Self-generated interior blow-up solutions in fractional elliptic equation with
absorption

Huyuan Chen, Patricio Felmer

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático
UMR2071 CNRS-UChile, Universidad de Chile

Casilla 170 Correo 3, Santiago, Chile.

<EMAIL_ADDRESS><EMAIL_ADDRESS>

and

Alexander Quaas

Departamento de Matemática, Universidad Técnica Federico Santa María

Casilla: V-110, Avda. España 1680, Valparaíso, Chile

<EMAIL_ADDRESS>

###### Abstract

In this paper we study positive solutions to problem involving the fractional
Laplacian

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega\setminus\mathcal{C},\\\\[5.69054pt]
u(x)=0,&x\in\Omega^{c},\\\\[5.69054pt] \lim_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}u(x)=+\infty,\end{array}\right.$ (0.1)

where $p>1$ and $\Omega$ is an open bounded $C^{2}$ domain in
$\mathbb{R}^{N}$, $\mathcal{C}\subset\Omega$ is a compact $C^{2}$ manifold
with $N-1$ multiples dimensions and without boundary, the operator
$(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is the fractional Laplacian.

We consider the existence of positive solutions for problem (0.1). Moreover,
we further analyze uniqueness, asymptotic behaviour and nonexistence.

Key words: Fractional Laplacian, Existence, Uniqueness, Asymptotic behavior,
Blow-up solution.

##  1 Introduction

In 1957, a fundamental contribution due to Keller in [11] and Osserman in [19]
is the study of boundary blow-up solutions for the non-linear elliptic
equation

$\left\\{\begin{array}[]{lll}-\Delta u+h(u)=0\ \ \rm{in}\ \
\Omega,\\\\[5.69054pt]
\lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty.\end{array}\right.$ (1.1)

They proved the existence of solutions to (1.1) when
$h:\mathbb{R}\to[0,+\infty)$ is a locally Lipschitz continuous function which
is nondecreasing and satisfies the so called Keller-Osserman condition. From
then on, the result of Keller and Osserman has been extended by numerous
mathematicians in various ways, weakening the assumptions on the domain,
generalizing the differential operator and the nonlinear term for equations
and systems. The case of $h(u)=u_{+}^{p}$ with $p=\frac{N+2}{N-2}$ is studied
by Loewner and Nirenberg [15], where in particular uniqueness and asymptotic
behavior were obtained. After that, Bandle and Marcus [2] obtained uniqueness
and asymptotic for more general non-linearties $h$. Later, Le Gall in [9]
established a uniqueness result of problem (1.1) in the domain whose boundary
is non-smooth when $h(u)=u_{+}^{2}$. Marcus and Véron [16, 18] extended the
uniqueness of blow-up solution for (1.1) in general domains whose boundary is
locally represented as a graph of a continuous function when $h(u)=u_{+}^{p}$
for $p>1$. Under this special assumption on $h$, Kim [12] studied the
existence and uniqueness of boundary blow-up solution to (1.1) in bounded
domains $\Omega$ satisfying $\partial\Omega=\partial\bar{\Omega}$. For another
interesting contributions to boundary blow-up solutions see for example
Kondratev, Nikishkin [13], Lazer, McKenna [14], Arrieta and Rodríguez-Bernal
[1], Chuaqui, Cortázar, Elgueta and J. García-Melián [4], del Pino and
Letelier [5], Díaz and Letelier [6], Du and Huang [7], García-Melián [10],
Véron [20], and the reference therein.

In a recent work, Felmer and Quaas [8] considered a version of Keller and
Osserman problem for a class of non-local operator. Being more precise, they
considered as a particular case the fractional elliptic problem

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=f(x),&x\in\Omega,\\\\[5.69054pt]
u(x)=g(x),&x\in\bar{\Omega}^{c},\\\\[5.69054pt] \lim_{x\in\Omega,\
x\to\partial\Omega}u(x)=+\infty,\end{array}\right.$ (1.2)

where $p>1$, $f$ and $g$ are appropriate functions and $\Omega$ is a bounded
domain with $C^{2}$ boundary. The operator $(-\Delta)^{\alpha}$ is the
fractional Laplacian which is defined as

$(-\Delta)^{\alpha}u(x)=-\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\delta(u,x,y)}{|y|^{N+2\alpha}}dy,\
\ x\in\Omega,$ (1.3)

with $\alpha\in(0,1)$ and $\delta(u,x,y)=u(x+y)+u(x-y)-2u(x)$.

In [8] the authors proved the existence of a solution to (1.2) provided that
$g$ explodes at the boundary and satisfies other technical conditions. In case
the function $g$ blows up with an explosion rate as $d(x)^{\beta}$, with
$\beta\in[-\frac{2\alpha}{p-1},0)$ and $d(x)=dist(x,\partial\Omega)$, it is
shown that the solution satisfies

$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\beta}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$

Here the explosion is driven by the external value $g$ and the external source
$f$ has a secondary role, not intervening in the explosive character of the
solution.

More recently, Chen, Felmer and Quaas [3] extended the results in [8] studying
existence, uniqueness and non-existence of boundary blow-up solutions when the
function $g$ vanishes and the explosion on the boundary is driven by the
external source $f$, with weak or strong explosion rate. Moreover, the results
are extended even to the case where the boundary blow-up solutions in driven
internally, when the external source and value, $f$ and $g$, vanish.
Existence, uniqueness, asymptotic behavior and non-existence results for blow-
up solutions of (1.2) are considered in [3]. In the analysis developed in [3],
a key role is played by the function $C:(-1,0]\to\mathbb{R},$ that governs the
behavior of the solution near the boundary. The function $C$ is defined as

$C(\tau)=\int^{+\infty}_{0}\frac{\chi_{(0,1)}(t)|1-t|^{\tau}+(1+t)^{\tau}-2}{t^{1+2\alpha}}dt$
(1.4)

and it possess exactly one zero in $(-1,0)$ and we call it $\tau_{0}(\alpha)$.
In what follows we explain with more details the results in the case of
vanishing external source and values, that is $f=0$ in $\Omega$ and $g=0$ in
$\bar{\Omega}^{c}$, which is the case we will consider in this paper. In
Theorem 1.1 in [3], we proved that whenever

$1+2\alpha<p<1-\frac{2\alpha}{\tau_{0}(\alpha)},$

then problem (1.2) admits a unique positive solution $u$ such that

$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{\frac{2\alpha}{p-1}}<+\infty.$

