#  Characterizing functional brain networks and emotional centers based on
Rasa theory of Indian aesthetics

Richa Tripathi  Indian Institute of Technology Gandhinagar, Gandhinagar-
382355, India  Dyutiman Mukhopadhyay  University College London, WC1E 6BT, UK
Indian Institute of Technology Gandhinagar, Gandhinagar- 382355, India
Chakresh Kumar Singh  Indian Institute of Technology Gandhinagar, Gandhinagar-
382355, India  Krishna Prasad Miyapuram  Indian Institute of Technology
Gandhinagar, Gandhinagar- 382355, India  Shivakumar Jolad  Indian Institute of
Technology Gandhinagar, Gandhinagar- 382355, India<EMAIL_ADDRESS>
###### Abstract

In Indian history of arts, Rasas are the aesthetics associated with any
auditory, visual, literary or musical piece of art that evokes highly
orchestrated emotional states. In this work, we study the functional response
of the brain to movie clippings meant to evoke the Rasas through network
analysis. We extract functional brain networks using coherence measures on EEG
recordings of film clips from popular Indian Bollywood movies representing
nine Rasas in the Indian Natyasastra. Structural and functional network
measures were computed for these brain networks, averaging over a range of
significant edge weights, in different brainwave frequency bands. We identify
segregation of neuronal wiring in the brain into modules using a community
detection algorithm. Further, using mutual information measure, we compare and
contrast the modular organizations of brain network corresponding to different
Rasas. Hubs identified using centrality measure reveal the network nodes that
are central to information propagation across all Rasas. We also observe that
the functional connectivity is suppressed when high-frequency waves such as
beta and gamma are dominant in the brain. The significant links causing
differences between the Rasa pairs are extracted statistically.

## Introduction

Emotions affect the body functioning in multiple ways such as by introducing
electro-physiological changes in the facial muscles, electrical activity in
the brain, changes in heartbeat rates and respiration [1]. These physiological
changes involve all the Sub-Systems of the body to a small or large extent and
hence affect our perception, thoughts and actions. Emotion states also have a
significant effect on the social development of the individuals[2]. Apart from
the cognitive origin, at the neurological level, it is believed that changes
in the density of neural stimulations [3] drive the emotions through agents
such as inborn releaser and external audio and visual stimulations. The
functional changes that induce emotions through neural segregation of
emotional and non-emotional pathways can be inferred using brain imaging and
EEG studies of individuals [4].

In the recent years, analysis of brain networks constructed from EEG and
advanced brain imaging techniques has contributed to the understanding of the
complex structure and functioning of the human brain [5, 6] and has given
clues to understanding its higher cognitive exhibits such as emotions and
reasoning abilities [7, 8, 9]. The functional brain networks studies have
given reasonable evidence against the locationist approach of functional
segregation of distinct brain regions corresponding to specific emotion
categories in favour of constructionist approach i.e. different emotions
involve brain circuits comprising of brain areas not specific to a particular
emotion [10, 8, 11]. Emotional stimuli affect large scale functional brain
networks which can be evaluated in terms of parameters such as node
betweenness and network efficiency [12]. The human brain can be decomposed
into multiple, distinct, and interacting networks such as salience network,
executive control network and task negative network, and emotional stimuli can
differentially affects these subnetworks [13, 12, 9]. For example, strong
changes in salience networks have been reported during watching an aversive
movie segment as compared to watching a neutral movie segment through fMRI
studies[14] .

Studies on emotion based on the analysis of single-electrode level based EEG
in the frequency domain have demonstrated the association of emotion with
asymmetric activity in the frontal brain in alpha band. Ekman and Davidson
(1993) [15] in their study found that voluntary facial expressions of smiles
of enjoyment produced high left frontal brain activity. On the contrary, Coan
et al. [16] reported that face expressions of fear produce low frontal
activation. Also, in a study by Sammler and colleagues [17], it was proposed
that increase in frontal midline theta power was associated with happy
emotion. Apart from using alpha and theta asymmetry for emotional
classification, power spectral features of EEG signals were extracted in five
frequency bands and used for emotion recognition using Deep Learning network
architecture with considerable accuracy[18]. It has also been demonstrated
that different patterns of functional connectivity are associated with
different emotional states in either single or combined frequency bands
ascertaining distinct response patterns of the central nervous system to
different emotional stimuli [7].

In previous studies on classifying emotions, researchers have largely focused
only on few contrasting dimensions such as threat-safe, sadness-happiness,
positive-negative-neutral and with a small number of recording sites of EEG
activity[19, 20, 7]. Also, the film-based studies on emotions and functional
response of brain has been previously carried out primarily based on the
Western classification of emotions which are happiness, sadness, fear, anger,
surprise, disgust [21, 22]. However, audio-visual stimuli such as film viewing
are capable of evoking sentiments which might be due to interplay the basic
emotions (dominant states) and transitory and temperamental states of
emotions. Indian Rasa theory, as described in the Natyasastra has outlined
these sentiments or Rasas as the superposition of these states of emotions
[23].

In this work, we study the functional response of brain captured through 128
channel EEG recordings to movie clippings eliciting the response corresponding
to the nine Rasas classified according to Indian Natyasastra. We have recorded
the response of participants to a wide range of audio-visual stimuli from
clippings of popular Indian Bollywood movies in Hindi. We have analysed the
patterns of brain activity in different frequency bands corresponding to
Brainwaves [24] and examined whether these activities would show specific
neural signatures across viewers. Different network measures were used to
characterises the structural and functional differences among the Rasa states.
Community structure corresponding to different Rasas were computed to
highlight the specificity to different brain areas or circuits and fill the
gap of brain network analysis. Modular organisations of networks corresponding
in each of the frequency bands were compared using information flow measure to
quantify the difference between the Rasas. Most central of the nodes were
identified using leverage centrality measure for each of the networks,
enabling us to find the most significant nodes for emotion perception and
segregate designated areas for particular emotion processing. The significant
links across all Rasas pairs were identified statistically to locate origins
of differences in functional networks using Honest Significant difference test
across emotion groups after the F-test as followed in a work by Lee (2014)[7].

## Background

###  Rasa and Natyasastra

A major source of the Indian system of classification of emotional states
comes from the ‘Natyasastra’, the ancient Indian treatise on the performing
arts, which dates back to 2nd Century AD (pg. LXXXVI: [23]). The ‘Natyasastra’
speaks about ‘sentiments’ or ‘Rasas’ (pg.102: [23]) which are produced when
certain ‘dominant states’ (sthayi bhava), ‘transitory states’ (vyabhicari
bhava) and ‘temperamental states’ (sattvika bhava) of emotions come together
(pgs.102, 105: [23]). This Rasa theory, which is still widely followed in
classical Indian performing arts, classifies eight Rasas or sentiments which
are: Sringara (erotic), Hasya (comic), Karuna (pathetic), Raudra (furious),
Veera (heroic), Bhayanaka (terrible), Bibhatsa (odious) and Adbhuta
(marvelous). There was a later addition of the ninth sentiment or Rasa called
Santa (peace) in later Sanskrit poetics (pg.102: [23]).

The eight dominant states (sthayi bhava) of emotions which give rise to these
corresponding eight Rasas or sentiments are: love (for Sringara), mirth (for
Hasya), sorrow (for Karuna), anger (for Raudra), energy (for Veera), terror
(for Bhayanaka), disgust (for Bibhatsa) and astonishment (for Adbhuta). Apart
from these there are thirty three ‘transitory states’ (vyabhicari bhava) and
eight ‘temperamental states’ (sattvika bhava) of emotions mentioned in the
Natyasastra which play a critical role in the generation of Rasas or
sentiments (pg.102: [23]).

