# Multiscale Fractal Analysis on EEG Signals
for Music-Induced Emotion Recognition

Kleanthis Avramidis, Athanasia Zlatintsi, Christos Garoufis and Petros Maragos
School of ECE, National Technical University of Athens, 15773 Athens, Greece
<EMAIL_ADDRESS><EMAIL_ADDRESS>{nzlat<EMAIL_ADDRESS>
###### Abstract

Emotion Recognition from EEG signals has long been researched as it can assist
numerous medical and rehabilitative applications. However, their complex and
noisy structure has proven to be a serious barrier for traditional modeling
methods. In this paper, we employ multifractal analysis to examine the
behavior of EEG signals in terms of presence of fluctuations and the degree of
fragmentation along their major frequency bands, for the task of emotion
recognition. In order to extract emotion-related features we utilize two novel
algorithms for EEG analysis, based on Multiscale Fractal Dimension and
Multifractal Detrended Fluctuation Analysis. The proposed feature extraction
methods perform efficiently, surpassing some widely used baseline features on
the competitive DEAP dataset, indicating that multifractal analysis could
serve as basis for the development of robust models for affective state
recognition.

###### Index Terms:

EEG, Multiscale Fractal Dimension, Multifractal Detrended Fluctuation
Analysis, Emotion Recognition

##  I Introduction

Machine Learning has made overwhelming progress in modeling rational
intelligence and the way humans perceive and act upon their environment. In
tasks such as Object Recognition [1] and Sequence Prediction [2], supervised
machine learning systems have accomplished to surpass, in many cases, the
human brain capabilities. However, there are still many challenges in
approaching emotion-driven intelligence, although it constitutes a fundamental
aspect of human’s perception and decision-making processes. The reason for
this is that emotions are highly subjective, and thus really difficult to be
labeled when expressed. A large number of studies attempt to include speech
[3, 4], text [5], as well as facial expressions [6] in building emotion
recognition systems. Nevertheless, there is a growing interest in emotion
tagging through physiological signals [7], since those are induced without our
active interference and thus depict more clearly the actual affective state.
Such methods have been popular in designing Human-Computer or Brain-Computer
Interfaces (BCI) that aid humans and adapt to personalized preferences.
Further, physiological signals have been used for medical purposes, among
others for the detection of epilepsy [8] and depression [9].

Among a variety of physiological signals, special consideration is given to
brain data. The electroencephalogram (EEG) is the most widely researched
signal of its kind and has been highly effective in detecting affective
states. A variety of time [10], frequency [11] and joint [12] domain features
have been extracted from EEG. Particular attention has been given to channel
connectivity features, such as mutual information [13] and differential
asymmetry [14], reporting the highest scores in literature. Nowadays though,
various types of deep neural networks have exceeded the performance of
traditional feature-oriented methods [15, 16].

However, processing EEG signals and extracting useful features remain core
challenges, since EEG, like most biological signals, is chaotic, nonlinear and
incorporates a large amount of noise, both from the recording equipment and
interfering physiological processes [17]. Because of the nature of such
signals [18] several nonlinear fractal methods have been proposed, one of them
being the Higuchi Fractal Dimension [19], which has been used extensively in
emotion recognition as an analysis tool [20, 21]. Due to their complexity
though, such signals do not always share the same structure over every time
scale, hence the fractal characteristics may vary and change dynamically or
accordingly to the examined scale. For this reason, in this paper we propose
the Multiscale Fractal Dimension [22] and Multifractal Detrended Fluctuation
Analysis [23] to examine the EEG signals and determine emotional information
buried in their fragmented structure. In order to demonstrate the efficiency
of the proposed multifractal features, we modify the BCI workflow [24] to
include our Feature Engineering algorithms (Fig. 1), obtaining competitive
results that surpass some widely used baseline methods.

The rest of the paper is organized as follows: Sec. 2 provides a detailed
description of the multifractal algorithms used in this study. In Sec. 3, we
analyze the structure of EEG signals in terms of stationarity and
fragmentation, which are important factors in fractal algorithms. Section 4
describes the feature extraction methodology, whereas in Sec. 5 we describe
the the context of our experiments and discuss their outcomes. Finally, in
Sec. 6 we conclude our analysis and propose further directions for future work
on the field.

Figure 1: Experiment Pipeline: I) EEG acquisition [25] II) EEG Preprocessing
[25] III) Stationarity and Fragmentation Analysis IV) Multifractal Feature
Extraction V) Binary Classification of concatenated Features

##  II Multifractal Methods

In signal analysis and geometry, the Fractal Dimension $D$ is an index of
complexity and fragmentation, comparing how the details of a signal’s pattern
change when measured at different scales. The fractal dimension of 1D signals
may vary between 1 and 2, and the larger the $D$ is, the larger the amount of
fragmentation of the signal. Alternatively, one could consider the Hurst
Exponent $H$ of a signal to analyse the global properties of its fluctuations.
Given a self-similar signal, e.g. fractional brownian motion, the fractal
dimension is derived from the Hurst exponent through $H=2-D$.

###  II-A Multiscale Fractal Dimension

Maragos [22] developed an efficient algorithm to determine the Multiscale
Fractal Dimension (MFD) of a signal by measuring the multiscale length of a
curve with disks of varying radius via morphological coverings. The cover is
created using 2D morphological set dilations of the signal graph $F$ by
multiscale versions $sB=\\{sb:b\in B\\}$ of a unit-scale convex symmetric set
$B$, $s\geq 0$ the scale:

$F\oplus sB=\\{z+sb\in\mathbb{R}^{2}:z\in F,b\in B\\}.$

Then, the cover area $A_{B}(s)=\mathrm{area}(F\oplus sB)$ is computed and the
fractal dimension $D$ is yielded by:

$D=\lim_{s\rightarrow 0}{\frac{\log[A_{B}(s)/s^{2}]}{\log[1/s]}}.$

It has been shown [22] that the above limit will not change if we approximate
$A_{B}(s)$ with 1D nonlinear convolutions instead of 2D set operations, which
enables its efficient calculation. In practice, $D$ can be estimated by a
least-squares line to find the slope of $\log[A_{B}(s)]$ versus $\log(s)$
since

$\log[A_{B}(s)]=(2-D)\log(s)+\mathrm{constant},$

assuming the power law $A_{B}(s)\approx s^{2-D}$ as $s\rightarrow 0$. We
therefore compute the slope of the data over a small scale window of $w$
scales that move along the scale axis $s$ $\\{s,s+1,...,s+w\\}$, creating a
profile of local MFDs $D(s,t)$ at each time $t$ (fractogram). The local slope
is now an estimate of $2-D$ and from this the fractal dimension $D$ can be
easily derived.

###  II-B Detrended Fluctuation Analysis

Detrended Fluctuation Analysis (DFA) [26] estimates the Hurst exponent $H$ in
time series data $x[n]$ of length $N$ by utilizing its cumulative sum
$y[n]=\sum_{n=1}^{N}{(x[n]-\mu_{x})}$. This profile is divided into $N_{s}$
non-overlapping windows $y[k,n],k=1,...,N_{s}$ of length $s$ and for every
window the local trend $r[k,n]$ is obtained through linear regression. We
denote $y_{d}[k,n]=y[k,n]-r[k,n]$ the detrended version of the k-th profile
segment. Then, the RMS value of each detrended segment is computed and
averaged across the segments:

$F(s)=\sqrt{\frac{1}{N_{s}}\sum_{k=1}^{N_{s}}{F_{k}^{2}(s)}},\quad
F_{k}(s)=\sqrt{\frac{1}{s}\sum_{n=1}^{s}{y_{d}[k,n]^{2}}}.$

The result of the above operations is a vector of $s$ values, one for each
chosen scale. The relationship between $F(s)$ and $s$ is described by the
power law $F(s)\propto s^{H}$, which determines $H$. Multifractal DFA (MFDFA)
[23] is essentially a generalization of DFA, where the computation of $F(s)$
includes $q$ moments:

$F_{q}(s)=\sqrt[q]{\frac{1}{N_{s}}\sum_{k=1}^{N_{s}}{F_{k}^{q}(s)}}.$

As a result, a separate line is computed for every value of the factor $q$,
with $q=2$ being the reduction to classical DFA. MFDFA could prove especially
useful in cases where the scaling exponents and complexities are dependent on
the scale, or change dynamically, in the context of time series.

##  III EEG Signals

###  III-A Dataset

We work on the competitive, but widely used DEAP Dataset [25], including data
from 32 subjects in their already preprocessed form. Each subject is exposed
to forty 60-seconds long music videos as stimuli, while having their EEG
recorded, along with other physiological signals. After watching each video,
the subject was instructed to rate their induced emotion in 5 dimensions:
valence (pleasantness), arousal (excitation), dominance, liking and
familiarity to the stimulus. We solely experiment with valence and arousal, as
they form a complete emotion space [25]. The rating ranges from 1 (weakest) to
9 (strongest). The EEG signals were recorded at a sampling rate of 512 Hz and
downsampled to 128 Hz. The 10-20 placement system was followed, using 32
electrodes.

###  III-B Stationarity & Hurst Exponent Estimation

Physiological signals like the EEG are widely researched as noisy and non-
stationary signals and commonly demand heavy pre-processing. The observed
structure is partly due to external stimuli or other physiological operations
and mainly indicates the complexity and the states of neural assemblies during
brain functioning [17]. In our experiments, it is crucial to determine the
stationarity of the signals in order to correctly interpret their fractal
properties. We apply the Augmented Dickey-Fuller (ADF) Test to a randomly
sampled set of EEG signals and, to our surprise, we derive evidence of strict
stationarity. Specifically, the examined signals appeared non-stationary only
at very low scales, up to windows of 100 samples or 0.8 seconds. The same
holds when we test the signal profiles, i.e. the cumulative sums. However, a
few signals exhibit non-stationarities at their major frequency bands. In
order to find the source of this stationarity, we reproduced the preprocessing
applied in [25] to a sample raw waveform. This included downsampling to 128
Hz, eye-artefact removal, filtering at 4–45Hz and averaging to the common
reference channel. After this procedure, it was found that the cause of
stationarity was the performed bandpass filtering.

Although fractal methods are used for analyzing time series that appear to
have long-memory correlations and non-stationary dynamics, they are not
restricted to such. Scale-free stationary processes can be viewed as
fractional Gaussian noise (fGn), while their increments typically construct
non-stationary processes in the form of fractional Brownian motion (fBm), of
the same Hurst exponent. Thus, the exponent estimation is crucial in
characterizing EEG signals for multifractal analysis [27] and can be
determined by monofractal DFA. If the estimated exponent is less than $H=1$,
then it characterizes a stationary process, which can be modeled as fGn with
that exponent. Otherwise, it is assumed to be produced by a non-stationary fBm
process with an exponent of $H-1$.

The EEG signals of the DEAP dataset provide a very low Hurst Exponent value
that approaches $0$, while their profiles and separate bands provide an
increased DFA-estimated exponent, still though below $0.2$ at most cases. The
results however alter when we examine the profiles of the filtered bands,
particularly alpha and beta, in which the exponent estimation shows a steady
increase. These values confirm the evidence from the conducted ADF Test that
EEG signals are negatively correlated and their fluctuations are smaller in
larger time windows, which is the typical behavior of fGn processes having
Hurst exponents below $0.5$.

##  IV Feature Extraction

Figure 2: Sample MFD profile of a signal along with the mean and standard
deviation features extracted from its 7 subsignals of 15 sec.