On the other hand, we proved that when $p\geq 1,$ then problem (1.2) does not
admit any solution $u$ such that

$0<\liminf_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}\leq\limsup_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau}<+\infty,$
(1.5)

for any $\tau\in(-1,0)\setminus\\{\tau_{0}(\alpha),-\frac{2\alpha}{p-1}\\}$.
We observe that the non-existence result does not include the case when $u$
has an asymptotic behavior of the form $d(x)^{\tau_{0}(\alpha)}$, where
$\tau_{0}(\alpha)$ is precisely where $C$ vanishes. We have a a special
existence result in this case, precisely if

$\max\\{1-\frac{2\alpha}{\tau_{0}(\alpha)}+\frac{\tau_{0}(\alpha)+1}{\tau_{0}(\alpha)},1\\}<p<1-\frac{2\alpha}{\tau_{0}(\alpha)},$

then, for any $t>0$, problem (1.2) admits a positive solution $u$ such that

$\lim_{x\in\Omega,x\to\partial\Omega}u(x)d(x)^{-\tau_{0}(\alpha)}=t.$

Motivated by these results and in view of the non-local character of the
fractional Laplacian we are interested in another class of blow-up solutions.
We want to study solutions that vanish at the boundary of the domain $\Omega$
but that explodes at the interior of the domain, near a prescribed embedded
manifold. From now on, we assume that $\Omega$ is an open bounded domain in
$\mathbb{R}^{N}$ with $C^{2}$ boundary, and that there is a $C^{2}$,
$(N-1)$-dimensional manifold $\mathcal{C}$ without boundary, embedded in
$\Omega$, such that, it separates $\Omega\setminus\mathcal{C}$ in exactly two
connected components. We denote by $\Omega_{1}$ the inner component and by
$\Omega_{2}$ the external component, that is
$\bar{\Omega}_{1}\cap\partial\Omega=\emptyset$ and
$\bar{\Omega}_{2}\cap\partial\Omega=\partial\Omega.$ Throughout the paper we
will consider the distance function

$D:\Omega\setminus\mathcal{C}\to\mathbb{R}_{+},\quad D(x)={\rm
dist}(x,\mathcal{C}).$ (1.6)

Let us consider the equations, for $i=1,2$,

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega_{i},\\\\[5.69054pt]
u(x)=0,&x\in\bar{\Omega}_{i}^{c},\\\\[5.69054pt] \lim_{x\in\Omega_{i},\
x\to\partial\Omega_{i}}u(x)=+\infty,\end{array}\right.$ (1.7)

which have solutions $u_{1}$ and $u_{2}$, for $i=1,2$ respectively, in the
appropriate range of the parameters. In the local case, that is, $\alpha=1$,
these two solutions certainly do not interact among each other, but when
$\alpha\in(0,1)$, due to the non-local character of the fractional Laplacian
and the non-linear character of the equation the solutions on each side of
$\Omega$ interact and it is precisely the purpose of this paper to study their
existence, uniqueness and non-existence.

In precise terms we consider the equation

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u(x)+|u|^{p-1}u(x)=0,&x\in\Omega\setminus\mathcal{C},\\\\[5.69054pt]
u(x)=0,&x\in\Omega^{c},\\\\[5.69054pt] \lim_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}u(x)=+\infty,\end{array}\right.$ (1.8)

where $p>1$, $\Omega$ and $\mathcal{C}\subset\Omega$ are as described above.
The explosion of the solution near $\mathcal{C}$ is governed by a function
$c:(-1,0]\to\mathbb{R},$ defined as

$c(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}+(1+t)^{\tau}-2}{t^{1+2\alpha}}dt.$
(1.9)

This function plays the role of the function $C$ used in [3], but it has
certain differences. In Section §2 we prove the existence of a number
$\alpha_{0}\in(0,1)$ such that $\alpha\in[\alpha_{0},1)$ the function $c$ is
always positive in $(-1,0)$, while if $\alpha\in(0,\alpha_{0})$ then there
exists exists a unique $\tau_{1}(\alpha)\in(-1,0)$ such that
$c(\tau_{1}(\alpha))=0$ and $c(\tau)>0$ in $(-1,\tau_{1}(\alpha))$ and
$c(\tau)<0$ in $(\tau_{1}(\alpha),0)$, see Proposition 2.1. We notice here
that $\tau_{1}(\alpha)>\tau_{0}(\alpha)$ if $\alpha\in(0,\alpha_{0})$.

Now we are ready to state our main theorems on the existence uniqueness and
asymptotic behavior of interior blow-up solutions to equation (1.8). These
theorems deal separately the case $\alpha\in(0,\alpha_{0})$ and
$\alpha\in[\alpha_{0},1)$.

###### Theorem 1.1

Assume that $\alpha\in(0,\alpha_{0})$ and the assumptions on $\Omega$ and
$\mathcal{C}$. Then we have:
$(i)$ If

$1+2\alpha<p<1-\frac{2\alpha}{\tau_{1}(\alpha)},$ (1.10)

then problem (1.8) admits a unique positive solution $u$ satisfying

$0<\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{\frac{2\alpha}{p-1}}\leq\limsup_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{\frac{2\alpha}{p-1}}<+\infty.$
(1.11)

$(ii)$ If

$\max\\{1-\frac{2\alpha}{\tau_{1}(\alpha)}+\frac{\tau_{1}(\alpha)+1}{\tau_{1}(\alpha)},1\\}<p<1-\frac{2\alpha}{\tau_{1}(\alpha)}.$
(1.12)

Then, for any $t>0$, there is a positive solution $u$ of problem (1.8)
satisfying

$\lim_{x\in\Omega\setminus{\mathcal{C}},x\to{\mathcal{C}}}u(x)D(x)^{-\tau_{1}(\alpha)}=t.$
(1.13)

$(iii)$ If one of the following three conditions holds

  * a)

$1<p\leq 1+2\alpha$ and $\tau\in(-1,0)\setminus\\{\tau_{1}(\alpha)\\}$,

  * b)

$1+2\alpha<p<1-\frac{2\alpha}{\tau_{1}(\alpha)}$ and
$\tau\in(-1,0)\setminus\\{\tau_{1}(\alpha),-\frac{2\alpha}{p-1}\\}$ or

  * c)

$p\geq 1-\frac{2\alpha}{\tau_{1}(\alpha)}$ and $\tau\in(-1,0)$,

then problem (1.8) does not admit any solution $u$ satisfying

$0<\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}u(x)D(x)^{-\tau}<+\infty.$
(1.14)

We observe that this theorem is similar to Theorem 1.1 in [3], where the role
of $\tau_{0}(\alpha)$ is played here by $\tau_{1}(\alpha)$. A quite different
situation occurs when $\alpha\in[\alpha_{0},1)$ and the function $c$ never
vanishes in $(-1,0)$. Precisely, we have

###### Theorem 1.2

Assume that $\alpha\in[\alpha_{0},1)$ and the assumptions on $\Omega$ and
$\mathcal{C}$. Then we have:
$(i)$ If $p>1+2\alpha$, then problem (1.8) admits a unique positive solution
$u$ satisfying (1.11).
$(ii)$ If $p>1,$ then problem (1.8) does not admit any solution $u$ satisfying
(1.14) for any $\tau\in(-1,0)\setminus\\{-\frac{2\alpha}{p-1}\\}$.

