# Personalized Filled-pause Generation with Group-wise Prediction Models

###### Abstract

In this paper, we propose a method to generate personalized filled pauses
(FPs) with group-wise prediction models. Compared with fluent text generation,
disfluent text generation has not been widely explored. To generate more
human-like texts, we addressed disfluent text generation. The usage of
disfluency, such as FPs, rephrases, and word fragments, differs from speaker
to speaker, and thus, the generation of personalized FPs is required. However,
it is difficult to predict them because of the sparsity of position and the
frequency difference between more and less frequently used FPs. Moreover, it
is sometimes difficult to adapt FP prediction models to each speaker because
of the large variation of the tendency within each speaker. To address these
issues, we propose a method to build group-dependent prediction models by
grouping speakers on the basis of their tendency to use FPs. This method does
not require a large amount of data and time to train each speaker model. We
further introduce a loss function and a word embedding model suitable for FP
prediction. Our experimental results demonstrate that group-dependent models
can predict FPs with higher scores than a non-personalized one and the
introduced loss function and word embedding model improve the prediction
performance.

Keywords: disfluency generation, filled-pause prediction, speaker grouping

languageresourceLanguage Resources

Personalized Filled-pause Generation with Group-wise Prediction Models

Yuta Matsunaga, Takaaki Saeki, Shinnosuke Takamichi, and Hiroshi
Saruwatari††thanks: This work was supported by JST, Moonshot R&D Grant Number
JPMJPS2011.
---
Graduate School of Information Science and Technology, The University of
Tokyo, Japan.

Abstract content

## 1\. Introduction

Disfluency generation aims to generate human-like disfluent texts [Qader et
al., 2018, Yang et al., 2020]. Compared with fluent text generation [Brown et
al., 2020], disfluent text generation has not been widely explored. Disfluency
includes filled pauses (FPs), rephrases, and word fragments [Elisabeth, 1994],
and it is known that the tendency to use them varies from speaker to speaker
[Shriberg, 1996, Elisabeth, 1994, Watanabe and Shirahata, 2019]. Disfluency
generation reproducing such individuality makes it possible to generate
personalized disfluent texts and can be applied to spontaneous speech
synthesis, which generates more human-like spontaneous speech than a reading-
style one. In this research, we focus on spontaneous speech synthesis and
address the disfluency generation reproducing individuality.

FPs are defined as words that have a filling-in role [Koiso et al., 2001], and
there are various words for FPs [Brown, 2017, Hirose et al., 2006]. Such FPs
are important because they have various effects on spontaneous speech. They
play an important role in speech generation: planning [Maclay and Osgood,
1959] and monitoring [Levelt, 1983]. They are also important to facilitate
communication: speakers can indicate that they are searching for words [Clark
and Fox Tree, 2002], and listeners can understand the word quickly [Fox Tree
and Schrock, 1999]. The use of FPs influences the perception of the speaker’s
personality for listeners [Gustafson et al., 2021]. We, therefore, focus on FP
generation to achieve FP-included spontaneous speech synthesis with these
various effects. In addition, it is known that the position [Shriberg, 1996]
and words [Elisabeth, 1994, Watanabe and Shirahata, 2019] of FPs differ from
speaker to speaker. Therefore, as shown in Figure 1, we propose a personalized
FP prediction method to reproduce not only whether each speaker uses FPs or
not, but also how each speaker uses them.

FP-included spontaneous speech synthesis consists of FP insertion and speech
synthesis. The FP insertion model predicts or selects the position and word of
FPs to generate disfluent texts from original fluent texts. The speech
synthesis model generates the acoustic features of the fluent and FP parts
from texts containing FPs. FP prediction is particularly difficult because of
the sparsity of positions [Ohta et al., 2007] and the bias of words (i.e. the
frequency difference between more and less frequently used FP words). It is
necessary to establish an FP prediction method to address these issues.

Figure 1: Personalized filled-pause (FP) prediction. In contrast to the
aforementioned conventional method that predicts only FPs generalized among
speakers, our proposed method further aims to predict FPs personalized to each
speaker.

In this paper, we propose a group-dependent FP prediction method using speaker
grouping that can highly reproduce individuality. First, we divide speakers
into groups on the basis of their tendency to use FPs. Then, we train group-
dependent models by fine-tuning on the basis of the non-personalized model
trained on multi-speaker data. This method does not require a large amount of
spontaneous speech data for each speaker, nor the time to train each speaker-
dependent model.

The experimental results clarify that all the group-dependent models based on
FP words have better F-scores: $0.456$ for positions and $0.288$ for words at
most, compared with the non-personalized model with F-scores of $0.376$ for
positions and $0.089$ for words. Moreover, almost all the models based on FP
positions have better F-scores: $0.461$ for positions and $0.277$ for words at
most than those of the non-personalized model.

In addition, we introduce a loss function that takes into account sparsity and
bias and a rich word embedding model, which improves F-scores for positions
and words. The key contributions of this work are as follows:

  * •

We propose a method that constructs group-dependent models by grouping
speakers on the basis of the tendency to use FPs and demonstrate that the
performance of almost all the models is better than the non-personalized
model. The group-dependent models and source implementation to train them are
available at the github
repository111https://github.com/ndkgit339/filledpause_prediction_group.

  * •

We introduce a loss function suitable for FP prediction and a rich word
embedding model and demonstrate that it improves the performance of the FP
prediction models.

## 2\. Related work

### 2.1. FP-included speech generation

Various studies have addressed FP-included speech generation from texts or
fluent speech. [Yan et al., 2021, Éva Székely et al., 2019a] have proposed
methods to synthesize FP-included disfluent speech from FP-excluded fluent
text. There is also a method to synthesize FP-included speech by using FPs’
information as input [Éva Székely et al., 2019b]. In [Adell et al., 2008], the
authors modeled the insertion of editing terms into fluent utterances and
local prosodic changes. In addition, a number of studies have created an
external module to predict FPs for FP-included speech synthesis [Wester et
al., 2015, Gustafson et al., 2021, Cong et al., 2021]. However, these studies
have not attempted to improve the performance of the prediction or reproduce
the FPs’ individuality.

### 2.2. FP-included text prediction

A number of studies have focused on FP-included text prediction from FP-
excluded fluent texts. [Qader et al., 2018] proposed an algorithm using a
probabilistic model to generate disfluent sentences from fluent sentences, but
the prediction of FP words is simple. To construct the spoken language model,
[Ohta et al., 2007] predicted the FPs’ positions and words in order, but the
scores of predicting each word are low. [Yamazaki et al., 2020] reported that
simultaneously predicting positions and words improves performance. [Tomalin
et al., 2015] also predicted FP word and position simultaneously using a
lattice-based n-gram model. However, these methods cannot reproduce the
diversity of FP words; the scores of the prediction of less frequent FPs are
still low. In contrast to these studies, we propose a method to train
prediction models reproducing the diversity of FP words.

## 3\. Method

To reproduce individuality in FP prediction, it is necessary to predict the
positions and words of FPs more precisely for each speaker. First, we
introduce a loss function that addresses the sparsity of positions and the
bias of words and a rich word embedding model. Furthermore, we propose a
method for building personalized prediction models.

### 3.1. Basic architecture and FP vocabulary

We construct a model that consists of two modules: a word embedding model and
an embedding-to-FPtag model [Yamazaki et al., 2020]. The word embedding model
generates word embeddings for each morpheme from a sequence of morphemes
segmented by morphological analysis of fluent text. We use a word embedding
model that has been pre-trained on large-scale Japanese text data. The
embedding-to-FPtag model predicts $14$ classes of no FP or $13$ FP words for
each morpheme embedding. In this paper, following the previous work of
[Yamazaki et al., 2020], we use a bidirectional long short-term memory (BLSTM)
[Graves and Schmidhuber, 2005] as an embedding-to-FPtag model and a cross
entropy loss.

We select the $13$ FP words by excluding any FP words used less frequently
($<$ $20$%) among all speakers using the Corpus of Spontaneous Japanese (CSJ
[Maekawa, 2003]) and cover approximately $83$% of the FPs used by each
speaker. Table 1 lists the FP words and example sentences.

Table 1: List of FP words and example utterances333In the English translation of the example utterances, we set FP positions before the English words corresponding to the next words of the Japanese FPs and used “uh” and “um”(if there were two or more FP words) as FP words.in CSJ (the lecture ID is A05M0058). | FP word
---
Japanese (English)
Example of an utterance
えー (ee) |  |  それをどうえー判断するか
---
c.f.) how you uh judge it
え (e) |  |  え大きく内容分けますと
---
c.f.) uh roughly divide contents
ま (ma) |  あの音声だとか言語ま そういう分野もございますし c.f.) there are also areas such as um speech, language, uh and so on
あの (ano)
あのー (anoo) |  まーこれあのー本当の意味 よく分からないんで c.f.) I’m not sure what uh this um really means
まー (maa)
えーと (eeto) |  |  えーとこれはあのー部屋の
---
c.f.) um this is uh the room
あ (a) |  |  これがあ副次的な効果として
---
c.f.) this is uh a side effect of the
あー (aa) |  |  それからあー建物の中にも
---
c.f.) then, in uh the building
ん (n) |  |  ん何て言うんですかね
---
c.f.) uh what can I say
んー (nn) |  |  んー模型を現場に持ち込んで
---
c.f.) bring uh a model to the site
えっと (etto) |  |  えっとただ小さい模型作って
---
c.f.) uh just make a small model
あーのー (aanoo) |  |  あーのー持っていかせないと
---
c.f.) uh have to let him take it

### 3.2. Weighted cross entropy loss

Since FPs tend to be sparse in positions and biased in words, there is a
problem in that the model predicts only no FPs or highly frequent FPs [Ohta et
al., 2007]. Therefore, we use weighted cross entropy loss to build a model to
predict even less frequent FPs precisely [Yan et al., 2021]. The loss weights
of the $14$ predicted classes are the reciprocals of their frequencies in the
training data so that the losses of less frequent FPs have large values.

### 3.3. Rich word embedding model

A previous study [Yamazaki et al., 2020] used fastText [Bojanowski et al.,
2017], which is a lightweight model that generates word representations, as
the word embedding model. Whereas fastText calculates the word embedding
without considering context information (e.g., position and neighboring
words), Bidirectional Encoder Representations from Transformers (BERT) [Devlin
et al., 2019] calculates the embeddings from the entire input texts
considering context information. As previous studies have shown the
relationship between subsequent clause length and FP usage [Watanabe and
Shirahata, 2019], there is a long context dependency for FP insertion. Since
BERT is more appropriate than fastText to capture this, we compare their
prediction performances.

### 3.4. Personalized FP prediction model

This study aims to develop an FP prediction method taking into account
individuality. We can construct speaker-dependent models using a sufficient
amount of spontaneous speech data of target speakers; however, it requires a
large amount of time and data to train each target speaker’s model. To address
this issue, as shown in Figure 2, we propose a group-dependent model training
method based on speaker grouping as an efficient way to train models that
reproduce the individuality of FPs. First, we train a non-personalized model
using a multi-speaker spontaneous speech corpus that contains transcriptions
and FP annotations. Then, we train group-dependent models by fine-tuning the
non-personalized model. We also train speaker-dependent models for comparison
with these models.

Group-dependent model. We use a grouping method like hierarchical clustering
to group speakers and perform fine-tuning to update the parameters of the
sequence-to-sequence model with the data of each group using the parameters of
the non-personalized model as initial values. We use the model of the group
that has an FP tendency closest to the target speaker for the inference. This
method does not train the prediction model of each target speaker. Therefore,
we can reduce the cost to collect a large amount of spontaneous speech data of
each speaker. Moreover, we do not require time to train the model of each
speaker.

Figure 2: Constructing group-dependent models on the basis of tendency of FP
usage

Speaker-dependent model. To compare with the group-dependent model, we adapt
the non-personalized model to the speakers. We use mid-size ($3.5$–$5.0$
hours) spontaneous speech data. We manually search lecture videos on the web
and transcribe and annotate texts in accordance with the rules of the CSJ. The
data includes transcribed fluent texts as well as FP words, FP tags, and
phrase timings. We train speaker-dependent models with this data by fine-
tuning the non-personalized model in the same way as the group-dependent
models.

## 4\. Experiment

### 4.1. Experimental setting

#### 4.1.1. Evaluation criteria

We first describe the criteria of the evaluations. To score the model to
predict FPs’ positions and words, we used precision, recall, F-score, and
specificity. First, for positions, we defined precision as the rate of
positions actually having FPs out of those predicted to have FPs and recall as
the rate of positions predicted to have FPs out of those actually having FPs.
Then, we calculated the F-scores from these scores. The specificity was the
rate of positions predicted to have no FPs out of the positions actually
having no FPs. For each FP’s word, we calculated precision, recall, F-score,
and specificity in the same way. To score the prediction for FP words, we used
the average of each word’s score weighted by the frequency of each FP.

#### 4.1.2. Training

In the following experiments, we used the data of $137$ speakers from the CSJ.
As the utterance unit, we used the breath groups separated by silence for
$0.2$ seconds or longer. We separated each breath group into morphemes using
Sudachi [Takaoka et al., 2018] for fastText and Juman++ [Morita et al., 2015]
for BERT. For the word embedding model, we used fastText with the dimension
set to $300$444https://fasttext.cc/docs/en/crawl-vectors.html and BERT with
the version of the LARGE WWM published by the Kurohashi-Chu-Murawaki labs at
Kyoto University555https://nlp.ist.i.kyoto-u.ac.jp/?ku_bert_japanese. These
models were pre-trained on large Japanese text data on Common Crawl and
Wikipedia for fastText and on Wikipedia for BERT. We used BLSTM as the
embedding-to-FPtag model and set the number of hidden layers and hidden size
to $1$ and $1024$, respectively. We apply gradient clipping with the maximum
of the norm set to $0.5$ and set the batch size to $32$. Unless otherwise
stated, we set the learning rate to 1.0×$10^{-5}$ and trained the models for
$60000$ steps.

#### 4.1.3. Cross validation

To evaluate the results of the prediction by the models, we applied cross
validation. We divided the speakers into $10$ sets. Then, we trained the
models using $9$ of these sets as training data with the remaining set for
evaluation. The average of the evaluation scores obtained by repeating this
process $10$ times was the evaluation score of the model. We considered the
missed scores as $0.0$ and calculated the average score.