Each 60-sec EEG segment is partitioned to its main frequency bands through
bandpass filtering with a 10th order Butterworth filter. We include alpha
(8-13 Hz), beta (14-29 Hz), and gamma (30-45 Hz) rhythms, as well as raw
signals in our analysis, since those have been acknowledged [28] as the most
emotion-sensitive. We select 12 left (Fp1, AF3, F7, F3, FC5, FC1, T7, C3, CP5,
CP1, P3, P7) and 12 right (Fp2, AF4, F4, F8, FC2, FC6, C4, T8, CP2, CP6, P4,
P8) channels that have shown competitive performance particularly when their
asymmetrical relation is examined [28].

###  IV-A Baseline Features

A set of widely used baseline features is extracted for comparative reasons
and to assess the combined efficiency of the proposed feature set. These
features are the Power Spectral Density (PSD) and the Higuchi Fractal
Dimension (HFD) [19]. PSD is computed across the entire signals through the
Welch method, resulting in 64 features per signal. The Higuchi Fractal
Dimension is extracted using PyEEG [29] and produces a scalar feature. To
derive a feature vector here, we first split each signal into windows of 15
seconds (1920 samples) with 50% overlap and then the HFD is determined for
each one of the 7 windows, resulting in a 7D vector.

TABLE I: Subject Dependent Task Accuracy in the form: Valence — Arousal

Features Channels Raw Signal Alpha Band Beta Band Gamma Band Combined PSD
0.642 — 0.652 0.598 — 0.645 0.629 — 0.639 0.635 — 0.620 0.631 — 0.648 HFD
Front 0.615 — 0.638 0.605 — 0.655 0.591 — 0.643 0.601 — 0.634 0.638 — 0.645
MFD Left 0.620 — 0.661 0.626 — 0.669 0.591 — 0.653 0.594 — 0.636 0.612 — 0.661
MFDFA 0.577 — 0.662 0.571 — 0.643 0.577 — 0.649 0.592 — 0.651 0.586 — 0.658
PSD 0.627 — 0.644 0.616 — 0.645 0.637 — 0.641 0.623 — 0.627 0.623 — 0.646 HFD
Front 0.606 — 0.644 0.604 — 0.655 0.595 — 0.633 0.572 — 0.627 0.623 — 0.644
MFD Right 0.607 — 0.655 0.605 — 0.652 0.566 — 0.652 0.602 — 0.641 0.597 —
0.657 MFDFA 0.587 — 0.655 0.573 — 0.641 0.603 — 0.650 0.573 — 0.620 0.586 —
0.652

TABLE II: Subject Independent Task Accuracy in the form: Valence — Arousal

Features Channels Raw Signal Alpha Band Beta Band Gamma Band Combined PSD
0.554 — 0.569 0.547 — 0.564 0.549 — 0.562 0.553 — 0.570 0.546 — 0.564 HFD
Front 0.541 — 0.601 0.552 — 0.588 0.541 — 0.616 0.545 — 0.584 0.585 — 0.621
MFD Left 0.553 — 0.606 0.566 — 0.631 0.545 — 0.618 0.554 — 0.580 0.559 — 0.615
MFDFA 0.569 — 0.630 0.546 — 0.600 0.545 — 0.598 0.532 — 0.545 0.553 — 0.608
PSD 0.553 — 0.580 0.557 — 0.560 0.558 — 0.573 0.552 — 0.579 0.555 — 0.575 HFD
Front 0.525 — 0.573 0.566 — 0.582 0.544 — 0.595 0.549 — 0.567 0.571 — 0.605
MFD Right 0.552 — 0.601 0.556 — 0.605 0.547 — 0.587 0.545 — 0.588 0.560 —
0.607 MFDFA 0.555 — 0.619 0.552 — 0.580 0.549 — 0.591 0.539 — 0.584 0.544 —
0.599

###  IV-B MFD Features

Since MFD is mainly used for short-time analysis [30], we again split each
signal into 15 sec. windows with 50% overlap. The proposed feature set
includes 30 linearly sampled features extracted out of each window’s MFD. The
respective features of each window are then summarized using 3 statistical
metrics: mean, median and standard deviation. In this way, we get a final 90D
feature vector incorporating the signal’s temporal variance. The signals are
analyzed at discrete scales of $s=1,...,274$ samples, thus the maximum scale
is at $s=1/7$ of the signals’ length. The fractogram of a sample signal along
with the variance of its 7 windows are shown in Fig. 2. The EEG fractograms
reveal a highly fragmented structure and a high fractal dimension $D>1.5$.
This finding is consistent with the low Hurst Exponent we got from monofractal
DFA.

Figure 3: MFDFA on an EEG: depicting 16 linear-like graphs for $F_{q}(s)$

###  IV-C MFDFA Features

We additionally acquire 30 features from processing the last half of each EEG
waveform through the computationally expensive MFDFA. We select 10 scales
ranging from 30 to 500 samples and $16$ q-moment values ranging from $-5$ to
$5$. The resulting representation is a set of 16 linear-like graphs of 10
values, as shown in Fig 3. 16 Hurst Exponent values are determined through
linear regression, one for each moment. The mass exponent $t$ is then derived
through $t(q)=qH(q)-1$. A monofractal signal with constant $H$ would produce a
linear graph, the EEG instead produces a curve that we utilize to produce the
signal’s multifractal spectrum $D$:

$D(q)=q^{\prime}h(q)-t(q^{\prime}),\quad
h(q_{n})=\frac{t(q_{n})-t(q_{n-1})}{q_{n}-q_{n-1}},$

where $n=1,...,15$, $q^{\prime}$ excludes the largest moment value, and $h(q)$
is the singularity exponent. The resulting curve, determined by 15 $h(q)$ and
15 $D(q)$ values, represents the MFDFA feature set.

##  V Experimental Evaluation

Experimental Protocol: We evaluate the features extracted from the
multifractal analysis on the emotion recognition task. The experimental
protocol can be divided into two categories: Subject Dependent, in which a
classifier is trained and tested on trials of a single participant, with the
final score being the average per-subject score, and Subject Independent,
where a classifier is trained on several participants and tested against
unseen trials. We shall mention that lower scores are typically reported [24]
for the latter, since EEG is highly subject-sensitive, thus we anticipate such
behavior also in our models.

In this work, we make use of a single classifier unifying features from all
available EEG channels. The model consists of a Standard Scaler, that
standardizes training features by removing their mean and scaling them to unit
variance, and a Support Vector Machine (SVM) with an RBF kernel. Experiments
consider single labels, i.e., valence or arousal, in binary format by setting
the threshold for binarization in the median score 5. We perform 5-fold cross
validation on stratified splits of the available data: approximately 56.5% of
all samples are of high valence and 59% of high arousal annotations.

Comparison to Baselines: The classification results for all features at the 2
distinct settings are summarized in Table 1 and 2. We notice the accuracy
difference between subject dependent and independent tasks, supporting the
claim that brain responses inherit mainly subjective characteristics. The EEG
PSD is shown to be efficient in the subject-dependent setting, where the raw
signal modality achieves 64.2% in valence and 65.2% in arousal. Interestingly,
these scores significantly drop in the subject-independent setting, where the
PSD emerges as the least efficient feature set, achieving only chance-level
scores in arousal, 6% below the top recorded accuracy of MFD. We can therefore
assume that the within-subject variability is concentrated more on separate
spectral characteristics of each participant and therefore, fractal analysis
is more robust across subjects.

Multifractal methods show indeed strong performance in both experiments,
surpassing chance levels and the baseline features in most cases. In contrast
to spectral features that are sensitive to valence, these features prove
efficient mainly in recognizing the arousal state, in which they achieve
around 5% higher scores. At the subject-dependent experiment particularly, MFD
of the alpha band and MFDFA at the raw signal yield 66.9% and 66.2%
respectively, whereas their highest subject-independent accuracy hits 63%. Our
results are in accordance with those reported in [31] for PSD and HFD, while
the top scores obtained by MFD and MFDFA surpass most of the ones reported
there. On the other hand, at subject-independent classification, multifractal
features perform comparably to those discussed at [32] for DEAP, although we
recognize the additional difficulty of eliminating all of the trials of a
tested participant from training.

TABLE III: MFD-HFD Arousal Accuracy

Feat/s Exp Raw Alpha Beta Gamma Comb Left Subject 0.663 0.670 0.657 0.637
0.656 Right Dep. 0.654 0.662 0.618 0.640 0.655 Left Subject 0.613 0.641 0.612
0.580 0.614 Right Indep. 0.604 0.610 0.591 0.582 0.615

Aggregated features: Multifractal features, and MFD in particular, clearly
outperform the Higuchi baseline in arousal and perform comparably in valence,
indicating that the multiscale variability of the EEG can capture latent
emotional information. Furthermore, the two kinds of fractal dimensions
provide even better scores in arousal when combined. As shown in Table 3, in
both subject dependent and independent settings we record higher accuracy than
the one we obtained from the individual features in the ablation study, mainly
when testing raw signals or the alpha band. The differences are significant in
the subject independent setting, whereas even the top scores obtained
previously are improved. The model can now predict arousal at 67% and 64% at
subject dependent and independent experiments respectively.

Other than that, it seems that the selection of a single feature type could be
adequate for affective state recognition, since we do not observe substantial
improvement for the “Combined” features, in which we measure the aggregated
performance of the three bands and the raw signal. We then compare the two
asymmetrical sets of channels, where the left hemisphere is more efficient in
terms of multifractal analysis, both in subject dependent and independent
setting, performing 2% better on average. In order to assess the asymmetrical
performance, we also experimented on aggregated trials, the results however
did not meet the scores obtained individually. It is evident though that
higher accuracy is acquired when we consider the difference or quotient of
those features, instead of just concatenating them. Finally, except for MFD
and HFD, aggregated sets between the mentioned feature types do not provide a
statistically significant improvement in recognition. We shall only mention
that some valence scores including PSD features surpass some of those we
report in the ablation study.

##  VI Conclusion

In this paper we analyzed the multiscale fractal structure of EEG and proposed
a feature extraction method utilizing two multifractal algorithms for emotion
recognition. The proposed features perform strongly against the baselines,
particularly in the challenging subject-independent setting and in arousal
recognition, indicating that arousal is correlated with the fragmented
structure of the EEG. Further improvements are achieved when the fractal
dimension features are aggregated, while the efficiency of the alpha frequency
band is underlined in all experiments. Our analysis showed that
multifractality and the anti-correlation properties should be considered when
processing EEG signals. Further work on EEG emotion analysis should consider
feature extraction algorithms for determining asymmetrical multifractal
properties, whereas an interesting direction would be the examination of the
correlation between brain signals and their stimuli.

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# Fractional Kirchhoff problem with Hardy potential: Breaking of resonance

B. Abdellaoui∗, A. Azzouz, A. Bensedik  B. Abdellaoui, Laboratoire d’Analyse
Nonlinéaire et Mathématiques Appliquées. Département de Mathématiques,
Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria. A. Azzouz,
Department of Mathematics, Faculty of Science,
University of Saida,
BP. 138, 20000, Saida, Algeria. A. Bensedik, Laboratoire Système dynamiques et
applications. Département de Mathématiques, Université Abou Bakr Belkaïd,
Tlemcen, Tlemcen 13000, Algeria.
E-mail addresses:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>.