Comparing Theorem 1.1 with Theorem 1.2 we see that the range of existence for
the absorption term is quite larger for the second one, no constraint from
above. The main difference with the results in [3], Theorem 1.1, with
vanishing $f$ and $g$ occurs when $\alpha$ is large and the function $c$ does
not vanish, allowing thus for existence for all $p$ large. This difference
comes from the fact that the fractional Laplacian is a non-local operator so
that in the interior blow-up, in each of the domains $\Omega_{1}$ and
$\Omega_{2}$ there is a non-zero external value, the solutions itself acting
on the other side of $\mathcal{C}$.

The proof of our theorems is obtained through the use of super and sub-
solutions as in [3]. The main difficulty here is to find the appropriate super
and sub-solutions to apply the iteration technique to fractional elliptic
problem (1.8). Here we make use of some precise estimates based on the
function $c$ and the distance function $D$ near $\mathcal{C}$.

This article is organized as follows. In section §2, we introduce some
preliminaries and we prove the main estimates of the behavior of the
fractional Laplacian when applied to suitable powers of the function $D$. In
section §3 we prove the existence of solution to problem (1.8) as given in
Theorem 1.1 and Theorem 1.2. Finally, in Section §4 we prove the uniqueness
and nonexistence statements of these theorems.

##  2 Preliminaries

In this section, we recall some basic results from [3] and obtain some useful
estimate, which will be used in constructing super and sub-solutions of
problem (1.8). The first result states as:

###### Theorem 2.1

Assume that $p>1$ and there are super-solution $\bar{U}$ and sub-solution
$\underline{U}$ of problem (1.8) such that

$\bar{U}\geq\underline{U}\ \ {\rm{in}}\
\Omega\setminus\mathcal{C},\quad\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}\underline{U}(x)=+\infty,\quad\bar{U}=\underline{U}=0\
\ {\rm{in}}\ \Omega^{c}.$

Then problem (1.8) admits at least one positive solution $u$ such that

$\underline{U}\leq u\leq\bar{U}\ \ {\rm{in}}\ \Omega\setminus\mathcal{C}.$

Proof. The procedure is similar to the proof of Theorem 2.6 in [3], here we
give the main differences.

Let us define $\Omega_{n}:=\\{x\in\Omega\,|\,D(x)>1/n\\}$ then we solve

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}u_{n}(x)+|u_{n}|^{p-1}u_{n}(x)=0,&x\in\Omega_{n},\\\\[5.69054pt]
u_{n}(x)=\underline{U},&x\in\Omega_{n}^{c}.\\\\[5.69054pt] \end{array}\right.$
(2.1)

To find these solutions of (2.1) we observe that for fix $n$ the method of
section 3 of [8] applies even if the domain is not connected since the
estimate of Lemma 3.2 holds with $\delta<1/2n$ (see also Proposition 3.2 part
ii) in [3]), form here sub and super-solution can be construct for the
Dirichlet problem and then existence holds for (2.1) by an iteration technique
(see also section 2 of [3] for that procedure). Then as in Theorem 2.6 in [3]
we have

$\underline{U}\leq u_{n}\leq u_{n+1}\leq\bar{U}\ \ \rm{in}\ \ \Omega.$

By monotonicity of $u_{n}$, we can define

$u(x):=\lim_{n\to+\infty}u_{n}(x),\ x\in\Omega\ \ {\rm{and}}\ \ u(x):=0,\
x\in\Omega^{c}.$

Which, by a stability property, is a solution of problem (1.8) with the
desired properties. $\Box$

In order to prove our existence result, it is crucial to have available super
and sub-solutions to problem (1.8). To this end, we start describing the
properties of $c(\tau)$ defined in (1.9), which is a $C^{2}$ function in
$(-1,0)$.

###### Proposition 2.1

There exists a unique $\alpha_{0}\in(0,1)$ such that
$(i)$ For $\alpha\in[\alpha_{0},1)$, we have $c(\tau)>0,$ for all
$\tau\in(-1,0);$
$(ii)$ For any $\alpha\in(0,\alpha_{0})$, there exists unique
$\tau_{1}(\alpha)\in(-1,0)$ satisfying

$c(\tau)\
\left\\{\begin{array}[]{lll}>0,&\rm{if}\quad\tau\in(-1,\tau_{1}(\alpha)),\\\\[5.69054pt]
=0,&\rm{if}\quad\tau=\tau_{1}(\alpha),\\\\[5.69054pt]
<0,&\rm{if}\quad\tau\in(\tau_{1}(\alpha),0)\end{array}\right.$ (2.2)

and

$\lim_{\alpha\to\alpha_{0}^{-}}\tau_{1}(\alpha)=0\quad\rm{and}\quad\lim_{\alpha\to
0^{+}}\tau_{1}(\alpha)=-1.$ (2.3)

Moreover, $\tau_{1}(\alpha)>\tau_{0}(\alpha)$, for all
$\alpha\in(0,\alpha_{0})$, where $\tau_{0}(\alpha)\in(-1,0)$ is the unique
zero of $C(\tau)$, defined in (1.4).