#### 4.1.4. Method of grouping speakers

We describe the method of grouping speakers. First, for clustering by FP word
usage, we calculated the rate of usage of each FP word by dividing the number
of each FP word usage by the total number of FP usage for each speaker. Then,
we applied hierarchical clustering using Euclidean distance and Ward’s method
[Jr., 1963]. We compared the results of the clustering by using a number of
distance thresholds and then set the threshold to $1.0$ which seems to have
the largest difference between clusters. We then classified the speakers into
$4$ classes. Next, for clustering by positional tendency, we use 1) head of
the sentence, 2) intra-sentence and boundary of breath group, 3) intra-
sentence and middle of breath group, and 4) end of the sentence as FP
positions, and calculated the rate of each FP position usage by dividing the
number of each FP position usage by the total number of FPs for each speaker.
Then, we applied hierarchical clustering using Euclidean distance and Ward’s
method. On the basis of the same criterion as the clustering by FP word, we
set the distance threshold to $1.7$ and then classified the speakers into four
classes on the basis of the distance.

### 4.2. Weighted cross entropy loss

To investigate whether the weighted cross entropy loss is effective for
predicting FPs, we first compared the prediction scores between the equal and
weighted losses. In this evaluation, for a word embedding model, we used
fastText which is the conventional model in [Yamazaki et al., 2020]. The
appropriate hyper-parameter settings with and without the weighted loss
function were different because the objective loss functions of the training
were different. In this evaluation, as hyper-parameters suitable for both
settings, we set the learning rate to 1.0×$10^{-3}$, multiplied $0.1$ every
$100000$ steps, and trained the model for $200000$ steps.

Table 2 lists the results. We can see that the F-scores are higher for both
positions and words in weighted than in equal, indicating that using loss
weights improved the performance of the prediction of positions and words. The
precision and recall of the model with weights are lower and higher,
respectively, for both positions and words. This suggests that introducing the
weights makes the model actively insert FPs and improves the recalls, but it
increases the number of mistakes. This is consistent with the result that the
position’s specificity is decreasing.

On the basis of this result, we used the weighted cross entropy loss in the
following experiments.

Table 2: Evaluation of weighted cross entropy loss Criterion | Equal | Weighted
---|---|---
Position | Precision | 0.307 | 0.292
Recall | 0.094 | 0.287
F-score | 0.143 | 0.288
Specificity | 0.997 | 0.989
Word | Precision | 0.088 | 0.078
Recall | 0.028 | 0.047
F-score | 0.042 | 0.054
Specificity | 0.999 | 0.999

### 4.3. Rich word embedding model

To investigate whether the BERT, a rich word embedding model is effective for
predicting FPs, we compared the prediction scores of fastText and BERT when
used as the word embedding model. In this evaluation, we used the hyper-
parameters described in Section 4.1.2.

Table 3 lists the results. We can see that BERT has higher F-scores than
fastText, which indicates that the prediction performance is improved by using
BERT as a word embedding model.

On the basis of this result, BERT was used as the word embedding model in the
following experiments.

Table 3: Comparison of fastText (lightweight embedding model) and BERT (rich embedding model) Criterion | fastText | BERT
---|---|---
Position | Precision | 0.237 | 0.254
Recall | 0.756 | 0.732
F-score | 0.360 | 0.376
Specificity | 0.961 | 0.964
Word | Precision | 0.069 | 0.070
Recall | 0.138 | 0.149
F-score | 0.065 | 0.089
Specificity | 0.996 | 0.994

### 4.4. Comparison of speaker-close and speaker-open prediction

We compared the prediction scores for speakers included (i.e. speaker-close)
and not included (i.e. speaker-open) in the training data. In this evaluation,
we used BERT as a word embedding model and the weighted cross entropy loss,
which were proven to perform better in previous evaluations. In the cross
validation of this evaluation, we also split the $9$ sets for training,
described in Section 4.1.3, to training and validation data in a ratio of
approximately $9$:$1$ with the speaker-close condition. We used that
validation data for speaker-close evaluation and the $1$ remaining set for
speaker-open evaluation.

Table 4 lists the results. We can see that the speaker-close prediction has
higher F-scores than speaker-open, indicating that the speaker-close
prediction has better performance for both position and word. However, the
difference between F-scores is only about $0.004$ for positions and $0.005$
for words, indicating that the speaker-open prediction achieves comparable
performance to the speaker-close one. Therefore, we can use the prediction
models to predict the FPs of the unseen speaker. Moreover, we can use the non-
personalized model to predict FPs of the unseen speaker when it is not
important to predict the personalized FPs but the non-personalized FPs are
required. The value of specificity is close to $1.0$, indicating that FPs are
rarely inserted in positions where there are actually no FPs. This is also
true for the other results.

Table 4: Comparison of speaker-open and speaker-close evaluation Criterion | Close | Open
---|---|---
Position | Precision | 0.264 | 0.263
Recall | 0.728 | 0.714
F-score | 0.387 | 0.383
Specificity | 0.966 | 0.966
Word | Precision | 0.073 | 0.071
Recall | 0.162 | 0.147
F-score | 0.096 | 0.091
Specificity | 0.994 | 0.994

### 4.5. Personalized FP prediction model

We first show the results of hierarchical clustering. Then, we describe the
prediction scores of the two types of personalized models: group-dependent
ones and speaker-dependent ones. Moreover, we present the scores of the
prediction of each FP and the distribution of the scores across speakers.
Finally, we describe the results of the prediction on lecture data of $2$
speakers. We trained the group-dependent models and speaker-dependent models
for $10000$ steps. In the cross validation of group-dependent models, we set
the number of speaker partitions to $5$ unlike the other experiments, since
the amount of data in each group was smaller than before. For speaker-
dependent models, as described in Section 3.4, we used the lecture data of the
University of Tokyo available on
YouTube666https://youtube.com/playlist?list=PLHxBhbJJasnfX6oBrkTygP8we61wEcRha
for $2$ speakers. The test data for each speaker was $20$ paragraphs. We split
the rest of the data into training and validation data in a ratio of
approximately $9$:$1$.

#### 4.5.1. Result of clustering

We describe the characteristics of the classes into which the speakers were
classified by clustering on the basis of their tendency to use FPs.

Figure 3: Rate of each FP word in each cluster by FP word usage Figure 4: Rate
of each FP position in each cluster by FP position usage

Figure 3 shows the results of calculating each FPs’ word rate for each class
by FP words. Cluster $1$, $2$, $3$, and $4$ contain $18$, $55$, $25$, and $39$
speakers respectively. Frequently used FPs are “ee” in Cluster $2$, “ano” and
“anoo” in Cluster $3$, “ma” and “maa’ in Cluster $4$, whereas ”ee” is
frequently used in Cluster $1$ and FPs other than ”ee” and ”e” are rarely
used. Figure 4 shows the results of calculating each FPs’ position rate for
each class by FP positions. Clusters $1$, $2$, $3$, and $4$ contain $50$,
$13$, $27$, $47$ speakers respectively. FPs are used more frequently in the
middle of the breath group in Cluster $1$, at the head of sentences in Cluster
$2$, and at the beginning of the breath group in Cluster $3$, whereas Cluster
$4$ shows an average tendency among all the classes.

#### 4.5.2. Evaluation of group-dependent models

To investigate the effectiveness of prediction by the proposed group-dependent
models, we compared the prediction scores of these models with those of the
non-personalized model.

Table 5 lists the results. We can see that the F-scores of all the group-
dependent models based on FP words are higher than that of the non-
personalized model for both positions and words, indicating that the grouping
of speakers by FP words proposed in this paper improves the performance of the
prediction. Table 6 shows that the F-scores of the group-dependent models are
higher than that of the non-personalized model, except for the position’s
F-score of Cluster $2$, indicating that the grouping of speakers by FP
positions proposed in this paper improves the performance for almost all the
group-dependent models.

Table 5: Evaluation of group-dependent models based on FP words (NP represents the non-personalized model.) Criterion (F-score) | NP | Group-dependent (word)
---|---|---
1 | 2 | 3 | 4
Position | 0.376 | 0.454 | 0.456 | 0.427 | 0.390
Word | 0.089 | 0.284 | 0.288 | 0.248 | 0.196
Table 6: Evaluation of group-dependent models based on FP positions (NP represents the non-personalized model.) Criterion (F-score) | NP | Group-dependent (position)
---|---|---
1 | 2 | 3 | 4
Position | 0.376 | 0.461 | 0.323 | 0.413 | 0.444
Word | 0.089 | 0.277 | 0.212 | 0.158 | 0.237

#### 4.5.3. Evaluation of prediction for each FP

To investigate whether the group-dependent models can reproduce the diversity
of FP words, we show the prediction scores of each FP.

Figure 5 shows the F-scores of the prediction on each FP word in the group-
dependent models by FP words and that in the group-dependent models by FP
positions. The horizontal axis from left to right represents more to less
frequent FP words, respectively, in the corpus. The word’s Cluster $2$ and
position’s Clusters $1$ and $4$ models have higher F-scores than the non-
personalized model for all the FP words. In contrast to the non-personalized
model, which predicts highly and less frequent FPs more and less precisely,
respectively, the aforementioned models have F-scores for less frequent FPs
close to that of frequent FPs, indicating that these models can reproduce the
diversity of FP words. The word’s Clusters $3$ and $4$, and position’s Cluster
$3$ models have better scores than the non-personalized model for FPs other
than ”ee,” and the scores for less frequent FPs are high, also indicating the
ability to reproduce the diversity. The model for word’s Cluster $1$ model has
lower scores than the non-personalized model for a number of FPs. In addition,
the position’s Cluster $2$ model has low F-scores of 0 for $8$ FPs.

Figure 5: F-score for each FP word on group-dependent model

#### 4.5.4. Evaluation of predictions for each speaker

To investigate the prediction performance of the group-dependent models for
each speaker, we show the distribution of the prediction scores across
speakers.

We describe the distribution of each speaker’s prediction score by the group-
dependent models. The cross validation was not performed, but the data was
divided into training, validation, and test data in an approximate ratio of
$3$:$1$:$1$ under the speaker-close condition, and the results were evaluated
using the test data. Figure 6 shows the results. The F-scores for each cluster
of positions and words are widely distributed, indicating that grouping
speakers improves the performance on average, but the tendency of improvement
differs from speaker to speaker. Therefore, further research is needed to
investigate which speakers have a worse prediction performance and construct
prediction models that perform well for them.

Figure 6: F-score for each speaker on group-dependent model

#### 4.5.5. Evaluation on lecture data

To investigate whether the models can be adapted to speakers, we evaluated the
prediction scores of the speaker-dependent models. We also compared the
prediction performance of the non-personalized, group-dependent, and speaker-
dependent models on the lecture data of $2$ speakers.

Tables 7 and 8 list the results of speakers A and B, respectively. We can see
that speaker-dependent models have lower scores than the non-personalized
model for both speakers, indicating that it is difficult to adapt the
prediction model to speakers. A possible reason is that the variation of the
usage tendency within each speaker is large and the tendency differs between
the training and evaluation data.

In Table 7, we can see that the score of the non-personalized model is the
highest, followed by the speaker-dependent model, and the group-dependent
models have the lowest score. In Table 8, we can see that the score of the
non-personalized model is the highest, followed by the group-dependent models,
and the speaker-dependent model has the lowest score. For speakers in which
the non-personalized model has the best performance and a high prediction
score was same for both speaker A and B, we can use the non-personalized model
for inference.

#### 4.5.6. Discussion

The experimental results demonstrated that grouping speakers on the basis of
their FP usage enabled the construction of group-dependent models with higher
prediction F-scores than the non-personalized model. One possible reason for
this is that grouping by FP usage reduces the variation in usage tendency
within the data group, which makes the model training easier. Another possible
reason is that the difference in tendency between the training and evaluation
data is reduced. This indicates the effectiveness of our experimental results
because the prediction model of the group close to the target speaker’s data
is actually used to predict unseen speakers.

In the evaluation using the lecture data, the F-score of the non-personalized
model was the highest. Considering the results in Figure 6, which shows that
the scores of the prediction models differed among speakers, suggesting that
the $2$ speakers in this experiment have particularly worse prediction
performance by the group-dependent models. Considering that speaker A has low
scores for all the models, one possible reason is that speaker A has a
particularly unusual usage tendency. Considering that speaker B has the best
score for the non-personalized model, one possible reason is that speaker B
has a usage tendency close to the common tendency among all the speakers. For
such speakers, we can use the non-personalized model for inference.

Table 7: Evaluation on lecture data of speaker A (NP represents the non-personalized model.) | Criterion
---
(F-score)
NP |  | Word
---
(Cluster 4)
| Position
---
(Cluster 2)
Speaker
Position | 0.243 | 0.137 | 0.114 | 0.146
Word | 0.061 | 0.016 | 0.018 | 0.057
Table 8: Evaluation on lecture data of speaker B (NP represents the non-personalized model.) | Criterion
---
(F-score)
NP |  | Word
---
(Cluster 4)
| Position
---
(Cluster 1)
Speaker
Position | 0.384 | 0.302 | 0.366 | 0.212
Word | 0.158 | 0.070 | 0.117 | 0.027

## 5\. Conclusion

To achieve FP prediction to reproduce individuality, we proposed a method to
construct group-dependent models with higher scores than the general model by
grouping speakers on the basis of their FP usage. This method made it possible
to predict the target speaker’s FP without learning the speaker-dependent
model of the target speaker each time. Furthermore, we introduced a weighted
loss function to address the sparsity of FP positions and the bias of FP words
and a rich word embedding model, and demonstrated that the performance of the
prediction was improved. However, since we found that the prediction
performance varied among speakers, we need to investigate which speakers have
worse prediction performance and address performance improvement for those
speakers. Moreover, our future work will involve synthesizing spontaneous
speech containing FPs predicted by the group-dependent models proposed in this
paper and subjectively evaluating individuality.

## 6\. Bibliographical References

## References

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# A new approach to the study of elliptic semilinear equations

Pilar Herreros  Departamento de Matemática, Pontificia Universidad Católica de
Chile, Casilla 306, Correo 22, Santiago, Chile<EMAIL_ADDRESS>
###### Abstract.

In this paper we define a new operator $J$ for the study of

$\Delta u+f(u)=0,\quad x\in\mathbb{R}^{N},N>2.$

Using $J$ we can easily see some qualitative properties of the solutions, for
example we can determine how many times $u$ changes sign, which are the values
of the local maxima and minima, and where $u$ changes concavity.

