Isolated singularities for fractional Lane-Emden equations

in the Serrin’s supercritical case

Huyuan<EMAIL_ADDRESS>and Feng<EMAIL_ADDRESS>
1Department of Mathematics, Jiangxi Normal University,

Nanchang, Jiangxi 330022, PR China

2Center for PDEs, School of Mathematical Sciences, East China Normal
University,

Shanghai, 200241, PR China

###### Abstract

In this paper, we give a classification of the isolated singularities of
positive solutions to the semilinear fractional elliptic equations

$(E)\qquad\qquad(-\Delta)^{s}u=|x|^{\theta}u^{p}\quad{\rm in}\ \
B_{1}\setminus\\{0\\},\qquad u=h\quad{\rm in}\ \ \mathbb{R}^{N}\setminus
B_{1},\qquad\qquad\quad$

where $s\in(0,1)$, $\theta\in(-2s,0]$, $p>\frac{N+\theta}{N-2s}$, $B_{1}$ is
the unit ball centered at the origin of $\mathbb{R}^{N}$ with $N>2s$. $h$ is a
nonnegative function in $\mathbb{R}^{N}\setminus B_{1}$. Our analysis of
isolated singularities of $(E)$ is based on an integral upper bounds and the
study of the Poisson problem with the fractional Hardy operators. It is worth
noting that our classification of isolated singularity holds in the Sobolev
super critical case $p>\frac{N+2s+2\theta}{N-2s}$ for $s\in(0,1]$ under
suitable assumption of $h$.

###### Contents

  1. 1 Introduction
  2. 2  Preliminary
    1. 2.1 Poisson problem for fractional Hardy problem
    2. 2.2 Uniformly integral upper bounds
    3. 2.3 Harnack inequality
  3. 3 Upper bounds
    1. 3.1  The case: $\theta=0$
    2. 3.2  The case: $\theta\in(-2s,0]$
  4. 4 Classification of isolated singularity
    1. 4.1  Some important estimates
    2. 4.2 The isolated singularities
  5. 5 Application to the decay at infinity

Keywords: Isolated singularities; Fractional Laplacian; Lane-Emden equation.

AMS Subject Classifications: 35R11; 35B40; 35J61.

##  1 Introduction

Let $s\in(0,1)$, $N>2s$ and $B_{r}(y)$ be the ball of radius $r>0$ centered at
$y$ in $\mathbb{R}^{N}$, $B_{r}:=B_{r}(0)$. Our concern of this paper is to
classify the isolated singular positive solutions of the fractional Lane-Emden
equation

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=|x|^{\theta}u^{p}&{\rm in}\
\,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ u=h&{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1},\end{array}\right.$ (1.1)

where $\theta\in(-2s,0]$, $p>\frac{N+\theta}{N-2s}$, $h$ is a nonnegative
function in $\mathbb{R}^{N}\setminus B_{1}$ and $(-\Delta)^{s}$ is the
fractional Laplacian defined by

$(-\Delta)^{s}u(x)=C_{N,s}\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}^{N}\setminus
B_{\epsilon}}\frac{u(x)-u(x+z)}{|z|^{N+2s}}dz$

with

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

being the normalized constant, see [25], $\Gamma$ being the usual Gamma
function. It is known that $(-\Delta)^{s}u(x)$ is well defined if $u$ is twice
continuously differentiable in a neighborhood of $x$ and contained in the
space
$L^{1}_{s}(\mathbb{R}^{N}):=L^{1}(\mathbb{R}^{N},\frac{dx}{(1+|x|)^{N+2s}})$.

When $s=1$ and $N\geq 3$, the isolated singularities of the Lane-Emden
equation

$\left\\{\begin{array}[]{lll}-\Delta u=|x|^{\theta}u^{p}&{\rm in}\ \
B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\ \,u=0&{\rm on}\ \,\partial
B_{1}\end{array}\right.$ (1.2)

has been studied extensively in the last decades. When $\theta=0$ and
$p\in(1,\,\frac{N}{N-2})$, Lions in [36] classified the singular solution by
building the connection with the weak solutions of

$-\Delta u=u^{p}+k\delta_{0}$

and the related positive solution $u_{k}$ has the asymptotic behavior

$\lim_{|x|\to 0^{+}}u_{k}(x)|x|^{N-2}=c_{N}k$

for some $k\geq 0$ and the normalized constant $c_{N}>0$ for $-\Delta$. In the
particular case $k=0$, the solutions have removable singularity at the origin.
In the Serrin’s critical and supercritical case $p\geq\frac{N}{N-2}$, however,
since it always has $k=0$, this method fails to classify the singularities.
When

$\theta\in(-2,2)\quad{\rm and}\quad\
p\in\Big{(}\frac{N+\theta}{N-2},\,\frac{N+2}{N-2}\Big{)}\setminus\Big{\\{}\frac{N+2+2\theta}{N-2}\Big{\\}},$

the singularity of solution for (1.2) was studied by Gidas-Spruck in [29] by
using analytic techniques from [5]. In this case, the positive singular
solutions have the singularity:

$u(x)=c_{p,\theta}|x|^{-\frac{2+\theta}{p-1}}(1+o(1))\quad{\rm as}\ \,|x|\to
0^{+}$

with the coefficient

$c_{p,\theta}=\big{(}\frac{2+\theta}{p-1}(N-2-\frac{2+\theta}{p-1})\big{)}^{\frac{1}{p-1}}.$
(1.3)

When

$\theta\in(-2,2)\quad{\rm and}\quad\ p=\frac{N+\theta}{N-2},$

the author in [4, 5] gave a beautiful classification: any positive solution
$u$ of (1.2) has removable singularity at origin or it has the asymptotic
behavior

$u(x)=\Big{(}\frac{(N-2)^{2}}{2+\theta}\Big{)}^{\frac{N-2}{2+\theta}}|x|^{2-N}(-\ln|x|)^{\frac{2-N}{2+\theta}}(1+o(1))\quad{\rm
as}\ \,|x|\to 0^{+}.$

Moreover, the existence of a singular solution is obtained by the phase plane
analysis. By the aid of these solutions, Pacard in[37] has constructed
positive solutions with the prescribed singular set. Later on, Caffarelli-
Gidas-Spruck in [9] (also see [35]) classified the singular solutions of (1.2)
when $\theta=0,\ p=\frac{N+2}{N-2}$ and its singular solutions satisfy

$u(x)=\varphi_{{}_{D}}(-\ln|x|)|x|^{-\frac{N-2}{2}}(1+o(|x|))\quad{\rm as}\
\,|x|\to 0^{+},$

where $\varphi_{D}$ may be the constant $c_{\frac{N+2}{N-2}}$ or be a periodic
function. In the Sobolev super critical case $\theta=0$ and
$p>\frac{N+2}{N-2}$, [30, 44] have studied the structure of positive radial
solutions of (1.2), [22] constructs some non-radially symmetric solutions. But
it is still open to give a full classification of isolated singularities for
singular solutions. We refer to [7, 23, 31, 39, 40, 43] for more singularities
for elliptic problem in various setting.

Motivated by various applications and relationships to the theory of PDEs,
there has been an increasing interests in Dirichlet problems with nonlocal
operators and the prototype of nonlocal operator is the fractional Laplacian.
The problems with fully nonlinear nonlocal operators are studied by [12, 13],
[11] connects the fractional problem with the second order degenerated problem
in a half space with one dimensional higher, and basic results about
regularities for the fractional problem could see [11, 38, 41] and blowing up
analysis for nonlocal problems could see [10, 18, 14]. Motivated by [36], the
isolated singularity of (1.1) with $\theta=0$ was studied in [16] for
$p\in(1,\frac{N}{N-2s})$ via the connection with the distributional solutions
of

$(-\Delta)^{s}u=u^{p}+k\delta_{0}\quad{\rm in}\ \,B_{1}\setminus\\{0\\},\qquad
u=0\ \ {\rm in}\ \,\mathbb{R}^{N}\setminus B_{1},$

for which the positive singularity could be described by fundamental solution
of fractional Laplacian. As well as for the Laplacian, this method fails for
the classification of isolated singularities for the Serrin’s critical and
supercritical case, i.e. $p\geq\frac{N}{N-2s}$. When $\theta=0$,
$p\in(\frac{N}{N-2s},\frac{N+2s}{N-2s}]$, [32, 10] build the platform of the
isolated singularity for positive solution to

$\left\\{\begin{array}[]{lll}\,\quad\ \ {\rm div}(t^{1-2s})\nabla U)=0&{\rm
in}\ \,B_{1}\times(0,1)\\\\[5.69054pt] \displaystyle\lim_{t\to
0^{+}}t^{1-2s}\partial_{t}U(x,t)=U^{p}(x,0)&{\rm on}\
\,B_{1}\setminus\\{0\\}\end{array}\right.$ (1.4)

and in the Sobolev’s critical case $p=\frac{N+2s}{N-2s}$, the behavior of non-
removable singular solution of (1.1) can be stated as following

$c_{2}|x|^{-\frac{N-2s}{2}}\leq u(x)\leq c_{1}|x|^{-\frac{N-2s}{2}}$

for some $c_{1}>c_{2}>0$, and then [46] gives a description of singular
solution near the singularity for $p\in(\frac{N}{N-2s},\frac{N+2s}{N-2s})$ as

$c_{4}|x|^{-\frac{2s}{p-1}}\leq u(x)\leq c_{3}|x|^{-\frac{2s}{p-1}}$

for some $c_{3}>c_{4}>0$. Also its estimates singularity for
$p=\frac{N}{N-2s}$ could see [45]. For the existence of isolated singular
solutions, [1, 2] constructed a sequence of isolated singular solution of

$(-\Delta)^{s}u=u^{p}\quad{\rm in}\ \,\mathbb{R}^{N}\setminus\\{0\\}$

for $p\in(\frac{N}{N-2s},\frac{N+2s}{N-2s})$, with fast decaying at infinity.
We can also see [3, 24] for the fractional Yamabe problem with isolated
singularities in the Sobolev’s critical case. Very recently, we provided in
[15] the classification of isolated singularities of (1.1) by a direct
analysis method and showed the existence of a sequence of singular solutions
of (1.1) with $\theta=0$ and $p=\frac{N}{N-2s}$.

Our aim in this paper is to classify the positive singularity of fractional
problem (1.1) in the Serrin’s supercritical case by a direct analysis method.
We first state our main results. To this end, we introduce some notations. For
$\tau\in(-N,2s)$, we denote

$\mathcal{C}_{s}(\tau)=2^{2s}\frac{\Gamma(\frac{N+\tau}{2})\Gamma(\frac{2s-\tau}{2})}{\Gamma(-\frac{\tau}{2})\Gamma(\frac{N-2s+\tau}{2})}.$
(1.5)

More properties about $\mathcal{C}_{s}(\tau)$ can be found in the next
section. In particular we notice that

$\mathcal{C}_{s}(\tau)>0\quad{\rm for}\ \ \tau\in(2s-N,0).$ (1.6)

Let

$\mathcal{K}_{p,\theta}:=\mathcal{C}_{s}\Big{(}-\frac{2s+\theta}{p-1}\Big{)}^{\frac{1}{p-1}},$
(1.7)

then it’s well defined since $-\frac{2s+\theta}{p-1}\in(2s-N,0)$ for
$p>\frac{N+\theta}{N-2s}$ and $\theta>-2s$. Here we use the notation
$\mathcal{K}_{p}=\mathcal{K}_{p,0}$

The classification of isolated singularities of (1.1) states as following.

###### Theorem 1.1

Assume that $s\in(0,1)$, $N>2s$, $\theta=0$

$p\in\Big{(}\frac{N}{N-2s},\frac{N+2s}{N-2s}\Big{)}$ (1.8)

and $h\in L^{1}_{s}(\mathbb{R}^{N}\setminus B_{1})$ is nonnegative.

Let $u_{0}$ be a positive solution of (1.1), then $u_{0}$ has either removable
singularity at the origin or there holds

$\displaystyle\frac{\mathcal{K}_{p}}{C_{0}}\leq\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s}{p-1}}\leq\mathcal{K}_{p}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s}{p-1}}\leq C_{0}\mathcal{K}_{p},$

where $C_{0}\geq 1$ is a constant from the Harnack inequality.

###### Theorem 1.2

Assume that $h$ is radially symmetric and nonnegative in
$\mathbb{R}^{N}\setminus B_{1}$, decreasing with respect to $|x|$ and
$h(1)\geq 0$,

$\theta=0\quad\ \&\quad\ p\geq\frac{N+2s}{N-2s}$ (1.9)

or

$\theta\in(-2s,0)\quad\ \&\quad\ p>\frac{N+\theta}{N-2s}.$ (1.10)

Let $u_{0}$ be a positive solution of (1.1) such that

$u_{0}(x)\geq h(1)\quad{\rm for\ any}\ \,x\in B_{1},$ (1.11)

then $u_{0}$ is radially symmetric, decreasing with respect to $|x|$.
Moreover, $u_{0}$ has either removable singularity at the origin or one has

$\frac{\mathcal{K}_{p,\theta}}{C_{0}}\leq\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq C_{0}\mathcal{K}_{p,\theta}.$
(1.12)

Our outline of the proofs is the following. The first step is to get an bound

$\limsup_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}<+\infty.$ (1.13)

To this end, we build the following important upper estimate:

###### Proposition 1.1

Assume that $\theta\in(-2s,+\infty)$, $p\geq\frac{N+\theta}{N-2s}$ and $u_{0}$
is a nonnegative classical solution of (1.1). Then we have
$|x|^{\theta}u_{0}^{p}\in L^{1}(B_{1},(1-|x|)^{s}dx)$ and there exists a
uniform constant $c_{5}>0$, independent of $r$ and $u_{0}$, such that

$\int_{B_{r}}|x|^{\theta}u_{0}^{p}dx\leq
c_{5}r^{N-\frac{\theta+2sp}{p-1}},\quad\forall\ r\in(0,\frac{1}{2}).$ (1.14)

With the help of Proposition 1.1, we can develop a direct blowing up technique
for $\theta=0$, in which we don’t have to transform the equation into an
extension problem, see [10, 34, 46], which doesn’t require the boundedness in
$L^{1}_{s}(\mathbb{R}^{N},dx)$, thanks to the local property. In our blowing
up procedure, if (1.13) fails, the essential part is to keep the scaled
singular solutions uniformly bounded in $L^{1}_{s}(\mathbb{R}^{N})$ for
$p>\frac{N}{N-2s}$ by (1.14) with $\theta=0$ and then we pass to the limit to
get a positive bounded solution for the limit equation and a contradiction
arises from the nonexistence of positive bounded solution for that limit
equation.

However, the blowing up technique fails for $\theta\not=0$, since the limit
equation is $(-\Delta)^{s}u=0$ in $\mathbb{R}^{N}$ when $\theta<0$, which has
positive solutions while for $\theta>0$ there is no limit equation at all.

Fortunately, when $\theta\in(-2s,0)$, we can proceed a direct method of moving
planes to obtain the properties of symmetry and monotonicity in the $|x|$
direction under the assumption on $h$ of symmetry and monotonicity, motivated
by [27]. As a consequence, these properties improve (1.14) to our desired
upper bound (1.17).

The second step is to get the estimate: for non-removable solution, there
holds

$\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}$ (1.15)

for $p\in(\frac{N+\theta}{N-2s},+\infty)$. These inequalities are motivated by
the Poisson problem involving the fractional Hardy operators, where the
coefficients of the Hardy potential is very sensitive to the blowing up rate
of the related solutions. Here we want to emphasize that the inequalities
(1.15) does not depends on the upper bounds and Harnack inequality.

Generally, we can draw a conclusion for general $\theta>-2s$:

###### Theorem 1.3

Assume that $\theta\in(-2s,+\infty)$, $p>\frac{N+\theta}{N-2s}$ and $u_{0}$ is
a nonnegative classical solution of (1.1) verifying that there exist
$C_{1}\geq 1$ and $r_{1}\in(0,1)$ such that for $x,y\in
B_{r_{1}}\setminus\\{0\\}$

$u_{0}(x)\leq C_{1}u_{0}(y)\quad{\rm for}\ \
\frac{1}{2}\leq\frac{|x|}{|y|}\leq 2,$ (1.16)

then

$\frac{\mathcal{K}_{p,\theta}}{C_{1}}\leq\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq C_{1}\mathcal{K}_{p,\theta}.$
(1.17)

Note that (1.16) is one type of Harnack inequality, which, together with the
pointwise upper bound, to obtain the lower bound $\displaystyle\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}>0$ for the non-removable solution.
To make clear of our results, we explain the relationship of integral upper
bound (1.14), pointwise upper bound $\displaystyle\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}<+\infty$ and Harnack inequality
(1.16):

  1. $(i)$

for $\theta=0$: integral upper bound + blow up analysis $\Rightarrow$
pointwise upper bound $\Rightarrow$ Harnack inequality;

  2. $(ii)$

for $\theta\in(-2s,0)$: integral upper bound + radial symmetry, increasing
monotonicity $\Rightarrow$ pointwise upper bound $\Rightarrow$ Harnack
inequality;

  3. $(iii)$

for $\theta\in(-2s,+\infty)$: integral upper bound + Harnack inequality
$\Rightarrow$ pointwise upper bound.

###### Remark 1.1

Under the assumption of $h$ in Theorem 1.2, the method the moving plane works,
all such nonnegative solutions of (1.1) are radially symmetric and decreasing
with respect to $|x|$.

If (1.1) has a positive solution, then

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=|x|^{\theta}u^{p}&{\rm in}\
\,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ u=h&{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1}\end{array}\right.$ (1.18)

admits the minimal nonnegative solution $u_{\rm min}$.

Any solution $u\in C^{s}(\mathbb{R}^{N})$ of (1.18) verifies that
$\displaystyle\inf_{x\in B_{1}}u(x)\leq h(1)$, thans to $u(1)=h(1)$. A
stronger version of (1.11) is

$\inf_{x\in B_{1}}u_{\rm min}(x)=h(1).$

Particularly,

$u_{\rm min}\equiv 0\quad{\rm if}\;\;h\equiv 0$

and

$u_{\rm min}>b\quad{\rm if}\quad h\equiv b\ \,{\rm for\ some\ }b>0.$

Under these assumptions, the isolated singularity of all nonnegative solution
of (1.1) could be classified.

