and r w is the unique point on which w ′ = 0. We argue in the same manner when w < 0, using the fact that in that case w is decreasing on [s, r w ] and increasing on [r w , t].

In the sequel we shall denote by w + the radial solution given by Proposition 4.1 of

|w ′ | α M a,A (r, w ′ , w ′′ ) = -|w| α w, w(0) = 1, w ′ (0) = 0.
We also denote by w -the radial solution of

|w ′ | α M a,A (r, w ′ , w ′′ ) = -|w| α w, w(0) = -1, w ′ (0) = 0.
The proof of the existence and uniqueness of w -is obtained by the same arguments used for w + . The results in Proposition 4.6 can be adapted to the case of w -, and then we also get that w -is oscillatory.

Eigenvalues and eigenfunctions

In this section we prove the existence of an infinite denumerable set of eigenvalues for the radial operator defined in equation (3.2). These eigenvalues are simple and isolated. We begin with some properties of the eigenfuntions.

Proposition 5.1. Suppose that u is a radial viscosity solution of

F (r, u ′ , u ′′ ) = -µ|u| α u in B(0, 1) u(1) = 0, u(0) > 0.
Then 0 is a local maximum for u, u is C 2 on a neighborhood (0, r o ) of zero, is C 1 on [0, r o ] and u ′ (0) = 0.

Proof of Proposition 5.1. First let us note that µ > 0, because if not the maximum principle would imply that u ≤ 0.

Since u is continuous there exists some neighborhood B(0, r o ) on which F (r, u ′ , u ′′ ) < 0. Then, using the comparison principle for such operators, and remarking that positive constants are sub-solutions, one gets that u(r) ≥ u(r 1 ) on B(0, r 1 ), if r 1 < r o . This implies in particular that u is decreasing from zero, and 0 is a local maximum. We now prove that u is C 1 around zero and C 2 on a neighborhood of 0, except on 0.

Let r 1 be the first zero of u. Then u > 0 on B(0, r 1 ) and λ + (B(0, r 1 )) = µ, by Proposition 2.6. Let w + be the C 1 solution in Proposition 4. 1 andβ + 1 its first zero (it exists according to Proposition 4.6). Define

v(r) = w + ( β + 1 r r 1
).

Then v > 0 on B(0, r 1 ) and v is an eigenfunction in B(0, r 1 ) for the eigenvalue (

β + 1 r 1 ) 2+α ; in particular, λ + (B(0, r 1 )) = µ = ( β + 1 r 1 ) 2+α
, and by the uniqueness of the first radial eigenfunction > 0 in Proposition 3.2, there exists some constant c > 0 such that u = cv on B(0, r 1 ). In particular u is C 2 on each point where u ′ is different from zero and C 1 everywhere on B(0, r 1 ). This proves in particular, since u is C 1 on B(0, r 1 ) and u has a maximum on 0, that u ′ (0) = 0.

Of course the symmetric result holds for u such that u(0) < 0. We now present an improvement of Proposition 2.5 which will be used in the proof of Corollary 5.4 Proposition 5.2. Suppose that s < t < 1. Suppose that there exist some eigenfunctions for the annulus B(0, 1)\B(0, s) and for B(0, 1)\B(0, t), which are C 2 on each point where their first derivative is different from 0, and C 1 anywhere, then λ ± (B(0, 1) \ B(0, s)) < λ ± (B(0, 1) \ B(0, t)).

Proof. Suppose by contradiction that λ ± (B(0, 1) \ B(0, s)) = λ ± (B(0, 1) \ B(0, t)), that we shall denote for simplicity by λ ± . Let ϕ and u be solutions of the equation F (r, ϕ ′ , ϕ ′′ ) + λ ± |ϕ| α ϕ = 0 which are C 2 on each point where their first derivative is different from 0, and C 1 anywhere, with ϕ = 0 on {r = 1} and {r = s}, and u = 0 on {r = 1} and {r = t}. To fix the ideas we also assume that ϕ and u are positive (and then we replace λ ± by λ + ).