We also use this functional to construct nonlinearities $f$ such that this
problem has at least two bound state solutions that change sign $j$ times, for
$j=1,\dots,k-1$. And another $f$ such that this problem has a unique ground
state solution, and at least two bound state solutions that change sign one
time.

##  1\. Introduction and main results

The main goal of this paper is to introduce a new approach for the study of
radial solutions of

$\displaystyle\begin{gathered}\Delta u+f(u)=0,\quad x\in\mathbb{R}^{N},N>2,\\\
\lim\limits_{|x|\to\infty}u(x)=0.\end{gathered}$ (1.3)

Any nonconstant solution to (1.3) is called a bound state solution. Bound
state solutions such that $u(x)\geq 0$ for all $x\in\mathbb{R}^{N}$, are
referred to as a first bound state solution, or a ground state solution. A
solution that changes sign $k-1$ times will be referred to as a $k^{th}$ bound
state.

We will work under the following assumptions on the nonlinearity $f$, that
determine the regularity and general shape of $f$:

  1. $(H_{1})$

     * •

$f\in C[0,\infty)\cap C^{1}(0,\infty)$, except possibly in a finite number of
points where it is not differentiable.

     * •

$f(0)=0$ and there exist $b\geq 0$ such that $f(s)>0$ for $s>b$, $f(s)\leq 0$
for $s\in[0,b]$ and moreover $f(s)<0$ on $(0,\epsilon)$ for some $\epsilon>0$.

     * •

We extend $f$ to $\mathbb{R}$ by $f(s)=-f(-s)$.

     * •

Using the notation

$F(s)=\int_{0}^{s}f(t)dt,$

there exists a unique $\beta\geq b$ such that $F(\beta)=0$.

In $1981$ Gidas, Ni and Nirenberg in [GNN] proved that all ground state
solutions to this problem are radially symmetric, and many other symmetry
results have followed (see, for example, [BN, FL, LN]). In this work we will
restrict ourselves to radial solutions. Therefore we will work with the radial
version of (1.3), that is

$\displaystyle\begin{gathered}u^{\prime\prime}+\frac{N-1}{r}u^{\prime}+f(u)=0,\quad
r>0,\quad N>2,\\\
u^{\prime}(0)=0,\quad\lim\limits_{r\to\infty}u(r)=0,\end{gathered}$ (1.6)

where ′ denotes differentiation with respect to $r$.

We will approach this by understanding the behaviour of solutions of the
initial value problem

$\displaystyle\begin{gathered}u^{\prime\prime}+\frac{N-1}{r}u^{\prime}+f(u)=0,\quad
r>0,\quad N>2,\\\ u(r_{0})=\alpha,\quad
u^{\prime}(r_{0})=\bar{\alpha}\end{gathered}$ (1.9)

for some $r_{0}\geq 0,\ \alpha>0$ and $\bar{\alpha}\in\mathbb{R}$.

Given a solution $u$ of (1.9), we introduce the operator

$J(u(r))=-\frac{u^{\prime}(r)}{r}.$

Different values of $J$, when compared to $f(u(r))$, mark different points in
the solution $u$ and help to understand the behaviour of such solutions. For
example, from de values of $J$, we can determine how many times $u$ changes
sign, which are the values of the local maxima and minima, and where $u$
changes concavity. We refer to Section 3 for more details.

The study of uniqueness (or multiplicity) of solutions to (1.6) involves the
comparison of solutions, usually solutions that have initial conditions close
to each other. As part of this new approach, we revisit some of the operators
used for the study of solutions and write them in terms of $J$. For this we
will require $f$ to satisfy a subcriticality condition:

  1. $(H_{2})$

$(F/f)^{\prime}(s)>(N-2)/(2N)$ for all $s>\beta$;

We use this condition to prove that the operators $J$ corresponding to
different solutions that change sign compare well. They do not intersect
unless they have negative energy. See Proposition 4.1.

As an application of this approach, in Section $5$ we construct functions $f$
that have multiple $k^{th}$-bound states. We will approach this problem using
a function $f$ defined by parts as

$\displaystyle
f_{\mu}(s)=\begin{cases}f_{1}(s)&s\leq\alpha_{1}=\alpha_{*}^{k}+\epsilon\\\
L(s)&\alpha_{1}\leq s\leq\alpha_{1}+\epsilon\\\
\lambda^{2}f_{2}\left(\frac{s}{\mu}\right)&s\geq\alpha_{1}+\epsilon\end{cases}$
(1.10)

where $f_{1}$ satisfies $(H_{1})$ and $(H_{2})$, $L(s)$ is the line from
$(\alpha_{1},f_{1}(\alpha_{1}))$ to
$(\alpha_{1}+\epsilon,\lambda^{2}f_{2}(\alpha_{1}+\epsilon))$, and
$\alpha_{1}$ and $\epsilon$ are given and will be chosen later. We will
further assume

  * $(H_{3})$

$(sf^{\prime}/f)(s)$ decreasing for all $s>b$, with
$(sf^{\prime}/f)(\beta)<\frac{N}{N-2}$.

  * $(H_{4})$

There is an initial condition $\alpha_{*}^{k}$ such that the solution to

$\displaystyle\begin{gathered}u^{\prime\prime}+\frac{N-1}{r}u^{\prime}+f_{1}(u)=0,\quad
r>0,\quad N>2,\\\ u(0)=\alpha_{*}^{k},\quad u^{\prime}(0)=0,\end{gathered}$
(1.13)

is a $k^{th}$-bound state solution.

In the first theorem we prove multiplicity of all $j$-bound state solutions,
for $j=0,1\dots,k$. For this we will consider $f_{2}$ that satisfy

  * $(H_{5})$

there is an initial condition $\widehat{\alpha}>\alpha_{*}^{k}$ such that the
solution to

$\displaystyle\begin{gathered}v^{\prime\prime}+\frac{N-1}{r}v^{\prime}+f_{2}(u)=0,\quad
r>0,\quad N>2,\\\ v(0)=\widehat{\alpha},\quad v^{\prime}(0)=0,\end{gathered}$
(1.16)

reaches $v(r_{0})=0$ with $v^{\prime}(r_{0})<0$.

######  Theorem A.

Assume that $f_{1}$, $\alpha_{*}^{k}$, $f_{2}$ and $\widehat{\alpha}$ satisfy
the assumptions above and
$0<\epsilon<\min\\{\beta/4,(\widehat{\alpha}-\alpha^{k}_{*})/2\\}$. There is a
positive constant $\bar{\mu}(\epsilon)$ such that, for any $\mu>\bar{\mu}$
there is a $\bar{\lambda}(\mu)$ such that for all $\lambda>\bar{\lambda}$,
problem (1.3) with $f_{\mu}$ given by (1.10) has at least two bound state
solutions that change sign $j$ times, for $j=1,\dots,k-1$.

The second theorem is perhaps more surprising, it shows that sometimes there
is multiplicity of $2^{nd}$-bound states but only one ground state solution.
For this we will consider a function of the form

$\displaystyle
f_{a}(s)=\begin{cases}f_{1}(s)&s\leq\alpha_{1}=\alpha_{*}^{k}+\epsilon\\\
L(s)&\alpha_{1}\leq s\leq\alpha_{1}+\epsilon\\\
\lambda^{2}(u+a)^{p}&s\geq\alpha_{1}+\epsilon\end{cases}$ (1.17)

with $f_{1}$ that satisfies $(H_{1})-(H_{4})$, and $p>\frac{N+2}{N-2}$.

######  Theorem B.

Assume that $f_{1}$, $\alpha_{*}^{k}$, $\alpha_{*}^{k+1}$ and $f_{a}$ as given
by (1.17), and $0<\epsilon<\beta/4$. Then there are constants $a$ and a
$\bar{\lambda}(a)$ such that for all $\lambda>\bar{\lambda}$, problem (1.3)
with $f=f_{a}$ has at least two bound state solutions that change sign $k$
times. Moreover, there are no bound state solutions that change sign $j$
times, for $j<k$, with initial condition $\alpha>\alpha_{*}^{k}$.

######  Corollary 1.1.

Assume that $f_{1}$, $\alpha_{*}^{1}$, $\alpha_{*}^{2}$ and $f_{a}$ as given
by (1.17), and $0<\epsilon<\beta/4$. and $f_{1}$ has a unique ground state
solution with initial condition in $(\beta,\alpha_{*}^{2})$. Then there are
positive constant $a$, $\epsilon$ and $\bar{\lambda}$ such that for all
$\lambda>\bar{\lambda}$, problem (1.3) with $f=f_{a}$ has a unique ground
state solution, and at least two bound state solutions that change sign one
time.

##  2\. Some properties of the solutions of the initial value problem

The aim of this section is to establish several properties of the solutions to
the initial value problem (1.9). Since $f$ is continuous, problem (1.9) has a
solution defined for all $r\geq 0$ for any $\alpha>\beta$ but it might not be
uniquely defined. It is straight forward to see that unique extendibility can
be lost only if $u$ reaches a double zero. In this case, we will extend the
solution as $0$, and consider it a bound state solution.

By standard theory of ordinary differential equations, the solution depends
continuously on the initial data in any compact subset of its domain of
definition.

Let us set

$Z_{1}(\alpha)=\sup\\{r>0\ |\ u(s,\alpha)>0\mbox{ and }u^{\prime}(s,\alpha)<0\
\mbox{ for all }s\in(0,r)\\}$

and define

$\displaystyle{\mathcal{N}_{1}}$ $\displaystyle=$
$\displaystyle\\{\alpha\in[\beta,\gamma_{*})\ :\
u(Z_{1}(\alpha),\alpha)=0\quad\mbox{and}\quad
u^{\prime}(Z_{1}(\alpha),\alpha)<0\\}$ $\displaystyle{\mathcal{G}_{1}}$
$\displaystyle=$ $\displaystyle\\{\alpha\in[\beta,\gamma_{*})\ :\
u(Z_{1}(\alpha),\alpha)=0\quad\mbox{and}\quad
u^{\prime}(Z_{1}(\alpha),\alpha)=0\\}$ $\displaystyle{\mathcal{P}_{1}}$
$\displaystyle=$ $\displaystyle\\{\alpha\in[\beta,\gamma_{*})\ :\
u(Z_{1}(\alpha),\alpha)>0\\},$

where $\beta$ is as defined in $(H_{1})$. If $Z_{1}(\alpha)=\infty$ we
consider $u(Z_{1}(\alpha),\alpha)=\lim_{r\to\infty}u(r,\alpha)$ that can only
be $0$ or $b$, thus $\alpha\in{\mathcal{G}_{1}}$ or ${\mathcal{P}_{1}}$
respectively.

We now extend these definitions by induction for $k\geq 2$. If
${\mathcal{N}_{k-1}}\not=\emptyset$, we set

$\displaystyle T_{k-1}(\alpha)=$
$\displaystyle\sup\\{r\in(Z_{k-1}(\alpha),D_{\alpha})\ :\
(-1)^{k}u^{\prime}(r,\alpha)\leq 0\\},$

and

$\displaystyle Z_{k}(\alpha)=\sup\\{r>T_{k-1}(\alpha)\ |\
(-1)^{k}u(s,\alpha)<0\mbox{ and }(-1)^{k}u^{\prime}(s,\alpha)>0\ $
$\displaystyle\mbox{ for all }s\in(T_{k-1}(\alpha),r)\\},$

if $T_{k-1}(\alpha)=\infty$, we set $Z_{k}(\alpha)=\infty$.

Figure 1. Solutions of (1.9) with initial condition in these sets

We now define

$\displaystyle{\mathcal{N}_{k}}$ $\displaystyle=$
$\displaystyle\\{\alpha\in\mathcal{N}_{k-1}\ :\
u(Z_{k}(\alpha),\alpha)=0\quad\mbox{and}\quad(-1)^{k}u^{\prime}(Z_{k}(\alpha),\alpha)>0\\},$
$\displaystyle{\mathcal{G}_{k}}$ $\displaystyle=$
$\displaystyle\\{\alpha\in\mathcal{N}_{k-1}\ :\
u(Z_{k}(\alpha),\alpha)=0\quad\mbox{and}\quad
u^{\prime}(Z_{k}(\alpha),\alpha)=0\\},$ $\displaystyle{\mathcal{P}_{k}}$
$\displaystyle=$ $\displaystyle\\{\alpha\in\mathcal{N}_{k-1}\ :\
(-1)^{k}u(Z_{k}(\alpha),\alpha)<0\\}.$

We will use some known properties of solutions of (1.9) and the sets defined
above, collected in the following Lemma. For a proof see for instance [CGHH1]
and references therein.

######  Lemma 2.1.

Assume that $f$ satisfies $(H_{1})$ and let $k\in\mathbb{N}$.

  1. (i)

The sets ${\mathcal{N}_{k}}$ and ${\mathcal{P}_{k}}$ are open in
$[\beta,\gamma_{*})$.

  2. (ii)

The boundary of $\mathcal{G}_{k}\cup\mathcal{P}_{k}$ is contained in
$\bigcup_{i=1}^{k}\mathcal{G}_{i}$.

  3. (iii)

Any solution $u$ of (1.9) has at most a finite number of sign changes.

We will also need a bound on $r$ and $r|u^{\prime}(r)|$ when the solution
crosses a given interval.

######  Lemma 2.2.

Let $v$ be the solution to the initial value problem

$\displaystyle\begin{gathered}v^{\prime\prime}+\frac{N-1}{r}v^{\prime}+g(v)=0,\quad
r>r_{\delta},\\\ v(r_{\delta})=\bar{\alpha}+\delta,\quad
v^{\prime}(r_{\delta})=v^{\prime}_{\delta}<0\end{gathered}$ (2.3)

where $g$ is a positive continuous function defined in
$[\bar{\alpha},\bar{\alpha}+\delta]$ and $r_{\delta}>0$. Let $\bar{r}$ be
defined by $v(\bar{r})=\bar{\alpha}$,
$||g||_{+}=\max\\{g(s):s\in[\bar{\alpha},\bar{\alpha}+\delta]\\}$. Consider
$\frac{\zeta}{N-2}>\delta$ and $B>1$ given by
$\delta\frac{B^{N-2}}{B^{N-2}-1}=\frac{\zeta}{N-2}$.