For the existence of minimal solution $u_{\rm min}$, it could be approached by
the sequence of functions $\\{v_{m}\\}_{m\in\mathbb{N}}$, which are the
solutions of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}v_{m}=|x|^{\theta}v_{m-1}^{p}&{\rm
in}\ \,B_{1},\\\\[5.69054pt] \qquad\ \ v_{m}=h&{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1}\end{array}\right.$

with $v_{0}$ being the $s$-harmonic extension of $h$ in $B_{1}$, if (1.1) has
a nonnegative solution, which is an upper bound for $\\{v_{m}\\}$.

Our methods could be extended to classify the singularity of prototype Lane-
Emden equation

$\left\\{\begin{array}[]{lll}-\Delta u=|x|^{\theta}u^{p}&{\rm in}\
\,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\ \ u=h&{\rm in}\ \,\partial
B_{1},\end{array}\right.$ (1.19)

in the super critical case $p>\frac{N+2+2\theta}{N-2}$ and $\theta>-2$, where
$h\geq 0$. We have the following results:

###### Theorem 1.4

Assume that $N\geq 3$, $\theta>-2$, $p>\frac{N+2+2\theta}{N-2}$, $v_{0}$ is a
positive solution of (1.19) and $c_{p,\theta}$ is given in (1.3).

$(i)$ If (1.16) holds for $v_{0}$, then $v_{0}$ has either removable
singularity at the origin or one has

$\frac{c_{p,\theta}}{C_{2}}\leq\liminf_{|x|\to
0^{+}}v_{0}(x)|x|^{\frac{2+\theta}{p-1}}\leq c_{p,\theta}\leq\limsup_{|x|\to
0^{+}}v_{0}(x)|x|^{\frac{2+\theta}{p-1}}\leq C_{2}c_{p,\theta},$ (1.20)

where $C_{2}\geq 1$ is a constant from the Harnack inequality.

$(ii)$ If $\theta\in(-2,0]$, $h$ is a nonnegative constant on $\partial B_{1}$
and

$v_{0}(x)\geq h,\quad\,\forall\,x\in B_{1},$ (1.21)

then $v_{0}$ is radially symmetric, decreasing with respect to $|x|$ and it is
either removable at the origin or has the singularity (1.20).

It is worth noting that the limit

$\lim_{|x|\to 0^{+}}v_{0}(x)|x|^{\frac{2+\theta}{p-1}}=c_{p,\theta}$

is not always true in the Sobolev supercritical case. In fact, when
$\frac{N+1}{N-3}<p<p_{c}(N-1)$, $p_{c}(N-1)$ being the Joseph-Lundgren
exponent, [22] shows that

$-\Delta u=u^{p}\quad{\rm in}\ \,\mathbb{R}^{N}\setminus\\{0\\}$

has infinitely many non-radially symmetric solutions with the form that

$u(r,\omega)=c_{p}r^{-\frac{2}{p-1}}w_{0}(\omega),\quad(r,\omega)\in\mathbb{R}\times\partial
B_{1},$

where $w_{0}$ is non-constant solution in $\partial B_{1}$. Together with our
theorem 1.4, we get a corollary that $c_{p,0}\in w_{0}[\partial B_{1}]$.

The remainder of this paper is organized as follows. In Section 2, we provide
preliminary estimates of the fractional Hardy Poisson problem, an essential
integral upper bound and Harnack inequality. In Section 3, we obtain the upper
bounds of positive solutions of (1.1) Section 4 is devoted to isolated
singularities of (1.1) and proofs of Theorem 1.1 and Theorem 1.2.

##  2  Preliminary

Our analysis of isolated singularities for solutions with fractional laplacian
is based on the fractional Hardy problem. In the sequel, we do the $C^{2}$
extension for $h$ in $B_{1}$, till denoting it by $h$, such that

$h=0\quad{\rm in}\ B_{\frac{3}{4}}.$

We denote

$U_{0}(x)=-(-\Delta)^{s}h(x)\quad{\rm for}\ x\in B_{1},$ (2.1)

which is bounded locally in $B_{1}$.

###  2.1 Poisson problem for fractional Hardy problem

Recall that for $\tau\in(-N,2s)$, we define

$\mathcal{C}_{s}(\tau)=2^{2s}\frac{\Gamma(\frac{N+\tau}{2})\Gamma(\frac{2s-\tau}{2})}{\Gamma(-\frac{\tau}{2})\Gamma(\frac{N-2s+\tau}{2})}=-\frac{c_{N,s}}{2}\int_{\mathbb{R}^{N}}\frac{|e_{1}+z|^{\tau}+|e_{1}-z|^{\tau}-2}{|z|^{N+2s}}\,dz,$
(2.2)

where $e_{1}=(1,0,\cdots,0)\in\mathbb{R}^{N}$ and there holds

$(-\Delta)^{s}|\cdot|^{\tau}=\mathcal{C}_{s}(\tau)|\cdot|^{\tau-2s}\quad{\rm
in}\ \ \mathbb{R}^{N}\setminus\\{0\\}.$ (2.3)

Note that $\mathcal{C}_{s}(\cdot)$ has two zeros point $\\{0,2s-N\\}$, i.e.

$\mathcal{C}_{s}(0)=\mathcal{C}_{s}(2s-N)=0.$

By [19, Lemma 2.3], we know that the function $\mathcal{C}_{s}$ is strictly
concave and has a unique maximum at $\big{\\{}\frac{2s-N}{2}\big{\\}}$ with
the maximal value
$2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}.$
Moreover,

$\mathcal{C}_{s}(\tau)=\mathcal{C}_{s}(2s-N-\tau)\quad{\rm for}\
\,\tau\in(-N,2s)$ (2.4)

and

$\lim_{\tau\to-N}\mathcal{C}_{s}(\tau)=\lim_{\tau\to
2s}\mathcal{C}_{s}(\tau)=-\infty.$ (2.5)

Particularly, we have that

$\mathcal{C}_{s}(\tau)>0\quad{\rm for}\
\tau\in(2s-N,0),\qquad\mathcal{C}_{s}(\tau)<0\quad{\rm for}\
\tau\in(-N,2s-N)\cup(0,2s).$ (2.6)

Now let

${\cal L}^{s}_{\mu}u(x)=(-\Delta)^{s}u(x)+\frac{\mu}{|x|^{2s}}u(x),$

where

$\mu\geq\mu_{0}:=-2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}.$

It is shown in [19] that the linear equation

$\mathcal{L}_{\mu}^{s}u=0\quad{\rm in}\ \ \mathbb{R}^{N}\setminus\\{0\\}$

has two distinct radial solutions

$\Phi_{\mu}(x)=\left\\{\begin{array}[]{lll}|x|^{\tau_{-}(\mu)}&{\rm
if}\;\;\mu>\mu_{0}\\\\[4.2679pt]
\phantom{}|x|^{-\frac{N-2s}{2}}\ln\left(\frac{1}{|x|}\right)&{\rm
if}\;\;\mu=\mu_{0}\end{array}\right.\quad{\rm and}\;\;\
\Gamma_{\mu}(x)=|x|^{\tau_{+}(\mu)},$

where $\tau_{-}(\mu)\leq\tau_{+}(\mu)$ verify that

$\displaystyle\tau_{-}(\mu)+\tau_{+}(\mu)=2s-N\quad\ {\rm for\ all}\ \
\mu\geq\mu_{0},$
$\displaystyle\tau_{-}(\mu_{0})=\tau_{+}(\mu_{0})=\frac{2s-N}{2},\quad\
\tau_{-}(0)=2s-N,\;\;\ \tau_{+}(0)=0,$
$\displaystyle\displaystyle\lim_{\mu\to+\infty}\tau_{-}(\mu)=-N\quad{\rm
and}\quad\lim_{\mu\to+\infty}\tau_{+}(\mu)=2s.$

In the sequel of the paper and when there is no ambiguity, we use the
notations $\tau_{+}=\tau_{+}(\mu)$, $\tau_{-}=\tau_{-}(\mu)$. Moreover, we
have that for $\mu\geq\mu_{0}$

$\mathcal{C}_{s}(\tau)+\mu>0\quad{\rm for}\ \ \tau\in(\tau_{-},\tau_{+})$

and

$\mathcal{C}_{s}(\tau)+\mu<0\quad{\rm for}\ \
\tau\in(-N,\tau_{-})\cup(\tau_{+},2s).$

Our analysis of singularities of positive solutions to (1.1) is based on the
study of the Poisson problem involving the fractional Hardy operator:

$\left\\{\begin{array}[]{lll}{\cal L}^{s}_{\mu}u=f&{\rm in}\ \
B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\ u=h&{\rm in}\ \
\mathbb{R}^{N}\setminus B_{1}.\end{array}\right.$ (2.7)

The dual of the operator ${\cal L}^{s}_{\mu}$ is a weighted fractional
Laplacian $(-\Delta)^{s}_{\Gamma_{\mu}}$ given by

$(-\Delta)^{s}_{\Gamma_{\mu}}v(x):=C_{N,s}\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}^{N}\setminus
B_{\epsilon}}\frac{v(x)-v(z)}{|x-z|^{N+2s}}\,\Gamma_{\mu}(z)dz.$ (2.8)

This expression is well defined for $x\in\mathbb{R}^{N}\setminus\\{0\\}$ if
$v\in L^{1}(\mathbb{R}^{N},\frac{\Gamma_{\mu}(x)}{1+|x|^{N+2s}}dx)$ and if $v$
is twice continuously differentiable in a neighborhood of $x$. From [19,
Proposition 3.1] there holds

$|(-\Delta)^{s}_{\Gamma_{\mu}}\xi(x)|\leq
c_{0}\min\\{\Lambda_{\mu}(x),|x|^{-N-2s}\\}\quad{\rm for}\ \,\xi\in
C^{2}_{c}(\mathbb{R}^{N}),\ \,\,x\in\mathbb{R}^{N}\setminus\\{0\\},$ (2.9)

where $c_{0}=c_{0}(s,\mu,\xi)>0$ is a constant and

$\Lambda_{\mu}(x)=\left\\{\begin{array}[]{lll}\displaystyle 1&{\rm if}\
\,\tau_{+}(s,\mu)>2s-1,\\\\[2.84526pt] |x|^{1-2s+\tau_{+}(s,\mu)}&{\rm if}\
\,\tau_{+}(s,\mu)<2s-1,\\\\[2.84526pt] \phantom{}\displaystyle
1+(-\ln|x|)_{+}&{\rm if}\ \,\tau_{+}(s,\mu)=2s-1.\end{array}\right.$ (2.10)

Then from [19, Theorem 1.4] we have

###### Theorem 2.1

Let $\mu\geq\mu_{0}$ and $f\in C^{\theta}_{loc}(\bar{B}_{1}\setminus\\{0\\})$
for some $\theta\in(0,1)$.

  1. (i)

(Existence) If $f\in L^{1}(B_{1},\Gamma_{\mu}(x)dx)$, then for every
$k\in\mathbb{R}$ there exists a solution $u_{k}\in
L^{1}(B_{1},\Lambda_{\mu}dx)$ of problem (2.7) satisfying the distributional
identity

$\int_{B_{1}}u_{k}(-\Delta)^{s}_{\Gamma_{\mu}}\xi\,dx=\int_{B_{1}}f\xi\,\Gamma_{\mu}dx+c_{s,\mu}k\xi(0)\quad{\rm
for\ all}\ \,\xi\in\mathcal{C}^{2}_{0}(B_{1}).$ (2.11)

  2. (ii)

(Existence and Uniqueness) If $f\in L^{\infty}(B_{1},|x|^{\rho}dx)$ for some
$\rho<2s-\tau_{+}(s,\mu)$, then for every $k\in\mathbb{R}$ there exists a
unique solution $u_{k}\in L^{1}(B_{1},\Lambda_{\mu}dx)$ of problem (2.7) with
the asymptotics

$\lim_{|x|\to 0^{+}}\>\frac{u_{k}(x)}{\Phi_{\mu}(x)}=k.$ (2.12)

Moreover, $u_{k}$ satisfies the distributional identity (2.11).

  3. (iii)

(Nonexistence) If $f$ is nonnegative and satisfies

$\int_{B_{1}}f\,\Gamma_{\mu}dx=+\infty,$ (2.13)

then the problem

$\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\mu}^{s}u=f\quad{\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[4.2679pt] \phantom{L_{\mu}\,}\displaystyle
u\geq 0\quad{\rm in}\ \,\mathbb{R}^{N}\setminus B_{1}\end{array}\right.$
(2.14)

has no nonnegative distributional solution $u\in
L^{\infty}_{loc}(\mathbb{R}^{N}\setminus\\{0\\})\cap
L^{1}_{s}(\mathbb{R}^{N},dx)$.

A direct consequence from Theorem 2.1 with $\mu=0$ is the nonexistence of
solutions for

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=f&{\rm in}\
\,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ u=h&{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1}.\end{array}\right.$ (2.15)

###### Corollary 2.1

Assume that the nonnegative function $f\in
C^{\beta}_{loc}(\bar{B}_{1}\setminus\\{0\\})$ for some $\beta\in(0,1)$ and
nonnegative function $h\in C^{\alpha}_{loc}(B_{2}\setminus B_{1})\cap
L^{1}(\mathbb{R}^{N}\setminus B_{1},\frac{dx}{(1+|x|)^{N+2s}})$ with
$\alpha\in(0,1)$. Then problem (2.15) has no positive solution, if

$\lim_{r\to 0^{+}}\int_{B_{1}\setminus B_{r}}f(x)dx=+\infty.$

Particularly, the above assumption is filled if there holds

$\liminf_{|x|\to 0^{+}}f(x)|x|^{N}>0.$

Next we introduce the comparison principle of ${\cal L}^{s}_{\mu}$.

###### Lemma 2.1

Assume that $\mu\geq\mu_{0}$, $O$ is a bounded $C^{2}$ domain containing the
origin and the functions $u_{i}\in C(\bar{O}\setminus\\{0\\})$ with $i=1,2$
verify in the classical sense

$\left\\{\begin{array}[]{lll}\displaystyle{\cal L}^{s}_{\mu}u_{1}\leq{\cal
L}^{s}_{\mu}u_{2}&{\rm in}\quad O\setminus\\{0\\},\\\\[5.69054pt]
\phantom{{\cal L}^{s}_{\mu}}\displaystyle u_{1}\leq u_{2}&{\rm in}\ \
\mathbb{R}^{N}\setminus O\end{array}\right.$ (2.16)

and satisfying

$\limsup_{|x|\to 0^{+}}u_{1}(x)|x|^{-\tau_{-}}\leq\liminf_{|x|\to
0^{+}}u_{2}(x)|x|^{-\tau_{-}}.$

Then

$u_{1}\leq u_{2}\quad{\rm in}\;\;O\setminus\\{0\\}.$

Proof. Let $u=u_{1}-u_{2}$ and then

${\cal L}^{s}_{\mu}u\leq 0\quad{\rm in}\ \,O\setminus\\{0\\}\quad{\rm
and}\quad\limsup_{|x|\to 0^{+}}u(x)\Phi_{s}^{-1}(x)\leq 0,$

then for any $\epsilon>0$, there exists $r_{\epsilon}>0$ converging to $0$ as
$\epsilon\to 0$ such that

$u\leq\epsilon\Phi_{\mu}\quad{\rm
in}\;\;\overline{B_{r_{\epsilon}}(0)}\setminus\\{0\\},$

where $\Phi_{\mu}(x)=|x|^{\tau_{-}}$ is the fundamental solution of ${\cal
L}^{s}_{\mu}$.

We see that

$u=0<\epsilon\Phi_{\mu}\quad{\rm in}\ \ \mathbb{R}^{N}\setminus O,$

then we have $u\leq\epsilon\Phi_{\mu}$ in $O\setminus\\{0\\}$ for any
$\epsilon>0$, which implies that $u\leq 0$ in $O\setminus\\{0\\}$. $\Box$

###### Lemma 2.2

Assume that $\mu>\mu_{0}$, the nonnegative function $g\in
C^{\beta}_{loc}(B_{1}\setminus\\{0\\})$ for some $\beta\in(0,1)$ and there
exists $\tau\in\mathbb{R}$ such that

$g(x)\geq|x|^{\tau-2s}\quad{\rm in}\ \,B_{\frac{1}{2}}\setminus\\{0\\}.$

Let $u_{g}\in C(\bar{B}_{1}\setminus\\{0\\})$ verify

$\left\\{\begin{array}[]{lll}\mathcal{L}_{\mu}^{s}u_{g}\geq g\quad\ \ {\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\,u_{g}\geq 0\quad\ \ {\rm
on}\ \,\mathbb{R}^{N}\setminus B_{1}.\end{array}\right.$ (2.17)

We have the following $(i)$ If $\tau\in(\tau_{-},\tau_{+})$, then there exists
$c_{6}>0$ such that

$u_{g}(x)\geq c_{6}|x|^{\tau}\quad{\rm in}\
\,B_{\frac{1}{2}}\setminus\\{0\\};$

$(ii)$ If $\tau>\tau_{+}$, then there exists $c_{7}>0$ such that

$u_{g}(x)\geq c_{7}|x|^{\tau_{+}}\quad{\rm in}\
\,B_{\frac{1}{2}}\setminus\\{0\\}.$

Proof.  $(i)$ For $\tau\in(2s-N,0)$, we have that

$(-\Delta)^{s}|x|^{\tau}=\mathcal{C}_{s}(\tau)|x|^{\tau-2s}\quad{\rm in}\
\,\mathbb{R}^{N}\setminus\\{0\\},$

where $\mathcal{C}_{s}(\tau)>0$ by (2.6).