###### Abstract.

In this paper we consider a fractional Kirchhoff problem with Hardy potential,

$\left\\{\begin{array}[]{rcll}M(\displaystyle\iint_{{I\\!\\!R}^{N}\times{I\\!\\!R}^{N}}\frac{|u(x)-u(y)|^{q}}{|x-y|^{N+qs}}\,dx\,dy)(-\Delta)^{s}u&=&\lambda\dfrac{u}{|x|^{2s}}+f(x,u)&\text{
in }\Omega,\\\ u&>&0&\text{ in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\Omega\subset{I\\!\\!R}^{N}$ is a bounded domain containing the origin,
$s\in(0,1)$, $q\in(1,2]$ with $N>2s$, $\lambda>0$ and $f$ is a measurable
nonnegative function with suitable hypotheses.

The main goal of this work is to get the existence of solution for the largest
class of $f$ without any restriction on $\lambda$. Our result covers also the
local case $s=1$.

###### Key words and phrases:

fractional Laplacian, Kirchhoff-type problem, Hardy potential.
2000 Mathematics Subject Classification:MSC 2000: 26A33,
34B15,34C25,58E30,35J65,35R11

∗ is partially supported by project MTM2016-80474-P, MINECO, Spain.

##  1\. Introduction

The present work is concerned with existence of positive solutions for a class
of fractional equation involving a Kirchhoff term and singular potential. More
precisely we consider the problem

${(\mathcal{P}_{s,q})}\left\\{\begin{array}[]{rcll}M\bigg{(}\displaystyle\iint_{{I\\!\\!R}^{N}\times{I\\!\\!R}^{N}}\frac{|u(x)-u(y)|^{q}}{|x-y|^{N+qs}}\,dx\,dy\bigg{)}(-\Delta)^{s}u&=&\lambda\dfrac{u}{|x|^{2s}}+f(x,u)&\text{
in }\Omega,\\\ u&>&0&\text{ in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\Omega\subset{I\\!\\!R}^{N}$ is a bounded domain containing the origin,
$s\in(0,1)$, $q\in(1,2]$ with $N>2s$, $\lambda>0$ and $f$ is a measurable
nonnegative function with suitable hypotheses. $M$ is a positive function
defined on $\mathbb{R}^{+}$.
Here, for $0<s<1$, the fractional Laplacian $(-\Delta)^{s}$ is defined by

(1.1) $(-\Delta)^{s}u(x):=a_{N,s}\mbox{ P.V.
}\int_{\mathbb{R}^{N}}{\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy},$

where

$a_{N,s}:=2^{2s-1}\pi^{-\frac{N}{2}}\frac{\Gamma(\frac{N+2s}{2})}{|\Gamma(-s)|}$

is the normalization constant such that the identity

$(-\Delta)^{s}u=\mathcal{F}^{-1}(|\xi|^{2s}\mathcal{F}u),\,\xi\in\mathbb{R}^{N}$

holds for all $u\in\mathcal{S}(\mathbb{R}^{N})$ where $\mathcal{F}u$ denotes
the Fourier transform of $u$ and $\mathcal{S}(\mathbb{R}^{N})$ is the Schwartz
class of tempered functions.

The problem $(\mathcal{P}_{s,q})$ is related to the following Hardy
inequality, proved in [5] and [16],

(1.2)
$\frac{a_{N,s}}{2}\int\int_{D_{\Omega}}{\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}}\,dx\,dy\geq\Lambda_{N,s}\int_{\Omega}{\frac{u^{2}}{|x|^{2s}}\,dx},\,u\in\mathcal{C}^{\infty}_{0}({I\\!\\!R}^{N}),$

where

${D_{\Omega}}:={I\\!\\!R}^{N}\times{I\\!\\!R}^{N}\setminus\big{(}\mathcal{C}\Omega\times\mathcal{C}\Omega\big{)}\,.$

and

(1.3)
$\Lambda_{N,s}=2^{2s}\dfrac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}.$

Notice that $\Lambda_{N,s}$ is optimal and not attained.

Nonlocal problems related to the Hardy potential were widely studied in the
last years. Before staring our main result, let us begin by some previous
results. In the case where $M\equiv 1$, then it is well known that problem
$(\mathcal{P}_{s,q})$ has a solution if and only if
$\lambda\leq\Lambda_{N,s}$. Moreover, for $\lambda\in(0,\Lambda_{N,s}]$ fixed
and taking into consideration the singular behavior near the origin of the
Hardy potential, it holds that a natural condition is imposed on $f$ in order
to get the existence of a solution. More precisely, if $f$ does not depends on
$u$ ($f(x,u):=f(x))$, then $(\mathcal{P}_{s,q})$ has a solution if and only if
$\displaystyle\int\limits_{\Omega}|x|^{-\gamma_{\lambda}}f(x)\,dx<\infty$
where $\gamma_{\lambda}$ will be explicitly stated in the next section.
If $f(x,u)=u_{+}^{p}$, then existence of solution holds if and only if
$p<p_{+}(\lambda)<2s$. We refer for instance to [2] and [10] for more details.

Where $M$ is not identically constant, then for the local case $s=1$ and
without the Hardy potential, the problem can be expressed in the general form

${(\mathcal{P}_{K})}\quad-M(\displaystyle\int_{\Omega}\left|\nabla
u\right|^{2}dx)\Delta u=g(x,u)\quad\text{in}\quad\Omega,\quad
u=0\quad\text{on}\quad\partial\Omega.$

$(\mathcal{P}_{K})$ is well known as Kirchhoff type problem and is widely
studied in the literature. The first motivation was the study of the
stationary solution to the classical equation

$\rho\dfrac{\partial^{2}u}{\partial
t^{2}}-\left(\frac{p_{0}}{h}+\frac{E}{2L}\int_{0}^{L}\left|\dfrac{\partial
u}{\partial x}\right|^{2}dx\right)\dfrac{\partial^{2}u}{\partial x^{2}}=0.$

This equation is an extension of the classical d’Alembert’s wave equation by
considering the effects of the changes in the length of the string during the
vibrations. In this equation, $L$ is the length of the string, $h$ is the area
of the cross-section, $E$ is the Young modulus of the material, $\rho$ is the
mass density and $P_{0}$ is the initial tension. The stationary
multidimensional Kirchhoff equation is given by

${(\mathcal{P}_{K})}\quad-M(\displaystyle\int_{\Omega}\left|\nabla
u\right|^{2}dx)\Delta u=g(x,u)\quad\text{in}\quad\Omega,\quad
u=0\quad\text{on}\quad\partial\Omega,$

which received much attention after the work of Lions [18], where a functional
setting was proposed to the problem.

The Kirchhoff function $M$ is assumed Lipschitz continuous, but not always
monotone, even if the model proposed by Kirchhoff in 1883 $M(t)=a+bt^{\beta}$
where $a,b\geq 0$ and $\beta\geq 0$ is monotone, see discussion in [20] and
[22]. If $M(0)=0$, the Kirchhoff problem is called degenerate, we may refer to
[8] and references therein for more details on this case.

In [3], using variational arguments the authors proved the existence of a
positive solution of ${(\mathcal{P}_{K})}$ if $M$ is a positive non-increasing
function and $g$ satisfies the Ambrosetti-Rabinowitz condition. The case where
$M$ is a non-decreasing function was considered in [4]. In [19], the author
collected some related problems to ${(\mathcal{P}_{K})}$ and unified the
presentation of the results. Especially, for $g$ such that $g(x,u)u\leq
a\left|u\right|^{2}+b\left|u\right|$ ($a,b>0$) he showed both existence and
uniqueness of solutions. In the case where $g(x,u)$ is asymptotically linear
at infinity with respect to $u$, some existence results are proved in [4]. The
singular case was considered in [12].

The nonlocal case $s\in(0,1)$, was considered recently by several authors,
including the critical and singular cases. We refer to [20, 21, 23] and
references therein for further details.

The novelty in our paper is to get some interactions between the Kirchhoff
term and the Hardy potential in order to get the existence of non negative
solution for a largest class of the datum $f$ and without any restriction on
the parameter $\lambda$. We will show that, under some growth hypothesis on
the Kirchhoff term, it is possible to break any resonance effect of the Hardy
term. The resut covers also the local case $s=1$.

Through this paper we will assume that $M$ is a continuous function such that:
(M1) $M:{I\\!\\!R}^{+}\to{I\\!\\!R}^{+}$ is increasing.
(M2) $\exists m_{0}>0;\quad M(t)\geq m_{0}\quad\forall t\in{I\\!\\!R}^{+}.$
(M3) $\underset{t\rightarrow\infty}{\lim}M(t)=+\infty$

We begin by analyzing the case where $f$ does not depend on $u$, namely
$f(x,u)=:=f(x)$. Then for $q=2$ and hence the problem has a variational
structure, we are able to proof the next result.

######  Theorem 1.1.

Assume that $f\in L^{2}(\Omega)$ is such that $f\gneqq 0$, then for all
$\lambda>0$, problem $(\mathcal{P}_{s,2})$ has a unique positive solution
$\bar{u}\in H_{0}^{s}(\Omega)$.

We mention here the strong regularity enjoyed by the solution. In the case
where $M\equiv 1$, such a regularity is false including for $f\in
L^{\infty}(\Omega)$ where $\lambda$ is closed to the Hardy constant
$\Lambda_{N,s}$.

In the general case $q<2$, we get the existence of a solution for a large
class of the datum $f$. More precisely we have

######  Theorem 1.2.

Let $\Omega\subset{I\\!\\!R}^{N}$ be a bounded domain such that $0\in\Omega$
and $1<q<\frac{N}{N-s}$. Suppose that $f\in L^{1}(\Omega)$ is such that
$\displaystyle\int_{\Omega}f|x|^{-\theta}dx<c$ for some positive constant
$\theta$. Then for all $\lambda>0$, the problem $(\mathcal{P}_{s,q})$ has a
minimal solution $\bar{u}$ such that $\bar{u}\in W^{s,\sigma}_{0}(\Omega)$ for
all $\sigma<\frac{N}{N-s}$.

In the second part of the paper we treat the case where $f(x,u)=u_{+}^{p}+\mu
g$. In this case and under suitable hypotheses on $\mu$ and $g$, we will show
the existence of a solution without any restriction on $\lambda$.

This paper is organized as follow. In Section 2, we give some results related
to fractional Sobolev spaces and some functional inequalities. The sense for
which solutions are defined is also presented, namely energy solutions and
weak solutions.

The proofs of the main existence results are given in Section 3. We begin by
the case of variational structure and at the end of this section, we use
approximating argument to treat the case of non variational structure.

##  2\. Preliminaries and functional setting

Let $\Omega\subset{I\\!\\!R}^{N}$, for $s\in(0,1)$ and $1\leq p<+\infty$, the
fractional Sobolev spaces $W^{s,p}(\Omega)$, are defined by

$W^{s,p}(\Omega)\equiv\Big{\\{}\phi\in
L^{p}(\Omega):\int_{\Omega}\int_{\Omega}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{N+ps}}dxdy<+\infty\Big{\\}}$

$W^{s,p}(\Omega)$ is a Banach space endowed with the following norm

$\|\phi\|_{W^{s,p}(\Omega)}=\Big{(}\displaystyle\int_{\Omega}|\phi(x)|^{p}dx\Big{)}^{\frac{1}{p}}+\Big{(}\displaystyle\int_{\Omega}\displaystyle\int_{\Omega}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{N+ps}}dxdy\Big{)}^{\frac{1}{p}}.$

We define the space $W_{0}^{s,p}(\Omega)$ as

$W_{0}^{s,p}(\Omega):=\left\\{u\in W^{s,p}({I\\!\\!R}^{N}):u=0\textup{ in
}{I\\!\\!R}^{N}\setminus\Omega\right\\}.$

If $\Omega$ is a bounded regular domain, then using Poincaré type inequality,
we can endow $W^{s,p}_{0}(\Omega)$ with the equivalent norm

$\|u\|_{W^{s,p}_{0}(\Omega)}:=\left(\iint_{D_{\Omega}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p},$

where

$D_{\Omega}:=({I\\!\\!R}^{N}\times{I\\!\\!R}^{N})\setminus(\mathcal{C}\Omega\times\mathcal{C}\Omega)=(\Omega\times{I\\!\\!R}^{N})\cup(\mathcal{C}\Omega\times\Omega).$

The next Sobolev inequality is proved in [13].