Using the same arguments as in Propositions 4.2 and 4.3, since ϕ(1) = u(1) = 0 and u ′ (1) < 0, ϕ ′ (1) < 0, by uniqueness there exists some constant c > 0 such that ϕ = cu as long as ϕ ′ or u ′ is different from zero. By Remark 3.1 there exists exactly one point r u on (t, 1) for which u ′ (r u ) = 0 and it is a global strict maximum for u on (t, 1). By uniqueness, ϕ ′ (r u ) = 0 and r u must also be a global strict maximum for ϕ on (t, 1). Then the equation satisfied by u and ϕ on the left of r u is equation (4.3). By local uniqueness of solutions to (4.3) one gets that u = cϕ on the left of r u and this is true as long as u ′ or ϕ ′ is different from 0, hence at least on (t, 1). We get a contradiction since u = 0 on {r = t} and ϕ(t) = 0.

We now prove the existence of a denumerable set of eigenvalues. The result in Proposition 4.6 implies that there exists a sequence β ± k of increasing zeros of w ± .

We now consider u ± k (r) = w ± (β ± k r). Then u ± k is an eigenfunction on B(0, 1) for the eigenvalue µ ± k := (β ± k ) α+2 and it has k -1 zeros inside the ball, say r i :=

β ± i β ± k , i ∈ [1, k -1]
. We need to prove that they are the only eigenvalues.

Proposition 5.3. The set of eigenvalues of the radial operator is the set {µ ± k : k ≥ 1}. These eigenvalues are simple in the following sense: Suppose that v is some eigenfunction for the eigenvalue µ ± k , which is C 1 and C 2 on each point where the first derivative is different from 0, then there exists some constant c > 0 such that v = cw ± ((µ ± k )

1 2+α •).
Proof of Proposition 5.3. Let µ be an eigenvalue. Let v be a corresponding eigenfunction, that we suppose to fix the ideas such that v(0) > 0.

Necessarily, since v is radial and

C 1 , v ′ (0) = 0. Let z(•) = v(µ -1 2+α •) v(0)
. Then z satisfies equation (4.2) and by uniqueness z = w + on [0, µ 1 2+α ). This implies that µ 1 2+α is one of the zeros of w. This proves also the simplicity of the eigenvalue µ. The fact that the eigenvalues are isolated is a consequence of the properties of the zeros of w + .

The following corollary is not necessary for the present paper; they will be useful for the bifurcation results announced in the final concluding section.

Corollary 5.4. There is uniqueness (up to a positive multiplicative constant) of the k-th eigenfunction. As a consequence one has µ - k < µ + k+1 and µ + k < µ - k+1 . Proof. It is sufficient to prove that β + k < β - k+1 and β - k < β + k+1 . We begin to prove that β + 1 < β - 2 . One has

λ -((β + 1 , β + 2 )) = 1 = λ -((0, β - 1 )
). Suppose first that β + 2 < β - 1 ; this contradicts Proposition 2.5. If β + 2 = β - 1 , one has a contradiction with Proposition 5.2.

We consider the case k ≥ 2. Suppose by contradiction that β - k < β + k+1 < β + k+2 ≤ β - k+1 , and in a first time we assume that β + k+2 < β - k+1 . In that case one would have

λ ǫ ((β + k+1 , β + k+2 )) = λ ǫ ((β - k , β - k+1 )) = 1
, where ǫ = sign(-1) k+1 ; this would then contradict Proposition 2.5. In a second time if we assume that β + k+2 = β - k+1 , this contradicts Proposition 5.2 In the same manner we should prove that β + k+1 < β - k+2 . For the sake of completeness we finish this section with some additional properties of the eigenvalues. This result is an analogue of one result in [START_REF] Busca | Nonlinear eigenvalues and bifurcation problems for Pucci's operator[END_REF]. Corollary 6.2. We assume that a < A. Then

λ + a,A,α ((c, b)) ≤ aϕ(α)(b -c) -2-α . Corollary 6.3. For all k ≥ 1 µ + k (α, a)(B(0, 1
)) ≤ aϕ(α)k 2+α , and µ - k (α, a)(B(0, 1)) ≤ aϕ(α)(k + 1) 2+α . Proof of Proposition 6.1. Let us note that one can also use the following result for general operators satisfying the hypothesis in Section 2, proved in [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF]: There exists some constant C which depends on a, A, N such that, if R is the radius of some ball included in Ω, then λ ± (Ω) ≤ C R α+2 . But we shall give a more precise estimate here. For the radial case, one can easily see that