We let

$w_{1}(x)=|x|^{\tau}-1\quad{\rm in}\ \ \mathbb{R}^{N}\setminus\\{0\\},$

which, direct computation, verifies that

${\cal L}^{s}_{\mu}w_{1}\leq(\mathcal{C}_{s}(\tau)+\mu)|x|^{\tau-2}\quad{\rm
in}\ B_{1}(0)\setminus\\{0\\},\qquad w_{1}\leq 0\ \ {\rm in}\ \
\mathbb{R}^{N}\setminus B_{1},$ (2.18)

where $\mathcal{C}_{s}(\tau)+\mu>0$ for $\tau\in(\tau_{-},\tau_{+})$ and
$\mu>\mu_{0}$.

Note that by the lower bound of $g$ there exists $t_{0}>0$ such that

${\cal L}^{s}_{\mu}(t_{0}u_{g})\geq
t_{0}g(x)\geq(\mathcal{C}_{s}(\tau)+\mu)|x|^{\tau-2s}={\cal
L}^{s}_{\mu}w_{1}\quad{\rm in}\ \,B_{1}\setminus\\{0\\}$

and

$\liminf_{|x|\to 0}u_{g}\Phi_{\mu}^{-1}(x)\geq 0=\lim_{|x|\to
0}w_{1}(x)\Phi_{\mu}^{-1}(x),\qquad t_{0}u_{g}\geq 0\geq w_{1}\quad{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1}.$

Then by Lemma 2.1, we have that

$t_{0}u_{g}\geq w_{1}\quad{\rm in}\ \,B_{1}\setminus\\{0\\}.$

$(ii)$ For $\tilde{\tau}\in\big{(}\tau_{+},\min\\{\tau,2s\\}\big{)}$ we set

$w_{2}(x)=|x|^{\tau_{+}}-|x|^{\tilde{\tau}}\quad{\rm in}\ \
\mathbb{R}^{N}\setminus\\{0\\},$

then

${\cal
L}^{s}_{\mu}w_{2}\leq-(\mathcal{C}_{s}(\tilde{\tau})+\mu)|x|^{\tilde{\tau}-2s}\leq-(\mathcal{C}_{s}(\tilde{\tau})+\mu)|x|^{\tau-2s}\quad{\rm
in}\ \,B_{1}\setminus\\{0\\}$

and

$w_{2}\leq 0\ \ {\rm in}\ \,\mathbb{R}^{N}\setminus B_{1},$

where $-(\mathcal{C}_{s}(\tilde{\tau})+\mu)>0$ for our choice of
$\tilde{\tau}$ and $\tilde{\tau}-2s\geq\tau-2s$. Lemma 2.1 implies that
$u_{g}\geq w_{2}$ in $B_{1}\setminus\\{0\\}$. $\Box$

###### Lemma 2.3

Let $\mu>\mu_{0}$, $g\in C^{\beta}_{loc}(B_{1}\setminus\\{0\\})$ with
$\beta\in(0,1)$ be a nonnegative function such that there exists
$\tau>\tau_{-}$ such that

$g(x)\leq|x|^{\tau-2s}\quad{\rm for}\ \,x\in B_{1}\setminus\\{0\\},$

and $h$ be a nonnegative function in $C^{\alpha}(B_{2}\setminus B_{1})\cap
L^{1}_{s}(\mathbb{R}^{N}\setminus B_{1})$ with $\alpha\in(0,1)$.

Let $u_{g}\in C(\bar{B}_{1}\setminus\\{0\\})$ verify

$\left\\{\begin{array}[]{lll}{\cal L}^{s}_{\mu}u_{g}\leq g\quad\ \ {\rm in}\ \
B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\,u_{g}=h\quad\ \ {\rm in}\ \
\mathbb{R}^{N}\setminus B_{1},\\\\[5.69054pt] \displaystyle\lim_{|x|\to
0^{+}}u_{g}(x)|x|^{-\tau_{-}}=0.\end{array}\right.$ (2.19)

Then $(i)$ if $\tau\in(\tau_{-},\tau_{+})$ then there exists $c_{8}>0$ such
that

$u_{g}(x)\leq c_{8}|x|^{\tau}\quad{\rm for}\ \,x\in B_{1}\setminus\\{0\\};$

$(ii)$ if $\tau>\tau_{+}$ then there exists $c_{9}>0$ such that

$u_{g}(x)\leq c_{9}|x|^{\tau_{+}}\quad{\rm for}\ \,x\in
B_{1}\setminus\\{0\\}.$

Proof.  From (2.1), a direct computation shows that for $x\in B_{\frac{1}{4}}$

$\displaystyle 0<U_{0}(x)$ $\displaystyle=$ $\displaystyle
c_{N,s}\int_{\mathbb{R}^{N}\setminus
B_{\frac{1}{2}}}\frac{h(y)}{|x-y|^{N+2s}}dy$ (2.20) $\displaystyle\leq$
$\displaystyle c_{N,s}\int_{\mathbb{R}^{N}\setminus
B_{\frac{1}{2}}}\frac{h(y)}{(|y|-\frac{1}{4})^{N+2s}}dy\leq
c_{10}\|h\|_{L^{1}_{s}(\mathbb{R}^{N})}$

by using the fact that

$|x-y|\geq|y|-\frac{1}{4}\quad{\rm for}\ \,|x|\leq\frac{1}{4}\ \,{\rm and}\
\,|y|\geq\frac{1}{2}.$

$(i)$ For $\tau\in(\tau_{-},\tau_{+})$, we have that

${\cal
L}^{s}_{\mu}|x|^{\tau}=(\mathcal{C}_{s}(\tau)+\mu)|x|^{\tau-2s}\quad{\rm for}\
\,x\in\mathbb{R}^{N}\setminus\\{0\\},$

where $\mathcal{C}_{s}(\tau)+\mu>0$ and $\tau-2s<0$.

Note that there exists $t_{3}>0$ such that

${\cal L}^{s}_{\mu}\big{(}t_{3}(u_{g}-h)\big{)}\leq
t_{3}(g(x)+U_{0})\leq(\mathcal{C}_{s}(\tau)+\mu)|x|^{\tau-2s}={\cal
L}^{s}_{\mu}|x|^{\tau}\quad{\rm for}\ \,x\in B_{\frac{1}{4}}\setminus\\{0\\},$
$\lim_{|x|\to 0}\big{(}u_{g}(x)-h(x)\big{)}|x|^{-\tau_{-}}(x)=0$

and

$t_{3}(u_{g}-h)_{+}\leq|x|^{\tau}\quad{\rm in}\ \ B_{1}\setminus
B_{\frac{1}{4}}\quad{\rm and}\quad t_{3}(u_{g}-h)=0<|x|^{\tau}\quad{\rm in}\ \
\mathbb{R}^{N}\setminus B_{1}.$

Then by Lemma 2.1, we have that

$t_{3}\big{(}u_{g}(x)-h(x)\big{)}\leq|x|^{\tau}\quad{\rm for}\ \,x\in
B_{1}\setminus\\{0\\}.$

$(ii)$ Take $\tilde{\tau}\in(\tau_{+},2s)\cap(\tau_{+},\tau]$ and

$\tilde{w}_{3}(x)=|x|^{\tau_{+}}-|x|^{\tilde{\tau}}+1\quad{\rm in}\ \
B_{1}\setminus\\{0\\}\quad{\rm and}\quad\tilde{w}_{3}(x)=1\quad{\rm in}\ \
\mathbb{R}^{N}\setminus B_{1}.$

Direct computation shows that for $x\in B_{\frac{1}{2}}\setminus\\{0\\}$

$\displaystyle{\cal L}^{s}_{\mu}\tilde{w}_{3}(x)$ $\displaystyle=$
$\displaystyle{\cal
L}^{s}_{\mu}\big{(}|x|^{\tau_{+}}-|x|^{\tilde{\tau}}\big{)}+c_{N,s}\int_{\mathbb{R}^{N}\setminus
B_{1}}\frac{|y|^{\tau_{+}}-|y|^{\tilde{\tau}}}{|x-y|^{N+2s}}dy$
$\displaystyle\geq$
$\displaystyle-(\mathcal{C}_{s}(\tilde{\tau})+\mu)|x|^{\tilde{\tau}-2s}-c_{N,s}\int_{\mathbb{R}^{N}\setminus
B_{1}}\frac{|y|^{\tilde{\tau}}}{|x-y|^{N+2s}}dy$ $\displaystyle\geq$
$\displaystyle-(\mathcal{C}_{s}(\tilde{\tau})+\mu)|x|^{\tilde{\tau}-2s}-2^{N+2s}c_{N,s}\int_{\mathbb{R}^{N}\setminus
B_{1}}\frac{|y|^{\tilde{\tau}}}{|y|^{N+2s}}dz,$

where $-(\mathcal{C}_{s}(\tilde{\tau})+\mu)>0$.

Thus, there exists $r_{0}\in(0,\frac{1}{2}]$ such that for $x\in
B_{r_{0}}\setminus\\{0\\}$

$\displaystyle{\cal L}^{s}_{\mu}\tilde{w}_{3}(x)$ $\displaystyle\geq$
$\displaystyle-\frac{1}{2}(\mathcal{C}_{s}(\tilde{\tau})+\mu)|x|^{\tau-2s}.$

As a consequence, there exists $t_{4}>0$ such that

$\displaystyle{\cal L}^{s}_{\mu}\big{(}t_{4}(u_{g}-h)\big{)}$
$\displaystyle\leq$ $\displaystyle t_{4}(g(x)+U_{0})$ $\displaystyle\leq$
$\displaystyle-\frac{1}{2}(\mathcal{C}_{s}(\tau)+\mu)|x|^{\tilde{\tau}-2s}\leq{\cal
L}^{s}_{\mu}\tilde{w}_{3}(x)\quad{\rm in}\quad B_{r_{0}}\setminus\\{0\\}$

and

$t_{4}(u_{g}-h)\leq\tilde{w}_{3}\quad{\rm in}\ \,\mathbb{R}^{N}\setminus
B_{r_{0}}.$

Together with

$\lim_{|x|\to 0^{+}}\big{(}t_{4}(u_{g}-h)(x)\big{)}|x|^{-\tau_{-}}=0,$

Lemma 2.1 is applied to obtain that

$t_{4}(u_{g}-h)\leq\tilde{w}_{3}\quad{\rm in}\ \,B_{r_{0}}\setminus\\{0\\}.$

Therefore, we obtain that

$u_{g}(x)\leq c_{9}|x|^{\tau_{+}}\quad{\rm for}\ \,x\in
B_{1}\setminus\\{0\\}.$

We complete the proof. $\Box$

###  2.2 Uniformly integral upper bounds

The integral upper bound (1.14) is essential for our analysis of isolated
singularities of (1.1) in the Serrin’s supercritical case.

Proof of Proposition 1.1.  Recall $U_{0}$ in (2.1), which is uniformly bounded
in $B_{\frac{1}{2}}$. Let $w=u_{0}-h$ in $\mathbb{R}^{N}$, then we have that

$(-\Delta)^{s}w=|x|^{\theta}u_{0}^{p}+U_{0}\quad{\rm in}\ \,B_{1},\qquad
w=0\quad{\rm in}\ \mathbb{R}^{N}\setminus B_{1}.$

From Theorem 2.1, we have that $|x|^{\theta}u_{0}^{p}+U_{0}\in L^{1}(B_{1})$,
so is $u_{0}$, thanks to the boundedness of $U_{0}$. Moreover, we have that

$\int_{B_{1}}w(-\Delta)^{s}\xi\,dx=\int_{B_{1}}(|x|^{\theta}u_{0}^{p}+U_{0})\xi\,dx\qquad{\rm
for\ all}\ \,\xi\in C^{2}_{0}(B_{1}).$

Let $\xi_{1}$ be the positive Dirichlet eigenfunction of $(-\Delta)^{s}$
related to the first eigenvalue, subject to the zero Dirichlet boundary
condition in $\mathbb{R}^{N}\setminus B_{1}$, i.e.

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=\lambda_{1}u&{\rm in}\
\,B_{1},\\\\[5.69054pt] \qquad\,u=0&{\rm in}\ \,\mathbb{R}^{N}\setminus
B_{1}.\end{array}\right.$ (2.21)

Then $\lambda_{1}>0$, $\xi_{1}$ is positive and $\xi_{1}\in
C^{2s}_{loc}(B_{1})\cap C^{s}(\mathbb{R}^{N})$. Moreover, by the standard
argument we can obtain that $\xi_{1}\in C^{2}_{0}(B_{1})$ and for some
$c_{10}>1$ there holds

$\frac{1}{c_{10}}\rho^{s}\leq\xi_{1}\leq c_{10}\rho^{s}\quad{\rm in}\
\,B_{1}.$

Using $\xi_{1}$ as a test function, we have that

$\displaystyle\int_{B_{1}}|x|^{\theta}u_{0}^{p}\xi_{1}\,dx+\int_{B_{1}}U_{0}\xi_{1}\,dx$
$\displaystyle=$ $\displaystyle\lambda_{1}\int_{B_{1}}w\xi_{1}\,dx$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{B_{1}}|x|^{\theta}w^{p}\xi_{1}\,dx\Big{)}^{\frac{1}{p}}\Big{(}\int_{B_{1}}\xi_{1}|x|^{-\frac{\theta}{p-1}}\,dx\Big{)}^{1-\frac{1}{p}},$

which implies

$\displaystyle\int_{B_{1}}|x|^{\theta}w^{p}\rho^{s}\,dx\leq c_{11},$

where $c_{11}>0$ depends on $h$ and we used the fact that
$-\frac{\theta}{p-1}>-N,$ thanks to $p\geq\frac{N+\theta}{N-2s}$. Let

$\xi_{r}(x)=\xi_{1}(r^{-1}x)\quad{\rm for}\ \,x\in\mathbb{R}^{N},$

then $\xi_{r}$ is the solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=\lambda_{1}r^{-2s}u&{\rm in}\
\,B_{r},\\\\[5.69054pt] \qquad\ \ u=0&{\rm in}\ \,\mathbb{R}^{N}\setminus
B_{r}.\end{array}\right.$ (2.22)

Let

$w=u_{0}-\tilde{u}_{0},$

where $\tilde{u}_{0}$ is the $s$-harmonic extension of $u_{0}$ in $B_{r}$,
i.e. the solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=0&{\rm in}\
\,B_{r},\\\\[5.69054pt] \qquad\ \ u=u_{0}&{\rm in}\ \,\mathbb{R}^{N}\setminus
B_{r}.\end{array}\right.$

Then we have that

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}w=|x|^{\theta}(w+\tilde{u}_{0})^{p}&{\rm
in}\ \,B_{r},\\\\[5.69054pt] \qquad\ \ w=0&{\rm in}\ \,\mathbb{R}^{N}\setminus
B_{r}\end{array}\right.$

and

$\int_{B_{r}}w(-\Delta)^{s}\zeta
dx=\int_{\mathbb{R}^{N}}|x|^{\theta}(w+\tilde{u}_{0})^{p}\zeta
dx,\quad\zeta\in\mathbb{X}_{r},$

which implies that

$\int_{B_{r}}u_{0}(-\Delta)^{s}\zeta
dx=\int_{\mathbb{R}^{N}}|x|^{\theta}u_{0}^{p}\zeta
dx+\int_{B_{r}}\tilde{u}_{0}(-\Delta)^{s}\zeta
dx,\quad\zeta\in\mathbb{X}_{r}.$

Now we take $\zeta=\xi_{r}$, we have that

$\lambda_{1}r^{-2s}\int_{B_{r}}u_{0}\xi_{r}dx=\int_{\mathbb{R}^{N}}|x|^{\theta}u_{0}^{p}\xi_{r}dx+\int_{B_{r}}\tilde{u}_{0}(-\Delta)^{s}\xi_{r}dx.$

Again since $-\frac{\theta}{p-1}>-N$, we have

$\displaystyle\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx$ $\displaystyle<$
$\displaystyle\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx+\int_{B_{r}}\tilde{u}_{0}(-\Delta)^{s}\xi_{r}dx$
$\displaystyle=$ $\displaystyle\lambda_{1}r^{-2s}\int_{B_{r}}u_{0}\xi_{r}dx$
$\displaystyle\leq$
$\displaystyle\lambda_{1}r^{-2s}\Big{(}\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx\Big{)}^{\frac{1}{p}}\Big{(}\int_{B_{r}}\xi_{r}|x|^{-\frac{\theta}{p-1}}dx\Big{)}^{1-\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}\|_{L^{\infty}(B_{1})}^{1-\frac{1}{p}}\lambda_{1}r^{(N-\frac{\theta}{p-1})(1-\frac{1}{p})-2s}\Big{(}\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx\Big{)}^{\frac{1}{p}},$

which implies that

$\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx<\|\xi_{1}\|_{L^{\infty}(B_{1})}\lambda_{1}^{\frac{p}{p-1}}r^{N-\frac{\theta}{p-1}-\frac{2sp}{p-1}}.$
(2.23)

Notice that

$\xi_{r}(x)=\xi_{1}(r^{-1}x)\geq\min_{z\in\bar{B}_{\frac{1}{2}}}\xi_{1}(z)\quad{\rm
in}\ B_{\frac{r}{2}},$

then

$\int_{B_{\frac{r}{2}}}|x|^{\theta}u_{0}^{p}dx<c_{12}\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx<c_{12}\|\xi_{1}\|_{L^{\infty}(B_{1})}\lambda_{1}^{\frac{p}{p-1}}r^{N-\frac{\theta}{p-1}-\frac{2sp}{p-1}}.$
(2.24)

Replace $r$ by $\frac{r}{2}$, we obtain (1.14). $\Box$

With the help of Proposition 1.1, we have the following sharp upper bound
under some suitable restrictions of $u_{0}$.