If $\zeta<r_{\delta}|v^{\prime}(r_{\delta})|$ then $\bar{r}<Br_{\delta}$ and

$\frac{\zeta}{B^{N-2}}\leq\bar{r}|v^{\prime}(\bar{r})|\leq
r_{\delta}|v^{\prime}(r_{\delta})|+\frac{B^{N}-1}{N}||g||_{+}r_{\delta}^{N}.$

###### Proof.

Since $v$ is a solution of (2.3), we have that for $r\in[r_{\delta},\bar{r}]$

$r^{N-1}v^{\prime}(r)=r_{\delta}^{N-1}v^{\prime}(r_{\delta})-\int_{r_{\delta}}^{r}t^{N-1}g(v(t))dt<r_{\delta}^{N-1}v^{\prime}(r_{\delta})$
(2.4)

dividing by $r^{N-1}$ and integrating over $[r_{\delta},\bar{r}]$ we get

$\delta=v(r_{\delta})-v(\bar{r})>\frac{r_{\delta}^{N-1}|v^{\prime}(r_{\delta})|}{N-2}\left(\frac{1}{r_{\delta}^{N-2}}-\frac{1}{\bar{r}^{N-2}}\right)$

implying

$\delta>\frac{r_{\delta}|v^{\prime}(r_{\delta})|}{N-2}\left(1-\frac{r_{\delta}^{N-2}}{\bar{r}^{N-2}}\right)>\frac{\zeta}{N-2}\left(1-\frac{r_{\delta}^{N-2}}{\bar{r}^{N-2}}\right)>\delta\frac{B^{N-2}}{B^{N-2}-1}\left(1-\frac{r_{\delta}^{N-2}}{\bar{r}^{N-2}}\right).$
(2.5)

Thus $\bar{r}<Br_{\delta}$. Moreover, from (2.4),

$\bar{r}|v^{\prime}(\bar{r})|\geq\Bigl{(}\frac{r_{\delta}}{\bar{r}}\Bigr{)}^{N-2}r_{\delta}|v^{\prime}(r_{\delta})|\geq\frac{1}{B^{N-2}}\zeta.$

Finally, using again (2.4) and since $r_{\delta}<\bar{r}<Br_{\delta}$, we
obtain

$\bar{r}|v^{\prime}(\bar{r})|\leq
r_{\delta}|v^{\prime}(r_{\delta})|+\frac{B^{N}-1}{N}||g||_{+}r_{\delta}^{N}\ $

∎

##  3\. The operator $J$

In this section we will define the operator $J$, and prove some basic facts
and how it can give information on a solution if the initial value problem
1.9. In this section we only assume that $f$ satisfies $(H_{1})$.

Let us set $\rho_{0}=0$,

$\rho_{i}(\alpha):=\sup\\{r>\rho_{i-1}\ |\ (-1)^{i}u^{\prime}(s,\alpha)>0\
\mbox{ for all }s\in(\rho_{i-1},r)\\},$

and $m_{i}=u(\rho_{i}(\alpha))$. Note that if $u(\rho_{i})=0$ with
$\rho_{i}<\infty$, we can extend the solution as $u=0$ and thus it is a bound
state solution.

Since solutions $u(r)$ are monotone decreasing in $[0,\rho_{1}(\alpha)]$, we
can study instead its inverse $r(s)$ for $s\in[m_{1},\alpha]$ which satisfies
the equation

$r^{\prime\prime}(s)-\frac{N-1}{r(s)}(r^{\prime}(s))^{2}-f(s)(r^{\prime}(s))^{3}=0.$
(3.1)

######  Definition 1.

For each solution $u(r)$ with $u(0)=\alpha$ and $u^{\prime}(0)=0$, and its
inverse $r(s)$ we define

$J(s)=\frac{|u^{\prime}|}{r}=\frac{-1}{r(s)r^{\prime}(s)}.$ (3.2)

It follows directly from equation (3.1) that $J$ satisfies

$J^{\prime}=\frac{1}{r^{2}}\left(N-\frac{f}{J}\right)=\frac{N}{Jr^{2}}\left(J-\frac{f}{N}\right),$
(3.3)

with $J(\alpha)=\frac{f(\alpha)}{N}$ and
$J^{\prime}(\alpha)=\frac{f^{\prime}(\alpha)}{N+2}<\frac{f^{\prime}(\alpha)}{N}$.
Thus $J>\frac{f}{N}$ for $s$ in some interval $(\alpha-\epsilon,\alpha)$.

Figure 2. $J$ of a solution with $\alpha\in\mathcal{P}_{1}$

After $\rho_{1}(\alpha)$ the solution $u$ will be monotonously increasing in
$[\rho_{1}(\alpha),\rho_{2}(\alpha)]$, we can study instead its inverse
$\bar{r}(s)$ for $s\in[m_{1},m_{2}]$. In this interval we define

$\bar{J}(s)=\frac{-1}{\bar{r}(s)\bar{r}^{\prime}(s)}$ (3.4)

and note that $\bar{J}(s)$ corresponds to $\frac{-u^{\prime}}{r}$. In this way
we can define $\mathcal{J}(r)$ along all the solution using $r(s)$ or
$\bar{r}(s)$ when appropriate. Note also that in both cases

$\mathcal{J}(r)=\frac{-u^{\prime}(r)}{r}=J(u(r))\ \mbox{ or }\ \bar{J}(u(r))$
(3.5)

thus $\mathcal{J}$ is differentiable in all $\mathbb{R}$.

Note also that if $v$ is the solution to (1.9) with $v(0)=-\alpha$, then
$\bar{J}_{u}(s)=-J_{v}(-s)$, thus all the analysis made for $J$ is also valid
for $\bar{J}$ with the appropriate sign changes.

In the next subsection we will prove that $J$ spirals counterclockwise, and
from their graphs we can identify ground states, bound states, and how many
times solutions cross $0$, among other things. First we will prove some basic
facts.

######  Lemma 3.1.

For any solution $r(s)$ on $[u(\rho_{1}(\alpha)),\alpha]$, $J(s)$ and $r(s)$
satisfy

  1. (i)

$J^{\prime}(s)>0$ if and only if $J(s)>\frac{f(s)}{N}$.

  2. (ii)

$r^{\prime\prime}(s)<0$ if and only if $J(s)>\frac{f(s)}{N-1}$.

  3. (iii)

$J(s)=0$ when $s=\rho_{1}(\alpha)$, and at this point $f(\rho_{1}(\alpha))\leq
0$.

  4. (iv)

If for some $\bar{s}$, $J(\bar{s})=\frac{f(\bar{s})}{N}\neq 0$, with
$J(s)>\frac{f(\bar{s})}{N}$ for $s>\bar{s}$, then
$f^{\prime}(\bar{s})<0=J^{\prime}(\bar{s})$.

###### Proof.

Follows directly from equation (3.3) and the fact that (3.1) can be written as

$r^{\prime\prime}(s)=(r^{\prime}(s))^{3}\left(f(s)-(N-1)J(s)\right).$

∎

Figure 3. $J(s)$ and $u(r)$ of a solution with $\alpha\in\mathcal{P}_{1}$

###  3.1. General behavior of $J$ and $u$

We will study some facts about the behaviour of $J$ and how it relates to the
behaviour of $u$. We can summarize this relation in the following proposition.

######  Proposition 3.2.

For each solution $u$ of (1.9), with $u(0)=\alpha$, the graph of the
functionals $J$ and $\bar{J}$, or equivalently the curve
$(u(r),\mathcal{J}(r))$ for $r\geq 0$, satisfies the following.

  1. (i)

It starts at $\left(\alpha,\frac{f(\alpha)}{N}\right)$, and spirals inwards,
counterclockwise, without self intersections.

  2. (ii)

It will first rotate around the origin, intersecting the $s$-axis outside
$(-\beta,\beta)$. In this process, it crosses the line $s=0$ $k$ times, $0\leq
k<\infty$, indicating that $\alpha\in\mathcal{N}_{k}$ and
$\alpha\notin\mathcal{N}_{k+1}$.

  3. (iii)

After crossing $s=0$ for the $k^{th}$ time, it rotates around $(-1)^{k}b$,
crossing the $s$-axis $j$ times, $0\leq j\leq\infty$. If $k=0$, we count the
initial point as $j=1$ and add the crossings after that.

    1. (a)

If $j=0$, then $J(-b)=0$ (or $\bar{J}(b)=0$), $u$ tends asymptotically to $\pm
b$, and $\alpha\in\mathcal{P}_{k+1}$.

    2. (b)

If $j=1$ then either $J(-b)=0$ (or $\bar{J}(b)=0$), $u$ tends asymptotically
to $\pm b$, and $\alpha\in\mathcal{P}_{k+1}$, or $J(0)=0$ and
$\alpha\in\mathcal{G}_{k+1}$, i.e. $u$ is a $k$-th bound state solution.

    3. (c)

If $j\geq 2$, $\alpha\in\mathcal{P}_{k+1}$ and after $j$ intersections $J$
ends in $J(\pm b)=0$ (or $\bar{J}(\pm b)=0$) and $u$ tends asymptotically to
$\pm b$. If $j=\infty$ it spirals around $\pm b$ forever.

Figure 4. $\mathcal{J}(s)$ and $u(r)$ for different initial conditions

###### Proof.

First, we will consider the classical energy functional
$I(r)=\frac{|u^{\prime}|}{2}+F(u)$, now in terms of $s$.

The energy $I(s)=\frac{r^{2}J^{2}}{2}+F$ has $I^{\prime}(s)=(N-1)J(s)$, so
$I(u(r))$ is decreasing with respect to $r$.

Also, if for $r_{1}<r_{2}$ we have $u(r_{1})=u(r_{2})$ and
$\mathcal{J}(r_{1})=\mathcal{J}(r_{2})$ then

$I(s(r_{1}))=\frac{r_{1}^{2}{\mathcal{J}}^{2}(s)}{2}+F(s)<\frac{r_{2}^{2}{\mathcal{J}}^{2}(s)}{2}+F(s)=I(s(r_{2})),$

a contradiction. Therefore $(u(r),\mathcal{J}(r))$ has no self intersections,
proving $(i)$.

Since $I(s)=J^{2}r^{2}(s)+F(s)<F(\alpha)$, $J$ must reach $0$ in
$(-\alpha,\alpha)$. Moreover, by Lemma 3.1$(iii)$ it must reach $0$ when
$f\leq 0$, therefore it has three options:

  1. (1)

$J(s)=0$ for some $s\in(0,b]$, then $\alpha\in\mathcal{P}_{1}$.

  2. (2)

$J(0)=0$, then $u$ is a ground state solution, $\alpha\in\mathcal{G}_{1}$.

  3. (3)

$J(0)>0$, then $\alpha\in\mathcal{N}_{1}$.

We will continue the study of the third case.

After $J$ reaches $0$, it has to reach $J(s)=0$ with $s\in(-\alpha,-b]$.

If $J(-b)=0$ then $u^{\prime\prime}=u^{\prime}=0$ and by unique continuation
$r=\infty$, therefore $u$ decreases asymptotically to $-b$.

If $J(m)=0$ with $m\in(-\alpha,-b)$, $u^{\prime}(r(m))=0$ with
$u^{\prime\prime}(r(m))=-f(m)>0$, a minimum, and we use $\bar{r}$ to study the
solution going up. Since $\bar{J}>f/N$, $\bar{J}^{\prime}<0$ until it reaches
$\bar{J}=f/N$ with $\bar{J}^{\prime}=0<f^{\prime}/N$, therefore they cross and
$\bar{J}^{\prime}>0$ there after.

As in Lemma 3.1 $(iv)$, if for some $\bar{s}$,
$\bar{J}(\bar{s})=\frac{f(\bar{s})}{N}$ again, then $f^{\prime}(\bar{s})<0$.
Therefore $\bar{J}<f/N$ until either $s=0$ or $\bar{J}=0$, since
$f^{\prime}>0$ or $f>0$ for $s<0$.

Therefore $\bar{J}$ has three options:

  1. (1)

$\bar{J}(s)=0$ for some $s\in[-b,0)$, then $\alpha\in\mathcal{P}_{2}$.

  2. (2)

$\bar{J}(0)=0$, then $u$ is a ground state solution,
$\alpha\in\mathcal{G}_{2}$.

  3. (3)

$\bar{J}(0)<0$, then $\alpha\in\mathcal{N}_{2}$.

For solutions in $\mathcal{N}_{2}$ we can repeat the same argument, until the
solution is in $\mathcal{G}_{k}$ or $\mathcal{P}_{k}$. This process ends in
finite steps since, by Lemma 2.1$(iii)$, there are no
$\alpha\in\mathcal{N}_{k}$ for all $k\in\mathbb{N}$.

$(iii)$ Solutions in $\mathcal{P}_{k}$:

We will write the argument for a solution in $\mathcal{P}_{1}$ for simplicity,
but it is the same for any $\mathcal{P}_{k}$.

If $J(b)=0$ then $u^{\prime\prime}=u^{\prime}=0$ and by unique continuation
$r=\infty$, therefore $u$ decreases asymptotically to $b$.

If $J(m)=0$ for $m\in(0,b)$, $u$ has a minimum and we use $\bar{r}$ to study
the solution going up. Since $\bar{J}>f/N$, $\bar{J}^{\prime}<0$ until it
reaches $\bar{J}=f/N$ with $\bar{J}^{\prime}=0<f^{\prime}/N$, therefore they
cross and $\bar{J}^{\prime}>0$ there after with $\bar{J}<f/N$ until $J=0$,
with $f>0$ ($s\geq b$).

If $\bar{J}(b)=0$ $u$ increases asymptotically to $b$. If $\bar{J}(M)=0$ for
$M\in(b,\alpha)$, $u$ has a maximum and we use $r$ to study the solution going
down. Since $J<f/N$, $J^{\prime}>0$ until it reaches $J=f/N$ with
$J^{\prime}=0>f^{\prime}/N$, therefore they cross and $J^{\prime}>0$ there
after with $J>f/N$ until $J=0$, with $f<0$ ($s\leq b$).

Repeating this argument, $J$ will spiral inward around $b$, without self
intersections, until either $J$ ends in $J(b)=0$, thus $u$ tends
asymptotically to $b$, or forever thus $u$ oscillates forever with decreasing
amplitude. ∎

We finish this section with the observation that the sets $\mathcal{N}_{k}$
are strictly nested, they are nested by definition and the following lemma
shows that $\mathcal{N}_{k}\neq\mathcal{N}_{k+1}$ if they are not empty. This
is done by showing that solutions close to a bound state solution that cross
$0$ will have negative energy before reaching $J=0$.