λ eq,α = inf u∈W 1,2+α 0 ((c,b)) b c |u ′ | 2+α (r)r (N -1)(1+α) dr b c |u| 2+α (r)r (N -1)(1+α) dr .
Let us consider the function u(r) = (rc)(br). We need to get an upper bound for

I = b c |2r -(c + b)| 2+α r (N -1)(1+α) dr,
and to get a lower bound for

J = b c (r -c) 2+α (b -r) 2+α r (N -1)(1+α) dr.
For the first integral we use the inequality r

(N -1)(1+α) ≤ 2 |1-(N -1)(1+α)| (r - c+b 2 ) (N -1)(1+α) + c+b 2 (N -1)(1+α) ).
In the following c(α, N ) is some constant which can vary from one line to another but is bounded for α ∈ [-1, M ]. We obtain that

J ≤ c(α, N ) b c |r - c + b 2 | 2+α+(N -1)(1+α) dr + c + b 2 (N -1)(1+α) b c |r - c + b 2 | 2+α dr ≤ c(α, N ) (b -c) 3+α+(N -1)(1+α) + (c + b) (N -1)(1+α) (b -c) 3+α ≤ c(α, N )(b -c) 3+α (c + b) (N -1)(1+α) .
To minorize I we use

r (N -1)(1+α) ≥ 2 -|1-(N -1)(1+α)| (r -c) (N -1)(1+α) + c (N -1)(1+α)
Let j be such that (r i ) 1≤i≤j-1 are the zeros of ϕ. By Remark 3.5 in Section 3, ϕ changes sign on each of them. As a consequence there exists δ > 0 such that, for all i ∈ [1, j -1], on [r iδ, r i + δ], ϕ has no other zero than r i and on [r i-1 + δ, r iδ] ϕ has no zero. From the fact that ϕ(r iδ)ϕ(r i + δ) < 0, one has for n large enough ϕ n (r iδ)ϕ n (r i + δ) < 0, and thus ϕ n has at least one zero in (r iδ, r i + δ). In the same manner there exists m > 0 such that |ϕ| > m on every interval [r i-1 + δ, r iδ], which implies by the uniform convergence of ϕ n towards ϕ that ϕ n cannot have a zero in this interval. As a consequence k ≥ j. Moreover, by the strict monotonicity of ϕ on [r iδ, r i + δ], ϕ n is also monotone for n large enough. This implies in particular the uniqueness of the zero of ϕ n on that interval. Finally, j = k.

There remains to prove that ϕ is "regular;" i.e., that ϕ is C 2 on each point where the first derivative is different from zero, and C 1 anywhere. Suppose that r < t are two successive zeros of ϕ; then for n large enough, there exists r n < t n two successive zeros of ϕ n which converge respectively to r, t. Moreover ϕ n (respectively ϕ) has constant sign on (r n , t n ) (respectively (r, t)). One can assume without loss of generality that this sign is negative.

We need to prove that ϕ is "regular " on [r, t]. Let r ′ n be the unique zero of ϕ ′ n on (r n , t n ). Then ϕ n is the unique fixed point on (r n , r ′ n ) of the operator T n defined as

T n (w)(r) = ϕ n (r ′ n ) - r r ′ n φ p ′ (1 + α n )µ + k (α n , a n , A) As N - n s r ′ n |w| αn w(t)t N - n dt ds,
where N - n = an(N -1)(1+αn)

A

. One can prove as is done in the proof of Proposition 4.4 that there exists some neighborhood (r ′ nδ, r ′ n ) of δ which does not depend on n such that, on the left of r n , ϕ ′ n < 0 and ϕ ′′ n > 0. In the same manner ϕ n is the unique fixed point of T n on (r ′ n , r ′ n + δ) defined as

T n (w)(r) = ϕ n (r ′ n ) - r r ′ n φ p ′ (1 + α n ) µ + k (α n , a n , A)
As N 0,n s r ′ n |w| αn w(t)t N 0,n dt ds,

where N 0,n = (N -1)(1 + α n ) and there exists some δ > 0 which does not depend on n such that, on (r ′ n , r ′ n + δ), ϕ ′ n > 0, and ϕ ′′ n > 0. Using Remark 3.1 there exists exactly one point r ′ such that ϕ is decreasing on (r, r ′ ) and increasing on (r ′ , t), hence, since ϕ n converges uniformly to ϕ, one gets that r ′ n converges to r ′ .