###### Corollary 2.2

Assume that $\theta>-2s$, $p\geq\frac{N+\theta}{N-2s}$ and $u_{0}$ is a
nonnegative classical solution of (1.1) verifying the Harnack inequality:
there exist $C_{0}>1$ and $r_{1}\in(0,1)$ such that for $x,y\in
B_{r_{1}}\setminus\\{0\\}$

$u_{0}(x)\leq C_{0}u_{0}(y)\quad{\rm for}\ \ 1\leq\frac{|x|}{|y|}\leq 2,$
(2.25)

then there exists a uniform $c_{13}>0$ independent of $r_{1}$ and $u$ such
that

$u_{0}(x)\leq c_{13}|x|^{-\frac{2s+\theta}{p-1}}\quad{\rm for}\ \,0<|x|<1.$
(2.26)

Proof.  Take $|x|=r$, then (2.25) implies that for any $y\in B_{r}\setminus
B_{\frac{r}{2}}$, there holds that

$u_{0}(y)\geq\frac{1}{C_{0}}u_{0}(x).$

From Proposition 1.1, we see that for any given $|x|=r$,

$\displaystyle C_{0}^{-p}\Big{(}1-(\frac{1}{2})^{N}\Big{)}|\partial
B_{1}|\,u_{0}^{p}(x)r^{N-\theta}\leq\int_{B_{r}\setminus
B_{\frac{r}{2}}}|y|^{\theta}u_{0}^{p}(y)dy<c_{14}r^{N-\frac{\theta}{p-1}-\frac{2sp}{p-1}},$

which implies that

$u_{0}(x)\leq c_{13}|x|^{-\frac{2s+\theta}{p-1}}.$

We complete the proof. $\Box$

###  2.3 Harnack inequality

Our aim in this subsection is to obtain the Harnack inequality for singular
solution of (1.1).

###### Proposition 2.1

Assume that $\theta>-2s$, $p\geq\frac{N+\theta}{N-2s}$ and $u_{0}$ is a
nonnegative solution of (1.1) such that

$\limsup_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}<+\infty.$

Then there exists $C_{0}>0$ such that for all $r\in(0,\frac{1}{2})$

$\sup_{x\in B_{2r}\setminus B_{r}}u_{0}(x)\leq C_{0}\Big{(}\inf_{x\in
B_{2r}\setminus B_{r}}u_{0}(x)+\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}\Big{)}.$

We assume more that $u_{0}$ is singular at the origin, then for all
$r\in(0,\frac{1}{2})$

$\sup_{x\in B_{2r}\setminus B_{r}}u_{0}(x)\leq C_{0}\inf_{x\in B_{2r}\setminus
B_{r}}u_{0}(x).$ (2.27)

Proof.  For fixed $r_{0}\in(\frac{1}{4},\frac{1}{2})$ and
$x_{0}\in\mathbb{R}^{N}$ verifying $|x_{0}|=r_{0}$, from there exists $C>0$
independent of $u_{0}$ such that for any $t\in(0,\frac{r_{0}}{4}]$

$\sup_{x\in B_{t}(x_{0})}u_{0}(x)<C.$

Let

$w(x)=u_{0}(x)(1-\eta_{0}(4x))-h(x)\ \ \ {\rm in}\ \ \mathbb{R}^{N},$

where $\eta_{0}$ be a smooth function such that $\eta_{0}=1$ in $B_{1}(0)$ and
$\eta_{0}=0$ in $\mathbb{R}^{N}\setminus B_{2}$, we recall that $h=0$ in
$B_{\frac{3}{4}}$. Then

$(-\Delta)^{s}w=|x|^{\theta}u_{0}^{p^{*}-1}w+(-\Delta)^{s}h+(-\Delta)^{s}(u_{0}\eta_{0}(4x))\quad{\rm
for}\ \ x\in B_{t}(x_{0}),$

where $(-\Delta)^{s}h>0$ in $B_{1}$. Note that $0<u_{0}^{p^{*}-1}\leq c_{15}$
for some $c_{15}>0$ independent of $u_{0}$ and

$\displaystyle 0$ $\displaystyle<$
$\displaystyle(-\Delta)^{s}h(x)+(-\Delta)^{s}(u_{0}(x)\eta_{0}(4x))$
$\displaystyle\leq$ $\displaystyle
c_{15}\Big{(}\|h\|_{L^{1}_{s}(\mathbb{R}^{N})}+\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}\Big{)}$
$\displaystyle\leq$ $\displaystyle
2c_{15}\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}.$

Then [42, Theorem 1.1] (also see [13, Theorem 11.1]) implies that

$\sup_{x\in B_{t}(x_{0})}u_{0}(x)\leq C_{3}\Big{(}\inf_{x\in
B_{t}(x_{0})}u_{0}(x)+\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}\Big{)},$

which infers

$\sup_{x\in B_{2r_{0}}\setminus B_{r_{0}}}u_{0}(x)\leq C_{3}\Big{(}\inf_{x\in
B_{2r_{0}}\setminus
B_{r_{0}}}u_{0}(x)+\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}\Big{)}$ (2.28)

by finite covering argument, the scaling property and the upper bound of
$u_{0}$.

Now we do the scaling:

$u_{t}(x)=t^{\frac{2s+\theta}{p-1}}u_{0}(tx)$

for $r\in(0,1]$. Then $u_{t}$ also verifies (1.1) and from the upper bound, we
have

$r^{2s}|x|^{\theta}u_{r}(x)^{p-1}\leq C\quad{\rm for}\ \ r_{0}t<|x|<2r_{0}t,$

where $C$ is dependent of $t$.

It follows by (2.28) that

$\sup_{x\in B_{2r}\setminus B_{r}}u_{0}(x)\leq C_{3}\Big{(}\inf_{x\in
B_{2r}\setminus B_{r}}u_{0}(x)+\|u_{0}\|_{L^{1}_{s}(\mathbb{R}^{N})}\Big{)},$

where $r=r_{0}t$. Thanks to

$\displaystyle\lim_{|x|\to 0^{+}}u_{0}(x)=+\infty,$

we obtain (2.27). We complete the proof. $\Box$

##  3 Upper bounds

###  3.1  The case: $\theta=0$

###### Proposition 3.1

Assume that $\theta=0$ and

$p\in\big{(}\frac{N}{N-2s},\frac{N+2s}{N-2s}\big{)}.$

Let $u_{0}$ be a nonnegative solution of (1.1), then there exists $c_{16}>0$
such that

$u_{0}(x)\leq c_{16}|x|^{-\frac{2s}{p-1}},\quad\forall\,x\in
B_{1}\setminus\\{0\\}.$ (3.1)

Proof.  The similar upper bound could see [46], where the fractional laplacian
is defined by the extension to a local setting. Here we provide the direct
blow-up analysis in the nonlocal setting.

Suppose by contradiction that there exists a sequence of points
$\\{x_{k}\\}\subset B_{\frac{1}{2}}\setminus\\{0\\}$ such that $|x_{k}|\to
0^{+}$ as $k\to+\infty$ and

$|x_{k}|^{\frac{2s}{p-1}}u_{0}(x_{k})\to+\infty\quad{\rm as}\ \,k\to+\infty.$

We can choose $x_{k}$ again having the property that

$|x_{k}|^{\frac{2s}{p-1}}u_{0}(x_{k})=\max_{x\in B_{1}\setminus
B_{|x_{k}|}}|x|^{\frac{2s}{p-1}}u_{0}(x)\to+\infty\quad{\rm as}\
\,k\to+\infty$ (3.2)

by the fact that the mapping $r\mapsto\max_{x\in B_{1}\setminus
B_{r}}|x|^{\frac{2s}{p-1}}u_{0}(x)$ is decreasing.

We denote

$\phi_{k}(x):=\big{(}\frac{|x_{k}|}{2}-|x-x_{k}|\big{)}^{\frac{2s}{p-1}}u_{0}(x)\quad{\rm
for}\ \,|x-x_{k}|\leq\frac{|x_{k}|}{2}.$

Let $\bar{x}_{k}$ be the maximum point of $\phi_{k}$ in
$B_{\frac{|x_{k}|}{2}}(x_{k})$, that is,

$\phi_{k}(\bar{x}_{k})=\max_{|x-x_{k}|\leq\frac{|x_{k}|}{2}}\phi_{k}(x).$

Set

$\nu_{k}=\frac{1}{2}\big{(}\frac{|x_{k}|}{2}-|\bar{x}_{k}-x_{k}|\big{)},$

then $0<2\nu_{k}<\frac{|x_{k}|}{2}$ and

$\frac{|x_{k}|}{2}-|x-x_{k}|\geq\nu_{k}\ \ {\rm for}\
\,|x-\bar{x}_{k}|\leq\nu_{k}.$

By the definition of $\phi_{k}$, for any $|x-\bar{x}_{k}|\leq\nu_{k}$,

$(2\nu_{k})^{\frac{2s}{p-1}}u_{0}(\bar{x}_{k})=\phi_{k}(\bar{x}_{k})\geq\phi_{k}(x_{k})\geq\nu_{k}^{\frac{2s}{p-1}}u_{0}(x),$

which implies that

$2^{\frac{2s}{p-1}}u_{0}(\bar{x}_{k})\geq u_{0}(x)\quad{\rm for\ any}\
|x-\bar{x}_{k}|\leq\nu_{k}.$

Moreover, we see that

$\displaystyle|\bar{x}_{k}|^{\frac{2s}{p-1}}u_{0}(\bar{x}_{k})\geq(2\nu_{k})^{\frac{2s}{p-1}}u_{0}(\bar{x}_{k})$
$\displaystyle=$ $\displaystyle\phi_{k}(\bar{x}_{k})\geq\phi_{k}(x_{k})$
$\displaystyle\geq$
$\displaystyle(\frac{|x_{k}|}{2})^{\frac{2s}{p-1}}u_{0}(x_{k})\to+\infty\quad{\rm
as}\ \,k\to+\infty$

by the fact that $|\bar{x}_{k}|\geq\frac{|x_{k}|}{2}\geq 2\nu_{k}$.

Denote

$W_{k}(y)=\frac{1}{m_{k}}u_{0}\big{(}m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\big{)},\quad\forall\,y\in\Omega_{k}\setminus\\{X_{k}\\},$

where

$m_{k}=u_{0}(\bar{x}_{k}),$
$\Omega_{k}:=\Big{\\{}y\in\mathbb{R}^{N}:\,m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\in
B_{1}\Big{\\}}$

and

$X_{k}=m_{k}^{\frac{p-1}{2s}}\bar{x}_{k}.$

Note that

$|X_{k}|=\big{(}u_{0}(\bar{x}_{k})|\bar{x}_{k}|^{\frac{2s}{p-1}}\big{)}^{\frac{p-1}{2s}}\to+\infty\quad{\rm
as}\ \,k\to+\infty.$

Thus, we have that for $y\in\Omega_{k}\setminus\\{X_{k}\\}$

$\displaystyle(-\Delta)^{s}W_{k}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{m_{k}^{p}}(-\Delta)^{s}u_{0}\big{(}m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{m_{k}^{p}}u_{0}^{p}\big{(}m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\big{)}$
$\displaystyle=$ $\displaystyle W_{k}^{p}(y),$

i.e.

$(-\Delta)^{s}W_{k}(y)=W_{k}^{p}(y)\quad{\rm for}\ \
x\in\Omega_{k}\setminus\\{X_{k}\\}.$ (3.3)

We claim that there is $c_{17}>0$ independent of $k$ such that

$\|W_{k}\|_{L^{1}_{s}(\mathbb{R}^{N})}\leq c_{17}$

and for any $\epsilon>0$, there exists $k_{1}>0$ and $R>0$ such that

$\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}W_{k}(y)(1+|y|)^{-N-2s}dy\leq\epsilon.$ (3.4)

In fact, since $|\bar{x}_{k}|\to 0$, we see that

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{N}\setminus\Omega_{k}}W_{k}(y)(1+|y|)^{-N-2s}dy$
$\displaystyle=$
$\displaystyle\frac{1}{m_{k}}\int_{\mathbb{R}^{N}\setminus\Omega_{k}}u_{0}\big{(}m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\big{)}(1+|y|)^{-N-2s}dy$
$\displaystyle=$ $\displaystyle m_{k}^{-p}\int_{\mathbb{R}^{N}\setminus
B_{1}}h(z)\big{(}m_{k}^{-\frac{p-1}{2s}}+|z+\bar{x}_{k}|\big{)}^{-N-2s}dz$
$\displaystyle\leq$ $\displaystyle
m_{k}^{-p}\int_{\mathbb{R}^{N}}h(z)(1+|z+\bar{x}_{k}|)^{-N-2s}dz$
$\displaystyle\leq$ $\displaystyle
u_{0}(\bar{x}_{k})^{-p}\int_{\mathbb{R}^{N}}h(z)(1+|z|)^{-N-2s}dz$
$\displaystyle\to$ $\displaystyle 0\quad{\rm as}\ \,k\to+\infty.$

Taking

$r_{k}=|\bar{x}_{k}|\,u_{0}(\bar{x}_{k})^{\frac{p-1}{2s}}\to+\infty\quad{\rm
as}\ \,k\to+\infty,$

we obtain that

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\int_{B_{r_{k}}(X_{k})}W_{k}(y)(1+|y|)^{-N-2s}dy$
$\displaystyle\leq$
$\displaystyle\frac{r_{k}^{-N-2s}}{m_{k}}\int_{B_{r_{k}}(X_{k})}u_{0}\big{(}m_{k}^{-\frac{p-1}{2s}}y-\bar{x}_{k}\big{)}dy$
$\displaystyle=$
$\displaystyle\frac{r_{k}^{-N-2s}m_{k}^{\frac{p-1}{2s}N}}{m_{k}}\int_{B_{|\bar{x}_{k}|}}u_{0}(z)dz$
$\displaystyle\leq$ $\displaystyle
r_{k}^{-N-2s}m_{k}^{\frac{p-1}{2s}N-1}\Big{(}\int_{B_{|\bar{x}_{k}|}}u_{0}^{p}(z)dz\Big{)}^{\frac{1}{p}}\Big{(}\int_{B_{|\bar{x}_{k}|}}dz\Big{)}^{1-\frac{1}{p}}$
$\displaystyle=$ $\displaystyle
c_{18}r_{k}^{-N-2s}m_{k}^{\frac{p-1}{2s}N-1}|\bar{x}_{k}|^{N\frac{p-1}{p}}\Big{(}\int_{B_{|\bar{x}_{k}|}}u_{0}^{p}(z)dz\Big{)}^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle
c_{18}r_{k}^{-N-2s}m_{k}^{\frac{p-1}{2s}N-1}|\bar{x}_{k}|^{\frac{pN}{p-1}+(N-\frac{2sp}{p-1})\frac{1}{p}}$
$\displaystyle=$ $\displaystyle c_{18}r_{k}^{-\frac{2s}{p-1}-2s}$
$\displaystyle\to$ $\displaystyle 0\quad{\rm as}\ \,k\to+\infty,$

where we used (1.14) and the fact that $\frac{\theta}{p-1}<N$.

Moreover, $W_{k}(y)\leq 1$ in $\Omega_{k}\setminus B_{r_{k}}$, then

$\displaystyle\int_{\Omega_{k}\setminus
B_{r_{k}}(X_{k})}W_{k}(y)(1+|y|)^{-N-2s}dy$ $\displaystyle\leq$
$\displaystyle\int_{\Omega_{k}\setminus B_{r_{k}}(X_{k})}(1+|y|)^{-N-2s}dy$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{R}^{N}}(1+|y|)^{-N-2s}dy$

and

$\displaystyle\int_{\big{(}\Omega_{k}\setminus
B_{r_{k}}(X_{k})\big{)}\setminus B_{R}(0)}W_{k}(y)(1+|y|)^{-N-2s}dy$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}(1+|y|)^{-N-2s}dy$ $\displaystyle\leq$ $\displaystyle c_{19}R^{-2s},$

where $c_{19}>0$ is independent of $k$. Now we conclude (3.4) and the claim is
proved.

Note that $0<W_{k}\leq 2^{\frac{2s}{p-1}}$ in $B_{\tilde{r}_{k}}$, where

$\tilde{r}_{k}=\nu_{k}m_{k}^{\frac{p-1}{2s}}=(\nu_{k}^{\frac{2+\theta}{p-1}}m_{k})^{\frac{p-1}{2s}}\to+\infty\quad{\rm
as}\ \,k\to+\infty,$

then for any $R>0$, there exists $k_{R}$ for any $k\geq k_{R}$

$\displaystyle\|W_{k}\|_{C^{2s+\alpha}(B_{{}_{R}})}$ $\displaystyle\leq$
$\displaystyle
c_{20}\Big{(}\|W_{k}\|_{L^{1}_{s}(\mathbb{R}^{N})}+\|W_{k}\|_{L^{\infty}(B_{{}_{2R}})}+\|W_{k}^{p}\|_{L^{\infty}(B_{{}_{2R}})}\Big{)}$
$\displaystyle\leq$ $\displaystyle
c_{20}\Big{(}\|W_{k}\|_{L^{1}_{s}(\mathbb{R}^{N})}+2^{1+\frac{2}{p-1}p}\Big{)},$

where $\alpha\in(0,s)$ and $c_{20}>0$.

By the arbitrary of $R$, up to subsequence, there exists a nonnegative
function $W_{\infty}\in L^{\infty}(\mathbb{R}^{N})$ such that as $k\to+\infty$

$W_{k}\to W_{\infty}\quad\ {\rm in}\ \
C^{2s+\alpha^{\prime}}_{loc}(\mathbb{R}^{N})\ \ \ {\rm and\ \,in}\ \
L^{1}_{s}(\mathbb{R}^{N})$ (3.5)

for some $\alpha^{\prime}\in(0,\alpha)$.