######  Lemma 3.3.

Let $\alpha_{*}^{k}$ be a $k^{th}$-bound state solution then there is
$\epsilon>0$ such that if
$\bar{\alpha}\in[\alpha_{*}^{k}-\epsilon,\alpha_{*}^{k}+\epsilon]\cap\mathcal{N}_{k}$
then $|u(\bar{\alpha},r)|<\beta$ for all $r>Z_{k}$.

###### Proof.

We will work with an odd $k$, the even case differ only in some signs. Suppose
the lemma is not true, then there exists a sequence $\alpha_{i}$ such that
$\alpha_{i}\rightarrow\alpha_{*}^{k}$ and $u(\alpha_{i},\bar{r}_{i})=-\beta$
for some $\bar{r}_{i}$.

Let $H(s)=r^{2(N-1)}I(s)$, then for $s\in(-\beta,\beta)$

$H^{\prime}(s)=r^{2(N-1)}(N-1)J(s)-2(N-1)\frac{r^{2(N-1)-2}}{J(s)}\left(\frac{r^{2}J^{2}}{2}+F\right)=-2(N-1)r^{2(N-1)-2}\frac{F(s)}{J(s)}>0.$

For $\alpha=\alpha_{*}^{k}$, since $H_{*}(s)>0$ and decreasing (backwards) the
limit $\lim_{s\rightarrow 0}H_{*}(s)=L$ exists and is greater or equal $0$. By
continuity and using the fact that $H_{i}^{\prime}(s)>0$ if $\beta<s<-\beta$ ,
we have that given $\epsilon>0$ there is $s^{*}>0$ such that for $\alpha_{i}$,
with large $i$, $H_{i}(s)<L+\epsilon$ for all $s\in[-\beta,s^{*}]$.

As, for $\alpha=\alpha^{*}$, $\lim_{s\rightarrow 0}I_{*}(s)=0$ we may choose
$s^{*}>0$ so that we also have $I_{i}(s)<\epsilon$ for $\alpha_{i}$, with
large $i$, for all $s\in[-\beta,s^{*}]$.

Let $F_{1}>0$, $F_{2}$ such that $-F_{2}\leq F(s)\leq-F_{1}$ if
$s\in[-\beta/2,-\beta/4]$. From $I_{i}(s)<\epsilon$ we obtain
$r_{i}^{2}J_{i}^{2}<2F_{2}$ for $s\in[-\beta/2,-\beta/4]$ if $i$ is large. So

$L+\epsilon>H_{i}(-\beta/4)=H_{i}(-\beta/2)+\int_{-\beta/2}^{-\beta/4}-2(N-1)r^{2(N-1)-2}\frac{F(s)}{J_{i}(s)}ds$
$\geq H_{i}(-\beta/2)+(r_{i}(-\beta/4))^{2(N-1)}\frac{F_{1}}{F_{2}}\beta/4.$

If $\lim_{s\to 0}r_{*}(s)=\infty$ then $r_{i}(-\beta/4)>r_{i}(0)\to\infty$, by
continuity of the solutions, and the last term tends to infinity as $i$ is
large. A contradiction.

If $r_{*}(0)=r^{*}<\infty$, since for $\alpha=\alpha^{*}$, $\lim_{s\rightarrow
0}J_{*}(s)=0$ we must have $L=0$ and, by continuity of the solutions,

$\epsilon\geq(r_{i}(-\beta/4))^{2(N-1)}\frac{F_{1}}{F_{2}}\beta/4\geq(r_{i}(0))^{2(N-1)}\frac{F_{1}}{F_{2}}\beta/4\to(r^{*})^{2(N-1)}\frac{F_{1}}{F_{2}}\beta/4$

that gives a contradiction if we choose $\epsilon$ small enough.

∎

###  3.2.  Behaviour of $J/f$

We want to study the behavior of $\frac{J}{f}$, for this we note that it
satisfies $\frac{J}{f}(\alpha)=1/N$,
$\left(\frac{J}{f}\right)^{\prime}(\alpha)=\frac{-2}{N(N+2)}\frac{f^{\prime}}{f}(\alpha)$
and

$\left(\frac{J}{f}\right)^{\prime}=\frac{-1}{Jr^{2}}\left[f^{\prime}r^{2}\left(\frac{J}{f}\right)^{2}-N\left(\frac{J}{f}\right)+1\right].$
(3.6)

Thus, $\left(\frac{J}{f}\right)^{\prime}=0$ for

$\psi_{1}(s)=\frac{N-\sqrt{N^{2}-4f^{\prime}r^{2}}}{2f^{\prime}r^{2}}\quad\mbox{
and }\psi_{2}(s)=\frac{N+\sqrt{N^{2}-4f^{\prime}r^{2}}}{2f^{\prime}r^{2}},$

if they exist, and $\left(\frac{J}{f}\right)^{\prime}>0$ if and only if
$\psi_{1}<\left(\frac{J}{f}\right)<\psi_{2}$. Note that $\psi_{1}$ and
$\psi_{2}$ depend on the solution, and they exist only when
$f^{\prime}r^{2}<N^{2}/4$.

In $\alpha$ we have $\psi_{1}(\alpha)=\frac{1}{N}=\frac{J}{f}$ and
$\psi_{2}(\alpha)=\infty$.

Also

$\psi_{1}^{\prime}(s)=\left(\psi_{1}-\frac{1}{N}\right)\frac{N}{\sqrt{N^{2}-4f^{\prime}r^{2}}}\left[\frac{f^{\prime\prime}}{f^{\prime}}-\frac{2}{Jr^{2}}\right],$

with
$\psi_{1}^{\prime}(\alpha)=\frac{-2}{N^{2}}\frac{f^{\prime}}{f}(\alpha)>\left(\frac{J}{f}\right)^{\prime}(\alpha).$

##  4\.  Comparing solutions

Let $u$ and $v$ be solutions to the initial value problem (1.9) with
$u(0)=u_{0}>v_{0}=v(0)$, and let $r_{u}$ and $r_{v}$ be the inverses of these
solutions. At $v_{0}$ we have $r_{u}>0=r_{v}$,
$r^{\prime}_{u}>-\infty=r^{\prime}_{v}$ and $J_{u}>\frac{f(v_{0})}{N}=J_{v}$.
We want to compare this solutions, and prove that when $f$ is subcritical
$J_{u}(s)>J_{v}(s)$ until $s=0$, or until their energy $I_{u},I_{v}<0$ and
thus they do not reach $0$.

To compare these solutions we will use the following Pohozaev type functional,
introduced by Erbe and Tang in [ET].

Let

$P(s)=r^{N}(s)\left(2N\frac{F}{f}(s)J(s)-r^{2}(s)J^{2}(s)-2F(s)\right),\quad$
(4.1)

with

$P^{\prime}(s)=\frac{dP}{ds}(s)=\left(2N\Bigl{(}\frac{F}{f}\Bigr{)}^{\prime}(s)-(N-2)\right){r^{N}(s)}J(s).$

Note that if $f$ satisfies $(H_{2})$ then $P^{\prime}(s)>0$ for all
$s\in[\beta,\alpha_{0}]$, and $P(\alpha_{0})=0$, thus $P(s)<0$.

We will use these ideas to compare solutions from a point onward, not
necessarily form $r=0$, so we will prove this results for solutions that, at a
point $\bar{s}$, have $r_{u}>r_{v}$, $J_{u}>J_{v}$ and $P_{u}<0\leq P_{v}$.

######  Proposition 4.1.

Let $f$ be a function that satisfies $(H_{1})$ and $(H_{2})$, and let $u$ and
$v$ be solutions to the initial value problem (1.9), that at some $\bar{s}$
satisfy $r_{u}(\bar{s})>r_{v}(\bar{s})$, $J_{u}(\bar{s})>J_{v}(\bar{s})$ and
$P_{u}(\bar{s})<0\leq P_{v}(\bar{s})$. Then
$r_{u}^{\prime}(s)>r_{v}^{\prime}(s)$ and $J_{u}(s)>J_{v}(s)$ for all
$s>-\beta$ with $I_{v}(s)\geq 0$.

In particular, if $v$ reaches $0$, then $r_{u}^{\prime}(s)>r_{v}^{\prime}(s)$
and $J_{u}(s)>J_{v}(s)$ for all $s\geq 0$.

Note that, if at some $s_{J}$ we have $J_{u}(s_{J})=J_{v}(s_{J})$ for the
first time ( $J_{u}>J_{v}$ in $(s_{J},\bar{s})$), then
$\frac{1}{r_{u}^{2}}\left(N-\frac{f}{J}\right)=J_{u}^{\prime}>J_{v}^{\prime}=\frac{1}{r_{v}^{2}}\left(N-\frac{f}{J}\right)$
and since $J>\frac{f}{N}$ we have $r_{u}(s_{J})<r_{v}(s_{J})$ and
$r_{u}^{\prime}(s_{J})<r_{v}^{\prime}(s_{J})$. Since
$r_{u}(\bar{s})>r_{v}(\bar{s})$ and
$r_{u}^{\prime}(\bar{s})>r_{v}^{\prime}(\bar{s})$, there must be an $s_{1}$
with $r_{u}(s_{1})=r_{v}(s_{1})$, at which point
$r_{u}^{\prime}(s_{1})>r_{v}^{\prime}(s_{1})$, and an $s_{2}$ with
$r_{u}^{\prime}(s_{2})=r_{v}^{\prime}(s_{2})$, where
$s_{J}<s_{2}<s_{1}<\bar{s}$. Note also that at $s_{2}$ we have
$I_{u}(s_{2})=I_{v}(s_{2})$.

We will prove this proposition by steps, showing that $s_{2}$ cannot exist in
different intervals, unless $I_{v}<0$.

######  Lemma 4.2.

Using the notation above, and the conditions of Proposition 4.1, there is no
$s_{2}$ in $[\beta,\bar{s}]$ and
$r^{2}_{u}J_{u}(\beta)>r_{v}^{2}J_{v}(\beta)$.

###### Proof.

Suppose there is a first (largest) $s_{2}\in[\beta,\bar{s}]$ with
$r_{u}^{\prime}(s_{2})=r_{v}^{\prime}(s_{2})$, then
$J_{u}(s_{2})>J_{v}(s_{2})$. There must be a $\sigma\in[s_{2},s_{1}]$ such
that $r_{u}^{2}J_{u}(\sigma)=r_{v}^{2}J_{v}(\sigma)$. Let
$D=\frac{r_{u}^{N-2}(\sigma)}{r_{v}^{N-2}(\sigma)}<1$, since
$r_{u}^{\prime}>r_{v}^{\prime}$ in $[s_{2},\bar{s}]$ we have $r_{u}^{N-2}\geq
Dr_{v}^{N-2}$ in this interval. Consider the function $(P_{u}-DP_{v})(s)$, it
satisfies $(P_{u}-DP_{v})(\bar{s})<0$ and by $(H_{2})$

$\displaystyle(P_{u}-DP_{v})^{\prime}(s)=\left(2N\Bigl{(}\frac{F}{f}\Bigr{)}^{\prime}-(N-2)\right)({r_{u}^{N}}J_{u}-D{r_{v}^{N}}J_{v})$
$\displaystyle>\left(2N\Bigl{(}\frac{F}{f}\Bigr{)}^{\prime}-(N-2)\right){r_{u}^{N-2}}(r_{u}^{2}J_{u}-r_{v}^{2}J_{v})>0$

in $[\sigma,v_{0}]$ therefore $(P_{u}-DP_{v})(\sigma)<0$ but

$\displaystyle(P_{u}-DP_{v})(\sigma)=r_{u}^{N-2}\left(2N\frac{F}{f}(r_{u}^{2}J_{u}-r_{v}^{2}J_{v})-(r_{u}^{4}J_{u}^{2}-r_{v}^{4}J_{v}^{2})-2F(s)(r_{u}^{2}-r_{v}^{2})\right)$
$\displaystyle=r_{u}^{N-2}\left(2F(s)(r_{v}^{2}-r_{u}^{2})\right)>0$

so we get a contradiction.

Terefore $r^{2}_{u}J_{u}(s)>r_{v}^{2}J_{v}(s)$ in $[\beta,\bar{s}]$.

∎

In the next step we will use the following functional introduced by Peletier
and Serrin in [PS]. When $I(s)>0$ we can define

$W(s)=r(s)\sqrt{r^{2}(s)J^{2}(s)+2F(s)}=r\sqrt{2I},\quad$ (4.2)

with

$W^{\prime}(s)=\frac{\partial W}{\partial
s}(s)=\frac{(N-2)r^{2}(s)J^{2}(s)-2F(s)}{r(s)J(s)\sqrt{r^{2}(s)J^{2}(s)+2F(s)}}.$

We note that the function
$h(s,x)=\frac{(N-2)x^{2}-2F(s)}{x\sqrt{x^{2}+2F(s)}}$ is decreasing with $x$
when $F<0$.

######  Lemma 4.3.

Using the notation above, and the conditions of Proposition 4.1, let
$s_{I}\geq-\beta$ be the minimum value where $I_{u},I_{v}\geq 0$. Then there
is no $s_{2}$ in $[s_{I},\beta]$ and if $s_{I}\leq 0$, then
$r_{u}^{2}(0)J_{u}(0)>r_{v}^{2}(0)J_{v}(0)$. If $s_{I}=-\beta$, then
$r_{u}^{2}(-\beta)J_{u}(-\beta)>r_{v}^{2}(-\beta)J_{v}(-\beta)$.

###### Proof.

Suppose there is a first (largest) $s_{2}\in[s_{I},\beta]$ with
$r_{u}^{\prime}(s_{2})=r_{v}^{\prime}(s_{2})$, then
$r_{u}(s_{2})<r_{v}(s_{2})$ and $J_{u}(s_{2})>J_{v}(s_{2})$. Then there must
be an $s_{1}>s_{2}$ with $r_{u}(s_{1})=r_{v}(s_{1})$.