From the definition of T n one sees that ϕ n converges uniformly on (r ′ - This implies that ϕ is a C 2 solution on (r ′ -3δ 4 , r ′ ). We do the same on (r ′ , r ′ + 3δ 4 ). We now consider the equation on (r, r ′ -δ 2 ). As soon as n is large enough in order that r > r ′ n-1 , on that interval ϕ n satisfies (ϕ ′ n , ϕ ′′ n ) = f n (ϕ n , ϕ ′ n ), where f n = (f 1,n , f 2,n ), f 1,n (r, y 1 , y 2 ) = y 2 , and It is clear that f n is uniformly Lipschitzian on (ϕ(r ′ ), ϕ(r)) × (ϕ ′ (r), ϕ ′ (r ′δ 2 )). Thus ϕ n converges in C 1 ( even C 2 ) to some solution ψ on (r, r ′ -δ 2 ) of (ψ ′ , ψ ′′ ) = f (ψ, ψ ′ ) with f = (f 1 , f 2 ), where f 1 and f 2 have been defined in 4.9 in Section 4. with the condition ψ(r) = 0, ψ ′ (r) = ϕ ′ (r).

This implies that ϕ is C 2 on (r, r ′ -δ 2 ). We can do the same on (r ′ + δ 2 , t) and get in that way the regularity of ϕ on [r ′ , t). In fact the proof contains the regularity of ϕ on an open neighborhood of [r, t]. Since this can be repeated on each interval delimited by two zeros of ϕ one gets the regularity of ϕ on B(0, 1). As a consequence of Proposition 5.3 we have obtained that µ = µ + k . Since µ + k (α n , a n ) has a unique cluster point we get that the entire sequence converges to µ + k .

Conclusion and supplementary results

Let K α,a be the operator defined on C(Ω) by: For f ∈ C(Ω), K α,a (f ) is the unique v ∈ C(Ω) solution of Fα,a (r, v ′ , v ′′ ) -|v|

α v = -f in Ω v = 0 on ∂Ω.
The operator K α,a is well defined since α > -1, and defining for µ positive given K α,a,µ (u) = K α,a ((µ + 1)|u| α u), one can note that the fixed points of K α,a,µ exist if µ is an eigenvalue, as some associated eigenfunction.

We will be able to derive from the continuity results in the last section some results about the degree of the operator K α,a,µ as a function of the position of µ with respect to the eigengalues µ ± k . Next we shall establish some bifurcation results for the equations defined as follows.

Let f be defined as (µ, s) → f (µ, s) which is "super-linear" in s uniformly with respect to µ in the sense that lim s→0 f (µ, s) |s| 1+α = 0. We also assume that f is locally bounded and continuous in all its variables.

Then we shall consider the problem Fα,a (r, u ′ , u ′′ ) + µ|u| α u + f (µ, u) = 0 in Ω u = 0 on ∂Ω, (

for which we shall prove bifurcation results, completing the results already obtained in [START_REF] Birindelli | Bifurcation for singular fully nonlinear equations, To appear in "On the notions of solution to nonlinear elliptic problems: results and developments[END_REF]. This will be the object of a forthcoming paper.

  there exists a continuous function ω with ω(0) = 0, such that if (X, Y ) ∈ S 2 and ζ ∈ R + satisfy and I is the identity matrix in R N , then for all (x, y) ∈ R N , x = y F (x, ζ(xy), X) -F (y, ζ(xy), -Y ) ≤ ω(ζ|x -y| 2 ).

Proposition 4 . 3 .

 43 If w ′ o = 0, for all w o , there exists a local unique solution to

3 .