For any $x\in\mathbb{R}^{N}$, take $R>4|x|$ and
$R\leq\min\big{\\{}r_{k},\tilde{r}_{k}\big{\\}}$ for $k$ large enough, note
that

$\displaystyle\frac{1}{C_{N,s}}(-\Delta)^{s}W_{k}(x)$ $\displaystyle=$
$\displaystyle{\rm
p.v.}\int_{B_{R}}\frac{W_{k}(x)-W_{k}(y)}{|x-y|^{N+2s}}dy+W_{k}(x)\int_{\mathbb{R}^{N}\setminus
B_{R}}\frac{1}{|x-y|^{N+2s}}dy$ $\displaystyle\ \
-\int_{\mathbb{R}^{N}\setminus B_{R}}\frac{W_{k}(y)}{|x-y|^{N+2s}}dy$
$\displaystyle=:$ $\displaystyle E_{1,k}(x)+E_{2,k}(x)-E_{3,k}(x),\ \,{\rm
respectively.}$

For any $\epsilon>0$, by convergence (3.5), we have that

$\Big{|}E_{1,k}(x)-{\rm
p.v.}\int_{B_{R}}\frac{W_{\infty}(x)-W_{\infty}(y)}{|x-y|^{N+2s}}dy\Big{|}<\frac{\epsilon}{3}$

and there exists $R_{1}>0$ such that $R>R_{1}$

$\Big{|}E_{2,k}(x)-W_{\infty}(x)\int_{\mathbb{R}^{N}\setminus
B_{R}}\frac{1}{|x-y|^{N+2s}}dy\Big{|}<\frac{\epsilon}{3}.$

Furthermore, by (3.4) we have that

$\displaystyle 0<E_{3,k}(x)$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{N}\setminus
B_{R}}W_{k}(y)(1+|y|)^{-N-2s}dy<\frac{\epsilon}{3}$

for $k>0$ and $R$ large enough.

Now we conclude that

$\lim_{k\to+\infty}(-\Delta)^{s}W_{k}(x)=(-\Delta)^{s}W_{\infty}(x)$

and $W_{\infty}$ is a bounded classical solution of

$(-\Delta)^{s}W_{\infty}=W_{\infty}^{p}\quad{\rm in}\ \,\mathbb{R}^{N}.$ (3.6)

Since $W_{\infty}(0)=1$, then $W_{\infty}>0$ in $\mathbb{R}^{N}$, thanks to
the nonnegative property of $W_{\infty}$. By [20, Theorem 3 ] or [21, Theorem
4.5], problem (3.6) has no bounded positive solution for
$p\in(1,\frac{N+2s}{N-2s})$. We complete the proof. $\Box$

###  3.2  The case: $\theta\in(-2s,0]$

When $\theta\in(-2s,0]$, we would derive the upper bound by the integral upper
bound and increasing monotonicity. We recall (1.9) and (1.10) , i.e.

$\qquad\theta=0\quad\&\quad p\geq\frac{N+2s}{N-2s}$

and

$\theta\in(-2s,0)\quad\&\quad p>\frac{N+\theta}{N-2s}.$

The upper bound states as following.

###### Proposition 3.2

Assume that (1.9) or (1.10) hold for $\theta$ and $p$, function $h$ is
radially symmetric in $\mathbb{R}^{N}\setminus B_{1}$ and decreasing with
respect to $|x|$ and $h(1)\geq 0$.

Let $u_{0}$ be a nonnegative solution of (1.1) satisfying (1.11), then there
exists $c_{22}>0$ such that

$0<u_{0}(x)\leq c_{22}|x|^{-\frac{2s+\theta}{p-1}},\quad\forall\,x\in
B_{1}\setminus\\{0\\}.$ (3.7)

In order to prove the upper bound, we first show the radial symmetry of
solution of (1.1). To this end, we use the moving plane method to show the
radial symmetry and monotonicity of positive solutions to equation (1.1).

We let

$u=u_{0}-h(1)\quad{\rm in}\ \ \mathbb{R}^{N},$

then $u\geq 0$ in $B_{1}$ by (1.11) and it verifies that

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=|x|^{\theta}\big{(}u+h(1)\big{)}^{p}&{\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ \,u=h-h(1)&{\rm in}\
\,\mathbb{R}^{N}\setminus B_{1}.\end{array}\right.$ (3.8)

###### Proposition 3.3

Let (1.9), (1.10) hold for $\theta$, $p$ respectively, $h$ be radially
symmetric in $\mathbb{R}^{N}\setminus B_{1}$ and decreasing with respect to
$|x|$ and $h(1)\geq 0$.

Let $u_{0}$ be a nonnegative solution of (1.1) satisfying (1.11), then
$u=u_{0}-h(1)$ is a positive solution of (3.8), $u$ is radially symmetric and
decreasing with respect to $|x|$.

Our method of moving planes is motivated by [27] to deal with the solution $u$
has possibly singular at the origin. For the methods of the moving planes for
integral equations, we refer to [21, 20]. For singular solutions, we will use
the direct moving plane method developing from [27] and we need the following
variant Maximum Principle for small domain.

###### Lemma 3.1

[27, Corollary 2.1] Let $O$ be an open and bounded subset of $\mathbb{R}^{N}$.
Suppose that $\varphi\in L^{\infty}(O)$ and $w\in
L^{\infty}(\mathbb{R}^{N})\cap C(\bar{O})$ is a classical solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}w(x)\geq\varphi(x)w(x),&x\in
O,\\\\[5.69054pt] \qquad\ \ w(x)\geq 0,&x\in O^{c}.\end{array}\right.$ (3.9)

Then there is $\delta>0$ such that whenever $|O^{-}|\leq\delta$, function $w$
has to be non-negative in $O$, where $O^{-}=\\{x\in O\ |\ w(x)<0\\}$.

For simplicity, we denote

$\Sigma_{\lambda}=\\{x=(x_{1},x^{\prime})\in\mathcal{O}_{1}\ |\
x_{1}>\lambda\\},$ (3.10) $u_{\lambda}(x)=u(x_{\lambda})\quad\mbox{and}\quad
w_{\lambda}(x)=u_{\lambda}(x)-u(x),$ (3.11)

where $\lambda\in(0,1)$ and $x_{\lambda}=(2\lambda-x_{1},x^{\prime})$ for
$x=(x_{1},x^{\prime})\in\mathbb{R}^{N}$ and
$\mathcal{O}_{1}=B_{1}\setminus\\{0\\}$. For any subset $A$ of
$\mathbb{R}^{N}$, we write $A_{\lambda}=\\{x_{\lambda}:\,x\in A\\}$.

The essential estimate in the procedure is to show that for any
$\lambda\in(0,1)$

$w_{\lambda}\geq 0\quad{\rm in}\ \,\Sigma_{\lambda}.$

On the contrary, we suppose that
$\Sigma_{\lambda}^{-}=\\{x\in\Sigma_{\lambda}\ |\
w_{\lambda}(x)<0\\}\not=\emptyset$ for $\lambda\in(0,1)$. Let us define

$w_{\lambda}^{+}(x)=\left\\{\begin{array}[]{lll}w_{\lambda}(x),&x\in\Sigma_{\lambda}^{-},\\\\[2.84526pt]
0,&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-}\end{array}\right.$ (3.12)

and

$w_{\lambda}^{-}(x)=\left\\{\begin{array}[]{lll}0,&x\in\Sigma_{\lambda}^{-},\\\\[2.84526pt]
w_{\lambda}(x),&x\in\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-}.\end{array}\right.$
(3.13)

Hence, $w_{\lambda}^{+}(x)=w_{\lambda}(x)-w_{\lambda}^{-}(x)$ for all
$x\in\mathbb{R}^{N}.$ It is obvious that
$(2\lambda,0,\cdots,0)\not\in\Sigma_{\lambda}^{-}$, since
$\displaystyle\lim_{|x|\to 0^{+}}u(x)=+\infty$.

###### Lemma 3.2

Assume that $\Sigma_{\lambda}^{-}\not=\emptyset$ for $\lambda\in(0,1)$, then

$(-\Delta)^{s}w_{\lambda}^{-}(x)\leq 0,\ \ \ \
\forall\,x\in\Sigma_{\lambda}^{-}.$ (3.14)

Proof. By direct computation, for $x\in\Sigma_{\lambda}^{-}$, we have

$\displaystyle\frac{1}{C_{N,s}}(-\Delta)^{s}w_{\lambda}^{-}(x)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\frac{w_{\lambda}^{-}(x)-w_{\lambda}^{-}(z)}{|x-z|^{N+2s}}dz=-\int_{\mathbb{R}^{N}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle-\int_{(\mathcal{O}_{1}\setminus(\mathcal{O}_{1})_{\lambda})\cup((\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1})}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz$
$\displaystyle-\int_{(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})\cup(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz-\int_{(\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz$
$\displaystyle=$ $\displaystyle-I_{1}-I_{2}-I_{3}.$

We estimate these integrals separately. Since $u=0\ {\rm in}\
(\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1}$ and $u_{\lambda}=0\ {\rm
in}\ \mathcal{O}_{1}\setminus(\mathcal{O}_{1})_{\lambda}$, then

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{(\mathcal{O}_{1}\setminus(\mathcal{O}_{1})_{\lambda})\cup((\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1})}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle\int_{(\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1}}\frac{u_{\lambda}(z)}{|x-z|^{N+2s}}dz-\int_{\mathcal{O}_{1}\setminus(\mathcal{O}_{1})_{\lambda}}\frac{u(z)}{|x-z|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle\int_{(\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1}}u_{\lambda}(z)\Big{(}\frac{1}{|x-z|^{N+2s}}-\frac{1}{|x-z_{\lambda}|^{N+2s}}\Big{)}dz\geq
0,$

since $u_{\lambda}\geq 0$ and $|x-z_{\lambda}|>|x-z|$ for all
$x\in\Sigma_{\lambda}^{-}$ and
$z\in(\mathcal{O}_{1})_{\lambda}\setminus\mathcal{O}_{1}.$

In order to decide the sign of $I_{2}$ we observe that
$w_{\lambda}(z_{\lambda})=-w_{\lambda}(z)$ for any $z\in\mathbb{R}^{N}$. Then,

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})\cup(\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz+\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z_{\lambda})}{|x-z_{\lambda}|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}}w_{\lambda}(z)(\frac{1}{|x-z|^{N+2s}}-\frac{1}{|x-z_{\lambda}|^{N+2s}})dz$
$\displaystyle\geq$ $\displaystyle 0,$

since $w_{\lambda}\geq 0$ in $\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}$
and $|x-z_{\lambda}|>|x-z|$ for all $x\in\Sigma_{\lambda}^{-}$ and
$z\in\Sigma_{\lambda}\setminus\Sigma_{\lambda}^{-}.$

Finally, since $w_{\lambda}(z)<0$ for $z\in\Sigma_{\lambda}^{-}$, we deduce

$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{(\Sigma_{\lambda}^{-})_{\lambda}}\frac{w_{\lambda}(z)}{|x-z|^{N+2s}}dz=\int_{\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z_{\lambda})}{|x-z_{\lambda}|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle-\int_{\Sigma_{\lambda}^{-}}\frac{w_{\lambda}(z)}{|x-z_{\lambda}|^{N+2s}}dz\geq
0.$

The proof is complete. $\Box$

Now we are ready to prove Proposition 3.3.

Proof of Proposition 3.3.  For simplicity, let $h(1)=t\geq 0$ and $u_{0}$ is
nontrivial, i.e. $u_{0}\geq t$ $u_{0}\not\equiv t$ in $B_{1}$.

In order to show the radial symmetry and decreasing monotonicity in $|x|$, we
divide the proof into four steps.

_Step 1:_ We prove that if $\lambda$ is close to $1$, then $w_{\lambda}>0$ in
$\Sigma_{\lambda}$. From the assumption (1.11), we see that $w_{\lambda}>0$ in
$\Sigma_{\lambda}$ for $\lambda=1$.

First we show that $w_{\lambda}\geq 0$ in $\Sigma_{\lambda}$, i.e.
$\Sigma^{-}_{\lambda}$ is empty. By contradiction, we assume that
$\Sigma^{-}_{\lambda}\not=\emptyset$. It must be in $B_{1}$ Now we apply
(3.14) and linearity of the fractional Laplacian to obtain that, for
$x\in\Sigma_{\lambda}^{-},$

$(-\Delta)^{s}w_{\lambda}^{+}(x)\geq(-\Delta)^{s}w_{\lambda}(x)=(-\Delta)^{s}u_{\lambda}(x)-(-\Delta)^{s}u(x).$
(3.15)

Combining with (3.15) and (3.12), for $x\in\Sigma_{\lambda}^{-}$, we have

$\displaystyle(-\Delta)^{s}w_{\lambda}^{+}(x)$ $\displaystyle\geq$
$\displaystyle-|x_{\lambda}|^{\theta}(u_{\lambda}(x)+t)^{p}+|x|^{\theta}(u(x)+t)^{p}$
$\displaystyle=$ $\displaystyle-\varphi(x)w_{\lambda}^{+}(x),$

where

$\displaystyle\varphi(x)$ $\displaystyle=$
$\displaystyle\frac{|x_{\lambda}|^{\theta}(u_{\lambda}(x)+t)^{p}-|x|^{\theta}(u(x)+t)^{p}}{u_{\lambda}(x)-u(x)}$
$\displaystyle=$
$\displaystyle\frac{|x_{\lambda}|^{\theta}-|x|^{\theta}}{u_{\lambda}(x)-u(x)}(u_{\lambda}(x)+t)^{p}+|x|^{\theta}\frac{(u_{\lambda}(x)+t)^{p}-(u(x)+t)^{p}}{u_{\lambda}(x)-u(x)}$
$\displaystyle<$
$\displaystyle|x|^{\theta}\frac{(u(x)+t)^{p}-(u_{\lambda}(x)+t)^{p}}{(u(x)+t)-(u_{\lambda}(x)+t)}$
$\displaystyle\leq$ $\displaystyle
2^{p}|x|^{\theta}(u(x)+t)^{p-1},\quad\forall\,x\in\Sigma_{\lambda}^{-}$

thanks to $\theta\in(-2s,0)$.

For
$x\in\Sigma_{\lambda}^{-}\subset\Sigma_{\lambda}\subset\mathbb{R}^{N}\setminus{B_{\lambda}}$,
$u_{\lambda}(x)<u(x)$. Moreover, there exists $M_{\lambda}>0$ such that

$\|u\|_{L^{\infty}(\mathbb{R}^{N}\setminus{B_{\lambda}})}\leq M_{\lambda},$

then there exists $c_{23}>0$ dependent of $\lambda$ such that

$\|\varphi\|_{L^{\infty}(\Sigma_{\lambda}^{-})}\leq c_{23}.$ (3.16)

Note that $M_{\lambda}\to\infty$ as $\lambda\to 0$ if
$\displaystyle\lim_{|x|\to 0^{+}}u(x)=\infty$.

Therefore, for $x\in\Sigma_{\lambda}^{-}$ and then

$(-\Delta)^{s}w_{\lambda}^{+}(x)\geq-\varphi(x)w_{\lambda}^{+}(x),\quad\forall\,x\in\Sigma_{\lambda}^{-}.$

Moreover, $w_{\lambda}^{+}=0$ in $(\Sigma_{\lambda}^{-})^{c}$. Choosing
$\lambda\in(0,1)$ close enough to $1$ we have $|\Sigma_{\lambda}^{-}|$ is
small and we apply Lemma 3.1 to obtain that

$w_{\lambda}=w_{\lambda}^{+}\geq 0\ \ \ \ \mbox{in}\ \ \Sigma_{\lambda}^{-},$

which is impossible. Thus,

$w_{\lambda}\geq 0\ \ \ \mbox{in}\ \ \Sigma_{\lambda}.$

If the function $\displaystyle\lim_{|x|\to 0^{+}}u(x)=+\infty$, then
$w_{\lambda}$ is positive near the point $(2\lambda,0,\cdots,0)$ and then
$w_{\lambda}\not\equiv 0$ in $\mathbb{R}^{N}$. If $u$ is bounded at the
origin, then the solution $u$ has removable singularity at the origin. In this
case, $u=0$ on $\partial B_{1}$ then $w_{\lambda}\not\equiv 0$.

Now we claim that for $0<\lambda<1$, $w_{\lambda}>0$ in $\Sigma_{\lambda}$.

Indeed, we assume on the contrary that there exists $x_{0}\in\Sigma_{\lambda}$
such that $w_{\lambda}(x_{0})=0,$ i.e. $u_{\lambda}(x_{0})=u(x_{0})$. Then

$\displaystyle(-\Delta)^{s}w_{\lambda}(x_{0})=(-\Delta)^{s}u_{\lambda}(x_{0})-(-\Delta)^{s}u(x_{0})=0.$
(3.17)

On the other hand, let
$K_{\lambda}=\big{\\{}(x_{1},x^{\prime})\in\mathbb{R}^{N}\ |\
x_{1}>\lambda\big{\\}}$. Noting $w_{\lambda}(z_{\lambda})=-w_{\lambda}(z)$ for
any $z\in\mathbb{R}^{N}$ and $w_{\lambda}(x_{0})=0$, we deduce

$\displaystyle(-\Delta)^{s}w_{\lambda}(x_{0})$ $\displaystyle=$
$\displaystyle-\int_{K_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2s}}dz-\int_{\mathbb{R}^{N}\setminus
K_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2s}}dz$ $\displaystyle=$
$\displaystyle-\int_{K_{\lambda}}\frac{w_{\lambda}(z)}{|x_{0}-z|^{N+2s}}dz-\int_{K_{\lambda}}\frac{w_{\lambda}(z_{\lambda})}{|x_{0}-z_{\lambda}|^{N+2s}}dz$
$\displaystyle=$
$\displaystyle-\int_{K_{\lambda}}w_{\lambda}(z)\Big{(}\frac{1}{|x_{0}-z|^{N+2s}}-\frac{1}{|x_{0}-z_{\lambda}|^{N+2s}}\Big{)}dz.$

The fact $|x_{0}-z_{\lambda}|>|x_{0}-z|$ for $z\in K_{\lambda}$ ,
$w_{\lambda}(z)\geq 0$ and $w_{\lambda}(z)\not\equiv 0$ in $K_{\lambda}$ yield

$(-\Delta)^{s}w_{\lambda}(x_{0})<0,$

which contradicts (3.17), completing the proof of the claim.