###  Western emotional classification and Rasa theory

According to Western version of emotion classification by Ekman, there are
basic or universal emotions [25, 21], [22] which are happiness, sadness, fear,
anger, surprise, disgust. However, there are also background emotion sets
which are: wellbeing-malaise; calm-tense; pain-pleasure; [26] as well as self-
referential social emotions which are: embarrassment, guilt, shame, jealousy,
envy, empathy, pride, admiration. [27, 28, 29, 30, 31, 32]. Also, there are
pioneering works of scientists like Lisa Feldman Barrett [33] who questions
Ekman’s concepts of discreteness of emotions. Apart from defining the
boundaries of universal emotions, research in emotion science also places
equal emphasis in trying to understand the interplay of the different
orchestrated processes that give rise to a basic emotion [33, 34, 35]. For
example, Panksepp [34, 36] defined primary processes of emotion (not the basic
or universal emotions of Ekman) as primary sub-neocortical processes of
emotion having their corresponding affective states which can be artificially
generated by brain stimulation in animals. Also, the four-step cascade process
proposed by Jerome Kagan [35] defines a provocative event which leads to
brain-change and subsequently leads to a feeling and the interpretation of the
feeling gives rise to an emotion.

We can find startling similarities between the Rasa theory (its concepts of
the generation of Rasas from the Bhavas) with the works of Panksepp and Kagan
[34, 36, 35, 37]. However, there had been very little previous work done on
the perception or brain science of emotional states based on Rasa theory
mostly due to the lack of awareness regarding the science of the Rasa theory
among the scientific community. One behavioural study was conducted by Hejmadi
[38], which investigated the identification of these emotions across cultures.
An image processing study was conducted in Ref. [39] for investigating the
variations in facial features based on nine Rasas. The study proposes a tool
for the design of intelligent emotion recognition system but does not offer a
psychophysical or brain-based perspective on how these individual emotions
differ in the way they affect our perceptual process. There have been no
brain-imaging based works on Rasa-theory comparable with similar works done on
emotions based on Western literature. We analyze through EEG, the patterns of
brain activity generated through nine different emotional responses based on
Indian Rasa theory while watching films and to find out whether these
activities would show specific neural signatures across viewers.

## Results

### Brain Network Characterization

Functional properties of complex networks are largely determined by the
statistical properties of its structure. We have calculated six well-known
network measures- Clustering Coefficient ($CC$), Characteristic Path Length
($L$), Network Density ($d$), Local and Global efficiency ($LE,GE$), and
Small-worldness index ($SW$) using Area Under Curve method (see Methods for
details).

The clustering coefficient ($CC$) is a measure of network segregation. The
high global clustering coefficient of the brain network quantifies functional
segregation in the brain for specialized processing to occur within densely
interconnected groups of brain regions. The characteristic path length $L$
(average shortest path length between all pairs of nodes in the network - see
e.g. [40]) on the other hand, is the most commonly used measure of functional
integration [41]. Network density $d$ measures the number of links present as
compared to the number of possible connections in the network. Global
efficiency of network quantifies information exchange across the whole network
where the information flow occurs on a global scale. The local efficiency
quantifies the network’s robustness to failure in the local node neighbourhood
i.e. measure of information flow between the neighbours of a node if it is
removed from the network.

Small-worldness index ($SW=\frac{C/C_{rand}}{L/L_{rand}}$ , see Methods)
measures the amount of deviation of actual networks from the random networks
[40]. Functional brain networks with high correlations retained show many
biologically relevant features such as "small-world" behavior[40, 42, 43],
existence of hubs and high network modularity. Small-world topology supports
distributed as well as segregated information processing at the same time and
is highly economical in terms of wiring cost involved in this information
processing [42].

In Fig. 1, we plot the six measures described above for all the Rasas across
all frequency bands. Network density $d>0.45$ for all the networks obtained
through AUC method shows that they are dense. Average shortest path length
lies between 1.2 to 1.8, indicating the ease of information propagation.
Network density decreases with increasing frequency, except in the gamma band-
where it becomes denser. Correspondingly we see a drop in path length.
Clustering coefficient shows small variations from 0.75 to 0.9. Gamma band has
greater spread than all other bands. Veera  and Bhayanaka shows the maximum
difference across $\delta,\theta,\alpha$, and $\beta$ bands. SW greater than
one for all Rasas, indicating deviation from absolute randomness as well as
regularity of connections in the brain. Global efficiency of the networks is
high - ranging from 0.7 to 0.87. Local efficiency is even higher- 0.88 to
0.95, indicating high robustness to the failure of nodes. The trend (increase
or decrease of network measures with frequency) is flipped at high frequency
in the $\gamma$ band. This is seen for all the Rasas and for each network
metric, suggesting a dynamic reorganisation of brain structure at high
frequency. Information flow at gamma frequency the brain has a special
structural organisation quite very different from the one at just lower
frequency band.

Across all the frequency bands, except $\gamma$ it is seen that the Rasas
Bhayanaka and Veera are at opposite extremities in terms of the values of
Network measures. The order of Rasas across the extent of network properties
enveloped by properties of these two Rasas is Bhayanaka, Karuna, Raudra,
Santa, Hasya, Sringara, Bibhatsa, Adbhuta and Veera. This order is not
followed in gamma bands, where Karuna and Hasya are at the ends of Rasas
spectrum.

Figure 1: Network measures Network Density (d), Path Length (L), Clustering
Coefficient (CC) , Small Worldness , Global Efficiency (GE), and Local
Efficiency (LE) (see text for details) of brain networks corresponding to
different Rasas across the five frequency bands ($\delta,\theta,\alpha,\beta$
and $\ \gamma$) along with their error bars.

### Edge Weight distribution

Edge weights denote the strength of connections between the nodes. The
cumulative edge weight distribution of all Rasas  is shown in supplementary
information Fig.SI2. We find that edge weights are considerably high for
$\delta$, $\theta$ and $\alpha$ bands for all networks and lower for $\alpha$
and $\beta$ bands. Through which we infer that functional connectivity of
brain for emotion perception is stronger in $\delta$, $\theta$, and $\gamma$
bands whereas the emotional pathways have lesser average information flow in
$\alpha$ and $\beta$ frequency regimes. We also observe that Raudra, Bhayanaka
and Hasya have higher correlations specially in $\delta$, $\theta$ an $\alpha$
bands as compared to other Rasas.

### Community structure

Quantitative analysis of brain using complex networks measures has revealed
the presence of highly connected hubs and significant modular
architecture[44], apart from showing small-worldness. Modules are functionally
specialised and spatially localised groups of nodes that function together in
unison to integrate the information globally that they process locally.

In Fig. 2, we show community structure of brain networks corresponding to all
the Rasas, with top 10$\%$ of the edges retained, extracted using the
modularity optimisation algorithm in Gephi. The communities were then
extracted from the complete networks (with all the edges retained, after
removing the 5 $\%$ weakest connections) corresponding to all the Rasas. All
the networks were highly modular with modularity, $Q\geq 0.58$, and had with
five to six communities in each case. Broadly, the community structure appears
similar across all networks, with one major community in the frontal lobes
(C1), two in the left and right hemisphere’s central/parietal brain regions
(C2, C3), one that encompasses a large area in the visual cortex in the
occipital areas (C4) and two other smaller communities along the left and
right temporal regions (C5, C6). The Bibhatsa Rasa shows split of the
community in the parietal lobe. The communities C4 and C5 are merged into one
for most of the graphs. However, the frequency band wise networks as shown in
the supplementary information (Fig. SI 3), generally shows them as separate
communities.

Figure 2: Brain Networks for various Rasas with 10 per cent highest edge
weight edges, organised into clear community structure. For all the Rasas the
network is dissociated into 5 to 6 communities. One can notice the changes in
the community structure from this figure, while there are a few clear
communities consistently present across all the networks. The modularity value
for network partition was above 0.5 for all the Rasas, depicting highly
modular structure at the level of strongest connections in the network.
Figure 3: Variation of Information (Eq. 3) matrices for various Rasas in
different frequency bands arranged in order overall, $\delta$, $\theta$,
$\alpha$, $\beta$ and $\gamma$, from left to right and top to bottom. It can
be seen that Rasas have more similar modular organization in $\delta$ and
$\alpha$ bands than in $\theta$, $\beta$ and $\gamma$ bands.