Proof. From (1.9), differentiating twice we find that

$c^{\prime\prime}(\tau)=\int_{0}^{+\infty}\frac{|1-t|^{\tau}(\log|1-t|)^{2}+(1+t)^{\tau}(\log(1+t))^{2}}{t^{1+2\alpha}}dt>0,$
(2.4)

so that $c$ is strictly convex in $(-1,0)$. We also see easily that

$c(0)=0\quad\mbox{and}\quad\lim_{\tau\to-1^{+}}c(\tau)=\infty.$ (2.5)

Thus, if $c^{\prime}(0)\leq 0$ then $c(\tau)>0$ for $\tau\in(-1,0)$ and if
$c^{\prime}(0)>0$, then there exists $\tau_{1}(\alpha)\in(-1,0)$ such that
$c(\tau)>0$ for $\tau\in(-1,\tau_{1}(\alpha))$, $c(\tau)<0$ for
$\tau\in(\tau_{1}(\alpha),0)$ and $c(\tau_{1}(\alpha))=0$. In order to
complete our proof, we have to analyze the sign of $c^{\prime}(0)$, which
depends on $\alpha$ and to make this dependence explicit, we write
$c^{\prime}(0)=T(\alpha)$. We compute $T(\alpha)$ from (1.9), differentiating
and evaluating in $\tau=0$

$T(\alpha)=\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt.$ (2.6)

We have to prove that $T$ possesses a unique zero in the interval $(0,1)$. For
this purpose we start proving that

$\lim_{\alpha\to 1^{-}}T(\alpha)=-\infty\quad\mbox{and}\quad\lim_{\alpha\to
0^{+}}T(\alpha)=+\infty.$ (2.7)

The first limit follows from the fact that $\log(1-s)\leq-s,$ for all
$s\in[0,1/4]$, and so

$\displaystyle\lim_{\alpha\to
1^{-}}\int_{0}^{\frac{1}{2}}\frac{\log(1-t^{2})}{t^{1+2\alpha}}dt\leq-\lim_{\alpha\to
1^{-}}\int_{0}^{\frac{1}{2}}t^{1-2\alpha}dt=-\infty$

and the fact that exists a constant $t_{0}$ such that

$\int_{\frac{1}{2}}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\leq
t_{0},\qquad\mbox{for all }\alpha\in(1/2,1).$

The second limit in (2.7) follows from

$\displaystyle\lim_{\alpha\to
0^{+}}\int_{2}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\geq\log
3\lim_{\alpha\to 0^{+}}\int_{2}^{+\infty}t^{-1-2\alpha}dt=+\infty$

and the fact that there exists a constant $t_{1}$ such that

$\int_{0}^{2}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}dt\leq t_{1},\quad\mbox{for
all }\alpha\in(0,1/2).$

On the other hand we claim that

$T^{\prime}(\alpha)=-2\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}\log
tdt<0,\ \ \ \alpha\in(0,1).$ (2.8)

In fact, since ${\log|1-t^{2}|}\log t$ is negative only for
$t\in(1,\sqrt{2})$, we have

$\displaystyle\int_{0}^{+\infty}\frac{\log|1-t^{2}|}{t^{1+2\alpha}}\log tdt$
$\displaystyle>$
$\displaystyle\int_{0}^{\sqrt{2}-1}\frac{\log(1-t^{2})}{t^{1+2\alpha}}\log
tdt+\int_{1}^{\sqrt{2}}\log(t^{2}-1)\log tdt$ $\displaystyle\geq$
$\displaystyle\int_{0}^{\sqrt{2}-1}\frac{-t^{2}}{t^{1+2\alpha}}\log
tdt+\int_{1}^{\sqrt{2}}\log(t-1)\log tdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{\sqrt{2}-1}t^{1-2\alpha}\log
tdt+\int_{0}^{\sqrt{2}-1}\log(1+t)\log tdt$ $\displaystyle\geq$
$\displaystyle-\int_{0}^{\sqrt{2}-1}t^{1-2\alpha}\log
tdt+\int_{0}^{\sqrt{2}-1}t\log tdt>0.$

Then, (2.7) and (2.8) the existence of the desired $\alpha_{0}\in(0,1)$ with
the required properties follows, completing $(i)$ and (2.2) in $(ii)$.

To continue with the proof of our proposition, we study the first limit in
(2.3). We assume that there exist a sequence $\alpha_{n}\in(0,\alpha_{0})$ and
$\tilde{\tau}\in(-1,0)$ such that

$\lim_{n\to+\infty}\alpha_{n}=\alpha_{0}\quad\mbox{
and}\quad\lim_{n\to+\infty}\tau_{1}(\alpha_{n})=\tilde{\tau}$

and so $c(\tilde{\tau})=0$. Moreover $c(0)=0$ and
$c^{\prime}(0)=T(\alpha_{0})=0$, contradicting the strict convexity of $c$
given by (2.4). Next we prove the second limit in (2.3). We proceed by
contradiction, assuming that there exist a sequence
$\\{\alpha_{n}\\}\subset(0,1)$ and $\bar{\tau}\in(-1,0)$ such that

$\lim_{n\to+\infty}\alpha_{n}=0\quad\mbox{and}\quad\tau_{1}(\alpha_{n})\geq\bar{\tau}>-1,\quad\mbox{for
all }n\in\mathbb{N}.$

Then there exist $C_{1},C_{2}>0$, depending on $\bar{\tau}$, such that

$\displaystyle\int^{2}_{0}|\frac{|1-t|^{\tau_{1}(\alpha_{n})}+(1+t)^{\tau_{1}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}|dt\leq
C_{1}$

and

$\displaystyle\lim_{n\to\infty}\int_{2}^{+\infty}\frac{|1-t|^{\tau_{1}(\alpha_{n})}+(1+t)^{\tau_{1}(\alpha_{n})}-2}{t^{1+2\alpha_{n}}}dt\leq-
C_{2}\lim_{n\to\infty}\int_{2}^{+\infty}\frac{1}{t^{1+2\alpha_{n}}}dt=-\infty.$

Then $c(\tau_{1}(\alpha_{n}))\to-\infty$ as $n\to+\infty$, which is impossible
since $c(\tau_{1}(\alpha_{n}))=0.$

We finally prove the last statement of the proposition. Since
$\tau_{0}(\alpha)\in(-1,0)$ is such that $C(\tau_{0}(\alpha))=0$ and we have,
by definition, that

$c(\tau)=C(\tau)+\int_{1}^{+\infty}\frac{(t-1)^{\tau}}{t^{1+2\alpha}}dt,$

we find that $c(\tau_{0}(\alpha))>0$, which together with (2.2), implies that
$\tau_{0}(\alpha)\in(-1,\tau_{1}(\alpha)).$ $\Box$