If $s_{1}\in[\beta,\alpha_{0}]$ then $r_{u}(\beta)<r_{v}(\beta)$ and by Lemma
4.2

$W_{u}(\beta)=r_{u}^{2}J_{u}=\frac{r_{u}}{|r_{u}^{\prime}|}>\frac{r_{v}}{|r_{v}^{\prime}|}=W_{v}(\beta).$

If $s_{1}\in[s_{I},\beta]$, since $J_{u}(s_{1})>J_{2}(s_{1})$,

$W_{u}(s_{1})=r_{u}\sqrt{r_{u}^{2}J_{u}^{2}+2F}>r_{v}\sqrt{r_{v}^{2}J_{v}^{2}+2F}=W_{v}(s_{1}).$

Let $\bar{s}=\min\\{s_{1},\beta\\}$, then in $[s_{2},\bar{s}]$ we have
$r_{u}J_{u}=\frac{1}{|r_{u}^{\prime}|}>\frac{1}{|r_{v}^{\prime}|}=r_{v}J_{v}$
and since $h(s,x)$ decreases with $x$,

$(W_{u}-W_{v})^{\prime}(s)=h(s,r_{u}J_{u})-h(s,r_{v}J_{v})<0$

thus $W_{u}(s_{2})>W_{v}(s_{2})$. On the other hand

$W_{u}-W_{v}(s_{2})=(r_{u}-r_{v})\sqrt{\frac{1}{|r_{u}^{\prime}|^{2}}+2F(s)}<0$

and we have a contradiction.

Therefore there is no $s_{2}$ in $[s_{I},\beta]$ and if $s_{I}\leq 0$ then

$r_{u}^{2}J_{u}(0)=W_{v}(0)>W_{v}(0)=r_{v}^{2}J_{v}(0).$

Similarly, if $s_{I}=-\beta$ then
$r_{u}^{2}J_{u}(-\beta)=W_{v}(-\beta)>W_{v}(-\beta)=r_{v}^{2}J_{v}(-\beta).$ ∎

###### Proof of Proposition 4.1.

Using the notation above, by Lemmas 4.2 and 4.3 there is no
$s_{2}\in[s_{I},\bar{s}]$, and therefore $r_{u}^{\prime}(s)>r_{v}^{\prime}(s)$
and $J_{u}(s)>J_{v}(s)$ for all $s>s_{I}$. If $s_{I}>-\beta$ then
$I_{u}(s_{I})=\frac{1}{2|r_{u}^{\prime}(s_{I})|^{2}}+F(s_{I})>\frac{1}{2|r_{v}^{\prime}(s_{I})|^{2}}+F(s_{I})=I_{v}(s_{I})=0$
and the first statement follows.

To prove the second statement we note that if $v$ reaches $0$ then
$I_{v}(0)>0$, therefore $s_{I}\leq 0$ and the conclusion is obtained for all
$s\geq 0$. ∎

##  5\. The effect of magnitude changes

We want to study the behaviour of the solutions when $f$ has a magnitude
change. We will approach this problem using a function $f$ defined by parts as
in (1.10) and (1.17).

In [CGHH2] the author in collaboration with C. Cortázar and M. García-Huidobro
studied this effect by considering functions of the form

$\displaystyle f(s)=\begin{cases}f_{1}(s)&s\leq\alpha_{1}\\\
L(s)&\alpha_{1}\leq s\leq\alpha_{1}+\epsilon\\\
\lambda^{2}f_{2}\left(s\right)&s\geq\alpha_{1}+\epsilon\end{cases}$ (5.1)

proving that for appropriate $\alpha_{1}$, big enough $\lambda$ and small
enough $\epsilon$, problem 1.6 has at least two ground state solutions.

We could expect that, if we choose $\alpha_{1}$ just above a bound state
solution with one sign change (a $2^{nd}$-bound state), then for big enough
$\lambda$ we will get both a second $2^{nd}$-bound state solution and a second
ground state solution. Surprisingly this does not always happen, as we can see
in Theorem $B$.

We will begin by understanding the behaviour of $u$ and $J$ below an
$\alpha_{1}$, if it reaches $\alpha_{1}$ with fixed $|u^{\prime}|r$ and small
enough $r$. We will see that the behaviour will depend strongly on the value
$|u^{\prime}(\alpha_{1})|r(\alpha_{1})$.

######  Proposition 5.1.

Let $f_{1}$ be a function that satisfies $(H_{1})$ and $(H_{2})$, and
$g_{\lambda}$, $\lambda\in(1,\infty)$ a family of functions with
$g_{\lambda}(s)=f_{1}(s)$ for $s<\alpha_{1}$. Let $u_{\lambda}$ be a family of
solutions to (1.9) with $f=g_{\lambda}$ that reach $\alpha_{1}$ with

$\eta^{-}<J_{\lambda}(\alpha_{1})r_{\lambda}^{2}(\alpha_{1})<\eta^{+}<N\frac{f}{f^{\prime}}(\alpha_{1}),$
$\quad\lim_{\lambda\to\infty}J_{\lambda}(\alpha_{1})r_{\lambda}^{2}(\alpha_{1})=\eta^{-}\quad\mbox{and
}\quad\lim_{\lambda\to\infty}r_{\lambda}(\alpha_{1})=0,$

where $\eta^{-},\eta^{+}>0$ are constants. For any $\alpha_{-},\alpha_{+}$
such that
$\beta<\alpha_{-}<\alpha_{1}-\frac{\eta^{-}}{N-2}<\alpha_{+}<\min\\{\alpha_{1},\alpha_{1}-\frac{\eta^{-}}{N-2}+\frac{2N}{N-2}\frac{F}{f}(\beta)\\}$,
there exist $\lambda_{0}$ and $s_{0}<\alpha_{1}$ such that the solutions
$u_{-}$ and $u_{+}$ with initial conditions $\alpha_{-}$ and $\alpha_{+}$ have

$J_{-}(s)<J_{\lambda}(s)<J_{+}(s)\quad\mbox{ for }\ s<s_{0},$

for all $\lambda>\lambda_{0}$, as long as their respective energies are
positive.

We will start with two lemmas that will help in the proof, the first one on
the behaviour of $\frac{J_{\lambda}}{f}$.

######  Lemma 5.2.

Let $u_{\lambda}$ be a family of solutions as in Proposition 5.1, then there
is $\lambda_{1}$ such that for $\lambda>\lambda_{1}$ we have
$\left(\frac{J_{\lambda}}{f}\right)^{\prime}(\alpha_{1})>0$ and
$\frac{J_{\lambda}}{f}$ has a first minimum at $s_{\lambda}$.

Moreover, $\lim_{\lambda\to\infty}r_{\lambda}(s_{\lambda})=0$,
$\lim_{\lambda\to\infty}\frac{J_{\lambda}}{f}(s_{\lambda})=\frac{1}{N}$ and
$\lim_{\lambda\to\infty}s_{\lambda}=\alpha_{1}-\frac{\eta^{-}}{N-2}$.

###### Proof.

We begin by observing that $J_{\lambda}(\alpha_{1})\to\infty$ with $\lambda$
thus

$\left(\frac{J_{\lambda}}{f}\right)^{\prime}(\alpha_{1})=\frac{1}{fr^{2}_{\lambda}}\left[N-\frac{f}{J_{\lambda}}-J_{\lambda}r_{\lambda}^{2}\frac{f^{\prime}}{f}\right]>\frac{1}{fr^{2}_{\lambda}}\left[\frac{f^{\prime}}{f}\left(N\frac{f}{f^{\prime}}-\eta^{+}\right)-\frac{f}{J_{\lambda}}\right]$

will be positive for $\lambda>\lambda_{1}$ for some $\lambda_{1}$. Therefore,
since $\frac{J_{\lambda}}{f}$ decreases near $b$, it must have a minimum, and
we denote by $s_{\lambda}$ the largest $s$ where a minimum is obtained.

Using Lemma 2.2 with $\bar{\alpha}+\delta=\alpha_{1}$ and
$\delta<\frac{\eta^{-}}{N-2}$ we can find a $B>1$ such that
$r_{\lambda}(\alpha_{1}-\delta)<Br_{\lambda}(\alpha_{1})$ and

$\frac{1}{B^{N-2}}\eta^{-}\leq
J_{\lambda}r^{2}_{\lambda}(\alpha_{1}-\delta)\leq
J_{\lambda}r^{2}_{\lambda}(\alpha_{1})+\frac{B^{N}-1}{N}f(\alpha_{1})r_{\lambda}^{N}(\alpha_{1}).$

Therefore $r_{\lambda}(\alpha_{1}-\delta)\to 0$ and
$J_{\lambda}r^{2}_{\lambda}(\alpha_{1}-\delta)$ is bounded, thus
$J_{\lambda}(\alpha_{1}-\delta)\to\infty$.

Using that $(Jr^{2})^{\prime}=(N-2)-\frac{f}{J}$ we can see that

$\displaystyle J_{\lambda}r^{2}_{\lambda}(\alpha_{1})-(N-2)\delta<\
J_{\lambda}r^{2}_{\lambda}(\alpha_{1}-\delta)$
$\displaystyle=J_{\lambda}r^{2}_{\lambda}(\alpha_{1})-\int_{\alpha_{1}-\delta}^{\alpha_{1}}(N-2)-\frac{f}{J_{\lambda}}\
ds$
$\displaystyle<J_{\lambda}r^{2}_{\lambda}(\alpha_{1})-(N-2)\delta+\frac{f}{J_{\lambda}}(\alpha_{1}-\delta)\delta$

and since $J_{\lambda}(\alpha_{1}-\delta)\to\infty$ and
$J_{\lambda}r^{2}_{\lambda}(\alpha_{1})\to\eta^{-}$ we get that
$J_{\lambda}r^{2}_{\lambda}(\alpha_{1}-\delta)\to\eta^{-}-(N-2)\delta$ for any
$\delta<\frac{\eta^{-}}{N-2}$. Given $\epsilon>0$ small , we choose
$\alpha_{1}-\frac{\eta^{-}}{N-2}<\alpha_{\epsilon}<\alpha_{1}-\frac{\eta^{-}-\epsilon}{N-2}$
and $\lambda_{2}>\lambda_{1}$ such that
$J_{\lambda}r^{2}_{\lambda}(\alpha_{\epsilon})<\epsilon$ for all
$\lambda>\lambda_{2}$.

For each solution $u_{\lambda}$ let $s_{\lambda}^{3}$ be the value where
$\frac{J}{f}(s_{\lambda}^{3})=\frac{1}{N-3}$, or $s_{\lambda}^{3}=s_{\lambda}$
if $\frac{J}{f}(s_{\lambda})>\frac{1}{N-3}$, thus $s_{\lambda}\leq
s_{\lambda}^{3}<\alpha_{\epsilon}$.

In $[s_{\lambda}^{3},\alpha_{\epsilon}]$,
$\frac{J_{\lambda}}{f}(s)\geq\frac{1}{N-3}$, therefore
$(J_{\lambda}r_{\lambda}^{2})^{\prime}(s)=(N-2)-\frac{f}{J_{\lambda}}>1$ and

$\alpha_{\epsilon}-s_{\lambda}^{3}<J_{\lambda}r_{\lambda}^{2}(\alpha_{\epsilon})-J_{\lambda}r_{\lambda}^{2}(s_{\lambda}^{3})<J_{\lambda}r_{\lambda}^{2}(\alpha_{\epsilon})<\epsilon.$

Therefore $\alpha_{\epsilon}-s_{\lambda}^{3}$ and
$J_{\lambda}r_{\lambda}^{2}(s_{\lambda}^{3})$ tend to $0$. Note that since
$J_{\lambda}r_{\lambda}^{2}$ tends to $0$ and $J_{\lambda}\geq f/(N-3)$ is
bounded, $r_{\lambda}(s_{\lambda}^{3})$ tends to $0$. Also, since we can take
any $\epsilon$, $s_{\lambda}^{3}$ tends to
$\alpha_{1}-\frac{\eta^{-}-\epsilon}{N-2}$.

If $\lim_{\lambda\to\infty}\frac{J_{\lambda}}{f}(s_{\lambda})\neq\frac{1}{N}$,
let $L^{-}$ be such that
$\frac{J_{\lambda}}{f}(s_{\lambda})>L^{-}>\frac{1}{N}$ for all
$\lambda>\lambda_{3}$, for some $\lambda_{3}>\lambda_{2}$. Note that
$L^{-}<\frac{1}{N-3}$.

Let $\bar{\epsilon}=N-\frac{1}{L^{-}}$, if $s_{\lambda}^{3}\neq s_{\lambda}$
then in $[s_{\lambda},s_{\lambda}^{3}]$

$(J_{\lambda}r_{\lambda}^{\bar{\epsilon}})^{\prime}(s)=r_{\lambda}^{\bar{\epsilon}-2}\left((N-\bar{\epsilon})-\frac{f}{J_{\lambda}}\right)\geq
0$

therefore
$J_{\lambda}r_{\lambda}^{\bar{\epsilon}}(s_{\lambda})<J_{\lambda}r_{\lambda}^{\bar{\epsilon}}(s_{\lambda}^{3})$
that tends to $0$. Since $J_{\lambda}(s_{\lambda})$ is bounded away from $0$,
$r_{\lambda}(s_{\lambda})$ tends to $0$.

Using the analysis of Section 3.2, we see that for $\lambda>\lambda_{0}$ we
have $\psi_{1}^{\lambda}<\frac{J_{\lambda}}{f}<\psi_{1}^{\lambda}$ and
$\frac{J_{\lambda}}{f}(s)$ will decrease (going backwards) until it reaches
$\psi_{1}^{\lambda}$. Therefore the minimum will be achieved at an
$s_{\lambda}$ with

$\frac{J_{\lambda}}{f}(s_{\lambda})=\psi_{1}^{\lambda}=\frac{N-\sqrt{N^{2}-4f^{\prime}r_{\lambda}^{2}}}{2f^{\prime}r_{\lambda}^{2}}.$

Since $r_{\lambda}(s_{\lambda})$ tends to $0$, with $f^{\prime}$ bounded, we
have

$\lim_{\lambda\to\infty}\frac{J_{\lambda}}{f}(s_{\lambda})=\lim_{\lambda\to\infty}\frac{N-\sqrt{N^{2}-4f^{\prime}r_{\lambda}^{2}}}{2f^{\prime}r_{\lambda}^{2}}=\frac{1}{N},$
(5.2)

a contradiction.

Therefore
$\lim_{\lambda\to\infty}\frac{J_{\lambda}}{f}(s_{\lambda})=\frac{1}{N}$, and
equation (5.2) gives that $\lim_{\lambda\to\infty}r_{\lambda}(s_{\lambda})=0$.