### Distance between Networks

To compare the similarities and differences in the network’s modular
organization (communities), we use the Variation of information (VI) measure
[45] (See Eq. 3 in Methods). The VI measure signifies the difference in the
modular organisation of the networks, summarising the differences in
segregation and network wiring. In Fig. 3, we show the Variation of
Information measure for different Rasas in different frequency bands. Networks
that are farthest from each other are shown in brighter cell colour ($VI\simeq
0.5$), and those which are closest are shaded dark ($VI\simeq 0.0$). Six VI
measure matrices were obtained - one corresponding to overall spectrum and
five other corresponding to each of frequency bands (see Fig. 3). We have
evaluated the distance matrix between all Rasas based on the total power to
visualise the overall differences in network partitions. In the supplementary
information (Fig. SI 1), we have compared the modular organization of each
emotion across all frequencies i.e with one emotion at a time.

For all the pairs of Rasas, the VI falls in the range 0.22 to 0.56. The colour
bar (common for all matrices) range is chosen to represent the maximum
variation in VI. We observe that, for the full network, the dynamic range of
VI variation is much small, hence we cannot infer much about the similarity or
dissimilarity of different network organizations. In the lowest frequency
$\delta$ band - Bibhatsa, Hasya and Karuna are closest to each other. Maximum
difference is between Adbhuta and Bhayanka. For $\theta$ band- Bhayanaka,
Bibhatsa , Sringara are relatively closest. Others are far from each other.
The $\alpha$ band is the most distinctive frequency band where all the Rasas
are much closer to each other (except the Karuna). The $\alpha$ band activity
for all the Rasas , except Karuna seems indistinguishable from each other. For
$\beta$ band, the pairs Adbhuta, Bhayanaka, Raudra, and Sringara are the
closer to each other, and Hasya and Adbhuta are the farthest. In the $\gamma$
band, Veera is farthest from Santa, Hasya and Bhayanaka. Adbhuta is much
closer to Bhayanaka and Karuna.

Figure 4: Brain maps showing Honest Significant Differences of coherences for
groups, (a)Adb-Kar (b)Bha-Sri (c)Bib-Has (d)Has-Rau (e)Rau-Sri and (f) Sri-
Veer, with higher (Blue) and lower (Red) coherences for the group on the left.
The values in the panel on the left of each figure is the Variation of
information measure corresponding to the two Rasas in all frequency bands and
in overall spectrum.

### Honest significant edges among different Rasas

To assess whether a particular edge weight in the brain network of a given
Rasa is on average higher, or lower compared to other groups, we performed
F-test on all the edge weights against the null hypothesis that all Rasas
(groups) yield the same mean edge weight. The edge weights that pass the
F-test were the ones considered significant across all groups (with
significance $\alpha\leq 0.05$). But, this does not reveal which individual
group means contribute to this significance. To identify where the group
differences lie, Tukey’s Honest Significant Difference (HSD) was performed as
a post-hoc analysis on the edge weights that were proven significant by the
F-test. Tukey’s test enables us to determine two Rasas/ groups, corresponding
to each of these significant edge weights.

We extracted significant edges for all 36 pairs of un-thresholded networks.
The pairs are shown in the Fig. 4 shows significant edges for six such pairs.
The edges represent higher (Blue) and lower (Red) coherence for the group on
the left in each pair. The group means of edge weights that contribute to the
significant difference is spread over the entire space for some and it is
highly localized for others.Between Adbhutam and Karunyam , edges in the
central and parietal left regions show significant positive differences.
Bhayanka and Sringara have differences in the frontal and temporal left
regions. Raudra and Sringara has differences in right frontal and left
parietal regions.

A detailed table with all the significant connections for all the pairs is in
the supplementary information (Table SI 4). In the figure, a panel with
corresponding VI measure value for the corresponding network pair for overall
spectrum and the individual frequency bands are provided.

Figure 5: Brain map showing nodes with (a)highest positive leverage
centralities (b) highest negative leverage centralities for all the Rasas.
Pink coloured nodes are the one which are consistently central for at least
eight Rasas, other colors are for Rasas as shown in the color panel above.

### Hub Identification Using Centrality measure

Hubs in the brain networks are nodes that are critical to information flow
across the network and are identified based on various centrality measures
such as Degree Centrality, Eigen-Vector Centrality[46], Betweenness Centrality
[47], and Leverage Centrality[48]. As far as Brain networks are concerned,
work by Joyce et al. suggests that Leverage centrality [48] is computationally
cheaper and more accurate for identifying hubs than other centrality measures,
as revealed by their Receiver Operating Characteristic curves (see Methods for
details).

Leverage centrality also incorporates information about local node
neighbourhood, such that a node with high positive leverage centrality is more
impactful to its neighbours so that its neighbours draw more information from
it than any other nodes in their neighbourhood. In contrast, a negative
leverage centrality node is influenced more by its neighbours than being an
influencer node. In the present study, we designate central nodes as nodes
with the largest positive and negative leverage centralities.

In Fig. 5, are shown a brain map with coloured nodes having the highest (top
12) positive and highest negative (top 12) leverage centralities. Colour
coding is based on the Rasas as shown in the figure. Nodes shown in pink are
consistently central across all the Rasas. For the positive centrality case,
they all lie in the periphery of the brain map, mostly in the parietal brain
regions indicating that most important information relay centres for the
perception of emotions are located at the parietal regions of the brain. The
central nodes with most negative leverage centralities are in the frontal and
central brain regions demonstrating that nodes drawing maximum information
from their neighbours are in these regions (In the Supplementary Information,
we provide two tables (Table SI 2, Table SI 3) with labels of nodes with
highest positive and negative leverage centralities for all the Rasas ).

## Discussion

The Rasas discussed in this work evoke emotions which are related to, but are
different from the western classification of emotions such as anger, fear,
happiness, and sadness. But there is very little work done on the science of
Rasas and how they are perceived in the brain. To the best of our knowledge,
this is the first attempt to design a network analysis probe to identify
neural signatures for different Rasas mentioned in Indian Natyasastra.

We have studied changes in the functional connectivity and modular network
organisation in different frequency bands, concerning different Rasa states,
elicited in participants through watching of movie clips. We have
characterized how the emotional states are similar or dissimilar by measuring
network segregation in different frequency bands. We find that the community
structure in the alpha band is most similar across all Rasa genres and low
synchrony of nodes in higher frequency bands. Previous studies have reported
higher correlations for negative or stress full visual stimuli[49] for lower
frequency bands. In line with this, our results indicate higher coherence in
these bands for Raudra and Bhayanaka Rasas.

Our coherence analysis has also shown higher synchronization among nodes in
lower frequency bands delta, theta and alpha and weaker correlations in beta
and gamma bands for all the networks. We have identified the community
structure of the brain for different Rasas at the level of strongest links
through which we can track the community evolution such as the birth, growth,
merger and collapse of communities. We find the existence of four dominant
communities consistently across all Rasas and localisation of hubs on brain
map. We have also located the origins of coherence differences across Rasa
pairs in edges that are significantly different for these groups. Finally, we
identified most central nodes in all the networks using Leverage centrality
and located them on the scalp, showing crucial centres for information
processing.

There are a few limitations to this work. Our observations are based on
analysis of the signal space of electrodes placed on the scalp which does not
have a clear source mapping inside the brain. Western classification of
emotions such as those by Ekman, have been studied for decades using various
imaging methods and analytical tools by researchers in various disciplines.
However, we find very limited literature on quantitative study of Rasas based
on brain imaging or EEG. Lack of reference limits standardization of our
results. But we hope that our work will trigger interest in the research
community to explore the scientific study of Rasas further.