Next we prove the main proposition in this section, which is on the basis of
the construction of super and sub-solutions. By hypothesis on the domain
$\Omega$ and the manifold $\mathcal{C}$, there exists $\delta>0$ such that the
distance functions $d(\cdot)$, to $\partial\Omega$, and $D(\cdot)$, to
$\mathcal{C}$, are of class $C^{2}$ in $B_{\delta}$ and $A_{\delta}$,
respectively, and $dist(A_{\delta},B_{\delta})>0$, where
$A_{\delta}=\\{x\in\Omega\ |\ D(x)<\delta\\}$ and $B_{\delta}=\\{x\in\Omega\
|\ d(x)<\delta\\}$. Now we define the basic function $V_{\tau}$ as follows

$V_{\tau}(x):=\left\\{\begin{array}[]{lll}D(x)^{\tau},&x\in
A_{\delta}\setminus\mathcal{C},\\\\[5.69054pt] d(x)^{2},&x\in
B_{\delta},\\\\[5.69054pt] l(x),&x\in\Omega\setminus(A_{\delta}\cup
B_{\delta}),\\\\[5.69054pt] 0,&x\in\Omega^{c},\end{array}\right.$ (2.9)

where $\tau$ is a parameter in $(-1,0)$ and the function $l$ is positive such
that $V_{\tau}$ is of class $C^{2}$ in $\mathbb{R}^{N}\setminus\mathcal{C}$.

###### Proposition 2.2

Let $\alpha_{0}$ and $\tau_{1}(\alpha)$ be as in Proposition 2.1.
$(i)$ If $(\alpha,\tau)\in[\alpha_{0},1)\times(-1,0)$ or
$(\alpha,\tau)\in(0,\alpha_{0})\times(-1,\tau_{1}(\alpha)),$ then there exist
$\delta_{1}\in(0,\delta]$ and $C>1$ such that

$\frac{1}{C}D(x)^{\tau-2\alpha}\leq-(-\Delta)^{\alpha}V_{\tau}(x)\leq
CD(x)^{\tau-2\alpha},\ \ x\in A_{\delta_{1}}\setminus\mathcal{C}.$

$(ii)$ If $(\alpha,\tau)\in(0,\alpha_{0})\times(\tau_{1}(\alpha),0),$ then
there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that

$\frac{1}{C}D(x)^{\tau-2\alpha}\leq(-\Delta)^{\alpha}V_{\tau}(x)\leq
CD(x)^{\tau-2\alpha},\ \ x\in A_{\delta_{1}}\setminus\mathcal{C}.$

$(iii)$ If $(\alpha,\tau)\in(0,\alpha_{0})\times\\{\tau_{1}(\alpha)\\},$ then
there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that

$|(-\Delta)^{\alpha}V_{\tau}(x)|\leq CD(x)^{\min\\{\tau,2\tau-2\alpha+1\\}},\
\ x\in A_{\delta_{1}}\setminus\mathcal{C}.$

This proposition and its proof has many similarities with Proposition 3.2 in
[3], but it has also important differences so we give a complete proof of it.

Proof. By compactness of $\mathcal{C}$, we just need to prove that the
corresponding inequality holds in a neighborhood of any point
$\bar{x}\in\mathcal{C}$ and, without loss of generality, we may assume
$\bar{x}=0$. For a given $0<\eta\leq\delta$, we define

$Q_{\eta}=(-\eta,\eta)\times B_{\eta}\subset\mathbb{R}\times\mathbb{R}^{N-1},$

where $B_{\eta}$ denotes the ball centered at the origin and with radius
$\eta$ in $\mathbb{R}^{N-1}$. We observe that $Q_{\eta}\subset\Omega.$ Let
$\varphi:\mathbb{R}^{N-1}\to\mathbb{R}$ be a $C^{2}$ function such that
$(z_{1},z^{\prime})\in\mathcal{C}\cap Q_{\delta}$ if and only if
$z_{1}=\varphi(z^{\prime})$. We further assume that $e_{1}$ is normal to
$\mathcal{C}$ at $\bar{x}$ and then there exists $C>0$ such that
$|\varphi(z^{\prime})|\leq C|z^{\prime}|^{2}$ for $|z^{\prime}|\leq\delta$.
Thus, choosing $\eta>0$ smaller if necessary we may assume that
$|\varphi(z^{\prime})|<\frac{\eta}{2}$ for $|z^{\prime}|\leq\eta$. In the
proof of our inequalities, we will consider a generic point along the normal
$x=(x_{1},0)\in A_{\eta/4}$, with $0<|x_{1}|<\eta/4$. We observe that
$|x-\bar{x}|=D(x)=|x_{1}|$. By definition we have

$-(-\Delta)^{\alpha}V_{\tau}(x)=\frac{1}{2}\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy+\frac{1}{2}\int_{\mathbb{R}^{N}\setminus
Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy.$ (2.10)

It is not difficult to see that the second integral is bounded by
$Cx_{1}^{\tau}$, for an appropriate constant $C>0$, so that we only need to
study the first integral, that from now on we denote by $\frac{1}{2}E(x_{1})$.

Our first goal is to obtain positive constants $c_{1},c_{2}$ so that lower
bound for $E(x_{1})$

$E(x_{1})\geq
c_{1}c(\tau)|x_{1}|^{\tau-2\alpha}-c_{2}|x_{1}|^{\min\\{\tau,2\tau-2\alpha+1\\}}$
(2.11)

holds, for all $|x_{1}|\leq\eta/4$. For this purpose we assume that
$0<\eta\leq\delta/2$, then for all $y=(y_{1},y^{\prime})\in Q_{\eta}$ we have
that $x\pm y\in Q_{\delta}$, so that

$D(x\pm y)\leq|x_{1}\pm y_{1}-\varphi(\pm y^{\prime})|,\quad\mbox{for\ all}\ \
y\in Q_{\eta}.$

From here and the fact that $\tau\in(-1,0)$, we have that

$E(x_{1})=\int_{Q_{\eta}}\frac{\delta(V_{\tau},x,y)}{|y|^{N+2\alpha}}dy\geq\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy+\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy,$
(2.12)

where the functions $I$ and $J$ are defined, for $y\in Q_{\eta}$, as

$I(y)=|x_{1}-y_{1}|^{\tau}+|x_{1}+y_{1}|^{\tau}-2x_{1}^{\tau}$ (2.13)

and

$J(y)=|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}.$ (2.14)

In what follows we assume $x_{1}>0$ (the case $x_{1}<0$ is similar). For the
first term of the right hand side in (2.12), we have