To prove the last limit we note that $(r_{\lambda}^{2})^{\prime}=-\frac{2}{J}$
is bounded for $s\in[s_{\lambda},s_{\lambda}^{3}]$, therefore from the
previous statement we get that $s_{\lambda}^{3}-s_{\lambda}$ tend to $0$, and
since $s_{\lambda}^{3}\to\alpha_{1}-\frac{\eta^{-}}{N-2}$ we get
$\lim_{\lambda\to\infty}s_{\lambda}=\alpha_{1}-\frac{\eta^{-}}{N-2}$.

∎

To finish the proof of Proposition 5.1 we will use the functional $P$ to
compare solutions. For this we recall that $P^{\prime}>0$ for $s<\alpha_{1}$,
thus $P$ decreases.

######  Lemma 5.3.

Let $u_{\lambda}$ be a family of solutions as in Proposition 5.1, then for any
$\bar{\alpha}_{-},\bar{\alpha}_{+}$ such that
$\beta<\bar{\alpha}_{-}<\alpha_{1}-\frac{\eta^{-}}{N-2}<\bar{\alpha}_{+}<\min\\{\alpha_{1},\alpha_{1}-\frac{\eta^{-}}{N-2}+\frac{2N}{N-2}\frac{F}{f}(\beta)\\}$,
there is a $\lambda_{3}>0$ such that for all $\lambda>\lambda_{3}$,

$P_{\lambda}(\bar{\alpha}_{-})<0<P_{\lambda}(\bar{\alpha}_{+}).$

###### Proof.

Let
$\alpha_{1}-\frac{\eta^{-}}{N-2}<\bar{\alpha}_{+}<\min\\{\alpha_{1},\alpha_{1}-\frac{\eta^{-}}{N-2}+\frac{2N}{N-2}\frac{F}{f}(\beta)\\}$,
by the proof of Lemma 5.2, taking $\bar{\alpha}_{+}$ as $\alpha_{\epsilon}$
for some $\epsilon<2N\frac{F}{f}(\beta)$, when $\lambda\to\infty$,
$r_{\lambda}(\bar{\alpha}_{+})\to 0$,
$J_{\lambda}r^{2}_{\lambda}(\bar{\alpha}_{+})<\epsilon$ is bounded, and
$J_{\lambda}(\bar{\alpha}_{+})\to\infty$. Therefore

$P_{\lambda}(\bar{\alpha}_{+})=r_{\lambda}^{N}\left(\left(2N\frac{F}{f}-J_{\lambda}r_{\lambda}^{2}\right)J_{\lambda}-2F\right)$

will be positive if $J_{\lambda}(\bar{\alpha}_{+})$ is big enough, therefore
there is a $\bar{\lambda}_{3}$ such that $P_{\lambda}(\bar{\alpha}_{+})>0$ for
$\lambda>\bar{\lambda}_{3}$.

On the other hand, note that

$\frac{P}{Jr^{N}}=2\frac{F}{f}\left({N}-\frac{f}{J}\right)-Jr^{2}<2\frac{F}{f}\left({N}-\frac{f}{J}\right)$

has

$\left(\frac{P}{Jr^{N}}\right)^{\prime}=2\left(\frac{F}{f}\right)^{\prime}\left({N}-\frac{f}{J}\right)-2\frac{F}{f}\left(\frac{f}{J}\right)^{\prime}-(N-2)+\frac{f}{J}$
$>\frac{2}{N}\frac{f}{J}-2\frac{F}{f}\left(\frac{f}{J}\right)^{\prime}$

Let $\bar{\alpha}_{-}<\alpha_{1}-\frac{\eta^{-}}{N-2}$, then integrating over
$[\bar{\alpha}_{-},s_{\lambda}]$, where $\left(\frac{f}{J}\right)^{\prime}<0$,
and assuming $\frac{J}{f}<1/(N-1)$ we get

$\left(\frac{P}{Jr^{N}}\right)(\bar{\alpha}_{-})\leq\left(\frac{P}{Jr^{N}}\right)(s_{\lambda})-\frac{2(N-1)}{N}(s_{\lambda}-\bar{\alpha}_{-})\leq
2\frac{F}{f}(s_{\lambda})\left({N}-\frac{f}{J}(s_{\lambda})\right)-\frac{2(N-1)}{N}(s_{\lambda}-\bar{\alpha}_{-})$

that will be negative for $\lambda>\lambda_{3}$, for some
$\lambda_{3}>\bar{\lambda}_{3}$.

∎

###### Proof of Proposition 5.1.

Let $s_{\lambda}$ as in Lemma 5.2, since
$\lim_{\lambda\to\infty}r_{\lambda}(s_{\lambda})=0$ and
$s_{\lambda}\leq\alpha_{1}-\frac{\eta^{-}}{N-2}$, for any
$\alpha_{1}-\frac{\eta^{-}}{N-2}<\alpha_{+}<\min\\{\alpha_{1},\alpha_{1}-\frac{\eta^{-}}{N-2}+\frac{2N}{N-2}\frac{F}{f}(\beta)\\}$,
$u_{\lambda}$ intersects the solutions $u_{+}$ at a point
$\sigma^{\lambda}_{+}$. Moreover, for large enough $\lambda$ we will have
$\sigma^{\lambda}_{+}>\alpha_{1}-\frac{\eta^{-}}{N-2}$. Let $\lambda_{3}$ be
as in Lemma 5.3, and chose $\lambda_{0}>\lambda_{3}$ such that
$\sigma^{\lambda}_{+}>\alpha_{1}-\frac{\eta^{-}}{N-2}$ for all
$\lambda>\lambda_{0}$.

Then, at $\sigma_{+}^{\lambda}$ we have
$r_{\lambda}(\sigma_{+}^{\lambda})=r_{+}(\sigma_{+}^{\lambda})$ with
$r_{\lambda}(s)<r_{+}(s)$ for $s<\sigma_{+}^{\lambda}$ near,
$r^{\prime}_{\lambda}(\sigma_{+}^{\lambda})<r^{\prime}_{+}(\sigma_{+}^{\lambda})$,
and by Lemma 5.3
$P_{\lambda}(\sigma_{+}^{\lambda})>0>P_{+}(\sigma_{+}^{\lambda})$.

We can use Proposition 4.1 to prove $J_{\lambda}<J_{+}$ until they reach $0$
or $I<0$.

Let $\beta<\alpha_{-}<\alpha_{1}-\frac{\eta^{-}}{N-2}$, and $u_{-}$ the
solution with $u_{-}(0)=\alpha_{-}$. Then its inverse $r_{-}$ satisfies
$r_{\lambda}(\alpha_{-})>0=r_{-}(\alpha_{-})$ ,
$J_{\lambda}(\alpha_{-})<0=J_{-}(\alpha_{-})$ and by Lemma 5.3
$P_{\lambda}(\alpha_{-})<0=P_{-}(\alpha_{-})$ for all $\lambda>\lambda_{0}$.
Therefore we can use Proposition 4.1 to conclude the proof.

∎

######  Corollary 5.4.

Given $f_{1}$ that satisfies $(H_{1}),(H_{2})$ and an
$\alpha_{1}>\alpha_{*}^{1}$. If there is a
$(N-2)(\alpha_{1}-\alpha_{*}^{1})<K<N\frac{f}{f^{\prime}}(\alpha_{1})$ then
there is a $\delta>0$ such that if $u$ is a solution that reaches $\alpha_{1}$
with $r(\alpha_{1})<\delta$ and $|u^{\prime}(r(\alpha_{1}))r(\alpha_{1})|=K$,
then $u(r)>0$ for all $r\in[0,\infty]$.

Note that if $f_{1}$ satisfies $(H_{3})$ there is always such $K$.

###### Proof.

Let $u_{\lambda}(r)$ be the solution of 1.9 with
$u(\frac{1}{\lambda})=\alpha_{1}$ and $u^{\prime}(\frac{1}{\lambda})=\lambda
K$. Then it satisfies the hypothesis of Proposition 5.1 and we can choose
$\alpha_{1}-\frac{K}{N-2}<\alpha_{+}<\alpha_{*}^{1}$ such that for some
$\lambda_{0}$ and $s_{0}<\alpha_{1}$ the solutions satisfy

$J_{\lambda}(s)<J_{+}(s)\quad\mbox{ for }\ s<s_{0},$

for all $\lambda>\lambda_{0}$. Since $\alpha_{+}<\alpha_{*}^{1}$, $u_{+}$
cannot reach $0$, and there is an $s_{+}>0$ with $J_{+}(s_{+})=0$. Then either
$J_{\lambda}(s)$ reach $0$ or $I(s)<0$ with $s>s_{+}$. In either case, the
solution $u_{\lambda}$ do not reach $0$. ∎

###  5.1. Proof of the main Theorems

To prove existence of a $k^{th}$-bound state solutions we recall that the sets
$\mathcal{N}_{k}$ and $\mathcal{P}_{k}$ are open subsets of
$\mathcal{N}_{k-1}$, therefore all boundary points must be bound state
solutions. If there are $\alpha_{N}\in\mathcal{N}_{k}$ and
$\alpha_{P}\in\mathcal{P}_{k}$, then there must be a boundary point of
$\mathcal{N}_{k}$ in $(\alpha_{P},\alpha_{N})$ (or $(\alpha_{N},\alpha_{P})$).
This boundary point must be a bound state solution, and by Lemma 3.3 it must
be in $\mathcal{N}_{k-1}$, thus it is a $k^{th}$-bound state solution.

To prove Theorem A we want to choose $\alpha_{1}$ and $\epsilon$ in such a way
that the solutions with initial condition $\alpha_{+}$ from Proposition 5.1 do
not reach $0$. That is, we need $\alpha_{+}<\alpha_{*}^{1}$. Then, there will
be an $\alpha>\alpha_{*}^{k}+\epsilon$ with $\alpha\in\mathcal{P}_{1}$ and
$\alpha_{*}^{k}+\epsilon\in\mathcal{N}_{k+1}$, therefore there must be a
$j^{th}$-bound state solution separating $\mathcal{N}_{j}$ and
$\mathcal{P}_{J}$ for each $j\leq k$. Before choosing $\alpha_{+}$ we need to
control what happens to the solutions when $f=L$.

######  Lemma 5.5.

Given $\epsilon>0$ and $\alpha_{x}>\alpha_{1}+\epsilon$, for each $\lambda$
let $u_{\lambda}$ be the solution of 1.9 with $f$ as in (1.10) with constant
$\lambda^{2}$ and $u_{\lambda}(0)=\alpha_{x}$, and $r_{\lambda}$ their
respective inverses. Let $v$ be the solution of $\eqref{ivp}$ with $f=f_{2}$
and $v(0)=\alpha_{x}$. If
$\zeta:=J_{v}r_{v}^{2}(\alpha_{1}+\epsilon)<N\frac{f}{f^{\prime}}(\alpha_{1}+\epsilon)$,
then $J_{\lambda}r_{\lambda}^{2}(\alpha_{1}+\epsilon)=\zeta$ is independent of
$\lambda$ and $J_{\lambda}r_{\lambda}^{2}(\alpha_{1})\geq\zeta-(N-2)\epsilon$
tends to $\zeta-(N-2)\epsilon$ when $\lambda\to\infty$.

###### Proof.

For each $\lambda$, the function ${w_{\lambda}}(r)={v}(\lambda r)$ satisfies:

$w_{\lambda}^{\prime\prime}(r)+\frac{(N-1)}{r}w_{\lambda}^{\prime}(r)={\lambda^{2}}\left(v^{\prime\prime}(\lambda
r)+\frac{(N-1)}{\lambda r}v^{\prime}(\lambda
r)\right)=-{\lambda^{2}}f(w_{\lambda}(r))$

therefore $w_{\lambda}=u_{\lambda}$ in $[\alpha_{1}+\epsilon,\alpha_{x}]$.
Moreover, $r^{v}=\lambda r_{\lambda}^{u}$ and
$v^{\prime}(r^{v})=v^{\prime}(\lambda
r_{\lambda}^{u})=u_{\lambda}^{\prime}(r_{\lambda}^{u})/\lambda$ hence
$J_{\lambda}(r_{\lambda}^{u})^{2}=r_{\lambda}^{u}|u_{\lambda}^{\prime}(r_{\lambda}^{u})|=\zeta$
is independent of $\lambda$, and $J_{\lambda}=\lambda^{2}J_{v}$.

Since $\zeta<N\frac{f}{f^{\prime}}$

$\left(\frac{J_{\lambda}}{f}\right)^{\prime}(\alpha_{1}+\epsilon)=\frac{1}{fr^{2}_{\lambda}}\left[N-\frac{f}{J_{\lambda}}-J_{\lambda}r_{\lambda}^{2}\frac{f^{\prime}}{f}\right]=\frac{1}{fr^{2}_{\lambda}}\left[\frac{f^{\prime}}{f}\left(N\frac{f}{f^{\prime}}-\zeta\right)-\frac{f}{\lambda^{2}J_{v}}\right]$

will be positive for $\lambda>\lambda_{1}$ for some $\lambda_{1}$.

Using Lemma 2.2 with $\bar{\alpha}=\alpha_{1}+\epsilon$, $\delta=\epsilon$,
$g=f$ and $\zeta=J_{v}r_{v}^{2}(\alpha_{1}+\epsilon)$ we get that for $B>1$
with
$\epsilon\frac{B^{N-2}}{B^{N-2}-1}=\frac{J_{v}r_{v}^{2}(\alpha_{1}+\epsilon)}{N-2}$,
$r_{\lambda}^{u}(\alpha_{1})<Br_{\lambda}^{u}(\alpha_{1}+\epsilon)$ and

$\frac{\zeta}{B^{N-2}}\leq
J_{\lambda}(r_{\lambda}^{u})^{2}(\alpha_{1})\leq\zeta+\frac{B^{N}-1}{N}||g||_{+}(r_{\lambda}^{u}(\alpha_{1}+\epsilon))^{N}.$

Therefore when $\lambda\to\infty$ we get $r_{\lambda}(\alpha_{1})\to 0$ and
$J_{\lambda}r^{2}_{\lambda}(\alpha_{1})$ is bounded away from $0$, thus
$J_{\lambda}(\alpha_{1})\to\infty$.

Using that $(Jr^{2})^{\prime}=(N-2)-\frac{f}{J}$ we can see that

$\zeta-(N-2)\epsilon<J_{\lambda}r^{2}_{\lambda}(\alpha_{1})=\zeta-\int_{\alpha_{1}+\epsilon}^{\alpha_{1}}(N-2)-\frac{f}{J_{\lambda}}\
ds<\zeta-(N-2)\epsilon+\frac{f}{J_{\lambda}}(\alpha_{1})\epsilon$

and since $J_{\lambda}(\alpha_{1})\to\infty$ and
${f}(\alpha_{1})=f_{1}(\alpha_{1})$ we get that
$J_{\lambda}r^{2}_{\lambda}(\alpha_{1})\to\zeta-(N-2)\epsilon$.