## Materials and Methods

Prior to conducting EEG experiments, ethical clearance was taken from the
Institute Ethical Committee (IEC) of Indian Institute of Technology
Gandhinagar. Informed consent was obtained from all the participants before
conducting experiments.

### Subjects

Participants were 20 healthy, right-handed students from Indian Institute of
Technology Gandhinagar (mean age: 26 years, 16 males; 4 females). All of the
participants were proficient in Hindi and English languages. They were all
informed about the task and were asked to remain attentive while watching the
film clip. We did an independent ranking of many movie clips corresponding to
each of the Rasas and only those were selected which were ranked best suited
to evoke a particular response among the Rasas.

### Movie Clips

Complex, naturalistic stimuli like film-viewing evoke highly reliable brain
activity across viewers as per current research works [50]. In this work, we
used nine film clips from Bollywood films (popular Indian Hindi language
cinema) made between 1970’s to the current time (see Table SI 1). These film
clips were chosen to elicit the nine Rasas among the viewers, based on the
opinion of experts in Natyashastra. Independent rating of the movies was done
on a small number of subjects. The length of the movie clips varied from 42s
to 2 mins 37s. Based on Natyasastra [51] human Rasas are classified into nine
categories [22] as given in the background section

### EEG Experimental Procedure

We conducted the EEG experiment on the participants during which they were
asked to watch chosen film clips representative of nine different Rasa Genres.
The electrical activity of the brain was recorded using 128 channel high-
density Geodesic EEG Systems™ with a sampling frequency of 250Hz. A
representative diagram of the node placement (along with node numbers from 1
to 128) used in the present work for brain network visualization, is shown in
Fig. 6 (a). The figure also depicts the electrode allocations to anatomical
regions of the brain. The following abbreviations are used: F, L O and T stand
for Frontal, Parietal, Occipital and Temporal lobes; L and R represent Left
and Right regions. In Fig. 2, brain maps for the nine Rasas of the overall
spectrum are shown.

The participants were shown all the film clips in random order. A white
fixation cross with a black screen for ten secs was shown before each clip.
Initial few seconds of the time series recordings were neglected before
analysis to avoid major fluctuations due to adjustment. Signals beyond 60 Hz
frequencies were filtered out to avoid noise effects. The design of the
experiment was done in E-prime™ and synced with Net-station™ acquisition
software. The entire study on the collected data from the EEG experiment is
summarised in the Fig. 6.

(a)

(b)

Figure 6: (a) A brain map of the electrode positions of the EEG cap. All the
Brain networks and extracted communities in the present work follow this Node
placement. (b) Workflow structure governing EEG data recording, pre-
processing, network construction and analysis

### Construction of Brain Network

Standard time series analysis measures such as Coherence [52], Phase
Synchronization [53], Mutual Information [54], and Granger Causality [55]
offer a huge depth into understanding the synchrony and information
propagation in the functional brain. We use Electroencephalographic Coherence
as a metric for deriving functional brain connectivity i. e. the degree of
association between any two brain regions (whose electrical activity is
recorded by electrodes in the signal space). Holding values between 0 (nil
coherence) and 1 (complete coherence), it compares similarity of the power
spectra (measured in microvolts squared, $\mu V^{2}$ ) of the time-series
recorded by these electrodes. The regions showing highest coherence are
assumed to be the most synchronized functionally and vice-versa. The coherence
measure (Eq. 1) between two time series $X$ and $Y$ is defined as,

$C_{XY}(f)=\frac{|G_{XY}(f)|^{2}}{|G_{XX}(f)G_{YY}(f))|},$ (1)

where $G_{XY}(f)$ is the cross-power spectral density and $G_{XX}(f)$ and
$G_{YY}(f)$ are the respective auto-power spectral densities [56].

The time series data (in microvolts, $\mu$V) was transformed to frequency
domain via Fast Fourier Transform (FFT) and the power in different Brainwave
frequency bands [24]: $\delta$ (1-4 Hz), $\theta$ (4 - 8 Hz), $\alpha$ (8 - 12
Hz), $\beta$ (12 - 40 Hz) and $\gamma$ (40 - 100 Hz), corresponding to
different brain waves was calculated. The Coherence spectra, thus decomposed
into five frequency bands resulted in five $128\times 128$ coherence matrices
corresponding to each of the nine stimuli. The frequency decomposition of the
spectrum was made to estimate the power in each of these bands and capture the
brain state when a particular emotion was evoked. Apart from powers in
individual bands we also used average coherence across the whole spectrum in
the analysis. The coherence matrix was mapped to a weighted adjacency matrix,
and corresponding brain network was constructed with $N=128$ nodes. In this
network, the nodes are the electrode locations on the scalp and edges are
connections between them measured by the coherence values.

Each of the electrodes from the EEG cap records the mean activity from a brain
region and has contributions from the neighbouring brain areas as well, which
may be spatially distant from the recording point. The resulting functional
brain network is a dense network, retaining all the edges ($\mathcal{O}\sim
N^{2}$) with non-zero edge weights. Also, in such functional networks, weak
and non-significant links may represent spurious connections or false
statistical dependencies. Such links tend to obscure the topology of stronger
and significant connections and should ideally be discarded, by applying hard,
or weighted thresholding. In the present work, the edges weights (coherences)
representing functional correlations are Fischer Z-transformed, averaged over
all the participants and then thresholded to retain only significant
connections. To this end, 5 $\%$ of the weakest connections were omitted from
the network, for all the nine cases. All the network analysis was done on
these thresholded networks.

### Network Analysis

Network analysis of the functional networks performed in the present study
consists of five elements as summarised in the Fig. 7. They are explained in
the following sections.

Figure 7: Block diagram of the methods used in the Network analysis of Brain
Networks.

#### Small-worldness index

To distinguish between true functional connections from random connections
among vertices, networks metrics must be compared to the corresponding mean
values of a benchmark random graph with the same number of nodes and edges. We
measured the small-worldness index of the networks defined as
$[C/C_{rand}]/[L/L_{rand}]$, where $C_{rand}$ and $L_{rand}$ are the mean
clustering coefficient and the characteristic path length of an ensemble of
random networks with fixed number of nodes $N$ and edges $\langle
E\rangle=E_{B}$ of the Brain network.

#### Area Under Curve method

The network measures and metrics were evaluated on the un-thresholded
networks, using the Area Under the Curve (AUC) approach[57]. In the AUC
procedure two cutoffs (one upper, one lower)on the edge strengths are
consistently determined for each network, such that a range of important
connections are retained, and only important network structure is dealt with.
The lower bound in this threshold range, $\kappa_{-}$, is an edge weight at
which network is fully connected and the upper bound, $\kappa_{+}$ is the
threshold at which the network just gets disconnected. The network is
binarised at each threshold within this range, retaining only edges with
weights larger than the threshold value and eliminating the remaining ones and
network metric ($M$) is evaluated at each connection density. This results in
a curve showing the variation of $M$ with edge weights lying in the threshold
range. The curve was then integrated over all these thresholds to yield the
AUC value of the metric $M$.

#### Community detection and Graph Visualization

For visualisation of dominant communities in the brain networks of all the
Rasas, we retained top 10 $\%$ of the edges and extracted communities on the
obtained network. We used Gephi software[58] which utilises a modularity
optimisation algorithm[59] for detection of communities from the brain
networks and Geolayout scheme for node positioning. The community structure of
networks in different frequency bands for all the Rasas is provided as
animated graphic files in the supplementary material.

#### Leverage Centrality

Leverage centrality (defined in Eq 2),

$l_{v}=\frac{1}{k_{v}}\sum_{N_{v}}\frac{k_{v}-k_{w}}{k_{v}+k_{w}}$ (2)

measures the relationship between the degree of a vertex $k_{v}$ and the
degree of each of its neighbors $k_{w}$, averaging over all the neighbors
$N_{v}$. For the weighted network, as in our case, degrees are weighted
degrees.