######  Theorem 2.1.

(Fractional Sobolev inequality) Assume that $0<s<1,p>1$ in ordrer to get
$ps<N$. Then, there exists a positive constant $S\equiv S(N,s,p)$ such that
for all $v\in C_{0}^{\infty}({I\\!\\!R}^{N})$,

$\iint_{{I\\!\\!R}^{2N}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{N+ps}}\,dxdy\geq
S\Big{(}\displaystyle\int_{\mathbb{R}^{N}}|v(x)|^{p_{s}^{*}}dx\Big{)}^{\frac{p}{p^{*}_{s}}},$

where $p^{*}_{s}=\dfrac{pN}{N-ps}$.

In the particular case $p=2$, $H^{s}(\Omega):=W^{s,2}(\Omega)$ turns out to be
a Hilbert spaces. If $\Omega=\mathbb{R}^{N}$, the Fourier transform provides
yet an alternative definition, more precisely, the Plancherel identity leads
to

$\int_{\mathbb{R}^{N}}|\xi|^{2s}|\mathcal{F}(u)(\xi)|^{2}d\xi={\frac{a_{N,s}}{2}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy.$

The operator $(-\Delta)^{s}$ defined for $u\in\mathcal{S}(\mathbb{R}^{N})$ by

$(-\Delta)^{s}u:=a_{N,s}\mbox{ P.V.
}\int_{\mathbb{R}^{N}}{\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy},\,\,\qquad
s\in(0,1),$

can be extended by density from $\mathcal{S}(\mathbb{R}^{N})$ to
$H^{s}(\mathbb{R}^{N})$. In this way, the associated scalar product can be
reformulated as

$\begin{split}\langle
u,v\rangle_{H^{s}(\mathbb{R}^{N})}&:=\langle(-\Delta)^{s}u,v\rangle+(u,v)\\\
&:=P.V.\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}{\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,dx\,dy}+\int_{\mathbb{R}^{N}}uv\,dx.\end{split}$

In the case of bounded domain if we denote by
$H^{-s}(\Omega):=[H_{0}^{s}(\Omega)]^{*}$ the dual space of
$H_{0}^{s}(\Omega)$, then

$(-\Delta)^{s}:H_{0}^{s}(\Omega)\rightarrow H^{-s}(\Omega),$

is a continuous operator.

We state, by the next definition, the sense in which the solution is defined.

######  Definition 2.2.

Assume that $h\in L^{1}(\Omega)$ and consider the problem

(2.1) $\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u&=&h&\text{ in }\Omega,\\\
u&=&0&\text{ in }{I\\!\\!R}^{N}\backslash\Omega,\\\ \end{array}\right.$

we say that $u\in L^{1}(\Omega)$ is a weak solution to (2.1) if $u=0$ in
${I\\!\\!R}^{N}\backslash\Omega$ and for all $\psi\in\mathbb{X}_{s}$, we have

$\int\limits_{\Omega}u((-\Delta)^{s}\psi)dx=\int\limits_{\Omega}h\psi dx.$

where

$\mathbb{X}_{s}\equiv\Big{\\{}\psi\in\mathcal{C}({I\\!\\!R}^{N})\,|\,\text{supp}(\psi)\subset\overline{\Omega},\,\,(-\Delta)^{s}\psi(x)\hbox{
pointwise defined and }|(-\Delta)^{s}\psi(x)|<C\hbox{ in }\Omega\Big{\\}}.$

The next existence result is proved in [17], [11] and [1].

######  Theorem 2.3.

Assume that $h\in L^{1}(\Omega)$, then problem (2.1) has a unique weak
solution $u$ obtained as the limit of the sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$, where $u_{n}$ is the unique solution of the
approximating problem

(2.2) $\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u_{n}&=&h_{n}(x)&\mbox{ in
}\Omega,\\\ u_{n}&=&0&\mbox{ in
}{I\\!\\!R}^{N}\backslash\Omega,\end{array}\right.$

with $h_{n}=T_{n}(h)$ and $T_{n}(\sigma)=\max(-n,\min(n,\sigma))$. Moreover,

(2.3) $T_{k}(u_{n})\to T_{k}(u)\hbox{ strongly in
}H^{s}_{0}(\Omega),\quad\forall k>0,$ (2.4) $u\in
L^{\theta}(\Omega)\,,\qquad\forall\ \theta\in\big{[}1,\frac{N}{N-2s}\big{)}\,$

and

(2.5) $\big{|}(-\Delta)^{\frac{s}{2}}u\big{|}\in
L^{r}(\Omega)\,,\qquad\forall\ r\in\big{[}1,\frac{N}{N-s}\big{)}\,.$

Furthermore

(2.6) $u_{n}\to u\mbox{ strongly in }W^{s,q_{1}}_{0}(\Omega)\mbox{ for all
}q_{1}<\frac{N}{N-s}.$

As a subsequent of this existence result, we get a compactness result which
will be systematically used in the sequel.

######  Theorem 2.4.

Assume that $1<q<\frac{N}{N-s}$ be fixed and consider the operator
$D:L^{1}(\Omega)\to W^{s,q}_{0}(\Omega)$ defined by $D(h)=u$ where $u$ is the
unique solution to the following problem

$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u&=&h(x)&\mbox{ in }\Omega,\\\
u&=&0&\mbox{ in }{I\\!\\!R}^{N}\backslash\Omega,\end{array}\right.$

then $D$ is a compact operator, moreover, there exists a positive constant
$C\equiv C(\Omega,s,N,q)$ such that

$||u||_{W^{s,q}_{0}(\Omega)}\leq C||h||_{L^{1}(\Omega)}.$

Since we are considering problem involoving the Hardy potential, we have
tospecify the behavior of any solution in the neighborhood of the singular
point $0$.

For $\lambda\leq\Lambda_{N,s}$ fixed, we define $\alpha>0$ by the identity

(2.7)
$\lambda=\lambda(\alpha)=\lambda(-\alpha)=\dfrac{2^{2s}\,\Gamma(\frac{N+2s+2\alpha}{4})\Gamma(\frac{N+2s-2\alpha}{4})}{\Gamma(\frac{N-2s+2\alpha}{4})\Gamma(\frac{N-2s-2\alpha}{4})}.$

Denote

(2.8) $\gamma:=\dfrac{N-2s}{2}-\alpha,$

then $0<\gamma\leq\dfrac{N-2s}{2}$ and $|x|^{-\gamma}$ is the unique solution
to the equation

$(-\Delta)^{s}u=\lambda\frac{u}{|x|^{2s}},$

with $(-\Delta)^{s/2}(|x|^{-\gamma})\in L^{2}_{loc}({I\\!\\!R}^{N})$. Form [2]
we have the next result.

######  Theorem 2.5.

Consider the problem

(2.9)
$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u&=&\lambda\dfrac{u}{|x|^{2s}}+f(x)&\text{
in }\Omega,\\\ u&>&0&\text{ in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\Omega\subset{I\\!\\!R}^{N}$ is a bounded domain containing the origin,
$s\in(0,1)$ with $N>2s$ and $0<\lambda\leq\Lambda_{N,s}$. Then problem (2.9)
has a nonnegative solution if and only if
$\displaystyle\int\limits_{\Omega}f|x|^{-\gamma}dx<\infty$. In this case the
unique solution of (2.9) satisfies $u(x)\geq C|x|^{-\gamma}$ in $B_{r}(0)$.

In what follows, $C$ denotes any positive constant that does not depend on the
solution and that can change from line to other.

##  3\. Existence results in the case $f(x,u):=f(x)$.

In what follows and for the sake of simplicity we set

$||u||_{s,q}^{q}=\iint_{D_{\Omega}}\dfrac{|u(x)-u(y)|^{q}}{|x-y|^{N+qs}}\,dxdy,$

where

$D_{\Omega}:=({I\\!\\!R}^{N}\times{I\\!\\!R}^{N})\setminus(\mathcal{C}\Omega\times\mathcal{C}\Omega).$

Along this section we will assume that the function $M$ satisfies the
conditions $(M1),(M2)$ stated in the introduction.

Recall that we are considering the next problem

${(\mathcal{P}_{s,2})}\left\\{\begin{array}[]{rcll}M(||u||_{s,2}^{2})(-\Delta)^{s}u&=&\lambda\dfrac{u}{|x|^{2s}}+f(x)&\text{
in }\Omega,\\\ u&>&0&\text{ in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Our first existence result consists in the following.

######  Theorem 3.1.

Assume that $f\in L^{2}(\Omega)$ is such that $f\gneqq 0$, then for all
$\lambda>0$, the problem $(\mathcal{P}_{s,2})$ has a unique positive solution
$\bar{u}$ in $H_{0}^{s}(\Omega)$.

Proof. Let $f\in L^{2}(\Omega)$ and fix $\lambda>0$, we will proceed by
approximation.

Consider the truncated problem

${(\mathcal{P}^{n}_{s})}\left\\{\begin{array}[]{rcll}M(||u_{n}||_{s,2}^{2})(-\Delta)^{s}u_{n}&=&\dfrac{\lambda
T_{n}(u_{n}^{+})}{|x|^{2s}+\frac{1}{n}}+f(x)&\text{ in }\Omega,\\\
u_{n}&>&0&\text{ in }\Omega,\\\ u_{n}&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

It is clear that solutions of problem $(\mathcal{P}^{n}_{s})$ are critical
points of the functional $J_{n}$ given by

$J_{n}(v)=\dfrac{1}{2}\widehat{M}(\left\|v\right\|_{s}^{2})-\lambda\int_{\Omega}\dfrac{\widehat{T}_{n}(v^{+})}{\left|x\right|^{2s}+\frac{1}{n}}dx-\int_{\Omega}fvdx,$

where

$\widehat{M}(t)=\int^{t}_{0}M(\tau)d\tau,\qquad\widehat{T}_{n}(t)=\int^{t}_{0}T_{n}(\tau)d\tau.$

Taking into consideration the behavior of $M$ and setting

$D_{n}=\inf_{H_{0}^{s}(\Omega)\backslash\\{0\\}}J_{n}(u),$

we reach that $D_{n}$ is achieved and then we get the existence of $u_{n}\in
H_{0}^{s}(\Omega)$ that solves $(\mathcal{P}^{n}_{s})$. We claim that the
sequence $\\{u_{n}\\}_{n}$ is bounded in $H_{0}^{s}(\Omega)$. We argue by
contradiction. Assume that $||u_{n}||_{s,2}\to\infty$ as $n\to\infty$. Taking
$u_{n}$ as a test function in $(\mathcal{P}^{n}_{s})$ and using Hardy and
Hölder inequalities, it follow that

(3.1)
$\begin{array}[]{ll}M(||u_{n}||_{s,2}^{2})||u_{n}||_{s,2}^{2}\leq\lambda\displaystyle\int_{\Omega}\dfrac{\left|u_{n}\right|^{2}}{\left|x\right|^{2s}}dx+\displaystyle\int_{\Omega}fu_{n}dx\\\
\leq\dfrac{\lambda}{\Lambda_{N,s}}\left\|u_{n}\right\|^{2}_{s,2}+\left\|f\right\|_{L^{2}(\Omega)}||u||_{s,2}.\end{array}$

Thus

(3.2)
$||u||^{2}_{s,2}\left(M\left(||u||^{2}_{s,2}\right)-\frac{\lambda}{\Lambda_{N,s}}\right)\leq
2^{-1}\left(||u||^{2}_{s,2}+\left\|f\right\|^{2}_{L^{2}(\Omega)}\right),$

from which we have,

(3.3)
$2||u_{n}||^{2}_{s,2}(M(||u||^{2}_{s,2})-\frac{\lambda}{\Lambda_{N,s}}-1)\leq\left\|f\right\|^{2}_{L^{2}(\Omega)}.$

By $(M2)$, we deduce that

$M(||u_{n}||^{2}_{s,2})-\frac{\lambda}{\Lambda_{N,s}}-1\to\infty\mbox{ as
}n\to\infty.$

Hence we reach a contradiction with (3.3). Thus we conclude that
$\\{u_{n}\\}_{n}$ is bounded in $H_{0}^{s}(\Omega)$ and the claim follows.