∎

###### Proof of Theorem A.

Let $\alpha_{1}=\alpha_{*}^{k}+\epsilon$ and choose
$\alpha_{1}+\epsilon-\alpha_{*}^{1}<d<\alpha_{1}+\epsilon-\beta$. Let $v(r)$
be as in condition $(H_{5})$, with $v(0)=\widehat{\alpha}>\alpha_{1}+\epsilon$
and let $r_{v}(s)$ be its inverse. Let
$K_{m}=min_{s\in[0,r_{v}(\alpha_{1}+\epsilon)]}J_{v}r_{v}^{2}$, and note that
$K_{m}>0$.

For each $\mu>0$, the function ${w_{\mu}}(r)=\mu{v}(r)$ satisfies:

$w_{\mu}^{\prime\prime}(r)+\frac{(N-1)}{r}w_{\mu}^{\prime}(r)={\mu}\left(v^{\prime\prime}(r)+\frac{(N-1)}{r}v^{\prime}(r)\right)=-{\mu}f(v)=-{\mu}f(w_{\mu}/\mu).$

Let $r_{\mu}$ be de inverse of $w_{\mu}$, then $r_{\mu}(s)=r_{v}(s/\mu)$ and
$J_{\mu}r_{\mu}^{2}(\alpha_{1}+\epsilon)=\mu
J_{v}r_{v}^{2}((\alpha_{1}+\epsilon)/\mu)>\mu K_{m}$. Therefore we can choose
$\mu_{0}$ such that for $\mu>\mu_{0}$ the solution
$J_{\mu}r_{\mu}^{2}(\alpha_{1}+\epsilon)>(N-2)d$.

For each fixed $\mu$, let $w_{\mu}(\alpha,r)$ be the solutions with initial
condition $w_{\mu}(\alpha,0)=\alpha$. Since
$J_{\mu}r_{\mu}^{2}(\alpha,\alpha_{1}+\epsilon)$ depends continuously on
$\alpha$ and $J_{\mu}r_{\mu}^{2}(\alpha_{1}+\epsilon,\alpha_{1}+\epsilon)=0$,
there must be an $\bar{\alpha}$ such that
$J_{\mu}r_{\mu}^{2}(\bar{\alpha},\alpha_{1}+\epsilon)=(N-2)d$. By condition
$(H_{3})$ we have
$J_{\mu}r_{\mu}^{2}(\bar{\alpha},\alpha_{1}+\epsilon)<(N-2)(\alpha_{1}+\epsilon-\beta)<N\frac{f}{f^{\prime}}(\alpha_{1}+\epsilon).$

Let $u_{\lambda}(r)$ be the solution of $\eqref{ivp}$ with $f_{\mu}$ as in
(1.10) and $u_{\lambda}(0)=\bar{\alpha}$. Then by Lemma 5.5, with
$\zeta=J_{\mu}r_{\mu}^{2}(\bar{\alpha},\alpha_{1}+\epsilon)$, we have
$J_{\lambda}r_{\lambda}^{2}(\alpha_{1})>\zeta-(N-2)\epsilon$ tends to
$\zeta-(N-2)\epsilon$ when $\lambda\to\infty$.

We can now choose $\alpha_{+}$ with
$\alpha_{1}+\epsilon-d<\alpha_{+}<\alpha_{*}^{1}$ and use Proposition 5.1 to
find $\bar{\lambda}$ such that the solution $u_{+}$ with initial condition
$\alpha_{+}$ satisfies

$J_{\lambda}(s)<J_{+}(s)\quad\mbox{ for }\ s<s_{0}$

if the energy is positive.

Since $\alpha_{+}<\alpha_{*}^{1}$, $u_{+}$ cannot reach $0$, and there is an
$s_{+}>0$ with $J_{+}(s_{+})=0$. Then either $J_{\lambda}(s)$ reach $0$ or
$I(s)<0$ for some $s>s_{+}$. In either case, the solution $u_{\lambda}$ do not
reach $0$. By the argument above, there must be a $j^{th}$-bound state
solution with initial condition in $(\alpha_{*}^{k}+2\epsilon,\bar{\alpha})$
for each $j\leq k$. ∎

To prove Theorem $B$ we will use that for a continuous family of functions,
the constants $\lambda_{0}$ and $s_{0}$ in Proposition 5.1 can be chosen to
depend continuously on the initial condition of the solution.

######  Lemma 5.6.

Let $f_{1}$ be a function that satisfies $(H_{1})$ and $(H_{2})$, and
$g_{\lambda}$, $\lambda\in(1,\infty)$ a family of functions that depends
continuously on $\lambda$, with $g_{\lambda}(s)=f_{1}(s)$ for $s<\alpha_{1}$.
Let $<[\gamma_{-},\gamma_{+}]\subset(\alpha_{1},\infty)$ and
$u_{\lambda}(\alpha,r)$ be a family of solutions to (1.9) with $f=g_{\lambda}$
that reach $\alpha_{1}$ with

$\eta^{-}(\alpha)<J_{\lambda}(\alpha,\alpha_{1})r_{\lambda}^{2}(\alpha,\alpha_{1})<\eta^{+}(\alpha)<N\frac{f}{f^{\prime}}(\alpha_{1}),$
$\quad\lim_{\lambda\to\infty}J_{\lambda}(\alpha,\alpha_{1})r_{\lambda}^{2}(\alpha,\alpha_{1})=\eta^{-}(\alpha)\quad\mbox{and
}\quad\lim_{\lambda\to\infty}r_{\lambda}(\alpha,\alpha_{1})=0,$

For any $\alpha_{-},\alpha_{+}$ such that

$\beta<\alpha_{-}<\alpha_{1}-\frac{\sup{\eta^{-}(\alpha)}}{N-2}<\alpha_{1}-\frac{\inf{\eta^{-}(\alpha)}}{N-2}<\alpha_{+}<\alpha_{1},$

there exist $\lambda_{0}$ and $s_{0}<\alpha_{1}$ such that the solutions
$u_{-}$ and $u_{+}$ with initial conditions $\alpha_{-}$ and $\alpha_{+}$ have

$J_{-}(s)<J_{\lambda}(\alpha,s)<J_{+}(s)\quad\mbox{ for }\ s<s_{0},$

for all $\lambda>\lambda_{0}$, and all $\alpha\in(\gamma_{-},\gamma_{+})$ as
long as their respective energies $I(\alpha,s)>0$.

###### Proof.

A careful inspection of the proof of Proposition 5.1, and the lemmas in
section $5$, shows that the constants $\lambda_{1}(\alpha)$,
$s_{\lambda}(\alpha)$, $\sigma_{+}^{\lambda}(\alpha)$, etc. are (or can be
chosen to be) continuously dependant on $\alpha$. ∎

###### Proof of Theorem B.

We will work with an odd $k$, the even case differ only in some signs. By
Lemma 3.3 there is an $\tilde{\alpha}^{k}>\alpha_{*}^{k}$ such that all
solutions of (1.9) with $f=f_{1}$ with
$u(0)\in(\alpha_{*}^{k},\tilde{\alpha}^{k}]$ reach a minimum with negative
energy. Therefore for each solution there is an $\tilde{s}\in(-\beta,0)$ with
$J(\tilde{s})>0$ and $I(\tilde{s})<0$.

It is known that solutions $v(\alpha,r)$ of (1.9) with $f=s^{p}$ and
$v(\alpha,0)=\alpha$ converges to a singular solution $\bar{v}$ on bounded
sets when the initial condition $\alpha$ tends to $\infty$. From Miyamoto and
Naito [MN] we obtain this for $0<r_{0}\leq r\leq r_{1}$ sufficiently small,
from continuity of the solutions we can extend this to compact sets. The
singular solution $\bar{v}$ is the classical solution in $(0,\infty)$ such
that $\lim\limits_{r\to 0}\bar{v}(r)=\infty$, (see Serrin and Zou [SZ]), it is

$\bar{v}(r)=C(N,p)r{{}^{\frac{-2}{p-1}}},\quad\mbox{where}\quad
C(N,p)=\Bigl{(}\frac{2}{p-1}\Bigl{(}N-2-\frac{2}{p-1}\Bigl{)}\Bigl{)}^{1/(p-1)}.$

If we consider $J_{v}r^{2}_{v}(\alpha,1)$ for $\alpha>1$, then
$J_{v}r^{2}_{v}(\alpha,1)\to J_{\bar{v}}r^{2}_{\bar{v}}(1)=\frac{2}{p-1}$,
therefore it is bounded and there is a
$K_{1}=\sup_{\alpha\in(1,\infty)}J_{v}r^{2}_{v}(\alpha,1)$. Note also that
$v(\alpha,r)=\alpha v(1,\alpha^{\frac{p-1}{2}}r)$ thus
$r_{v}(\alpha,s)=\alpha^{-\frac{p-1}{2}}r_{v}(1,\frac{s}{\alpha})$ and
$J_{v}r^{2}_{v}(\alpha,s)=\alpha J_{v}r^{2}_{v}(1,\frac{s}{\alpha})$.
Therefore
$K_{s}=\sup_{\alpha\in(s,\infty)}J_{v}r^{2}_{v}(\alpha,s)={K_{1}}{s}$.

Let
$d=\alpha_{*}^{k+1}+2\epsilon-\frac{\tilde{\alpha}^{k}+\alpha_{*}^{k}}{2}$,
$\bar{s}=(N-2)d/K_{1}$ and set $a=\bar{s}-(\alpha_{*}^{k+1}+2\epsilon)$. Then
a solution $u(\alpha,r)$ of (1.9) with $f=(s+a)^{p}$ is of the form
$u(\alpha,r)=v(\alpha+a,r)-a$ and
$J_{u}r_{u}^{2}(\alpha,s)=J_{v}r_{v}^{2}(\alpha+a,s+a)$ and
$J_{u}r_{u}^{2}(\alpha,\alpha_{*}^{k+1}+2\epsilon)=J_{v}r_{v}^{2}(\alpha+a,\bar{s})$
thus

$K=\sup_{\alpha\in(\alpha_{*}^{k+1}+2\epsilon,\infty)}J_{u}r^{2}_{u}(\alpha,\alpha_{*}^{k+1}+2\epsilon)=\sup_{\alpha\in(\bar{s},\infty)}J_{v}r^{2}_{v}(\alpha,\bar{s})=\bar{s}K_{1}=(N-2)d$

We can now choose $\alpha_{+}$ and $\alpha_{-}$ with
$\alpha_{*}^{k}<\alpha_{-}<\alpha_{*}^{k+1}+2\epsilon-d=\frac{\tilde{\alpha}^{k}+\alpha_{*}^{k}}{2}<\alpha_{+}<\min\\{\tilde{\alpha}^{k},\alpha_{*}^{k+1}+\epsilon,\alpha_{*}^{k+1}+\epsilon-\frac{K}{N-2}+\frac{2N}{N-2}\frac{F}{f}(\beta)\\}$.

Let $u$ be any solution of problem (1.9), with $f$ as in (1.17) and
$\alpha>\alpha_{*}^{k+1}+2\epsilon$. Then by Lemma 5.5, with
$\zeta=J_{u}r_{u}^{2}(\alpha_{*}^{k+1}+2\epsilon)\leq K$, we have
$J_{\lambda}r_{\lambda}^{2}(\alpha_{*}^{k+1}+\epsilon)>\zeta-(N-2)\epsilon$
tends to $\zeta-(N-2)\epsilon$ when $\lambda\to\infty$. Since
$\alpha_{-}<\alpha_{*}^{k+1}+2\epsilon-d<\alpha_{*}^{k+1}+\epsilon-\frac{\zeta-(N-2)\epsilon}{N-2}$
we can use Proposition 5.1 to find $\bar{\lambda}(\alpha)$ and show that the
solution $u_{-}$ with initial condition $\alpha_{-}$ have

$J_{-}(s)<J_{\lambda}(s)\quad\mbox{ for }\ s<s_{0},$

for all $\lambda>\bar{\lambda}(\alpha)$, as long as their respective energies
are positive. In particular, $0<J_{-}(0)<J_{\lambda}(0)$ thus $u$ reaches $0$.

Since when $\alpha\to\infty$ the solutions $u(\alpha,r)$ converge to some
$\bar{u}(r)$ on compact sets, there will be a $\bar{\lambda}_{\infty}$ such
that we can choose $\bar{\lambda}(\alpha)<\bar{\lambda}_{\infty}$ for
$\alpha>A$. Since for $\alpha=\alpha_{*}^{k+1}+\epsilon$ the solutions are
independent of $\lambda$ and satisfy the hypothesis of Proposition 4.1, we can
choose $\lambda_{*}$ such that we can choose
$\bar{\lambda}(\alpha)<\bar{\lambda}_{*}$ for $\alpha<B$. From Lemma 5.6 the
choice of $\bar{\lambda}(\alpha)$ can be done uniformly for $\alpha\in[B,A]$.
Therefore choosing $\bar{\lambda}$ as de largest of these three we get that
for $\lambda>\bar{\lambda}$ al solutions with initial condition
$\alpha>\alpha_{*}^{k+1}+\epsilon$ cut $0$ at least once.

On the other hand, since $K$ is a supremum, there is an $\alpha_{\Box}$ such
that
$J_{u}r^{2}_{u}(\alpha_{\Box},\alpha_{*}^{k+1}+2\epsilon)>(N-2)(\alpha_{*}^{k+1}+2\epsilon-\tilde{\alpha}^{1})$.
Then by Lemma 5.5, and Proposition 5.1 we can find
$\bar{\lambda}(\alpha_{\Box})$ and show that the solution $u_{+}$ with initial
condition $\alpha_{+}$ have

$J_{+}(s)>J_{\lambda}(s)\quad\mbox{ for }\ s<s_{0},$

for all $\lambda>\lambda_{0}$, as long as their respective energies are
positive. In particular, $J_{\lambda}(\tilde{s})<J_{+}(\tilde{s})<0$ thus $u$
can not reach $0$ a second time. By the argument at the beginning of this
subsection, there must be a $k^{th}$-bound state solution with initial
condition in $(\alpha_{*}^{k}+2\epsilon,\alpha_{\Box})$.

∎

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