#### Variation of Information

Between two network partitions $X$ and $Y$, the $VI$ is defined in terms of
the conditional entropy as below:

$VI(X,Y)=H(X/Y)+H(Y/X).$ (3)

The conditional entropy $H(X/Y)$ of partition $X$ given $Y$, with $M_{x}$ and
$M_{y}$ modules can be computed as :

$H(X/Y)=-\sum_{i=1}^{M_{x}}\sum_{j=1}^{M_{y}}p(x_{i},y_{i})\log
p(x_{i}/y_{j}),$ (4)

where $p(x_{i},y_{j})=n_{ij}/N$ is the joint probability of randomly selecting
a node that belongs to modules $X_{i}$ and $Y_{j}$ and
$p(X_{i}|Y_{j})=n_{ij}/b_{j}$ is the conditional probability that a node
belongs to module $X_{i}$ in partition $X$, given that it is in module $Y_{j}$
in partition $Y$, and $b_{j}$ is the number of nodes in community $Y_{j}$ of
partition $Y$. $n_{ij}$ is the number of nodes that are simultaneously present
in module $X_{i}$ and $Y_{j}$.

#### Honest Significant Difference across Rasas

In our case, each of the Rasas was a group formed by twenty networks
(corresponding to twenty participants coherence matrices) and mean of a
particular edge weight was calculated over these twenty networks. The edge
weight values in the coherence matrices corresponding to these groups were the
ones tested for significance using the above two tests. So, practically we ran
F-test $128\choose 2$ (number of edges) times. The edges that were significant
according to the F-test were subject to Tukey’s test, to reveal which edge
weight is significant across any two groups, with significance $\alpha\leq
0.05$. So, the two groups with significantly different mean were calculated
for each edge weight, and in this group pair, one has a significantly higher
or lower mean edge weight than the other. These are depicted for six pairs in
the Fig. 4 in the results section.

## Data availability

All the figures generated and used in the study, Gephi network files,
coherence matrices, and tables are uploaded on a Google drive folder
(Rasas$\\_$BrainNetwork$\\_$Data). Raw EEG data can be provided upon request
to authors. Details of the movie clippings (start and end times) are given in
the Supplementary Information.

## Acknowledgments

We would like to thank Dipanjan Roy from National Brain Research Center
(NBRC), India and Pranjali Kulkarni, Indian Institute of Technology
Gandhinagar (IITGn) for insightful suggestions into this study. The study is a
part of the Department of Science and Technology (India) sponsored Cognitive
Science Research Initiative (DST-CSRI) project (Project number:
SR/CSRI/PDF-08/2013, Dated 30/10/2014). RT would like to thank Amit Reza from
IITGn for helpful discussions. We thank IITGn for providing access to the
High-Performance Computing services.

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# Infinitely many solutions for the Dirichlet problem involving the
p-Laplacian in annulus

Anderson L. A. de Araujo  Departamento de Matemática, Universidade Federal de
Viçosa, 36570-000, Viçosa (MG), Brazil<EMAIL_ADDRESS>
###### Abstract.

We present a result of existence of infinitely many solutions for the
Dirichlet problem involving the p-Laplacian in annular domains, when $p\leq
N$, contouring the failure of compactness of $W^{1,p}(\Omega)$ in
$C^{0}(\overline{\Omega})$ applying a variable change.

###### Key words and phrases:

Dirichlet problem; Ordinary differential equations; p-Laplacian; Multiple
solutions

###### 2010 Mathematics Subject Classification:

35J20; 35J25; 35J60

.

##  1\. Introduction

This paper is to investigate the following autonomous Dirichlet problem in
annulus

(1.1)
$\left\\{\begin{array}[]{cll}\displaystyle-\Delta_{p}\,u=f(u)&\textup{in}&\Omega_{a,b},\\\
\displaystyle u=0&\textup{on}&\partial\Omega_{a,b}\end{array}\right.$

where $\Omega_{a,b}=\\{x\in\mathbb{R}^{N}:a<|x|<b\\}$ with $0<a<b$ constants
in $\mathbb{R}$,

$1<p\leq N,$

$\Delta_{p}u=div(|\nabla\,u|^{p-2}\nabla\,u)$ and $f:\mathbb{R}\to\mathbb{R}$
is a continuous function.

In order to study the solution of (1.1), one can make a standard change of
variables. In the $N>p$, be $t=-\frac{A}{r^{(N-p)/(p-1))}}+B$ and $v(t)=u(r)$,
where

$A=\frac{(ab)^{\frac{N-p}{p-1}}}{b^{\frac{N-p}{p-1}}-a^{\frac{N-p}{p-1}}}\,\,\,\mbox{and}\,\,\,B=\frac{b^{\frac{N-p}{p-1}}}{b^{\frac{N-p}{p-1}}-a^{\frac{N-p}{p-1}}},$

then the problem (1.1) transforms into the boundary value problem for the
nonautonomous ODE

(1.2)
$\left\\{\begin{array}[]{lll}\displaystyle(|v^{\prime}(t)|^{p-2}v^{\prime}(t))^{\prime}+q(t)f(v(t))=0&\textup{in}&(0,1),\\\
\displaystyle v(0)=v(1)=0,&&\end{array}\right.$

where

(1.3)
$q(t)=\left(\frac{p-1}{N-p}\right)^{p}\frac{A^{\frac{(p-1)p}{N-p}}}{(B-t)^{\frac{p(N-1)}{N-p}}}.$

In the case $p=N$, one sets $r=a(\frac{b}{a})^{t}$ and $v(t)=u(r)$, obtaining
again the problem (1.2), now with

$q(t)=\left[a(\frac{b}{a})^{t}\ln\frac{b}{a}\right]^{p}.$

Note that in both cases, the function $q(t)$ is well defined, continuous and
bounded between positive constants in the interval $[0,1]$, that is, there
exist $q_{1},q_{0}>0$ such that

$0<q_{0}\leq q(t)\leq q_{1}.$

For our purpose, we shall restrict our attention to the ordinary boundary
value problem (1.2), where the function $q(t)$ is continuous and positive on
the interval $[0,1]$, while for $f$ we consider the assumptions below. A weak
solution of (1.2) is any $v\in W^{1,p}_{0}(0,1)$ such that

$\int_{0}^{1}|v^{\prime}(t)|^{p-2}v^{\prime}(t)w^{\prime}(t)dt-\int_{0}^{1}q(t)f(v(t))w(t)dt=0,$

for each $w\in W^{1,p}_{0}(0,1)$.

We are interested in the existence of infinitely many non-negative weak
solutions for problem (1.1), or equivalently, to the problem (1.2). Precisely,
if $F(\xi)=\int_{0}^{\xi}f(t)dt$, our aim is to prove the following results:

######  Theorem 1.1.

Assume that $f(0)=0$, $q(t)\geq q_{0}>0$ in $[0,1]$ and $\inf_{\xi\geq
0}F(\xi)\geq 0$. Moreover, suppose that there exist two sequences
$\\{a_{k}\\}_{k\in\mathbb{N}}$ and $\\{b_{k}\\}_{k\in\mathbb{N}}$ in
$]0,+\infty[$, with $a_{k}<b_{k}$, $\lim_{k\to+\infty}b_{k}=+\infty$, such
that

(i):

$\lim_{k\to+\infty}\frac{b_{k}}{a_{k}}=+\infty$;

(ii):

$\max_{[a_{k},b_{k}]}f\leq 0$ for all $k\in\mathbb{N}$;

(iii):

Suppose that

$\frac{\sigma(p,q_{0})}{p(\sup_{t\in(0,1)}dist(t,\\{0,1\\}))^{p}}<\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^{p}}<+\infty,$

where

$\sigma(p,q_{0})=\inf_{\mu\in]0,1[}\frac{1}{q_{0}\mu(1-\mu)^{p-1}}.$

Then, problem (1.2) (respectively the problem (1.1)) admits an unbounded
sequence of non-negative weak solutions in $W^{1,p}_{0}(0,1)$ (respectively in
$W^{1,p}_{0}(\Omega_{a,b})$).