$\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy=x_{1}^{\tau-2\alpha}\int_{Q_{\frac{\eta}{x_{1}}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz.$

On one hand we have that, for a constant $c_{1}$, we have

$\displaystyle\int_{\mathbb{R}^{N}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz=2c(\tau)\int_{\mathbb{R}^{N-1}}\frac{1}{(|z^{\prime}|^{2}+1)^{\frac{N+2\alpha}{2}}}dz^{\prime}=c_{1}c(\tau),$

and, on the other hand, for constants $C_{2}$ and $C_{3}$ we have

$\displaystyle|\int_{-\frac{\eta}{x_{1}}}^{\frac{\eta}{x_{1}}}\int_{|z^{\prime}|\geq\frac{\eta}{x_{1}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz|$
$\displaystyle\leq$
$\displaystyle\int_{-\frac{\eta}{x_{1}}}^{\frac{\eta}{x_{1}}}(|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}+2)dz_{1}\int_{|z^{\prime}|\geq\frac{\eta}{x_{1}}}\frac{dz^{\prime}}{|z^{\prime}|^{N+2\alpha}}\leq
C_{2}x_{1}^{2\alpha}$

and

$\displaystyle|\int_{|z_{1}|\geq\frac{\eta}{x_{1}}}\int_{\mathbb{R}^{N-1}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}dz|$
$\displaystyle\leq$ $\displaystyle
2\int_{\frac{\eta}{x_{1}}}^{+\infty}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}+2}{z_{1}^{1+2\alpha}}dz_{1}\int_{\mathbb{R}^{N-1}}\frac{1}{(1+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz^{\prime}\leq
C_{3}x_{1}^{2\alpha}.$

Consequently, for an appropriate constant $c_{2}$

$|\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy-
c_{1}c(\tau)x_{1}^{\tau-2\alpha}|\leq c_{2}x_{1}^{\tau}.$ (2.15)

Next we estimate the second term of the right hand side in (2.12). Since

$\int_{Q_{\eta}}\frac{J(-y)}{|y|^{N+2\alpha}}dy=\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy,$

we only need to estimate

$\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy=\int_{B_{\eta}}\int^{\eta}_{-\eta}\frac{|x_{1}+y_{1}-\varphi(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}.$
(2.16)

We notice that $|x_{1}+y_{1}-\varphi(y^{\prime})|\geq|x_{1}+y_{1}|$ if and
only if

$\varphi(y^{\prime})(x_{1}+y_{1}-\frac{\varphi(y^{\prime})}{2})\leq 0.$

From here and (2.16), we have

$\displaystyle\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle\geq$
$\displaystyle\int_{B_{\eta}}\int^{-x_{1}+\frac{\varphi_{+}(y^{\prime})}{2}}_{-\eta}\frac{|x_{1}+y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\eta}_{-x_{1}+\frac{\varphi_{-}(y^{\prime})}{2}}\frac{|x_{1}+y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle E_{1}(x_{1})+E_{2}(x_{1}),$

where $\varphi_{+}(y^{\prime})=\max\\{\varphi(y^{\prime}),0\\}$ and
$\varphi_{-}(y^{\prime})=\min\\{\varphi(y^{\prime}),0\\}$. We only estimate
$E_{1}(x_{1})$ ($E_{2}(x_{1})$ is similar). Using integration by parts, we
obtain

$\displaystyle E_{1}(x_{1})$ (2.17) $\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{x_{1}-\eta}\frac{|y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|y_{1}|^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{0}_{x_{1}-\eta}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau}-y_{1}^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{-\varphi_{+}(y^{\prime})^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{(\eta-
x_{1}+\varphi_{+}(y^{\prime}))^{\tau+1}-(\eta-
x_{1})^{\tau+1}}{(\eta^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle-\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{0}_{x_{1}-\eta}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}-(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle+\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{-2^{-\tau}\varphi_{+}(y^{\prime})^{\tau+1}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{\varphi_{+}(y^{\prime})^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\frac{\varphi_{+}(y^{\prime})}{2}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}+y_{1}^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle\geq$
$\displaystyle\frac{-2^{-\tau}}{\tau+1}\int_{B_{\eta}}\frac{\varphi_{+}(y^{\prime})^{\tau+1}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\min\\{\frac{\varphi_{+}(y^{\prime})}{2},x_{1}\\}}_{0}\frac{(\varphi_{+}(y^{\prime})-y_{1})^{\tau+1}+y_{1}^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle A_{1}(x_{1})+A_{2}(x_{1}).$

In order to estimate $A(x_{1})$, we split $B_{\eta}$ in $O=\\{y^{\prime}\in
B_{\eta}:|\frac{\varphi_{+}(y^{\prime})}{2}-x_{1}|\geq\frac{x_{1}}{2}\\}$ and
$B_{\eta}\setminus O$. On one hand we have

$\displaystyle\int_{O}\frac{|y^{\prime}|^{2\tau+2}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle
x_{1}^{2\tau-2\alpha+1}\int_{B_{\eta/{x_{1}}}}\frac{|z^{\prime}|^{2\tau+2}}{(1/4+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dz^{\prime}$
$\displaystyle\leq$ $\displaystyle C(x_{1}^{2\tau-2\alpha+1}+x_{1}^{\tau}).$

On the other hand, for $y^{\prime}\in B_{\eta}\setminus O$ we have that
$|y^{\prime}|\geq c_{1}\sqrt{x_{1}}$, for some constant $c_{1}$, and then

$\displaystyle\int_{B_{\eta}\setminus
O}\frac{|y^{\prime}|^{2\tau+2}}{((\frac{\varphi_{+}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle\int_{B_{\eta}\setminus
B_{c_{1}\sqrt{x_{1}}}}|y^{\prime}|^{2\tau+2-N-2\alpha}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle C(x_{1}^{\tau-\alpha+\frac{1}{2}}+1).$

Thus, for some $C>0$,

$\displaystyle A_{1}(x_{1})\geq-Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
(2.18)

Next we estimate $A_{2}(x_{1})$:

$\displaystyle A_{2}(x_{1})$ $\displaystyle\geq$
$\displaystyle\frac{2(N+2\alpha)}{\tau+1}\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{\varphi_{+}(y^{\prime})^{\tau+1}(y_{1}-x_{1})}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dy_{1}dy^{\prime}$
$\displaystyle\geq$ $\displaystyle
C\int_{B_{\eta}}\int^{x_{1}}_{0}\frac{|y^{\prime}|^{2\tau+2}(y_{1}-x_{1})}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dy_{1}dy^{\prime}$
$\displaystyle\geq$ $\displaystyle
Cx_{1}^{2\tau-2\alpha+1}\int_{B_{\eta/{x_{1}}}}\int^{1}_{0}\frac{|z^{\prime}|^{2\tau+2}(z_{1}-1)}{((z_{1}-1)^{2}+|z^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}dz_{1}dz^{\prime}$
$\displaystyle\geq$ $\displaystyle-
C_{1}x_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}},$

for some $C,C_{1}>0$. From here, (2.17) and (2.18) we obtain, for some $C>0$

$E_{1}(x_{1})\geq-Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$

Using the similar estimate for $E_{2}(x_{1})$, we obtain

$\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy\geq-
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.19)

Thus, from (2.12), (2.15), (2.19) and noticing that these inequalities also
hold with $x_{1}<0$ with the obvious changes, we conclude the lower bound for
$E(x_{1})$ we gave in (2.11). Our second goal is to get an upper bound for
$E(x_{1})$ and for this, we first recall Lemma 3.1 in [3] to obtain

$D(x\pm y)^{\tau}\leq(x_{1}\pm
y_{1}-\varphi(y^{\prime}))^{\tau}(1+C|y^{\prime}|^{2}),\,\,\mbox{for
all}\quad|x_{1}|\leq\eta/4,y=(y_{1},y^{\prime})\in Q_{\eta}.$

From here we see that

$\displaystyle E(x_{1})$ $\displaystyle\leq$
$\displaystyle\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}dy+\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy$
(2.20)
$\displaystyle+C\int_{Q_{\eta}}\frac{I(y)+J(y)+J(-y)}{|y|^{N+2\alpha}}|y^{\prime}|^{2}dy.$

We denote by $E_{3}(x_{1})$ the third integral above. The first integral was
studied in (2.15), so we study the second integral and that we only need to
consider the term $J(y)$, since the other is completely analogous. We see that
$|x_{1}+y_{1}-\varphi(y^{\prime})|\leq|x_{1}+y_{1}|$ if and only if

$\varphi(y^{\prime})(x_{1}+y_{1}-\frac{\varphi(y^{\prime})}{2})\geq 0.$

As before, we will consider only the case $x_{1}>0$, since the other one is
analogous. From (2.16) we have

$\displaystyle\int_{Q_{\eta}}\frac{J(y)}{|y|^{N+2\alpha}}dy$
$\displaystyle\leq$
$\displaystyle\int_{B_{\eta}}\int^{-x_{1}+\frac{\varphi_{-}(y^{\prime})}{2}}_{-\eta}\frac{|x_{1}+y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\eta}_{-x_{1}+\frac{\varphi_{+}(y^{\prime})}{2}}\frac{|x_{1}+y_{1}-\varphi_{+}(y^{\prime})|^{\tau}-|x_{1}+y_{1}|^{\tau}}{(y_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$ $\displaystyle F_{1}(x_{1})+F_{2}(x_{1}).$

Next we estimate $F_{1}(x_{1})$ ($F_{2}(x_{1})$ is similar), using integration
by parts

$\displaystyle F_{1}(x_{1})$ $\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int_{x_{1}-\eta}^{\frac{\varphi_{-}(y^{\prime})}{2}}\frac{|y_{1}-\varphi_{-}(y^{\prime})|^{\tau}-|y_{1}|^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\int_{B_{\eta}}\int^{\varphi_{-}(y^{\prime})}_{x_{1}-\eta}\frac{(\varphi_{-}(y^{\prime})-y_{1})^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle+\int_{B_{\eta}}\int^{\frac{\varphi_{-}(y^{\prime})}{2}}_{\varphi_{-}(y^{\prime})}\frac{(y_{1}-\varphi_{-}(y^{\prime}))^{\tau}-(-y_{1})^{\tau}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy_{1}dy^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((x_{1}-\varphi_{-}(y^{\prime}))^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{(\eta-
x_{1}+\varphi_{-}(y^{\prime}))^{\tau+1}-(\eta-
x_{1})^{\tau+1}}{(\eta^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle-\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\varphi_{-}(y^{\prime})}_{x_{1}-\eta}\frac{(\varphi_{-}(y^{\prime})-y_{1})^{\tau+1}-(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle+\frac{1}{\tau+1}\int_{B_{\eta}}[\frac{2^{-\tau}(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}+\frac{-(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((x_{1}-\varphi_{-}(y^{\prime}))^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}]dy^{\prime}$
$\displaystyle+\frac{N+2\alpha}{\tau+1}\int_{B_{\eta}}\int^{\frac{\varphi_{-}(y^{\prime})}{2}}_{\varphi_{-}(y^{\prime})}\frac{(y_{1}-\varphi_{-}(y^{\prime}))^{\tau+1}+(-y_{1})^{\tau+1}}{((y_{1}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}+1}}(y_{1}-x_{1})dy_{1}dy^{\prime}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\tau+1}\int_{B_{\eta}}\frac{2^{-\tau}(-\varphi_{-}(y^{\prime}))^{\tau+1}}{((\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}=B(x_{1}).$

Since $(\frac{\varphi_{-}(y^{\prime})}{2}-x_{1})^{2}\geq x_{1}^{2}$, we have

$\displaystyle B(x_{1})$ $\displaystyle\leq$
$\displaystyle\frac{2^{-\tau}}{\tau+1}\int_{B_{\eta}}\frac{(-\varphi_{-}(y^{\prime}))^{\tau+1}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}$
$\displaystyle\leq$ $\displaystyle
C\int_{B_{\eta}}\frac{|y^{\prime}|^{2\tau+2}}{(x_{1}^{2}+|y^{\prime}|^{2})^{\frac{N+2\alpha}{2}}}dy^{\prime}\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}},$

for some $C>0$ independent of $x_{1}$. Thus we have obtained that

$F_{1}(x_{1})\leq Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.21)

Similarly, we can get an analogous estimate for $F_{2}(x_{1})$ and these two
estimates imply

$\int_{Q_{\eta}}\frac{J(y)+J(-y)}{|y|^{N+2\alpha}}dy\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.22)