Therefore, we get the existence of $\bar{u}\in H_{0}^{s}(\Omega)$ such that,
up to a subsequence, $u_{n}\rightharpoonup\bar{u}$ weakly in
$H_{0}^{s}(\Omega)$, and $u_{n}\longrightarrow\bar{u}$ strongly in
$L^{q}(\Omega)$ for all $1\leq q<2^{*}_{s}$.

We claim that there exists a positive constant $\rho$ such that up to a
subsequence denoted by $u_{n}$, we have

(3.4) $\frac{\lambda}{M(||u_{n}||_{s,2}^{2})}<\Lambda_{N,s}-\rho\mbox{ for all
}n.$

By virtue of $(M1)$ and since $||u||_{s,2}\leq C$, we obtain,

$\liminf_{n\rightarrow\infty}M(||u||_{s,2}^{2}):=\mu>0.$

Therefore, we get the existence of a subsequence denoted by $\\{u_{n}\\}_{n}$
such that,

$M(||u_{n}||_{s,2}^{2})\rightarrow\mu\mbox{ as }n\to\infty.$

Obviously, $u_{n}\rightharpoonup\bar{u}$ weakly in $H_{0}^{s}(\Omega)$, thus
by passing to the limit in the problem of $u_{n}$, it follows that $\bar{u}$
solves

$\mu(-\Delta)^{s}\bar{u}=\lambda\frac{\bar{u}}{\left|x\right|^{2s}}+f.$

Thus $\bar{u}\in H_{0}^{s}(\Omega)$, satisfies $\bar{u}\geq 0$ in $\Omega$ and

$(-\Delta)^{s}\bar{u}=\dfrac{\lambda}{\mu}\dfrac{\bar{u}}{|x|^{2s}}+\dfrac{f(x)}{\mu}\text{
in }\Omega.$

From the result of [2], we conclude that $\lambda/\mu\leq\Lambda_{N,s}$.

Let us show that $\lambda/\mu<\Lambda_{N,s}-\rho$ for some positive constant
$\rho>0$.

If $\lambda/\mu=\Lambda_{N,s}$, then $\bar{u}$ is solution of the equation

$(-\Delta)^{s}\bar{u}=\Lambda_{N,s}\dfrac{\bar{u}}{\left|x\right|^{2s}}+\dfrac{f(x)}{\mu}\quad\text{in}\hskip
4.26773pt\Omega,$

Using again the result of [2], one can see that $\bar{u}(x)\geq
C|x|^{-\frac{N-2s}{2}}$ in a small ball $B_{\eta}(0)$. Thus

$\int_{B_{\eta}(0)}\bar{u}^{2^{*}_{s}}dx\geq
C\int_{B_{\eta}(0)}|x|^{-N}dx=\infty,$

which is a contradiction with the fact that $\bar{u}\in H_{0}^{s}(\Omega)$.
Hence we get the existence of a positive constant $\rho$ such that

(3.5) $\frac{\lambda}{M(||u_{n}||_{s,2}^{2})}<\Lambda_{N,s}-\rho.$

To conclude, we prove the strong convergence of the sequence $\\{u_{n}\\}_{n}$
in $H^{s}_{0}(\Omega)$.
It is clear that $u_{n}$ solves

$(-\Delta)^{s}u_{n}=\dfrac{\lambda}{M(||u_{n}||_{s,2}^{2})}\dfrac{u_{n}}{\left(\left|x\right|+n^{-1}\right)^{2s}}+\dfrac{f(x)}{M(||u_{n}||_{s,2}^{2})}.$

Using $u_{n}-\bar{u}$ as a test function, we obtain that

(3.6)
$\displaystyle\int_{\Omega}(-\Delta)^{s}u_{n}(u_{n}-\bar{u})dx=\dfrac{\lambda}{M(||u_{n}||_{s,2}^{2})}\displaystyle\int_{\Omega}\dfrac{u_{n}(u_{n}-\bar{u})}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx+\dfrac{1}{M(||u_{n}||_{s,2}^{2})}\displaystyle\int_{\Omega}f(x)(u_{n}-\bar{u})dx.$

Since the term $\dfrac{1}{M(||u_{n}||_{s,2}^{2})}$ is bounded away from zero
and $u_{n}-\bar{u}\to 0$ strongly in $L^{2}(\Omega)$, it holds that

$\dfrac{1}{M(||u_{n}||_{s,2}^{2})}\int_{\Omega}f(x)(u_{n}-\bar{u})dx\to
0\mbox{ as }n\to\infty.$

We deal now with the term
$\displaystyle\int_{\Omega}\dfrac{u_{n}\left(u_{n}-\bar{u}\right)}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx$.

We have

$\displaystyle\int_{\Omega}\dfrac{u_{n}(u_{n}-\bar{u})}{(|x|+n^{-1})^{2s}}dx$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\Omega}\dfrac{\left(u_{n}-\bar{u}\right)^{2}}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx+\displaystyle\int_{\Omega}\dfrac{\bar{u}\left(u_{n}-\bar{u}\right)}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\Omega}\dfrac{\left(u_{n}-\bar{u}\right)^{2}}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx+\displaystyle\int_{\Omega}\dfrac{u_{n}-\bar{u}}{\left(\left|x\right|+n^{-1}\right)^{s}}\dfrac{\bar{u}}{\left(\left|x\right|+n^{-1}\right)^{s}}dx.$

Taking into consideration that
$\dfrac{\bar{u}}{{(|x|+n^{-1})^{s}}}\to\dfrac{\bar{u}}{|x|^{s}}$ strongly in
$L^{2}(\Omega)$ and that
$\dfrac{u_{n}-\bar{u}}{{\left(\left|x\right|+n^{-1}\right)^{s}}}\rightharpoonup
0$ weakly in $L^{2}\left(\Omega\right),$ (as $n\to\infty$), we obtain

$\displaystyle\int_{\Omega}\dfrac{u_{n}-\bar{u}}{\left(\left|x\right|+n^{-1}\right)^{s}}\dfrac{\bar{u}}{\left(\left|x\right|+n^{-1}\right)^{s}}dx\to
0\mbox{ as }n\to\infty.$

Consequently, and using again the Hardy inequality, we get

(3.7)
$\int_{\Omega}\dfrac{u_{n}\left(u_{n}-\bar{u}\right)}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx=\int_{\Omega}\dfrac{\left(u_{n}-\bar{u}\right)^{2}}{\left(\left|x\right|+n^{-1}\right)^{2s}}dx+o(1)\leq\dfrac{1}{\Lambda_{N,s}}\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}+o(1).$

Now, by the weak convergence, it holds that

$\displaystyle\int_{\Omega}(-\Delta)^{s}u_{n}(u_{n}-\bar{u})dx=\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}+o(1).$

Therefore, by estimates (3.6) and (3.7), it follows that

$\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}\leq\dfrac{\lambda}{\Lambda_{N,s}M(||u_{n}||_{s,2}^{2})}\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}+o(1).$

Recall that by (3.5), we have
$\dfrac{\lambda}{M(||u_{n}||_{s,2}^{2})}<\Lambda_{N,s}-\rho$ for all $n$. Thus

$\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}<\displaystyle\dfrac{\Lambda_{N,s}-\rho}{\Lambda_{N,s}}\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}+o(1).$

Since $0<\displaystyle\dfrac{\Lambda_{N,s}-\rho}{\Lambda_{N,s}}<1$, then
$\left\|u_{n}-\bar{u}\right\|_{s,2}^{2}\longrightarrow 0$ as $n\to\infty$ and
then $u_{n}\longrightarrow\bar{u}$ strongly in $H^{s}_{0}(\Omega)$.

As a consequence it follows that $M(||u_{n}||_{s,2})\to M(||u||_{s,2})$.
Passing to the limit in problem $(\mathcal{P}^{n}_{s})$, we obtain that
$\bar{u}$ solves $(\mathcal{P}_{s,2})$.

It remains to proof yet the uniqueness part.

Assume that $u,v$ are two positive solutions to problem $(\mathcal{P}_{s,2})$.
To show the uniqueness we have just to prove that
$M(||u||_{s,2}^{2})=M(||v||_{s,2}^{2})$. Assume by contradiction that
$M(||u||_{s,2}^{2})<M(||v||_{s,2}^{2})$. Setting $u_{1}=M(||u||_{s,2}^{2})u$
and $v_{1}=M(||v||_{s,2}^{2})v$, it holds that

$(-\Delta)^{s}v_{1}=\frac{\lambda}{M(||v||_{s,2}^{2})}\,\dfrac{v_{1}}{|x|^{2s}}+f(x),$

and

$(-\Delta)^{s}u_{1}=\frac{\lambda}{M(||u||_{s,2}^{2})}\,\dfrac{u_{1}}{|x|^{2s}}+f(x).$

Since $\frac{\lambda}{M(||v||_{s,2}^{2})}<\frac{\lambda}{M(||u||_{s,2}^{2})}$
and $u_{1}=v_{1}=0$ in $\mathbb{R}^{N}\setminus\Omega$, then $v_{1}<u_{1}$ in
$\Omega$. Thus

$M(||v||_{s,2}^{2})v<M(||u||_{s,2}^{2})u\mbox{ in }\Omega.$

Recall that, by hypothesis, $M(||u||_{s,2}^{2})<M(||v||_{s,2}^{2})$, thus
$v<u$ in $\Omega$. Going back to the equation of $v$ and using $v$ as a test
function, it follows that

$M(||v||_{s,2}^{2})||v||_{s,2}^{2}=\lambda\int\limits_{\Omega}\dfrac{v^{2}}{|x|^{2s}}dx+\int\limits_{\Omega}fvdx\leq\lambda\int\limits_{\Omega}\dfrac{u^{2}}{|x|^{2s}}dx+\int\limits_{\Omega}fudx=M(||u||_{s,2}^{2})||u||_{s,2}^{2}$

Since $M(t^{2})t^{2}$ is a strict monotone function, we conclude that
$||v||_{s,2}^{2}<||u||_{s,2}^{2}$ and then $M(||v||_{s,2}^{2})\leq
M(||u||_{s,2}^{2})$, a contradiction with the main hypothesis. Hence the
uniqueness result follows.