######  Remark 1.2.

The constant $\sigma(p,q_{0})$ is well defined. To see this, let
$\sigma:(0,1)\to\mathbb{R}$ be the function defined by
$\sigma(x)=\frac{1}{x(1-x)^{p-1}}$. As
$\sigma^{\prime}(x)=\frac{-(1-x)^{p-2}(1-px)}{x^{2}(1-x)^{2(p-1)}}$ we have
$\sigma^{\prime}(x)<0$ if $0<x<\frac{1}{p}$, $\sigma^{\prime}(\frac{1}{p})=0$
and $\sigma^{\prime}(x)>0$ if $\frac{1}{p}<x<1$. Therefore,

$\sigma(p,q_{0})=\inf_{\mu\in]0,1[}\frac{1}{q_{0}\mu(1-\mu)^{p-1}}=\frac{1}{q_{0}p}.$

######  Theorem 1.3.

Assume that $f(0)=0$, $q(t)\geq q_{0}>0$ in $[0,1]$ and $\inf_{\xi\geq
0}F(\xi)\geq 0$. Moreover, suppose that there exist two sequences
$\\{a_{k}\\}_{k\in\mathbb{N}}$ and $\\{b_{k}\\}_{k\in\mathbb{N}}$ in
$]0,+\infty[$, with $a_{k}<b_{k}$, $\lim_{k\to+\infty}b_{k}=+\infty$, such
that

(i):

$\lim_{k\to+\infty}\frac{b_{k}}{a_{k}}=+\infty$;

(ii):

$\max_{[a_{k},b_{k}]}f\leq 0$ for all $k\in\mathbb{N}$;

(iii):

Suppose that

$\frac{\sigma(p,q_{0})}{p(\sup_{t\in(0,1)}dist(t,\\{0,1\\}))^{p}}<\limsup_{\xi\to
0^{+}}\frac{F(\xi)}{\xi^{p}}<+\infty,$

where

$\sigma(p,q_{0})=\inf_{\mu\in]0,1[}\frac{1}{q_{0}\mu(1-\mu)^{p-1}}.$

Then, problem (1.2) (respectively the problem (1.1)) admits an sequence of
non-negative weak solutions, which strongly converges to $0$ in
$W^{1,p}_{0}(0,1)$ (respectively in $W^{1,p}_{0}(\Omega_{a,b})$).

The statement and the proves of Theorems 1.1 and 1.3 are very similar the
Theorems $1.1$ and $1.2$ of F. Cammaroto, A. Chinnì, B. Di Bella [2] which is
the principal motivation and reference of this paper. But in [2] the authors
supposed that $p>N$ and a standard argument, chiefly based on the compact
embedding $W^{1,p}(\Omega)$ in $C^{0}(\overline{\Omega})$ while in this paper
we consider $p\leq N$ and soon, the above mentioned immersion is not
satisfied. For the sake of completeness, we decided to remake the proves of
Theorems 1.1 and 1.3 to explicit the most important changes in specific
arguments compared with [2].

The existence of infinitely many solutions of problem (1.1) in general bounded
domains $\Omega$ has been studied extensively. Among them, the ones which are
closest to the present article are certainly [2, 3, 4] and references therein.
The approach used in [2] is based on a recent variational principle obtained
by Ricceri in [6], while in [3, 4] is based on the method of lower and upper
solutions.

A more general problem that (1.1) was studied in Bonanno and Bisci [1] in
general bounded domains $\Omega$, and the authors also supposed that $p>N$
following the same arguments of [2]. Also examples and applications are given
in comparison with [2]. Following ours results, we can prove analogous results
to the results of [1], in $\Omega_{a,b}$, when $p\leq N$.

The ours approach is based on the changed of variables, problem (1.2), and the
recent variational principle obtained by Ricceri in [6]. The following result
is a direct consequence of Theorem $2.5$ of [6].

######  Proposition 1.4.

Let $X$ be a reflexive real Banach space, and let $\Phi$,
$\Psi:X\to\mathbb{R}$ be two sequentially weakly lower semicontinuous and
Gâteaux differentiable functional. Assume also that $\Psi$ is (strongly)
continuous and satisfies $\lim_{\|x\|\to+\infty}\Psi(x)=+\infty$. For each
$r>\inf_{X}\Psi$, put

$\varphi(r)=\inf_{x\in\Psi^{-1}(]-\infty,r[)}\frac{\Phi(x)-\inf_{\overline{(\Psi^{-1}(]-\infty,r[))_{w}}}\Phi}{r-\Psi(x)},$

where $\overline{(\Psi^{-1}(]-\infty,r[))_{w}}$ is the closure of
$\Psi^{-1}(]-\infty,r[)$ in the weak topology. Fixed $\lambda\in\mathbb{R}$,
then

(1):

if $\\{r_{n}\\}_{\in\mathbb{N}}$ is a real sequences with
$\lim_{n\to+\infty}r_{n}=+\infty$ such that $\varphi(r_{n})<\lambda$, for each
$n\in\mathbb{N}$, the following alternative holds: either $\Phi+\lambda\Psi$
has a global minimum, or there exists a sequence $\\{x_{n}\\}$ of critical
points of $\Phi+\lambda\Psi$ such that
$\lim_{n\to+\infty}\Psi(x_{n})=+\infty$.

(2):

if $\\{s_{n}\\}_{\in\mathbb{N}}$ is a real sequences with
$\lim_{n\to+\infty}s_{n}=(\inf_{X}\Psi)^{+}$ such that
$\varphi(s_{n})<\lambda$, for each $n\in\mathbb{N}$, the following alternative
holds: either there exists a global minimum of $\Psi$ which is a local minimum
of $\Phi+\lambda\Psi$, or there exists a sequence $\\{x_{n}\\}$ of pairwise
distinct critical points of $\Phi+\lambda\Psi$, with
$\lim_{n\to+\infty}\Psi(x_{n})=\inf_{X}\Psi$, which weakly converges to a
global minimum of $\Psi$.

Before stating our main result, we wish to point out that in the sequel
$f(x)=0$ for each $x\in]-\infty,0[$. We shall consider the Sobolev space
$W^{1,p}_{0}(0,1)$ endowed with the norm

$\|v\|:=\left(\int_{0}^{1}|v^{\prime}(t)|^{p}dt\right)^{1/p}.$

We recall that there exists a constant $c>0$ such that

(1.4) $\sup_{x\in[0,1]}|v(t)|\leq c\|v\|$

for each $v\in W^{1,p}_{0}(0,1)$.

##  2\. Proofs

###### Proof of Theorem 1.1.

Let us apply part ($a$) of Proposition 1.4. To this end choose
$X=W^{1,p}_{0}(0,1)$ and for each $v\in X$, put

$\Phi(v)=-\int_{0}^{1}\left(\int_{0}^{u(t)}q(t)f(s)ds\right)dt$

and

$\Psi(v)=\|v\|^{p}.$

It is well known that the critical points in $X$ of the functional
$\Phi+(1/p)\Psi$ are precisely the weak solutions of problem (1.2). By the
compact embedding of $W^{1,p}_{0}(0,1)$ in $C([0,1])$, it is not difficult
ensures that the functionals $\Phi$ and $\Psi$ are Gâteaux differentiable and
sequentially weakly lower semicontinuous, moreover $\Psi$ is obviously
(strong) continuous and coercive.

In our case the function $\varphi$ of Proposition 1.4 is defined by setting

$\varphi(r)=\inf_{\|v\|^{p}<r}\frac{\sup_{\|v\|^{p}<r}\int_{0}^{1}q(t)F(v(t))dt-\int_{0}^{1}q(t)F(v(t))dt}{r-\|v\|^{p}}$

for each $r\in]0,+\infty[$.