_Step 2:_ We prove $\lambda_{0}:=\inf\\{\lambda\in(0,1)\ |\ w_{\lambda}>0\ \
\rm{in}\ \ \Sigma_{\lambda}\\}=0$. If not, we set $\lambda_{0}>0$. Hence,
$w_{\lambda_{0}}\geq 0$ in $\Sigma_{\lambda_{0}}$ and
$w_{\lambda_{0}}\not\equiv 0$ in $\Sigma_{\lambda_{0}}$. The claim in Step 1
implies $w_{\lambda_{0}}>0$ in $\Sigma_{\lambda_{0}}$.

Letting $\epsilon\in(0,\lambda_{0}/4)$ small enough, we now show that
$w_{\lambda_{\epsilon}}>0$ in $\Sigma_{\lambda_{\epsilon}}$ for
$\lambda_{\epsilon}=\lambda_{0}-\epsilon$.

To this end, we set $D_{\mu}=\\{x\in\Sigma_{\lambda}\ |\
dist(x,\partial\Sigma_{\lambda})\geq\mu\\}$ for $\mu>0$ small. Since
$w_{\lambda}>0$ in $\Sigma_{\lambda}$ and $D_{\mu}$ is compact, there exists
$\mu_{0}>0$ such that $w_{\lambda}\geq\mu_{0}$ in $D_{\mu}$. By the continuity
of $w_{\lambda}(x)$, for $\epsilon>0$ small enough and
$\lambda_{\epsilon}=\lambda-\epsilon,$ we have that
$w_{\lambda_{\epsilon}}(x)\geq 0\ \ \rm{in}\ \ D_{\mu}$. Therefore,
$\Sigma_{\lambda_{\epsilon}}^{-}\subset\Sigma_{\lambda_{\epsilon}}\setminus
D_{\mu}$ and $|\Sigma_{\lambda_{\epsilon}}^{-}|$ is small if $\epsilon$ and
$\mu$ are small. Using (3.14) and proceeding as in Step 1, we have for all
$x\in\Sigma_{\lambda_{\epsilon}}^{-}$ that

$\displaystyle(-\Delta)^{s}w_{\lambda_{\epsilon}}^{+}(x)$ $\displaystyle=$
$\displaystyle(-\Delta)^{s}u_{\lambda_{\epsilon}}(x)-(-\Delta)^{s}u(x)-(-\Delta)^{s}w_{\lambda_{\epsilon}}^{-}(x)$
$\displaystyle\geq$
$\displaystyle(-\Delta)^{s}u_{\lambda_{\epsilon}}(x)-(-\Delta)^{s}u(x)$
$\displaystyle\geq$ $\displaystyle-\varphi(x)w_{\lambda_{\epsilon}}^{+}(x).$

From (3.16) $\varphi(x)$ is controlled by some constant dependent of
$\lambda$.

Since $w_{\lambda_{\epsilon}}^{+}=0$ in
$(\Sigma_{\lambda_{\epsilon}}^{-})^{c}$ and
$|\Sigma_{\lambda_{\epsilon}}^{-}|$ is small, for $\epsilon$ and $\mu$ small,
Lemma 3.1 implies that $w_{\lambda_{\epsilon}}\geq 0$ in
$\Sigma_{\lambda_{\epsilon}}$. Combining with $\lambda_{\epsilon}>0$ and
$w_{\lambda_{\epsilon}}\not\equiv 0$ in $\Sigma_{\lambda_{\epsilon}}$, we
obtain $w_{\lambda_{\epsilon}}>0$ in $\Sigma_{\lambda_{\epsilon}}$. This
contradiction arises from the definition of $\lambda_{0}$.

Therefore, we have that $\lambda_{0}=0$.

_Step 3:_ By Step 2, we have $\lambda_{0}=0$, which implies that
$u(-x_{1},x^{\prime})\geq u(x_{1},x^{\prime})$ for $x_{1}\geq 0.$ Using the
same argument from the other side, we conclude that $u(-x_{1},x^{\prime})\leq
u(x_{1},x^{\prime})$ for $x_{1}\geq 0$ and then
$u(-x_{1},x^{\prime})=u(x_{1},x^{\prime})$ for $x_{1}\geq 0.$ Repeating this
procedure in all directions we see that $u$ is radially symmetric.

Finally, we prove $u(r)$ is strictly decreasing in $r\in(0,1)$. Let us
consider $0<x_{1}<\widetilde{x}_{1}<1$ and let
$\lambda=\frac{x_{1}+\widetilde{x}_{1}}{2}$. As proved above we have

$w_{\lambda}(x)>0\ \ \mbox{for}\ \ x\in\Sigma_{\lambda}.$

Then

$\displaystyle 0<w_{\lambda}(\widetilde{x}_{1},0,\cdots,0)$ $\displaystyle=$
$\displaystyle
u_{\lambda}(\widetilde{x}_{1},0,\cdots,0)-u(\widetilde{x}_{1},0,\cdots,0)$
$\displaystyle=$ $\displaystyle
u(x_{1},0,\cdots,0)-u(\widetilde{x}_{1},0,\cdots,0),$

i.e $u(x_{1},0,\cdots,0)>u(\widetilde{x}_{1},0,\cdots,0).$ From the radial
symmetry of $u$ and decreasing in the direction $\frac{x}{|x|}$, we can
conclude the monotonicity of $u$. $\Box$

Proof of Proposition 3.2.  Note that

$u=u_{0}-h(1)$

and Proposition 3.3 shows that $u$ is radially symmetric and decreasing with
respect to $|x|$, so is $u_{0}$. Then (2.25) holds for $u_{0}$ and we apply
Corollary 2.2 to obtain that

$u_{0}(x)\leq c_{23}|x|^{-\frac{2s+\theta}{p-1}}\quad{\rm for}\ \,x\in
B_{1}\setminus\\{0\\}.$

We complete the proof. $\Box$

##  4 Classification of isolated singularity

###  4.1  Some important estimates

###### Proposition 4.1

Assume that $h\in L^{1}_{s}(\mathbb{R}^{N}\setminus B_{1})$,
$\theta\in(-2s,+\infty)$ and

$p\in\Big{(}\frac{N+\theta}{N-2s},\frac{N+2s+2\theta}{N-2s}\Big{)}.$

Let $u_{0}$ be a positive solution of (1.1) verifying

$\limsup_{|x|\to 0^{+}}u_{0}(x)=+\infty,$

then $u_{0}$ satisfies that

$\displaystyle\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}},$

where $\mathcal{K}_{p,\theta}$ is given in (1.7).

Proof.  We recall that

$\mathcal{K}_{p,\theta}=\mathcal{C}_{s}\big{(}-\frac{2s+\theta}{p-1}\big{)}^{\frac{1}{p-1}}.$

For $p\in\big{(}\frac{N+\theta}{N-2s},\frac{N+2s+2\theta}{N-2s})$, there holds
that

$-\frac{2s+\theta}{p-1}\in(2s-N,\frac{2s-N}{2})$

and

$\tau_{-}(-\mathcal{K}_{p,\theta}^{p-1})=-\frac{2s+\theta}{p-1}\quad{\rm
and}\quad\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=2s-N+\frac{2s+\theta}{p-1}.$

From Proposition 3.1, we have that

$\limsup_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}<+\infty.$

Set

$k=\liminf_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}.$ (4.1)

Part 1:  we claim

$k\leq\mathcal{K}_{p,\theta}.$

In fact, if

$k\in(\mathcal{K}_{p,\theta},+\infty],$

letting

$\epsilon_{0}=\frac{k-\mathcal{K}_{p,\theta}}{2\mathcal{K}_{p,\theta}},$

then there exists $r_{1}\in(0,1)$ such that

$u_{0}(x)\geq\mathcal{K}_{p,\theta}\big{(}1+\epsilon_{0}\big{)}|x|^{-\frac{2s+\theta}{p-1}}\quad{\rm
for}\ \ 0<|x|<r_{1},$

then for $0<|x|<r_{1}$,

$\displaystyle u_{0}^{p-1}(x)$ $\displaystyle\geq$
$\displaystyle\mathcal{K}_{p,\theta}^{p-1}\big{(}1+\epsilon_{0}\big{)}^{p-1}|x|^{-\theta-2s}>\mathcal{K}_{p,\theta}^{p-1}\big{(}1+(p-1)\epsilon_{0}\big{)}|x|^{-\theta-2s}.$

Therefore, $u_{0}$ verifies that

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u_{0}=\frac{\mathcal{K}_{p,\theta}^{p-1}+\frac{p-1}{2}\epsilon_{0}}{|x|^{2s}}u_{0}+f_{1}&{\rm
in}\ \ B_{r_{1}}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ u_{0}\geq 0&{\rm
in}\ \ \mathbb{R}^{N}\setminus B_{r_{1}},\end{array}\right.$ (4.2)

where

$\mu_{p}=-\Big{(}\mathcal{K}_{p,\theta}^{p-1}+\frac{p-1}{2}\epsilon_{0}\Big{)}<-\mathcal{K}_{p}^{p-1}$

and

$\displaystyle f_{1}(x)$ $\displaystyle=$
$\displaystyle|x|^{\theta}\Big{(}u_{0}^{p-1}-\frac{\mathcal{K}_{p,\theta}^{p-1}+\frac{p-1}{2}\epsilon_{0}}{|x|^{2s+\theta}}\Big{)}u_{0}(x)$
$\displaystyle\geq$
$\displaystyle\frac{p-1}{2}\epsilon_{0}\mathcal{K}_{p,\theta}^{p}|x|^{-\frac{(2s+\theta)p}{p-1}}.$

Observe that

$\tau_{+}(\mu_{p})<\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=2s-N+\frac{2s+\theta}{p-1}$

that is

$\tau_{+}(\mu_{p})-\frac{(2s+\theta)p}{p-1}<-N,$

which implies that

$f_{1}\not\in L^{1}(B_{r_{1}},\Gamma_{\mu_{p}}dx)$

and a contradiction arises by Theorem 2.1, from which problem (4.4) has no
such positive solution. Therefore we obtain that $k\leq\mathcal{K}_{p}.$

Part 2: Set

$\kappa=\limsup_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}$ (4.3)

and we prove that

$\kappa\geq\mathcal{K}_{p,\theta}.$

In fact, if

$\kappa<\mathcal{K}_{p,\theta},$

letting

$\epsilon_{1}=\min\Big{\\{}\frac{\mathcal{K}_{p,\theta}-\kappa}{2\mathcal{K}_{p,\theta}},\,\frac{1}{4}\Big{\\}},$

then there exists $r_{2}\in(0,1)$ such that

$u_{0}(x)\leq\mathcal{K}_{p,\theta}\big{(}1-\epsilon_{1})|x|^{-\frac{2s+\theta}{p-1}}\quad{\rm
for}\ \ 0<|x|<r_{2},$

and for $0<|x|<r_{2}$

$\displaystyle u_{0}^{p-1}(x)$ $\displaystyle\leq$
$\displaystyle\mathcal{K}_{p,\theta}^{p-1}\big{(}1-\epsilon_{1})^{p-1}|x|^{-\theta-2s}.$

Let

$\tilde{\mu}_{p}=-\mathcal{K}_{p,\theta}^{p-1}\big{(}1-\epsilon_{1})^{p-1}>-\mathcal{K}_{p,\theta}^{p-1},$

then

$\tau_{-}(\tilde{\mu}_{p})<-\frac{2s+\theta}{p-1}.$

Then $u_{0}$ is a positive solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u_{0}=\frac{\mathcal{K}_{p,\theta}^{p-1}\big{(}1-\epsilon_{1})^{p-1}}{|x|^{2s}}u_{0}+f_{2}&{\rm
in}\ \,B_{r_{2}}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \ u_{0}\geq 0&{\rm
in}\ \,\mathbb{R}^{N}\setminus B_{r_{2}},\\\\[5.69054pt]
\displaystyle\lim_{|x|\to
0^{+}}u_{0}(x)|x|^{-\tau_{-}(\tilde{\mu}_{p})}=0,\end{array}\right.$ (4.4)

where

$\displaystyle
f_{2}(x)=|x|^{\theta}\Big{(}u_{0}^{p-1}-\frac{\mathcal{K}_{p,\theta}^{p-1}\big{(}1-\epsilon_{1})^{p-1}}{|x|^{2s+\theta}}\Big{)}u_{0}(x)\leq
0.$

Then Lemma 2.3 with $\mu=\tilde{\mu}_{p}$ implies that

$\limsup_{|x|\to 0^{+}}u_{0}(x)|x|^{-\tau_{+}(\tilde{\mu}_{p})}<+\infty,$

where

$\tau_{+}(\tilde{\mu}_{p})>\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=2s-N+\frac{2s+\theta}{p-1}>-\frac{2s+\theta}{p-1}.$

Now we take the value $\tau_{0}=\tau_{+}(\tilde{\mu}_{p})<0$, then

$(-\Delta)^{s}u_{0}(x)\leq c_{24}|x|^{\tau_{0}p+\theta}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\}$

and letting

$\tau_{1}:=p\tau_{0}+\theta+2s,$

if $\tau_{1}>0$,

$u_{0}(x)\leq d_{1}\quad{\rm in}\ \,B_{r_{2}}\setminus\\{0\\},$

which contradicts that $u_{0}$ is non-removable at the origin. Then we are
done.

If $\tau_{1}=0$, applying Lemma 2.3 with $\mu=0$ to obtain that

$u_{0}(x)\leq d_{1}\Big{(}\ln\frac{r_{0}}{|x|}+1\Big{)}.\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\},$

then for $\epsilon>0$

$u_{0}(x)\leq\tilde{d}_{1}|x|^{-\epsilon}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\},$

which implies that

$(-\Delta)^{s}u_{0}(x)\leq c_{24}|x|^{\theta-\epsilon p}\quad{\rm in}\
\,B_{r_{2}}.\setminus\\{0\\}$

For $\epsilon>0$ small enough, we apply Lemma 2.3 with $\mu=0$ to obtain that

$u_{0}(x)\leq d_{1}\quad{\rm in}\ \,B_{r_{2}}\setminus\\{0\\}$

and we are done.

If $\tau_{1}<0$, applying Lemma 2.3 with $\mu=0$ to obtain that

$u_{0}(x)\leq d_{1}|x|^{\tau_{1}}\quad{\rm in}\ \,B_{r_{2}}\setminus\\{0\\}$

and

$(-\Delta)^{s}u_{0}(x)\leq c_{24}|x|^{\tau_{1}p+\theta}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\},$

thus, letting

$\tau_{2}:=p\tau_{1}+\theta+2s,$

if $\tau_{2}\geq 0$, we can get that $u_{0}$ is bounded at the origin by the
same proof of the case $\tau_{1}\geq 0$ and we are done.

If $\tau_{2}<0$, iteratively, we can prove that

$u_{0}(x)\leq d_{1}(|x|^{\tau_{n}}+1)\quad{\rm in}\
\,B_{r_{n}}\setminus\\{0\\},$

where

$\tau_{n}:=p\tau_{n-1}+\theta+2s,\quad n=1,2,\cdots.$

Note that

$\tau_{1}-\tau_{0}=(p-1)\tau_{0}+\theta+2s>0,$

thanks to $\tau_{0}>-\frac{2s}{p-1}$, and

$\displaystyle\tau_{n}-\tau_{n-1}$ $\displaystyle=$ $\displaystyle
p(\tau_{n-1}-\tau_{n-2})$ $\displaystyle=$ $\displaystyle
p^{n-1}(\tau_{1}-\tau_{0})\to+\infty\quad{\rm as}\ \,n\to+\infty,$

then there exists $n_{1}\in\mathbb{N}$ such that $\tau_{n_{1}}\geq 0$, by the
same argument of the case $\tau_{1}\geq 0$, we then obtain that $u_{0}$ is
bounded at the origin which contradicts the assumption

$\limsup_{|x|\to 0^{+}}u_{0}(x)=+\infty.$

Thus, we conclude that

$\displaystyle\liminf_{|x|\to
0^{+}}u(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u(x)|x|^{\frac{2s+\theta}{p-1}}.$

We complete the proof. $\Box$

Now we deal with the Sobolev’s critical and supercritical case.

###### Proposition 4.2

Assume that $h\in L^{1}_{s}(\mathbb{R}^{N}\setminus B_{1})$,
$\theta\in(-2s,+\infty)$ and

$p\geq\frac{N+2s+2\theta}{N-2s}.$

Let $u_{0}$ be a positive solution of (1.1) verifying

$\limsup_{|x|\to 0^{+}}u_{0}(x)=+\infty,$

then

$\displaystyle\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq\mathcal{K}_{p,\theta}\leq\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}.$

Proof.  Part 1: we claim that

$\liminf_{|x|\to 0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\leq
c_{*}^{-\frac{1}{p-1}}\mathcal{K}_{p,\theta}.$ (4.5)

Note that for $p\geq\frac{N+2s+2\theta}{N-2s}$,

$-\frac{2s+\theta}{p-1}\geq\frac{2s-N}{2}.$

Let

$u_{p}(x)=\mathcal{K}_{p,\theta}|x|^{-\frac{2s+\theta}{p-1}},$

where

$\mathcal{K}_{p,\theta}=\mathcal{C}_{s}(-\frac{2s+\theta}{p-1})^{\frac{1}{p-1}}.$

Note that

$(-\Delta)^{s}u_{p}=|x|^{\theta}u_{p}^{p}\quad{\rm in}\
\,\mathbb{R}^{N}\setminus\\{0\\},$

which could be written as

${\cal
L}_{-\mathcal{K}_{p,\theta}^{p-1}}^{s}u_{p}:=(-\Delta)^{s}u_{p}-\mathcal{K}_{p,\theta}^{p-1}|x|^{-2s}u_{p}=0\quad{\rm
in}\ \,\mathbb{R}^{N}\setminus\\{0\\}$

and

$\tau_{-}(-\mathcal{K}_{p,\theta}^{p-1})=2s-N+\frac{2s+\theta}{p-1}\quad{\rm
and}\quad\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=-\frac{2s+\theta}{p-1}.$

By contradiction, we assume that $u_{0}$ is a positive solution of (1.1) such
that

$\liminf_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}>\mathcal{K}_{p,\theta}.$

Then there exist $\epsilon_{0}>0$ and $r_{0}>0$ such that

$|x|^{\theta}u_{0}^{p-1}(x)\geq(\mathcal{K}_{p,\theta}^{p-1}+2\epsilon_{0})|x|^{-2s}\quad{\rm
for\ any}\ 0<|x|<r_{0}.$

Therefore, $u_{0}$ is a super solution of

${\cal L}_{\mu_{1}}^{s}u_{0}\geq\epsilon_{0}|x|^{\theta}u_{0}^{p}\quad{\rm
in}\ B_{r}\setminus\\{0\\},$

where $r\in(0,r_{0})$ and

$\mu_{1}=-(\mathcal{K}_{p,\theta}^{p-1}+\epsilon_{0})<-\mathcal{K}_{p,\theta}^{p-1}.$

By [19, Proposition 1.2], the mapping $\mu\in(\mu_{0},0)\mapsto\tau_{+}(\mu)$
is continuous and strictly increasing, then

$\tau_{+}(\mu_{1})<\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=-\frac{2s+\theta}{p-1}\quad{\rm
for}\ \ p\geq\frac{N+2s+2\theta}{N-2s}.$

By Lemma 2.2 with $\mu=\mu_{1}$, there exists $c>0$ such that

$u_{0}(x)\geq c_{25}|x|^{\tau_{+}(\mu_{1})}\quad{\rm in}\
B_{r}\setminus\\{0\\}.$

Let $\tau_{0}=\tau_{+}(\mu_{1})$. If $\tau_{0}\leq-\frac{N+\theta}{p}$, where
$-\frac{N+\theta}{p}>\frac{2s-N}{2}$ for $p\geq\frac{N+2s+2\theta}{N-2s}$.
Then $|x|^{\theta}u_{0}^{p}\not\in L^{1}(B_{1})$, a contradiction arises from
Theorem 2.1 part $(iii)$ with $\mu=0$. Then we are done.