######  Remarks 3.2.

It is not difficult to show that the above existence and the uniqueness result
holds also for all $f\in L^{\frac{2N}{N+2s}}(\Omega)$.

As a consequence of the previous construction, we have the next regularity
result.

######  Proposition 3.3.

Assume that $f\in L^{\frac{2N}{N+2s}}(\Omega)$, then for all
$\sigma<2_{s}^{*}$, there exists a positive constant $C:=C(N,\Omega,\sigma)$
such that if $u$ is the solution to problem $(\mathcal{P}_{s,2})$, we have

$||u||_{L^{\sigma}(\Omega)}\leq C||f||_{L^{\frac{2N}{N+2s}}(\Omega)}.$

###### Proof.

Since $u$ solves $(\mathcal{P}_{s,2})$, then
$\frac{\lambda}{M(||u||_{s,2}^{2})}\leq\Lambda_{N,s}$ and

$(-\Delta)^{s}u=\frac{\lambda}{M(||u||_{s,2}^{2})}\,\dfrac{u}{|x|^{2s}}+\frac{f(x)}{M(||u||_{s,2}^{2})}\leq\Lambda_{N,s}\dfrac{u}{|x|^{2s}}+\frac{1}{C}f(x).$

Setting $w$ to be the unique solution to the problem

$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}w&=&\Lambda_{N,s}\dfrac{w}{|x|^{2s}}+\frac{1}{C}f(x)&\text{
in }\Omega,\\\ w&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

then $u\leq w$ in $\Omega$. Notice that from [2] we obtain that, for all
$\sigma<2^{*}_{s}$,

$||w||_{L^{\sigma}(\Omega)}\leq
C||\frac{1}{C}f||_{L^{\frac{2N}{N+2s}}(\Omega)}\leq
C||f||_{L^{\frac{2N}{N+2s}}(\Omega)}.$

Hence we conclude. ∎

To complete this part, we have the next compactness result.

######  Proposition 3.4.

Let $\\{f_{n}\\}_{n}\subset L^{\frac{2N}{N+2s}}(\Omega)$ be such that
$f_{n}\geq 0$ and $||f_{n}||_{L^{\frac{2N}{N+2s}}(\Omega)}\leq C$ for all $n$.
Define $u_{n}$, the unique solution to problem $(\mathcal{P}_{s,2})$ with
$f:=f_{n}$. Then there exists a $u\in H^{s}_{0}(\Omega)$, such that, us to a
subsequence, $u_{n}\to u$ strongly in $H^{s}_{0}(\Omega)$.

###### Proof.

Let $w_{n}$ be the unique solution to the problem

$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}w_{n}&=&\Lambda_{N,s}\dfrac{w_{n}}{|x|^{2s}}+\frac{1}{C}f_{n}(x)&\text{
in }\Omega,\\\ w_{n}&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

then $u_{n}\leq w_{n}$ and for all $\sigma<2^{*}_{s}$,

$||w_{n}||_{L^{\sigma}(\Omega)}\leq C||f_{n}||_{L^{\frac{2N}{N+2s}}(\Omega)}.$

Hence $||u_{n}||_{L^{\sigma}(\Omega)}\leq C$ for all $n$. If
$M(||u_{n}||_{s,2}^{2})\leq 2\frac{\lambda}{\Lambda_{N,s}}$, then
$||u_{n}||_{s,2}^{2}\leq C$. Moreover, if $M(||u_{n}||_{s,2}^{2})\geq
2\frac{\lambda}{\Lambda_{N,s}}$, then using $u_{n}$ as a test function in the
problem solved by $u_{n}$, it follows that

$M(||u_{n}||_{s,2}^{2})||u_{n}||_{s,2}^{2}\leq\frac{\lambda}{\Lambda_{N,s}}||u_{n}||_{s,2}^{2}+\int_{\Omega}f_{n}u_{n}dx.$

Thus

$(M(||u_{n}||_{s,2}^{2})-\frac{\lambda}{\Lambda_{N,s}})||u_{n}||_{s,2}^{2}\leq\int_{\Omega}f_{n}u_{n}dx.$

Hence, using Hölder and Sobolev inequalities, we conclude that

$||u_{n}||_{s,2}^{2}\leq C\mbox{ for all }n.$

Hence $\\{u_{n}\\}_{n}$ is bounded in $H^{s}_{0}(\Omega)$. Therefore, we get
the existence of $u\in H^{s}_{0}(\Omega)$ such that, up to a subsequence,
$u_{n}\rightharpoonup u$ weakly in $H^{s}_{0}(\Omega)$, $u_{n}\to u$ strongly
in $L^{\sigma}(\Omega)$ for all $\sigma<2^{*}_{s}$ and
$(M(||u_{n}||_{s,2}^{2}):=\alpha_{n}\to\alpha_{0}\geq C$. Notice that $u_{n}$
solves

$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u_{n}&=&\frac{\lambda}{M(||v||_{s,q}^{q})}\frac{u_{n}}{|x|^{2s}}+\dfrac{f_{n}}{M(||u_{n}||_{s,q}^{q})}&\text{
in }\Omega\\\ u_{n}&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Setting
$h_{n}(x):=\frac{\lambda}{M(||v||_{s,q}^{q})}\frac{u_{n}}{|x|^{2s}}+\dfrac{f_{n}}{M(||u_{n}||_{s,q}^{q})}$,
it holds that $||h_{n}||_{L^{1}(\Omega)}\leq C$. Therefore by the compactness
result in [1], we reach that, up to a subsequence, $u_{n}\to u$ strongly in
$W^{s,q}_{0}(\Omega)$ for all $q<\frac{N}{N-s}$. Thus by interpolation, it
holds that $u_{n}\to u$ strongly in $W^{s,q}_{0}(\Omega)$ for all $q<2$ and
the result follows. ∎

###  3.1. Some results related to non variational problems.

In this subsection we will assume that $q<2$. Consider the problem

(3.8)
$\left\\{\begin{array}[]{rcll}M(\left\|u\right\|_{s,q}^{q})(-\Delta)^{s}u&=&f&\text{
in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

It is clear that problem (3.8) can not be viewed and treated in a variational
setting. We begin by proving the next proposition.

######  Proposition 3.5.

Assume that $f\in L^{2}(\Omega)$ and $1<q<2$, then problem (3.8) has a unique
solution $u_{q}$ in $H^{s}_{0}(\Omega)$.

Proof. To show the existence of a solution we will use the Schauder’s Fixed
Point Theorem. Let $E=W_{0}^{s,q}(\Omega)$ and consider the operator
$T:E\longrightarrow E$ defined by $T(v)=u$ where $u$ is the unique solution
for the problem

$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}u&=&\dfrac{f}{M(||v||_{s,q}^{q})}&\text{
in }\Omega\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Taking into consideration the properties of $M$ and classical results on
regularity of the fractional Laplacian we conclude that $T$ is well defined
and

(3.9)
$||u||_{H^{s}_{0}(\Omega)}\leq\dfrac{c_{0}}{M(||v||_{s,q}^{q})}\|f\|_{L^{2}(\Omega)}\leq\dfrac{c_{0}}{m_{0}}\|f\|_{L^{2}(\Omega)}.$

Hence

$||u||_{E}\leq\dfrac{c_{1}}{m_{0}}\left\|f\right\|_{L^{2}(\Omega)}$

for some positive constant $c_{1}$ depending only on $\Omega,N$ and $s$.
Choosing $r=m^{-1}_{0}c_{1}||f||_{L^{2}(\Omega)}$, it holds that
$T(B_{r}(0))\subset B_{r}(0)$, where $B_{r}(0)$ is the ball centered at zero
with radius $r>0$ of $E$.
It is not difficult to show that $T$ is continuous. To finish we have just to
prove that $T$ is compact.

Let $\\{v_{n}\\}_{n}$ be a bounded sequence in $E$. Define $u_{n}=T(v_{n})$,
then from (3.9) it holds that $\\{u_{n}\\}_{n}$ is bounded in
$H^{s}_{0}(\Omega)$ and a priori in $E$. Hence, up to a subsequence, we get
the existence of $\hat{u}\in H^{s}_{0}(\Omega)$ such that
$u_{n}\rightharpoonup\hat{u}$ weakly in $H^{s}_{0}(\Omega)$.

Setting $g_{n}=\dfrac{f}{M(\left\|v_{n}\right\|_{s,q}^{q})}$, then
$||g_{n}||_{L^{1}(\Omega)}\leq C$ for all $n$. Hence from the compactness
result in Theorem 2.4 it holds that $u_{n}\to\hat{u}$ strongly in
$W^{s,p}_{0}(\Omega)$ for all $p<\frac{N}{N-s}$. Taking into consideration
that the sequence $\\{u_{n}\\}_{n}$ is bounded in $H^{s}_{0}(\Omega)$ and
using Vitali’s Lemma, we conclude that $u_{n}\to\hat{u}$ strongly in
$W^{s,\sigma}_{0}(\Omega)$ for all $\sigma<2$, in particular $u_{n}\to\hat{u}$
strongly in $E$. Hence $T$ is compact.

Thus by Schauder’s Fixed Point Theorem, we get the existence of $\tilde{u}\in
E$ such that $T(\tilde{u})=\tilde{u}$ and then $\tilde{u}$ solves problem
(3.8). It is not difficult to show that $u_{q}\in H^{s}_{0}(\Omega)$.

We prove now the uniqueness part. Suppose that $u_{1}$ and $u_{2}$ are two
solutions to the problem (3.8), then

$M(\left\|u_{1}\right\|_{s,q}^{q})\left(-\Delta\right)^{s}u_{1}=f=M(\left\|u_{2}\right\|_{s,q}^{q})\left(-\Delta\right)^{s}u_{2}.$

Thus

$(-\Delta)^{s}\bigg{(}M(||u_{1}||_{s,q}^{q})u_{1}-M(||u_{2}||_{s,q}^{q})u_{2}\bigg{)}=0.$

Since $M(||u_{i}||_{s,q}^{q})u_{i}\in H^{s}_{0}(\Omega)$ for $i=1,2$, then

(3.10)
$M(\left\|u_{1}\right\|_{s,q}^{q})u_{1}=M(\left\|u_{2}\right\|_{s,q}^{q})u_{2}.$

Therefore we get
$M(||u_{1}||_{s,q}^{q})||u_{1}||_{s,q}=M(||u_{2}||_{s,q}^{q})||u_{2}||_{s,q}$.
Recall that $q>1$, then by $(M1)$ and $(M2)$, the function $t\longmapsto
M(t^{q})t$ is increasing and then $||u_{1}||_{s,q}=||u_{2}||_{s,q}$. Going
back to (3.10), we obtain $u_{1}=u_{2}$.

In the case where $1<q<\frac{N}{N-s}$, we are able to show that problem (3.8)
has a unique solution for all $f\in L^{1}(\Omega)$.

######  Theorem 3.6.

Assume that $f\in L^{1}(\Omega)$ and $1<q<\frac{N}{N-s}$, then problem (3.8)
has a unique solution $\bar{u}$ in $W_{0}^{s,\sigma}(\Omega)$ for all
$\sigma<\frac{N}{N-s}$. In particular $\bar{u}\in W_{0}^{s,q}(\Omega)$.
Moreover defining $T:L^{1}(\Omega)\to W^{s,q}_{0}(\Omega)$ with $u=T(f)$ being
the unique solution to problem (3.8), then $T$ is a compact operator.