Now, put $r_{k}=\left(\frac{b_{k}}{c}\right)^{p}$, we wish to prove that
$\varphi(r_{k})<\frac{1}{p}$ for each $k\in\mathbb{N}$. To this aim, it
suffices to prove that, for each $k\in\mathbb{N}$, there exists a function
$v_{k}\in X$, with $\|v_{k}\|^{p}<r_{k}$, such that

$\sup_{\|v\|^{p}<r_{k}}\int_{0}^{1}q(t)F(v(t))dt-\int_{0}^{1}q(t)F(v_{k}(t))dt<\frac{1}{p}(r_{k}-\|v_{k}\|^{p}).$

From ($iii$) we can choose a constant $h$ such that

(2.1)
$\frac{\sigma(p,q_{0})}{p(\sup_{t\in(0,1)}dist(t,\\{0,1\\}))^{p}}<h<\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^{p}}$

and so there exists $t_{0}\in(0,1)$ such that
$\left(\frac{\sigma(p,q_{0})}{ph}\right)^{1/p}<dist(t_{0},\\{0,1\\})$.
Therefore, we can fix $\gamma$ satisfying

$\left(\frac{\sigma(p,q_{0})}{ph}\right)^{1/p}<\gamma<dist(t_{0},\\{0,1\\}).$

Observe that by ($ii$),

(2.2) $\max_{[0,a_{k}]}F=\max_{[0,b_{k}]}F.$

Now, fix $k\in\mathbb{N}$ and consider the function $v_{k}\in X$ defined by
setting

$v_{k}(t)=\left\\{\begin{array}[]{lll}\displaystyle
0&\textup{if}&t\in(0,1)\backslash(t_{0}-\gamma,t_{0}+\gamma),\\\
\displaystyle\xi_{k}&\textup{if}&t\in(t_{0}-\frac{\gamma}{2},t_{0}+\frac{\gamma}{2}),\\\
\displaystyle\frac{2\xi_{k}}{\gamma}(\gamma-|t-t_{0}|)&\textup{if}&t\in(t_{0}-\gamma,t_{0}+\gamma)\backslash(t_{0}-\frac{\gamma}{2},t_{0}+\frac{\gamma}{2})\\\
\end{array}\right.$

with $\xi_{k}\in]0,a_{k}]$ such that

(2.3) $F(\xi_{k})=\max_{\eta\in[0,a_{k}]}F(\eta).$

In view of ($i$), we can choose $k\in\mathbb{N}$ so that

(2.4) $\frac{b_{k}}{a_{k}}>\frac{2c}{\gamma}\left(\gamma\right)^{1/p}$

fro all $k>\overline{k}$.

Fix $k>\overline{k}$. By (2.4) we have

$\begin{array}[]{rcl}\|v_{k}\|^{p}&=&\displaystyle\int_{0}^{1}|v_{k}^{\prime}(t)|^{p}dt=\int_{(t_{0}-\gamma,t_{0}+\gamma)\backslash(t_{0}-\frac{\gamma}{2},t_{0}+\frac{\gamma}{2})}\frac{2^{p}\xi_{k}^{p}}{\gamma^{p}}dt\\\
&=&\displaystyle\frac{2^{p}\xi_{k}^{p}}{\gamma^{p}}(|(t_{0}-\gamma,t_{0}+\gamma)|-|(t_{0}-\frac{\gamma}{2},t_{0}+\frac{\gamma}{2})|)=\frac{2^{p}\xi_{k}^{p}}{\gamma^{p}}(2\gamma-\gamma)\\\
&=&\displaystyle\frac{2^{p}\xi_{k}^{p}}{\gamma^{p-1}}<r_{k}.\end{array}$

Because of (1.4), (2.2) and (2.3), for each $v\in X$, fulfilling
$\|v\|^{p}\leq r_{k}$, one has

$\sup_{x\in[0,1]}|v(t)|\leq cr_{k}^{1/p}$

as well as

$F(v(t))\leq\max_{\eta\in[0,cr_{k}^{p}]}F(\eta)=F(\xi_{k})\,\,\,\mbox{in}\,\,\,(0,1).$

Next, since $\lim_{k\to+\infty}\frac{r_{k}}{\xi_{k}^{p}}=+\infty$, there
exists $k_{0}\in\mathbb{N}$ such that

(2.5)
$\frac{r_{k}}{\xi_{k}^{p}}>p\limsup_{\xi\to+\infty}\frac{F(\xi)}{\xi^{p}}\left(\int_{0}^{1}q(t)dt-\int_{t_{0}-\frac{\gamma}{2}}^{t_{0}+\frac{\gamma}{2}}q(t)dt\right)+\frac{2^{p}}{\gamma^{p-1}}$

for all $k>k_{0}$. Hence, using (2.5), we get

$\begin{array}[]{r}\displaystyle\sup_{\|v\|^{p}<r_{k}}\int_{0}^{1}q(t)F(v(t))dt-\int_{0}^{1}q(t)F(v_{k}(t))dt\\\
\displaystyle\leq
F(\xi_{k})\int_{0}^{1}q(t)dt-\int_{t_{0}-\frac{\gamma}{2}}^{t_{0}+\frac{\gamma}{2}}q(t)F(\xi_{k})dt\\\
\displaystyle\leq
F(\xi_{k})\left(\int_{0}^{1}q(t)dt-\int_{t_{0}-\frac{\gamma}{2}}^{t_{0}+\frac{\gamma}{2}}q(t)dt\right)\\\
\displaystyle<\frac{1}{p}(r_{k}-\|v_{k}\|^{p})\end{array}$

for each $k>k^{*}\geq k_{0}$.

Since $\lim_{k\to+\infty}r_{k}=+\infty$, the previous inequality assures that
the conclusion ($1$) of Proposition 1.4 can be used and either the functional
$\Phi+(1/p)\Psi$ has a global minimum, or there exists a sequence
$\\{v_{k}\\}_{k\in\mathbb{N}}$ of solutions of problem (1.2) such that
$\lim_{k\to+\infty}\|v_{k}\|=+\infty$.

The other step is to verify that the functional $\Phi+(1/p)\Psi$ has no global
minimum. Taking into account (2.1), one has, for each $k\in\mathbb{N}$.

$\sup_{\eta\geq k}\frac{F(\eta)}{\eta^{p}}>h$

and so there exists $\eta_{k}\geq k$ such that

$\frac{F(\eta_{k})}{\eta_{k}^{p}}>h.$

Now, if we consider a function $w_{k}\in X$ defined by setting

$w_{k}(t)=\left\\{\begin{array}[]{lll}\displaystyle
0&\textup{if}&x\in(0,1)\backslash(t_{0}-\gamma,t_{0}+\gamma),\\\
\displaystyle\eta_{k}&\textup{if}&x\in(t_{0}-\overline{\mu}\gamma,t_{0}+\overline{\mu}\gamma),\\\
\displaystyle\frac{\eta_{k}}{\gamma(1-\overline{\mu})}(\gamma-|t-t_{0}|)&\textup{if}&x\in(t_{0}-\gamma,t_{0}+\gamma)\backslash(t_{0}-\overline{\mu}\gamma,t_{0}+\overline{\mu}\gamma)\\\
\end{array}\right.$