If $\tau_{0}\in\big{(}-\frac{N+\theta}{p},-\frac{2s+\theta}{p-1}\big{)}$, then

$(-\Delta)^{s}u_{0}(x)\geq
d_{0}^{p}|x|^{p\tau_{0}+\theta}=d_{0}^{p}|x|^{\tau_{1}-2s}\quad{\rm in}\
\,B_{r_{0}}\setminus\\{0\\},$

where

$\tau_{1}:=p\tau_{0}+\theta+2s.$

If $\tau_{1}\leq-\frac{N+\theta}{p}$, we are done.

If $\tau_{1}\in\big{(}-\frac{N+\theta}{p},-\frac{2s+\theta}{p-1}\big{)}$, by
Lemma 2.2 with $\mu=0$, we have that

$u_{0}(x)\leq d_{1}|x|^{\tau_{1}}\quad{\rm in}\ \,B_{r_{0}}\setminus\\{0\\}.$

Iteratively, we recall that

$\tau_{j}:=p\tau_{j-1}+\theta+2s,\quad j=1,2,\cdots.$

If $\tau_{j}\leq-\frac{N+\theta}{p}$ we are done, otherwise it follows from
Lemma 2.2 that

$u_{0}(x)\geq d_{j+1}|x|^{\tau_{j+1}},$

where

$\tau_{j+1}=p\tau_{j}+\theta+2s<\tau_{j}.$

Thanks to $\tau_{1}-\tau_{0}=(p-1)\tau_{0}+\theta+2s<0$, we have that

$\tau_{j}-\tau_{j-1}=p(\tau_{j-1}-\tau_{j-2})=p^{j-1}(\tau_{1}-\tau_{0})\to-\infty\
\,{\rm as}\ j\to+\infty,$

then there exists $j_{0}\in\mathbb{N}$ such that

$\tau_{j_{0}}>-\frac{N+\theta}{p}\quad{\rm
and}\quad\tau_{j_{0}+1}\leq-\frac{N+\theta}{p}.$

For this, we have that

$\displaystyle(-\Delta)^{s}u_{0}(x)\geq
d_{j_{0}}^{p}|x|^{\tau_{j_{0}+1}-2s}\quad{\rm in}\ \
B_{r_{0}}\setminus\\{0\\},$

then $u_{0}(x)\geq d_{j_{0}+1}|x|^{\tau_{j_{0}+1}},$ and a contradiction
arises.

Part 2:  we claim

$\limsup_{|x|\to
0^{+}}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}\geq\mathcal{K}_{p,\theta}.$

By contradiction, we assume that $u_{0}$ is a positive super solution of (1.1)
in $\Omega\setminus\\{0\\}$ such that

$\limsup_{|x|\to+\infty}u_{0}(x)|x|^{\frac{2s+\theta}{p-1}}<\mathcal{K}_{p,\theta}.$

Then there exist $\epsilon_{1}\in(0,\frac{1}{2}\mathcal{K}_{p,\theta}^{p-1})$
and $r>0$ such that

$|x|^{\theta}u_{0}^{p-1}(x)\leq\big{(}\mathcal{K}_{p,\theta}^{p-1}-\epsilon_{1}\big{)}|x|^{-2s}\quad{\rm
for\ any}\ 0<|x|<r.$

Therefore, $u_{0}$ is a super solution of

${\cal L}_{\mu_{2}}^{s}u_{0}\leq 0\quad{\rm in}\ B_{r_{0}}\setminus\\{0\\},$

where

$\mu_{2}:=-\mathcal{K}_{p,\theta}^{p-1}+\epsilon_{1}>-\mathcal{K}_{p,\theta}^{p-1}.$

Notice that for $p\geq\frac{N+2\theta+2s}{N-2s}$, we have that

$\tau_{+}(-\mathcal{K}_{p,\theta}^{p-1})=-\frac{2s+\theta}{p-1}\geq\frac{2s-N}{2}>\tau_{-}\big{(}-\mathcal{K}_{p,\theta}^{p-1}\big{)}.$

By the strictly increasing monotonicity, we have that
$\tau_{+}(\mu_{2})>-\frac{2s+\theta}{p-1},$ thank to
$\mu_{2}>-\mathcal{K}_{p,\theta}^{p-1}$. By Lemma 2.3 with $\mu=\mu_{2}$, we
see that

$u_{0}(x)\leq c_{26}|x|^{\tau_{+}(\mu_{2})}\quad{\rm in}\
B_{r_{0}}\setminus\\{0\\}.$

Let $\tau_{0}=\tau_{+}(\mu_{2})>-\frac{2s+\theta}{p-1}$. Since
$\tau_{0}\in\big{(}-\frac{2s+\theta}{p-1},0\big{)}$, then

$(-\Delta)^{s}u_{0}(x)\leq
c_{27}|x|^{p\tau_{0}+\theta}=c_{27}|x|^{\tau_{1}-2s}\quad{\rm in}\
\,B_{r_{0}}\setminus\\{0\\},$

where

$\tau_{1}:=p\tau_{0}+\theta+2s.$

if $\tau_{1}>0$,

$u_{0}(x)\leq d_{1}\quad{\rm in}\ \,B_{r_{2}}\setminus\\{0\\},$

which contradicts that $u_{0}$ is non-removable at the origin. Then we are
done.

If $\tau_{1}=0$, applying Lemma 2.3 with $\mu=0$ to obtain that

$u_{0}(x)\leq d_{1}\Big{(}\ln\frac{r_{0}}{|x|}+1\Big{)}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\},$

then for $\epsilon>0$ small,

$u_{0}(x)\leq\tilde{d}_{1}|x|^{-\epsilon p}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\},$

which implies that

$(-\Delta)^{s}u_{0}(x)\leq c_{24}|x|^{\theta-\epsilon p}\quad{\rm in}\
\,B_{r_{2}}\setminus\\{0\\}.$

Then we apply Lemma 2.3 with $\mu=0$ to obtain that

$u_{0}(x)\leq d_{1}\quad{\rm in}\ \,B_{r_{2}}\setminus\\{0\\}$

for $\epsilon>0$ small enough and we are done.

If $\tau_{1}<0$, by Lemma 2.3, we have that

$u_{0}(x)\leq d_{1}|x|^{\tau_{1}}\quad{\rm in}\ \,B_{r_{0}}\setminus\\{0\\}.$

Iteratively, we reset that

$\tau_{j}:=p\tau_{j-1}+\theta+2s,\quad j=1,2,\cdots.$

Note that

$\tau_{1}-\tau_{0}=(p-1)\tau_{0}+\theta+2s>0$

thanks to $\tau_{0}>-\frac{2s+\theta}{p-1}$.

If $\tau_{j}p+\theta+2s\geq 0$ we are done by the same argument as
$\tau_{1}\geq 0$. If $\tau_{j}p+\theta+2s<0$ it follows by Lemma 2.2 that

$u_{0}(x)\geq d_{j+1}|x|^{\tau_{j+1}},$

where

$\tau_{j+1}=p\tau_{j}+2s+\theta<\tau_{j}.$

Note that

$\tau_{j}-\tau_{j-1}=p(\tau_{j-1}-\tau_{j-2})=p^{j-1}(\tau_{1}-\tau_{0})\to+\infty\
\,{\rm as}\ j\to+\infty,$

thus, there exists $j_{0}\in\mathbb{N}$ such that

$\tau_{j_{0}}<0\quad{\rm and}\quad\tau_{j_{0}+1}\geq 0,$

and $u_{0}$ is bounded at the origin. Thus a contradiction comes from the
assumption that

$\limsup_{|x|\to 0^{+}}u_{0}(x)=+\infty.$

We complete the proof. $\Box$

###  4.2 The isolated singularities

Proof of Theorem 1.1.  Let $u_{0}$ be a nonnegative solution of (1.1). For

$\theta=0\quad\&\quad p\in\Big{(}\frac{N}{N-2s},\frac{N+2s}{N-2s}\Big{)},$

Proposition 3.1 gives an upper bound

$u_{0}(x)\leq c_{16}|x|^{-\frac{2s}{p-1}},\quad\forall\,x\in
B_{1}\setminus\\{0\\}.$

Then $u_{0}$ verifies Harnack inequality (2.27) by Proposition 2.1. We now
apply Proposition 4.1 with
$p\in\big{(}\frac{N}{N-2s},\frac{N+2s}{N-2s}\big{)}$, we obtain that

$\displaystyle\frac{\mathcal{K}_{p}}{C_{0}}\leq\liminf_{|x|\to
0^{+}}u(x)|x|^{\frac{2s}{p-1}}\leq\mathcal{K}_{p}\leq\limsup_{|x|\to
0^{+}}u(x)|x|^{\frac{2s}{p-1}}\leq\mathcal{K}_{p}C_{0}.$

We complete the proof. $\Box$

Proof of Theorem 1.2.  When

$\theta=0\quad\&\quad p\geq\frac{N+2s}{N-2s}$

or

$\theta\in(-2s,0)\quad\&\quad p>\frac{N+\theta}{N-2s},$

and $h$ is radially symmetric, decreasing with respect to $|x|$, then the
positive solution $u_{0}$ is radially symmetric and decreasing with respect to
$r=|x|$, provided that $u_{0}(x)\geq h(1)$, then Proposition 3.2 provides an
upper bound

$u_{0}(x)\leq c_{22}|x|^{-\frac{2s+\theta}{p-1}},\quad\forall\,x\in
B_{1}\setminus\\{0\\}.$

Then our conclusion of $u_{0}$ follows by Proposition 2.1, Proposition 4.1 and
Proposition 4.2. $\Box$

Proof of Theorem 1.3.  For $\theta\in(-2s,+\infty)$, Corollary 2.2 gives an
upper bound

$u_{0}(x)\leq c_{22}|x|^{-\frac{2s+\theta}{p-1}},\quad\forall\,x\in
B_{1}\setminus\\{0\\}.$

Then our conclusion of $u_{0}$ follows by Proposition 4.1 and Proposition 4.2.
$\Box$

Proof of Theorem 1.4.  Let $v_{0}$ be a positive solution of (1.19). We claim
that

$v_{0}(x)\leq c_{28}|x|^{-\frac{2+\theta}{p-1}}\quad{\rm for}\ \,0<|x|<1/2$
(4.6)

for some $c_{28}>0$.

We first show that for $c_{29}>0$

$\int_{B_{r}}|x|^{\theta}v_{0}^{p}dx\leq
c_{29}r^{N-\frac{\theta+2p}{p-1}},\quad\forall\ r\in(0,\frac{1}{2}).$ (4.7)

In fact, let $r\in(0,1/2)$ and

$v=v_{0}-\tilde{v}_{0},$

where $\tilde{v}_{0}$ is the harmonic extension of $v_{0}$ in $B_{r}$, i.e.
the solution of

$\left\\{\begin{array}[]{lll}-\Delta u=0&{\rm in}\ \,B_{r},\\\\[5.69054pt]
\quad\ \ u=v_{0}&{\rm on}\ \,\partial B_{r}.\end{array}\right.$

Then we have that

$\left\\{\begin{array}[]{lll}-\Delta v=|x|^{\theta}(v+\tilde{v}_{0})^{p}&{\rm
in}\ \,B_{r}\setminus\\{0\\},\\\\[5.69054pt] \quad\ \ v=0&{\rm on}\ \,\partial
B_{r}\end{array}\right.$

and [36] shows that in the Serrin’s supercritical since
$p>\frac{N+\theta}{N-2}$,

$\int_{B_{r}}v(-\Delta)\eta
dx=\int_{B_{r}}|x|^{\theta}(v+\tilde{v}_{0})^{p}\eta dx,\quad\eta\in
C^{1.1}_{0}(B_{r}),$

which implies that for $\eta\in C^{1.1}_{0}(B_{r})$

$\displaystyle\int_{B_{r}}v_{0}(-\Delta)\eta dx$ $\displaystyle=$
$\displaystyle\int_{B_{r}}|x|^{\theta}v_{0}^{p}\eta
dx+\int_{B_{r}}\tilde{v}_{0}(-\Delta)\eta dx$ $\displaystyle=$
$\displaystyle\int_{B_{r}}|x|^{\theta}v_{0}^{p}\eta dx-\int_{\partial
B_{r}}v_{0}\nabla\eta\cdot xd\sigma(x),$

since the normal vector pointing outside of $B_{1}$ is $x\in\partial B_{1}$.

Now we take $\eta=\xi_{r}:=\xi_{1}(r^{-1}\cdot)$, where $\xi_{1}$ is the first
eigenvalue of $-\Delta$ in $B_{1}$ with the zero Dirichlet boundary condition.
Note that $\xi_{1}$ is radially symmetric and decreasing with respect to $|x|$
and then $\xi_{1}^{\prime}(1)<0$.

Then we have that

$\lambda_{1}r^{-2}\int_{B_{r}}v_{0}\xi_{r}dx=\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx-r^{-1}\xi_{1}^{\prime}(1)\int_{\partial
B_{r}}v_{0}d\sigma(x)>\int_{B_{r}}|x|^{\theta}u_{0}^{p}\xi_{r}dx.$

We observe that $-\frac{\theta}{p-1}>-N$ by our assumption that
$p>\frac{N+\theta}{N-2}$ and

$\displaystyle\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx$ $\displaystyle<$
$\displaystyle\lambda_{1}r^{-2}\int_{B_{r}}v_{0}\xi_{r}dx$ $\displaystyle\leq$
$\displaystyle\lambda_{1}r^{-2}\Big{(}\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx\Big{)}^{\frac{1}{p}}\Big{(}\int_{B_{r}}\xi_{r}|x|^{-\frac{\theta}{p-1}}dx\Big{)}^{1-\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}\|_{L^{\infty}(B_{1})}^{1-\frac{1}{p}}\lambda_{1}r^{(N-\frac{\theta}{p-1})(1-\frac{1}{p})-2}\Big{(}\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx\Big{)}^{\frac{1}{p}},$

which implies that

$\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx<\|\xi_{1}\|_{L^{\infty}(B_{1})}\lambda_{1}^{\frac{p}{p-1}}r^{N-\frac{\theta}{p-1}-\frac{2p}{p-1}}.$

Notice that

$\xi_{r}(x)=\xi_{1}(r^{-1}x)\geq\min_{z\in\bar{B}_{\frac{1}{2}}}\xi_{1}(z)\quad{\rm
in}\ B_{\frac{r}{2}},$

then

$\int_{B_{\frac{r}{2}}}|x|^{\theta}v_{0}^{p}dx<c_{30}\int_{B_{r}}|x|^{\theta}v_{0}^{p}\xi_{r}dx<c_{30}\|\xi_{1}\|_{L^{\infty}(B_{1})}\lambda_{1}^{\frac{p}{p-1}}r^{N-\frac{\theta}{p-1}-\frac{2p}{p-1}}.$

Replacing $r$ by $\frac{r}{2}$, we obtain (4.7).

Part $(i)$:  Since $v_{0}$ verifies (1.16), thus, taking $|x|=r$, for any
$y\in B_{r}\setminus B_{\frac{r}{2}}$, there holds that

$u_{0}(y)\geq\frac{1}{C_{1}}u_{0}(x).$

Now we deduce from (4.7) that for any given $|x|=r$,

$\displaystyle C_{0}^{-p}\Big{(}1-(\frac{1}{2})^{N}\Big{)}|\partial
B_{1}|\,v_{0}^{p}(x)r^{N-\theta}\leq\int_{B_{r}\setminus
B_{\frac{r}{2}}}|y|^{\theta}v_{0}^{p}(y)dy<c_{14}r^{N-\frac{\theta}{p-1}-\frac{2sp}{p-1}},$

which implies that for some $c_{31}\geq 1$

$\frac{1}{c_{31}}|x|^{-\frac{2+\theta}{p-1}}\leq v_{0}(x)\leq
c_{13}|x|^{-\frac{2+\theta}{p-1}}\quad{\rm for}\ \,0<|x|<\frac{1}{2}.$

By similar argument in Proposition 4.2 (some basic estimates could see [17]),
we obtain (1.20).