 3 Suppose now that r o ≥ r 1 is the first point after r 1 such that w ′ (r o ) = 0. By Remark 3.5 in Section 3, w(r o ) cannot be zero. If w(r o ) < 0, anticipating the behaviour of the possible solutions on the right of r o , we know by using the conclusion in Proposition 4.3 that one must have lim r→ro,r>ro d dr (|w ′ | α w ′ (r)) > 0, so the equation to solve on the right of r o is (4.4), and we get a local solution on the right of r o . The situation w(r o ) > 0 cannot occur, since this would imply that lim r→ro,r>ro d dr (|w ′ | α w ′ (r)) < 0 and w ′ could not be ≤ 0 on the left of r o and = 0 on r o . Proof of Proposition 4.2. We prove the result for equation (4.2), with k o = 1 and r o = 0; the changes to bring in the other cases are given shortly at the end of the proof.

3δ 4 ,

 4 r ′ ) to the solution ψ on that interval of T (ψ) = ψ, where T (w)(r) = ϕ(r ′ ) -α w(t)t N -dt ds.

f 2 ,

 2 n (r, y 1 , y 2 ) = M n -m n (y 2 )(N -1) r -|y 1 | αn y 1 |y 2 | αn, where M n and m n are respectively the functionsM n (x) = x A if x > 0 x an if x < 0, and m n (x) = Ax if x > 0 a n x if x < 0.

Proposition 5.5. The gap between the two first half eigenvalues is larger than between the second ones:

Proof of Proposition 5.5. Let ϕ ± i i = 1, 2 be the eigenfunctions associated with µ ± i with ϕ ± i (0) = ±1. Let r + be the first zero of ϕ + 2 , r -the first zero of ϕ - 2 . We prove that r -≥ r + . Indeed, suppose by contradiction that r -< r + , and define A + = {r : r + < r < 1} and A -= {r : r -< r < 1}, then A + ⊂ A -and then

We have obtained a contradiction.

Moreover, let us consider ψ(x) = ϕ + 2 (r + x). Then ψ is a radial solution on B(0, 1) of

, by the definition of the first half eigenvalue. In the same manner (r -) 2+α µ - 2 = µ - 1 , and then µ -

this yields the result.

6. The continuity of the spectrum with respect to the parameters.

In this section we let α ∈ (-1, ∞) and a ∈ [0, A] vary and for that reason we denote by Fα,a the operator F defined before. We denote by µ ± k (α, a) the corresponding eigenvalues. In order to prove the continuity of the map (α, a) → µ ± k (α, a), we begin by establishing the boundedness of the eigenvalues µ ± k (α, a) when α belongs to some compact set of (-1, ∞) and a ∈ [0, A]. Proposition 6.1. We suppose that a = A = 1. Let λ eq,α ((c, b)) be the first "radial" eigenvalue for the set B(0, b) \ B(0, c) and for the operator

Then there exists some continuous function ϕ(α), bounded on every compact set of [-1, ∞), such that

and thus

where, in the previous lines, B denotes the Euler function. We have obtained the result. Proof of Corollary 6.2. We use the inequality in Proposition 2.10

Proof of Corollary 5.4. Let us recall that we have denoted by (r i ) i the zeros of the eigenfunction ϕ + k . µ + k (B(0, 1)) coincides with λ + (B(0, r 1 )) and with λ + (B(0,

In each of the cases we get the result. Concerning µ - k we use the inequality

). Proof of Proposition 6.4. By Corollary 5.4, the sequence (µ ± k (α n , a n )) n is bounded, so we can extract from it a subsequence, denoted in the same manner for simplicity, such that µ ± k (α n , a n ) → µ, for some µ ∈ R + . We fix the integer k. Let ϕ n be such that ϕ n (0) = 1, and Fαn,an (r, ϕ

Using the compactness results in Corollary 2.8 one can extract from (ϕ n ) a subsequence, which will be denoted in the same manner for simplicity, which converges uniformly to a viscosity solution ϕ of Fα,a (r, ϕ ′ , ϕ ′′ ) + µ|ϕ| α ϕ = 0 in B(0, 1) ϕ(1) = 0.

By the uniform convergence, ϕ is not identically zero and ϕ(0) = 1. Thus µ is some eigenvalue. We must prove first that ϕ has k -1 zeros, secondly that ϕ is C 1 and C 2 on every point where the first derivative is different from zero.