Proof. We set $f_{n}=T_{n}(f)$, then $f_{n}\in L^{\infty}(\Omega)$. By Theorem
3.5, it follows that the problem

${(\mathcal{P}^{n}_{s,q})}\left\\{\begin{array}[]{rcll}M(\left\|u_{n}\right\|_{s,q}^{q})(-\Delta)^{s}u_{n}&=&f_{n}(x)&\text{
in }\Omega,\\\ u_{n}&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

has a unique solution $u_{n}$. We claim that
$M(\left\|u_{n}\right\|_{s,q}^{q})<C$ for all $n\geq 1$.

We argue by contradiction. Assume that
$M(\left\|u_{n}\right\|_{s,q}^{q})\longrightarrow+\infty$ as $n\to\infty$.
Then by the properties of $M$, we reach that
$\lim\limits_{n\to\infty}\left\|u_{n}\right\|_{s,q}=\infty$.

Define $g_{n}\equiv\dfrac{f_{n}}{M(\left\|u_{n}\right\|_{s,q}^{q})}$, then
$|g_{n}|\leq\dfrac{|f_{n}|}{m_{0}}$ and hence $||g_{n}||_{L^{1}(\Omega)}\leq
C$. Thus using Theorem 2.4 it holds that
$||u_{n}||_{W^{s,\sigma}_{0}(\Omega)}\leq\frac{C}{m_{0}}||f_{n}||_{L^{1}(\Omega)}\leq
C$ for all $n$ and for all $\sigma<\frac{N}{N-s}$. Choosing $\sigma=q$, we get
$||u_{n}||_{W^{s,q}_{0}(\Omega)}\leq C$, a contradiction with the main
hypothesis. Thus $M(\left\|u_{n}\right\|_{s,q}^{q})<C$ for all $n$ and then
claim follows.

Consequently $\left\|u_{n}\right\|_{s,q}\leq C$, therefore there exists
$\bar{u}\in W^{s,q}_{0}(\Omega)$ such that, up to a subsequence,
$u_{n}\rightharpoonup\bar{u}$ weakly in $W^{s,q}_{0}(\Omega)$.

Let us show that $u_{n}\longrightarrow\bar{u}$ strongly in
$W^{s,q}_{0}(\Omega)$.

Recall that

$(-\Delta)^{s}u_{n}=\dfrac{f_{n}}{M(\left\|u_{n}\right\|_{s,q}^{q})},$

by the boundedness of the sequence
$\\{\dfrac{f_{n}}{M(\left\|u_{n}\right\|_{s,q}^{q})}\\}_{n}$ in
$L^{1}(\Omega)$ and the compactness result in Theorem 2.4, we conclude that
$u_{n}\longrightarrow\bar{u}$ strongly in $W^{s,\sigma}_{0}(\Omega)$, for all
$\sigma<\dfrac{N}{N-s}$ and in particular $u_{n}\longrightarrow\bar{u}$
strongly in $W^{s,q}_{0}(\Omega)$

Hence $M(\left\|u_{n}\right\|_{s,q}^{q})\longrightarrow
M(\left\|\bar{u}\right\|_{s,q}^{q})$. It is clear that $\bar{u}$ solves the
problem (3.8). The uniqueness follows as in the proof of Theorem 3.5.

Define now $T:L^{1}(\Omega)\to W^{s,q}_{0}(\Omega)$ where $u=T(f)$ is the
unique solution to problem (3.8). It is clear that $T$ is well defined. To
show that $T$ is compact, we consider a sequence $\\{f_{n}\\}_{n}\subset
L^{1}(\Omega)$ and define $u_{n}=T(f_{n})$. Then

$(-\Delta)^{s}u_{n}=\dfrac{f_{n}}{M(\left\|u_{n}\right\|_{s,q}^{q})}.$

Now, repeating the same computations as above and using the compactness result
in Theorem 2.4 we deduce that, up to a subsequence,
$u_{n}\longrightarrow\bar{u}$ strongly in $W^{s,\sigma}_{0}(\Omega)$, for all
$\sigma<\frac{N}{N-s}$ and in particular $u_{n}\longrightarrow\bar{u}$
strongly in $W^{s,q}_{0}(\Omega)$. Hence the result follows.

We are now able to state the main result of this subsection.

######  Theorem 3.7.

Let $\Omega\subset{I\\!\\!R}^{N}$ be a bounded domain such that $0\in\Omega$
and $1<q<\frac{N}{N-s}$. Suppose that $f\in L^{1}(\Omega)$ is such that
$\displaystyle{\int_{\Omega}f\left|x\right|^{-\theta}dx}<c$ for some positive
constant $\theta$. Then for all $\lambda>0$, the problem

$(\mathcal{P}^{\lambda}_{s,q})\left\\{\begin{array}[]{rcll}M(\left\|u\right\|_{s,q}^{q})(-\Delta)^{s}u&=&\lambda\dfrac{u}{\left|x\right|^{2s}}+f&\text{
in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

has a solution $u$ such that $\bar{u}\in W^{s,\sigma}_{0}(\Omega)$ for all
$\sigma<\frac{N}{N-s}$.

Proof. Let $\lambda>0$ be fixed and define $f_{n}=T_{n}(f)$. Consider the
approximating problem

${(\mathcal{P}^{n}_{s,q})}\left\\{\begin{array}[]{rcll}M(||u_{n}||^{q}_{s,q})(-\Delta)^{s}u_{n}&=&\lambda\dfrac{T_{n}(u_{n})}{\left|x\right|^{2s}+\frac{1}{n}}+f_{n}(x)&\text{
in }\hskip 4.26773pt\Omega,\\\ u_{n}&=&0&\text{ in }\hskip
4.26773pt\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Using a direct variation of Theorem 2.4, we can show that Problem
$({P}^{n}_{s,q})$ has a solution $u_{n}\in H^{s}_{0}(\Omega)$.

Let begin by proving that $M(||u||^{q}_{s,q})<m_{1}$. We argue by
contradiction, if $M(||u_{n}||^{q}_{s,q})\longrightarrow+\infty$, then
$||u||^{q}_{s,q}\longrightarrow+\infty$. Observe that

(3.11)
$(-\Delta)^{s}u_{n}=\dfrac{\lambda}{M(||u_{n}||^{q}_{s,q})}\,\dfrac{T_{n}(u_{n})}{\left|x\right|^{2s}+\frac{1}{n}}+\dfrac{f_{n}}{M(||u_{n}||^{q}_{s,q})}$

For simplicity of typing we set

(3.12)
$g_{n}\equiv\dfrac{\lambda}{M(||u_{n}||^{q}_{s,q})}\,\dfrac{T_{n}(u_{n})}{|x|^{2s}+\frac{1}{n}}+\dfrac{f_{n}}{M(||u_{n}||^{q}_{s,q})}.$

We claim that $\bigg{\|}\dfrac{u_{n}}{|x|^{2s}}\bigg{\|}_{L^{1}(\Omega)}<C$
for some positive constant $C$.

To prove the claim we define $\varphi$ as the unique solution to the problem

(3.13)
$\left\\{\begin{array}[]{rcll}(-\Delta)^{s}\varphi&=&\beta\dfrac{\varphi}{\left|x\right|^{2s}}+1&\text{
in }\Omega,\\\ \varphi&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $0<\beta<\Lambda_{N,s}$ to be chosen later. Notice that $\varphi\in
L^{\infty}(\Omega\backslash B_{r}(0))$ and
$\varphi\approx|x|^{-\gamma_{\beta}}$ where

(3.14) $\gamma_{\beta}:=\dfrac{N-2s}{2}-\alpha,$

and $\alpha$ is given by (2.7) where $\lambda$ is substituted by $\beta$. Il
it not difficult to show that $\gamma_{\beta}\to 0$ as $\beta\to 0$.

Using $u_{n}$ as a test function in (3.13) and integrating over $\Omega$, it
holds that

$\beta\int_{\Omega}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx+\int_{\Omega}u_{n}dx=\dfrac{\lambda}{M(\left\|u_{n}\right\|_{s,q}^{q})}\int_{\Omega}\dfrac{u_{n}\varphi}{\left|x\right|^{2s}}dx+\dfrac{1}{M(\left\|u_{n}\right\|_{s,q}^{q})}\int_{\Omega}f_{n}\varphi
dx.$

Then,

$(\beta-\dfrac{\lambda}{M(\left\|u_{n}\right\|_{s,q}^{q})})\int_{\Omega}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx+\int_{\Omega}u_{n}dx\leq\dfrac{1}{m_{0}}\int_{\Omega}f_{n}\varphi
dx\leq\dfrac{1}{m_{0}}\int_{\Omega}f_{n}\left|x\right|^{-\gamma_{\beta}}dx.$

Now, since $M(||u_{n}||_{s,q}^{q})\longrightarrow+\infty$, we can choose
$n_{0}$ large enough such that
$\beta-\dfrac{\lambda}{M(||u_{n}||_{s,q}^{q})}>\varepsilon_{0}>0$ if $n\geq
n_{0}$. Hence, for $n\geq n_{0}$,

$\beta_{0}\int_{\Omega}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx+\int_{\Omega}u_{n}dx\leq\dfrac{1}{m_{0}}\int_{\Omega}f\left|x\right|^{-\theta_{\varepsilon}}dx.$

Recall that $\displaystyle{\int_{\Omega}f\left|x\right|^{-\theta}dx}<c$, then
we fix now $\beta$ small enough such that $\theta_{\varepsilon}\leq\theta$.
Thus $\displaystyle\int_{\Omega}f\left|x\right|^{-\theta_{\beta}}<\infty$.
Thus

(3.15) $\beta_{0}\int_{\Omega}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx+\int_{\Omega}u_{n}dx<m^{-1}_{0}c.$

From which we deduce that
$\bigg{\|}\dfrac{u_{n}}{|x|^{2s}}\bigg{\|}_{L^{1}(\Omega)}<m^{-1}_{0}c$ and
the claim follows.

Now going back to (3.11) and using the fact that the sequence
$\\{g_{n}\\}_{n}$ is bounded in $L^{1}(\Omega)$, we reach that the sequence
$\\{u_{n}\\}_{n}$ is bounded in $W^{s,\sigma}_{0}(\Omega)$ for all
$\sigma<\frac{N}{N-s}$, in particular for $\sigma=q$. Thus
$\left\|u_{n}\right\|_{s,q}<C_{1}$ and then
$M(\left\|u_{n}\right\|_{s,q}^{q})<\bar{C}$, a contradiction with the main
hypothesis. Thus $\\{M(||u_{n}||_{s,q}^{q})\\}_{n}$ is bounded in
${I\\!\\!R}^{+}$.

Hence, we get the existence of $\bar{u}\in W^{s,q}_{0}(\Omega)$ such that, up
to a subsequence, $u_{n}\rightharpoonup\bar{u}$ weakly in
$W^{s,q}_{0}(\Omega)$, $u_{n}\to\bar{u}$ strongly in $L^{\sigma}(\Omega)$ for
all $\sigma<q^{*}_{s}$ and $u_{n}\to\bar{u}\>\>a.e\mbox{ in }\Omega$.