where $\overline{\mu}\in]0,1[$ is such that

$\sigma(p,q_{0})=\frac{1}{q_{0}\overline{\mu}(1-\overline{\mu})^{p-1}},$

note that $\overline{\mu}$ is well defined by Remark 1.2. We have

$\begin{array}[]{l}\Phi(w_{k})+\frac{1}{p}\Psi(w_{k})\\\
=\displaystyle-\int_{0}^{1}q(t)F(w_{k}(t))dt+\frac{1}{p}\|w_{k}\|^{p}\\\
\leq\displaystyle-\int_{t_{0}-\overline{\mu}\gamma}^{t_{0}+\overline{\mu}\gamma}q(t)F(\eta_{k})dt+\frac{1}{p}\frac{\eta_{k}^{p}}{\gamma^{p}(1-\overline{\mu})^{p}}(2\gamma-2\overline{\mu}\gamma)\\\
\leq\displaystyle-2\overline{\mu}\gamma\,q_{0}F(\eta_{k})+\frac{2\gamma}{p}\frac{\eta_{k}^{p}}{\gamma^{p}(1-\overline{\mu})^{p-1}}\\\
=\displaystyle
2\overline{\mu}\gamma\,q_{0}(\frac{1}{p}\frac{\eta_{k}^{p}}{\gamma^{p}q_{0}\overline{\mu}(1-\overline{\mu})^{p-1}}-F(\eta_{k}))\\\
<\displaystyle
2\overline{\mu}\gamma\,q_{0}(\frac{1}{p}\frac{\eta_{k}^{p}}{\gamma^{p}q_{0}\overline{\mu}(1-\overline{\mu})^{p-1}}-h\eta_{k}^{p})=2\overline{\mu}\gamma\,q_{0}\eta_{k}^{p}\left(\frac{\sigma(p,q_{0})}{p\gamma^{p}}-h\right).\\\
\end{array}$

Since $h>\frac{\sigma(p,q_{0})}{p\gamma^{p}}$, we conclude that
$\lim_{k\to+\infty}\eta_{k}^{p}\left(\frac{\sigma(p,q_{0})}{p\gamma^{p}}-h\right)=-\infty$
and so the previous inequality shows that the functional $\Phi+(1/p)\Psi$ is
not bounded from below and then it has no global minimum.

Therefore, Proposition 1.4 assures that there is a sequence
$\\{v_{k}\\}_{k\in\mathbb{N}}\subset X$ of critical points of $\Phi+(1/p)\Psi$
such that $\lim_{n\to+\infty}\|v_{n}\|=+\infty$. As previously observed, every
function $v_{k}$ is a weak solution of (1.2).

Finally, we claim that each weak solution of problem (1.2) is non-negative in
$(0,1)$. Assume the contrary. Let $v$ be a weak solution of (1.2) such that
the set $A=\\{x\in(0,1):v(x)<0\\}$ is non-empty. By the continuity of $v$, $A$
is open and so $v|_{A}\in W^{1,p}_{0}(A)$. Then, for each $w\in
W^{1,p}_{0}(A)$,

$\int_{A}|v^{\prime}(t)|^{p-2}v^{\prime}(t)w^{\prime}(t)dt-\int_{A}q(t)f(v(t))w(t)dt=0.$

The assumptions on $f$ imply

$\int_{A}|v^{\prime}(t)|^{p-2}v^{\prime}(t)w^{\prime}(t)dt=0$

for each $w\in W^{1,p}_{0}(A)$ and so, in particular, one has

$\int_{A}|v^{\prime}(t)|^{p}dt=0,$

an absurd. This completes the proof. ∎

###### Proof of Theorem 1.3.

We take $X,\Phi,\Psi$ as in the proof of Theorem 1.1. In a similar way we
prove that $\varphi(s_{k})<\frac{1}{p}$ for each $k\in\mathbb{N}$, with
$s_{k}=\left(\frac{b_{k}}{c}\right)^{p}$. Now, fixed $h$ such that

$\frac{\sigma(p,q_{0})}{p(\sup_{t\in(0,1)}dist(t,\\{0,1\\}))^{p}}<h<\limsup_{\xi\to
0^{+}}\frac{F(\xi)}{\xi^{p}}.$

For each $k\in\mathbb{N}$, there exists $\eta_{k}\leq\frac{1}{k}$ such that
$\frac{F(\eta_{k})}{\eta_{k}^{p}}>h$. If we take $w_{k}$ as in the proof of
Theorem 1.1, of course the sequence $\\{w_{k}\\}$ strongly converges to 0 in
$X$ and $\Phi(w_{k})+(1/p)\Psi(w_{k})<0$ for all $k\in\mathbb{N}$. Since
$\Phi(0)+(1/p)\Psi(0)=0$, this means that $0$ is not a local minimum of
$\Phi+(1/p)\Psi$. Then, since $0$ is the only a global minimum of $\Psi$, the
part ($2$) of Proposition 1.4 ensures that there exists a sequence
$\\{v_{k}\\}_{k\in\mathbb{N}}\subset X$ of critical points of $\Phi+(1/p)\Psi$
such that $\lim_{k\to+\infty}\|v_{k}\|=0$ and this completes the proof. ∎

##  3\. Conclusions

We can also to investigate the following nonautonomous Dirichlet problem in an
annular domain

(3.1)
$\left\\{\begin{array}[]{cll}\displaystyle-\Delta_{p}\,u=g(|x|)f(u)&\textup{in}&\Omega_{a,b},\\\
\displaystyle u=0&\textup{on}&\partial\Omega_{a,b}\end{array}\right.$

where $\Omega_{a,b}=\\{x\in\mathbb{R}^{N}:a<|x|<b\\}$ with $0<a<b$ constants
in $\mathbb{R}$, $1<p\leq N$, $\Delta_{p}u=div(|\nabla\,u|^{p-2}\nabla\,u)$,
$f:\mathbb{R}\to\mathbb{R}$ is a continuous function and
$g:[a,b]\to\mathbb{R}$ is a continuous function such that $g(s)\geq g_{0}>0$.

In order to study the solution of (3.1), one can make a standard change of
variables, see Liu and Yang [5]. In the $N\geq p+1$, if
$t=-\frac{A}{r^{(N-p)/(p-1))}}+B$ and $v(t)=u(r)$, where

$A=\frac{(ab)^{\frac{N-p}{p-1}}}{b^{\frac{N-p}{p-1}}-a^{\frac{N-p}{p-1}}}\,\,\,\mbox{and}\,\,\,B=\frac{b^{\frac{N-p}{p-1}}}{b^{\frac{N-p}{p-1}}-a^{\frac{N-p}{p-1}}},$

then the problem (1.1) transforms into the boundary value problem for the
nonautonomous ODE

(3.2)
$\left\\{\begin{array}[]{lll}\displaystyle(|v^{\prime}(t)|^{p-2}v^{\prime}(t))^{\prime}+h(t)k(t)f(v(t))=0&\textup{in}&(0,1),\\\
\displaystyle v(0)=v(1)=0,&&\end{array}\right.$

where

$h(t)=g\left(\left(\frac{A}{B-t}\right)^{\frac{p-1}{N-p}}\right)$

and

$k(t)=\left(\frac{p-1}{N-p}\right)^{p}\frac{A^{\frac{(p-1)p}{N-p}}}{(B-t)^{\frac{p(N-1)}{N-p}}}.$

In the case $p=N$, one sets $r=b\left(\frac{a}{b}\right)^{t}$ and $v(t)=u(r)$,
obtaining again the problem (1.2), now with

$h(t)=g\left(b\left(\frac{a}{b}\right)^{t}\right)\,\,\mbox{and}\,\,k(t)=\left[b\left(\frac{a}{b}\right)^{t}\left(\ln\frac{b}{a}\right)^{-1}\right]^{p-1}.$

Note that in both cases, the function $h(t)$ and $q(t)$ are well defined,
continuous and bounded between positive constants in the interval $[0,1]$,
that is, there exists $q_{0},q_{1},q_{2},q_{3}>0$ such that

$0<q_{0}\leq h(t)\leq q_{1}\,\,\mbox{and}\,\,0<q_{2}\leq k(t)\leq q_{3}.$

Then, in (3.1) put $q(t)=h(t)k(t)$ and we can get analogous results as in
Theorems 1.1 and 1.3.

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