Part $(ii)$:  Let $u=v_{0}-h\geq 0$, under our assumptions that $h$ is a
nonnegative constant and $v_{0}\geq h$ in $B_{1}$, then $u$ is a classical
solution of

$\left\\{\begin{array}[]{lll}-\Delta u=|x|^{\theta}\big{(}u+h\big{)}^{p}&{\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \quad\ \ \,u=0&{\rm on}\
\,\partial B_{1}.\end{array}\right.$ (4.8)

For $\theta<0$, the moving plane works for the problem (4.8), see [6, 28] and
we get that $u$ is radially symmetric and decreasing with respect to $|x|$, so
is $v_{0}$. Then (4.7) implies (4.6).

Under the upper bound (4.6), we have the Harnack inequalities, for all
$r\in(0,\frac{1}{2})$

$\sup_{x\in B_{2r}\setminus B_{r}}v_{0}(x)\leq C_{1}\inf_{x\in B_{2r}\setminus
B_{r}}v_{0}(x)$

for some $C_{1}>0$. From the Harnack inequality in [40], we have that for some
constant $c_{32}\geq 1$

$\frac{1}{c_{32}}|x|^{-\frac{2+\theta}{p-1}}\leq v_{0}(x)\leq
c_{32}|x|^{-\frac{2+\theta}{p-1}}\quad{\rm for}\ \,0<|x|<\frac{1}{2}.$

Therefore, we obtain (1.20). $\Box$

##  5 Application to the decay at infinity

In this section, we apply the classification of isolated singularity to obtain
the decay at infinity for the positive solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=|x|^{\tilde{\theta}}u^{p}&{\rm
in}\ \,\mathbb{R}^{N}\setminus\bar{B}_{1},\\\\[5.69054pt] \qquad\ \ u=h&{\rm
in}\ \,\bar{B}_{1}.\end{array}\right.$ (5.1)

###### Theorem 5.1

Assume that

$p\in\Big{(}\frac{N+\tilde{\theta}}{N-2s},\,\frac{N+2s+\tilde{\theta}}{N-2s}\Big{)}\qquad{\rm
for}\quad\tilde{\theta}>-2s$ (5.2)

or

$p=\frac{N+2s+\tilde{\theta}}{N-2s}\qquad{\rm for}\quad\tilde{\theta}\geq 0,$
(5.3)

$h\in L^{1}(\bar{B}_{1})$ is a nonnegative and radially symmetric function
such that $h(1)\geq 0$, $|x|^{N-2s}h(x)$ is increasing with respect to $|x|$
in $B_{1}\setminus\\{0\\}$.

Let $u_{0}$ be a positive solution of (5.1) such that

$u_{0}(x)\geq h(1)|x|^{2s-N}\quad{\rm for}\
\,\forall\,x\in\mathbb{R}^{N}\setminus B_{1},$ (5.4)

then $u_{0}$ is radially symmetric, either for some $k\geq h(1)$

$\displaystyle\lim_{|x|\to+\infty}u_{0}(x)|x|^{N-2s}=k$

or

$\displaystyle\frac{\mathcal{K}_{p,\theta^{*}}}{C_{0}}\leq\liminf_{|x|\to+\infty}u_{0}(x)|x|^{\frac{2s+\tilde{\theta}}{p-1}}\leq\mathcal{K}_{p,\theta^{*}}\leq\limsup_{|x|\to+\infty}u_{0}(x)|x|^{\frac{2s+\tilde{\theta}}{p-1}}\leq
C_{0}\mathcal{K}_{p,\theta^{*}},$

where

$\theta^{*}=p(N-2s)-N-2s-\tilde{\theta}.$

Proof.  We use the Kelvin transformation:  let $u_{f}$ be a solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u=f&{\rm in}\
\,\mathcal{O},\\\\[5.69054pt] \qquad\ \ u=h&{\rm in}\ \
\mathbb{R}^{N}\setminus\mathcal{O},\end{array}\right.$ (5.5)

where $\mathcal{O}$ is an open set of $\mathbb{R}^{N}$ and $f$ is locally
Hölder continuous in $\mathcal{O}$. Denote

$u^{\sharp}(x)=|x|^{2s-N}u_{f}(\frac{x}{|x|^{2}})\quad{\rm for}\
x\in\mathbb{R}^{N}\setminus\\{0\\}$

and

$\mathcal{O}^{\sharp}=\Big{\\{}x\in\mathbb{R}^{N}:\frac{x}{|x|^{2}}\in\mathcal{O}\Big{\\}},$

then a direct computation shows that

$(-\Delta)^{s}u^{\sharp}(x)=|x|^{-2s-N}\big{(}(-\Delta)^{s}u_{f}\big{)}\left(\frac{x}{|x|^{2}}\right)\quad{\rm
for}\ x\in\mathcal{O}^{\sharp}.$

As a conclusion, $u^{\sharp}$ is a solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u^{\sharp}=|x|^{-2s-N}f^{*}&{\rm
in}\ \,\mathcal{O}^{\sharp},\\\\[5.69054pt] \qquad\ \
u^{\sharp}=|x|^{2s-N}h^{*}&{\rm in}\ \
\mathbb{R}^{N}\setminus\mathcal{O},\end{array}\right.$

where $f^{*}(x)=f(\frac{x}{|x|^{2}})$ and $h^{*}(x)=h(\frac{x}{|x|^{2}})$.

Let $u_{0}$ be a positive solution of (5.1). Taking
$f(x)=|x|^{\tilde{\theta}}u_{0}^{p}(x)$ and
$\mathcal{O}=\mathbb{R}^{N}\setminus B_{1}$ in (5.5), the function

$u^{\sharp}(x):=|x|^{2s-N}u_{0}(\frac{x}{|x|^{2}})$

is a positive solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u^{\sharp}=|x|^{\theta^{*}}(u^{\sharp})^{p}&{\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \
u^{\sharp}=|x|^{2s-N}h^{*}&{\rm in}\ \ \mathbb{R}^{N}\setminus
B_{1},\end{array}\right.$

where

$|x|^{2s-N}h^{*}\in L^{1}(\mathbb{R}^{N}\setminus B_{1},|x|^{-N-2s}dx),$

which is equivalent to $h\in L^{1}(B_{1})$.

Since

$|x|^{2s-N}h^{*}(x)=|y|^{N-2s}h(y)\quad{with}\ \,x=\frac{y}{|y|^{2}},$

then $x\in B_{1}\setminus\\{0\\}\mapsto|x|^{2s-N}h^{*}(x)$ is radially
symmetric and increasing with respect to $|x|$. Note that

$u_{0}(x)\geq h(1)|x|^{2s-N}\quad{\rm for}\ \,\forall
x\in\mathbb{R}^{N}\setminus B_{1}\quad\ \Longleftrightarrow\quad\
u^{\sharp}(y)\geq h(1)\quad{\rm for}\ \,y\in B_{1}\setminus\\{0\\}.$

To obtain the asymptotic behaviors, we would apply Theorem 1.2 with
$\theta^{*}=0$ and $p\geq\frac{N+2s}{N-2s}$ by (5.3); or with
$\theta^{*}\in(-2s,0)$ and $p\geq\frac{N+\theta^{*}}{N-2s}$ by (5.2). As a
consequence, we have that $u^{\sharp}$ is radially symmetric, decreasing with
respect to $|y|$, and is either removable at the origin or verifies

$\displaystyle\frac{\mathcal{K}_{p,\theta^{*}}}{C_{0}}\leq\liminf_{|x|\to
0^{+}}u^{\sharp}(x)|x|^{\frac{2s+\theta^{*}}{p-1}}\leq\mathcal{K}_{p,\theta^{*}}\leq\limsup_{|x|\to
0^{+}}u^{\sharp}(x)|x|^{\frac{2s+\theta^{*}}{p-1}}\leq
C_{0}\mathcal{K}_{p,\theta^{*}},$

which imply that $u_{0}$ is radially symmetric, either

$\displaystyle\lim_{|x|\to+\infty}u_{0}(x)|x|^{N-2s}=u^{\sharp}(0)$

or

$\displaystyle\frac{\mathcal{K}_{p,\theta^{*}}}{C_{0}}\leq\liminf_{|x|\to+\infty}u_{0}(x)|x|^{N-\frac{2sp+\theta^{*}}{p-1}}$
$\displaystyle\leq$ $\displaystyle\mathcal{K}_{p,\theta^{*}}$
$\displaystyle\leq$
$\displaystyle\limsup_{|x|\to+\infty}u_{0}(x)|x|^{N-\frac{2sp+\theta^{*}}{p-1}}\leq
C_{0}\mathcal{K}_{p,\theta^{*}}.$

We complete the proof. $\Box$

###### Theorem 5.2

Assume that $h\in L^{1}(B_{1})$ and

$p=\frac{N+2s+\tilde{\theta}}{N-2s}\qquad{\rm
for}\quad\tilde{\theta}\in(-2s,\,0).$ (5.6)

Let $u_{0}$ be a positive solution of (5.1) and $C_{0}\geq 1$ be the constant
from the Harnack inequality in Theorem 1.1 with $\theta=0$.

Then either for some $k>0$

$\displaystyle\lim_{|x|\to+\infty}u_{0}(x)|x|^{N-2s}=k$

or

$\displaystyle\frac{\mathcal{K}_{p}}{C_{0}}\leq\liminf_{|x|\to+\infty}u_{0}(x)|x|^{\frac{2s+\tilde{\theta}}{p-1}}\leq\mathcal{K}_{p}\leq\limsup_{|x|\to+\infty}u_{0}(x)|x|^{\frac{2s+\tilde{\theta}}{p-1}}\leq
C_{0}\mathcal{K}_{p}.$

Proof.  From the Kelvin transformation,

$u^{\sharp}(x):=|x|^{2s-N}u_{0}(\frac{x}{|x|^{2}})$

is a positive solution of

$\left\\{\begin{array}[]{lll}(-\Delta)^{s}u^{\sharp}=(u^{\sharp})^{p}&{\rm
in}\ \,B_{1}\setminus\\{0\\},\\\\[5.69054pt] \qquad\ \
u^{\sharp}=|x|^{2s-N}h^{*}&{\rm in}\ \ \mathbb{R}^{N}\setminus
B_{1},\end{array}\right.$

by our assumption (5.6) that $\tilde{\theta}=p(N-2s)-N-2s$.

From Theorem 1.1, $u^{\sharp}$ is either removable at the origin or it
verifies

$\displaystyle\frac{\mathcal{K}_{p}}{C_{0}}\leq\liminf_{|x|\to
0^{+}}u^{\sharp}(x)|x|^{\frac{2s}{p-1}}\leq\mathcal{K}_{p}\leq\limsup_{|x|\to
0^{+}}u^{\sharp}(x)|x|^{\frac{2s}{p-1}}\leq C_{0}\mathcal{K}_{p},$

then we have that either

$\displaystyle\lim_{|x|\to+\infty}u_{0}(x)|x|^{N-2s}=u^{\sharp}(0)$

or

$\displaystyle\frac{\mathcal{K}_{p}}{C_{0}}\leq\liminf_{|x|\to+\infty}u_{0}(x)|x|^{N-\frac{2sp}{p-1}}$
$\displaystyle\leq$ $\displaystyle\mathcal{K}_{p}$ $\displaystyle\leq$
$\displaystyle\limsup_{|x|\to+\infty}u_{0}(x)|x|^{N-\frac{2sp}{p-1}}\leq
C_{0}\mathcal{K}_{p},$

where we use $\theta^{*}=p(N-2s)-N-2s-\tilde{\theta}=0.$ We complete the
proof. $\Box$

Acknowledgements: Foundation of China, No. 12071189, by Jiangxi Province
Science Fund for Distinguished Young Scholars, No. 20212ACB211005, and by the
Jiangxi Provincial Natural Science Foundation, No. 20202ACBL201001, by the
Science and Technology Research Project of Jiangxi Provincial Department of
Education, No. GJJ200307, GJJ200325. F. Zhou is supported by Science and
Technology Commission of Shanghai Municipality (STCSM), Grant No. 18dz2271000
and also supported by NSFC (No. 11431005).

## References

  * [1] W. Ao, H. Chan, A. Dela-Torre, M. Fontelos, G. María del Mar; J. Wei, On higher-dimensional singularities for the fractional Yamabe problem: a nonlocal Mazzeo-Pacard program. Duke Math. J. 168(17), 3297–3411 (2019).
  * [2] W. Ao, H. Chan, M. González, J. Wei, Existence of positive weak solutions for fractional Lane-Emden equations with prescribed singular set. Calc. Var. PDEs 57(6), Art. 149, 25 pp (2018).
  * [3] W. Ao, A. Dela-Torre, M. González, J. Wei, A gluing approach for the fractional Yamabe problem with isolated singularities. J. Reine Angew. Math. 763, 25–78 (2020) .
  * [4] P. Aviles, On isolated singularities in some nonlinear partial differential equations. Indiana Univ. Math. J. 35, 773–791 (1983).
  * [5] P. Aviles, Local behavior of solutions of some elliptic equations. Comm. Math. Phys. 108, 177–192 (1987).
  * [6] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 22(1), 1–37 (1991).
  * [7] M.-F. Bidaut-Véron, L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math. 106 (3), 489–539 (1991).
  * [8] K. Bogdan, B. Dyda, The best constant in a fractional Hardy inequality. Math. Nachr. 284 (5-6), 629–638 (2011).
  * [9] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 271–297 (1989).
  * [10] L. Caffarelli, T. Jin, Y. Sire, J. Xiong, Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities. Arch. Rational Mech. Anal. 213(1), 245–268 (2014).
  * [11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eq. 32, 1245–1260 (2007).
  * [12] L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009).
  * [13] L. Caffarelli, L. Silvestre, Regularity results for nonlocal equations by approximation. Arch. Rational Mech. Anal. 200(1), 59–88 (2011).
  * [14] H. Chen, P. Felmer, A. Quaas, Large solutions to elliptic equations involving fractional Laplacian. Ann. Inst. H. Poincaré–AN, 32, 1199–1228 (2015).
  * [15] H. Chen, H. Hajaiej, On isolated singularities for fractional Lane-Emden equation in the Serrin’s critical case. arXiv:2109.07085 (2021).
  * [16] H. Chen, A. Quaas, Classification of isolated singularities of nonnegative solutions to fractional semilinear elliptic equations and the existence results. J. London Math. Soc. 97(2), 196–221 (2018).
  * [17] H. Chen, A. Quaas, F. Zhou. On nonhomogeneous elliptic equations with the Hardy-Leray potentials. J. Anal. Math. 144(1), 305–334 (2021).
  * [18] H. Chen, L. Véron, Semilinear fractional elliptic equations involving measures. J. Diff. Eq. 257(5), 1457–1486 (2014).
  * [19] H. Chen, T. Weth, The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions. Trans. Amer. Math. Soc. 374, 6881–6925 (2021).
  * [20] W. Chen, C. Li, B. Ou, Qualitative properties of solutions for an integral equation. Disc. Cont. Dyn. Syst. 12(2), 347–354 (2005).
  * [21] W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59(3), 330–343 (2006).
  * [22] E. N. Dancer, Z. Guo, J. Wei, Non-Radial Singular Solutions of the Lane-Emden Equation in $\mathbb{R}^{N}$. Indiana Univ. Math. J. No. 5, 1971–1996 (2012).
  * [23] J. Dávila, L. Dupaigne, A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation. J. Funct. Anal. 261(1), 218–232 (2011).
  * [24] A. Dela-Torre, M. González, Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry. Rev. Mat. Iberoam. 34(4), 1645–1678 (2018).
  * [25] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521-573 (2012).
  * [26] P. Felmer, A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226, 2712–2738 (2011).
  * [27] P. Felmer, Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian. Comm. Cont. Math. 16 (1), 1350023, 24 pp (2014).
  * [28] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979).
  * [29] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, 525–589 (1981).
  * [30] C. Gui, W.-M. Ni, X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^{n}$. Comm. Pure Appl. Math. 45(9), 1153–1181 (1992).
  * [31] Z. Han, Y. Li, E. Teixeira, Asymptotic behavior of solutions to the k-Yamabe equation near isolated singularities. Invent. Math. 182, 635–684 (2010).
  * [32] T. Jin, Y. Li, J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. 16(6), 1111–1171 (2014).
  * [33] T. Jin, Y. Li, J. Xiong, On a fractional Nirenberg problem, Part II: Existence of solutions. Int. Math. Res. Not. 6, 1555–1589 (2015).
  * [34] T. Jin, J. Xiong, Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities. Ann. Inst. H. Poincaré–AN 38(4), 1167–1216 (2021).
  * [35] N. Korevaar, R. Mazzeo, F. Pacard, R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135, 233–272 (1999).
  * [36] P. Lions, Isolated singularities in semilinear problems. J. Diff. Eq. 38, 441–450 (1980).
  * [37] F. Pacard, Existence and convergence of positive week solutions of $-\Delta u=u^{\frac{n}{n-2}}$ in bounded domain of $\mathbb{R}^{n}$, $n\geq 3$. Calc. Var. PDEs 1, 243–265 (1993).
  * [38] X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014).
  * [39] R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure Appl. Math., 41, 317–392 (1988).
  * [40] J. Serrin, Local behavior of solutions of quasi-linear equation. Acta. Math. 111, 247–302 (1964).
  * [41] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (1), 67–112 (2007).
  * [42] J. Tan, J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms. Disc. Cont. Dyn. Syst. 31(3), 975–983 (2011).
  * [43] L. Véron, Singularities of solutions of second order quasilinear equations. Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, viii+377 pp. (1996).
  * [44] X. Wang, On the Cauchy problem for reaction-diffusion equations. Trans. Amer. Math. Soc. 337(2), 549–590 (1993).
  * [45] J. Wei, K. Wu, Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent, Dis. Cont. Dyn. Sys. 10.3934/dcds.2022044 (2022).
  * [46] H. Yang, W. Zou, On isolated singularities of fractional semilinear elliptic equations. Ann. Inst. H. Poincaré–AN 38(2), 403–420 (2021).
  * [47] H. Zou, Symmetry of positive solutions of $\Delta u+u^{p}=0$ in $\mathbb{R}^{N}$. J. Diff. Eq. 120(1), 46–88 (1995).