Setting $\alpha_{n}:=M(\left\|u_{n}\right\|^{q}_{s,q})$, then we can assume
that $\alpha_{n}\longrightarrow\bar{\alpha}\geq m_{0}.$

Let us show that $\dfrac{u_{n}}{|x|^{2s}}\to\dfrac{u}{|x|^{2s}}$ strongly in
$L^{1}(\Omega)$. It is clear that
$\dfrac{u_{n}}{|x|^{2s}}\to\dfrac{\bar{u}}{|x|^{2s}}$ a.e. in $\Omega.$ Using
the fact that $u_{n}\to\bar{u}$ strongly in $L^{\sigma}(\Omega)$ for all
$\sigma<q^{*}_{s}$, we reach that

$\dfrac{u_{n}}{|x|^{2s}}\to\dfrac{\bar{u}}{|x|^{2s}}\mbox{ strongly in
}L^{1}(\Omega\backslash B_{R}(0)\mbox{ for all }R>0.$

Hence to conclude we have just to show that

$\dfrac{u_{n}}{|x|^{2s}}\to\dfrac{\bar{u}}{|x|^{2s}}\mbox{ strongly in
}L^{1}(B_{R}(0))\mbox{ for some }R>0.$

Recall that from (3.15), we have $\displaystyle\int_{\Omega}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx<C$ for all $n$. By the fact that $\varphi\geq
C_{\beta}|x|^{-\theta_{\varepsilon}}$ in $B_{R}(0)$, it holds that

$C_{\beta}R^{-\theta_{\beta}}\int_{B_{R}(0)}\dfrac{u_{n}}{\left|x\right|^{2s}}\leq
C\mbox{ for all }n.$

Fix $R>0$ to be chosen later and consider $E\subset B_{R}(0)$ a measurable
set. Then

$\displaystyle\int_{E}\dfrac{u_{n}}{\left|x\right|^{2s}}dx$ $\displaystyle=$
$\displaystyle\int_{E\cap B_{R}(0)}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx+\int_{E\backslash B_{R}(0)}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx$ $\displaystyle\leq$ $\displaystyle
C_{\beta}R^{\theta_{\beta}}+\int_{E\backslash B_{R}(0)}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx.$

Let $\varepsilon>0$, then we can choose $\delta>0$ such that if $|E|<\delta$
and $n\geq n_{0}$, then

$\int_{E\backslash B_{R}(0)}\dfrac{\varphi
u_{n}}{\left|x\right|^{2s}}dx\leq\frac{\varepsilon}{2}.$

It is clear that we can choose $R$ small enough such that

$C_{\beta}R^{\theta_{\beta}}\leq\frac{\varepsilon}{2}.$

Combining the above estimates, it holds that for $n\geq n_{0}$ and if
$|E|\leq\delta$, then

$\int_{E}\dfrac{u_{n}}{\left|x\right|^{2s}}dx\leq\varepsilon.$

Hence, using Vitali’s lemma, we reach that
$\dfrac{u_{n}}{|x|^{2s}}\to\dfrac{u}{|x|^{2s}}$ strongly in $L^{1}(\Omega)$.

Reminding the definition of $g_{n}$ in (3.12) and using the compactness result
in Theorem 2.4 we conclude, up to a subsequence, $u_{n}\to\bar{u}$ strongly in
$W^{\sigma,q}_{0}(\Omega)$ for all $\sigma<\frac{N}{N-s}$. Thus

$M(\left\|u_{n}\right\|^{q}_{s,q})\to
M(\left\|\bar{u}\right\|^{q}_{s,q})\mbox{ as }n\to\infty.$

Therefore we conclude that $\bar{u}$ satisfies

$(-\Delta)^{s}\bar{u}=\dfrac{\lambda}{M(\left\|\bar{u}\right\|_{s,q}^{q})}.\dfrac{\bar{u}}{\left|x\right|^{2s}}+\dfrac{f}{M(\left\|\bar{u}\right\|_{s,q}^{q})},$

and then $\bar{u}$ is solution of Problem $({P}^{\lambda}_{s,q})$.

######  Remark 1.

If $f\in L^{\sigma}(\Omega)$ for $\sigma>1$, then using Hölder inequality we
can prove that the condition stated in Theorem 3.7 holds for $f$ for some
$\theta>0$. Thus problem $({P}^{\lambda}_{s,q})$ has a solution for all $f\in
L^{\sigma}(\Omega)$ with $\sigma>1$ and for all $\lambda>0$.

##  4\. Existence results in the case $f(x,u):=u^{p}+g(x)$.

Let consider now the nonlinear problem

(4.1)
$\left\\{\begin{array}[]{rcll}M(\left\|u\right\|_{s,2}^{2})(-\Delta)^{s}u&=&\lambda\dfrac{u}{\left|x\right|^{2s}}+u^{p}+\mu
g&\text{ in }\Omega,\\\ u&>&0&\text{ in }\Omega,\\\ u&=&0&\text{ in
}\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

In the case where $M\equiv 1$, then, as in the local case, for
$\lambda<\Lambda_{N,s}$ fixed, it is possible to show the existence of a
critical value $p_{*}>2^{*}_{s}-1$ such that the problem (4.1) has a
nonnegative super-solution if and only if $p<p_{*}$. This can be obtained
using suitable radial computation, according the behavior of any super-
solution near the origin and a suitable test function, see [6] and [10].

The goal of this section is to check that, under suitable hypothesis on $g$
and $p$, the existence of a solution to problem (4.1) for all $\lambda>0$. The
first existence result is the following.

######  Theorem 4.1.

Let $g\in L^{\sigma}(\Omega)$ with $\sigma\geq\frac{2N}{N+2s}$ and suppose
that $p<2^{*}_{s}-1$, then there exists $\mu^{*}>0$ such that for all
$\lambda>0$ and for all $\mu<\mu^{*}$, problem (4.1) has a solution $u\in
H^{s}_{0}(\Omega)$.

###### Proof.

In the case $M=1$, existence of solution is proved under the condition
$\lambda\leq\Lambda_{N,s}$ and the comparison principle using a suitable
radial supersolution. However this argument is not applicable in our case and
we have to use a different argument in order to show the existence of a
solution. The main idea is to use a suitable fixed point argument.

Fix $p<r<2^{*}_{s}$, we can show the existence of $\mu^{*}>0$ such that if
$\mu<\mu^{*}$, then the algebraic equation

$C(l+\lambda^{\ast}\|f\|_{L^{\frac{2N}{N+2s}}(\Omega)})=l^{\frac{1}{p}},$

where $C$ is a universal constant that will be chosen later.

Define the set

$E:=\left\\{\varphi\in W_{0}^{s,1}(\Omega):||\varphi||_{L^{r}(\Omega)}\leq
l^{\frac{1}{2}}\right\\},$

Then $E$ is a closed convex set of $W_{0}^{s,1}(\Omega)$.

Now, we define $T:E\to W_{0}^{s,1}(\Omega)$ by $T(\varphi)=u$, where $u$ is
the weak solution to

(4.2)
$\left\\{\begin{array}[]{rcll}M(||u||_{s,2}^{2})(-\Delta)^{s}u&=&\lambda\dfrac{u}{\left|x\right|^{2s}}+\varphi_{+}^{p}+\mu
g(x)\,,&\mbox{ in }\Omega,\\\ u&=0\,,&\mbox{ in
}{I\\!\\!R}^{N}\setminus\Omega.\end{array}\right.$

It is clear that if $u=T(u)$, then $u$ solves problem (4.1). Hence, to prove
Theorem 4.1, we shall show that $T$ has fixed point belonging to
$W_{0}^{s,2}(\Omega)\cap\mathcal{C}^{0,\alpha}(\Omega)$ for some $\alpha>0$.

The proof will be given in several steps.

Step 1: $T$ is well defined.

Since $\varphi\in E$, then $\varphi_{+}^{p}+g\in L^{\frac{2N}{N+2s}}(\Omega)$.
Thus using Theorem 3.1 and the Remark 3.2, we get the existence and the
uniqueness of $u\in H^{s}_{0}(\Omega)$ such that $u$ solves the problem (4.2).
Hence $u\in W^{s,1}_{0}(\Omega)$ and then $T$ is well defined.

Step 2: $T(E)\subset E$.

Let $\varphi\in E$, we define $u=T(\varphi)$. By the regularity result in
Proposition 3.3, it follows that

(4.3) $\displaystyle||u||_{L^{r}(\Omega)}$ $\displaystyle\leq
C\big{\|}\varphi_{+}^{p}+\lambda g(x)\big{\|}_{L^{\frac{2N}{N+2s}}(\Omega)}$
$\displaystyle\leq
C(||\varphi||^{\frac{p}{r}}_{L^{r}(\Omega)}+\lambda^{\ast}\|f\|_{L^{\frac{2N}{N+2s}}(\Omega)})$
$\displaystyle\leq C(l+\lambda^{\ast}\|f\|_{L^{\frac{2N}{N+2s}}(\Omega)})\leq
l^{\frac{1}{p}}.$

Since, we have proved previously that $u\in W_{0}^{s,1}(\Omega)$, then using
the definition of $l$, we conclude that $u\in E$ and so, that $T(E)\subset E$.

Step 3: $T$ is continuous.

Let $\\{\varphi_{n}\\}_{n}\subset E$ be such that $\varphi_{n}\to\varphi$ in
$W_{0}^{s,1}(\Omega)$. Consider $u_{n}=T(\varphi_{n})$ and $u=T(\varphi)$.
Define

$h_{n}(x):=(\varphi_{n})_{+}^{p}+\lambda h(x),$

then to get the desired result, we have just to show that $h_{n}\to h$
strongly in $L^{\frac{2N}{N+2s}}(\Omega)$ where

$h(x):=(\varphi)_{+}^{p}+\lambda g(x).$

Using Sobolev inequality in $W^{s,1}_{0}(\Omega)$ and since
$\varphi_{n}\to\varphi$ strongly in $W_{0}^{s,1}(\Omega)$, then
$\varphi_{n}\to\varphi$ strongly in $L^{\frac{N}{N-1}}(\Omega)$. Since
$\\{\varphi_{n}\\}_{n}$ is bounded in $L^{r}(\Omega)$, then using an
interpolation arguments it holds that $\varphi_{n}\to\varphi$ strongly in
$L^{\sigma}(\Omega)$ for all $\sigma<r$ and in particular in
$L^{\frac{2N}{N+2s}}(\Omega)$. Hence $u_{n}\to u$ strongly in
$H^{s}_{0}(\Omega)$ and then the continuity of $T$ follows.

Step 4: $T$ is compact.

Let $\\{\varphi_{n}\\}\subset E$ be a bounded sequence in
$W_{0}^{s,1}(\Omega)$ and define $u_{n}=T(\varphi_{n})$. We will prove that,
up to a subsequence, $u_{n}\to u$ in $W_{0}^{s,1}(\Omega)$ for some $u\in
W_{0}^{s,1}(\Omega)$.

Recall that $\\{\varphi_{n}\\}_{n}\subset E$, then as in the proof of the Step
3, we reach that $\\{\varphi_{n}\\}_{n}$ is a bounded sequence in
$L^{\frac{2N}{N+2s}}(\Omega)$. Now, setting

$h_{n}(x):=(\varphi_{n})_{+}^{p}+\lambda g(x),$

we have that $\\{h_{n}\\}$ is a bounded sequence in
$L^{\frac{2N}{N+2s}}(\Omega)$. The result then follows from the compactness
result in Proposition 2.4.

As a conclusion, and since $E$ is a closed convex set of $W_{0}^{s,1}(\Omega)$
and, by the previous steps, we can apply the Schauder fixed point Theorem to
get the existence of $u\in E$ such that $T(u)=u$. Thus, we conclude that
problem (4.1) has a weak solution for all $0<\lambda\leq\lambda^{\ast}$. It is
clear that $u\in H^{s}_{0}(\Omega)$.

∎

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