Finally we obtain

$\displaystyle\int_{Q_{\eta}}\frac{I(y)}{|y|^{N+2\alpha}}|y^{\prime}|^{2}dy$
$\displaystyle=$ $\displaystyle
x_{1}^{\tau-2\alpha+2}\int_{Q_{\frac{\eta}{x_{1}}}}\frac{|1-z_{1}|^{\tau}+|1+z_{1}|^{\tau}-2}{|z|^{N+2\alpha}}|z^{\prime}|^{2}dz$
$\displaystyle\leq$ $\displaystyle Cx_{1}^{\min\\{\tau,\tau-2\alpha+2\\}}$

and, in a similar way,

$\int_{Q_{\eta}}\frac{J(y)|y^{\prime}|^{2}}{|y|^{N+2\alpha}}dy\leq
Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$

From the last two inequalities we obtain

$E_{3}(x_{1})\leq Cx_{1}^{\min\\{\tau,2\tau-2\alpha+1\\}}.$ (2.23)

Then, taking into account (2.20), (2.15), (2.22), (2.23) and considering also
the case $x_{1}<0$, we obtain

$E(x_{1})\leq
c_{1}c(\tau)|x_{1}|^{\tau-2\alpha}+c_{2}|x_{1}|^{\min\\{\tau,2\tau-2\alpha+1\\}}.$
(2.24)

From inequalities (2.11), (2.24) and Proposition 2.1 the result follows.
$\Box$

##  3 Existence of large solution

This section is devoted to use Proposition 2.2 to prove the existence of
solution of problem (1.8). To this purpose, our main goal is to construct
appropriate sub-solution and super-solution of problem (1.8) under the
hypotheses of Theorem 1.1 $(i)$, $(ii)$ and Theorem 1.2 $(i)$.

We begin with a simple lemma that reduces the problem to find them only in
$A_{\delta}\setminus\mathcal{C}$.

###### Lemma 3.1

Let $U$ and $W$ be classical ordered super and sub-solution of (1.8) in the
sub-domain $A_{\delta}\setminus\mathcal{C}$. Then there exists $\lambda$ large
such that $U_{\lambda}=U+\lambda\bar{V}$ and $W_{\lambda}=W-\lambda\bar{V}$,
are ordered super and sub-solution of (1.8), where $\bar{V}$ is the solution
of

$\left\\{\begin{array}[]{lll}(-\Delta)^{\alpha}\bar{V}(x)=1,&x\in\Omega,\\\\[5.69054pt]
\bar{V}(x)=0,&x\in\Omega^{c}.\end{array}\right.$ (3.1)

###### Remark 3.1

Here $U,W:I\\!\\!R^{N}\to\mathbb{R}$ are classical ordered of super and sub-
solution of (1.8) in the sub-domain $A_{\delta}\setminus\mathcal{C}$ if $U$
satisfies

$(-\Delta)^{\alpha}U+|U|^{p-1}U\geq 0\quad\mbox{in}\quad
A_{\delta}\setminus\mathcal{C}$

and $W$ satisfies the reverse inequality. Moreover, they satisfy

$U\geq W\ \ {\rm{in}}\
\Omega\setminus\mathcal{C},\quad\liminf_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}W(x)=+\infty,\quad
U=W=0\ \ {\rm{in}}\ \Omega^{c}.$

Proof. Notice that by the maximum principle $\bar{V}$ is nonnegative in
$\Omega$, therefore $U_{\lambda}\geq U$ and $W_{\lambda}\leq W$, so they are
still ordered. In addition $U_{\lambda}$ satisfies

$(-\Delta)^{\alpha}U_{\lambda}+|U_{\lambda}|^{p-1}U_{\lambda}\geq(-\Delta)^{\alpha}U+|U|^{p-1}U+\lambda>0,\quad\mbox{in}\quad\Omega\setminus\mathcal{C}.$

This inequality holds because of our assumption in
$A_{\delta}\setminus\mathcal{C}$ and the fact that
$(-\Delta)^{\alpha}U+|U|^{p-1}U$ is continuous in
$\Omega\setminus{A_{\delta}}$ and by taking $\lambda$ large enough.

By the same type of arguments we find that $W_{\lambda}$ is a sub-solution.
$\Box$

Proof of existence results in Theorem 1.1 $(i)$ and Theorem 1.2 $(i)$. We
define

$U_{\mu}(x)=\mu V_{\tau}(x)\ {\rm{and}}\ \ W_{\mu}(x)=\mu V_{\tau}(x),\
x\in\mathbb{R}^{N}\setminus\mathcal{C},$ (3.2)

where $V_{\tau}$ is defined in (2.9) with $\tau=-\frac{2\alpha}{q-1}$

1\. $U_{\mu}$ is Super-solution. By hypotheses of Theorem 1.1 $(i)$ and
Theorem 1.2 $(i)$, we notice that

$\tau\in(-1,0),\quad\rm{for}\ \alpha\in[\alpha_{0},1),$
$\tau\in(-1,\tau_{1}(\alpha)),\quad\rm{for}\ \alpha\in(0,\alpha_{0})$

and $\tau p=\tau-2\alpha$, then we use Proposition 2.2 part $(i)$ to obtain
that there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that

$\displaystyle(-\Delta)^{\alpha}U_{\mu}(x)+U^{p}_{\mu}(x)\geq-C\mu
D(x)^{\tau-2\alpha}+\mu^{p}D(x)^{\tau p},\quad x\in
A_{\delta_{1}}\setminus\mathcal{C}.$

Then there exist $\mu_{1}>1$ such that for $\mu\geq\mu_{1}$, we have

$(-\Delta)^{\alpha}U_{\mu}(x)+U^{p}_{\mu}(x)\geq 0,\ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$

2\. $W_{\mu}$ is Sub-solution.  We use Proposition 2.2 part $(i)$ to obtain
that there exist $\delta_{1}\in(0,\delta]$ and $C>1$ such that for $x\in
A_{\delta_{1}}\setminus\mathcal{C}$, we have

$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)$
$\displaystyle\leq$
$\displaystyle-\frac{\mu}{C}D(x)^{\tau-2\alpha}+\mu^{p}D(x)^{\tau p}$
$\displaystyle\leq$
$\displaystyle(-\frac{\mu}{C}+\mu^{p})D(x)^{\tau-2\alpha}.$

Then there exists $\mu_{3}\in(0,1)$ such that for all $\mu\in(0,\mu_{3})$, it
has

$(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)\leq 0,\ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$

To conclude the proof we use Lemma 3.1 and Proposition 2.2. $\Box$

Proof of Theorem 1.1 $(ii)$. For any given $t>0$, we denote

$U(x)=tV_{\tau_{1}(\alpha)}(x),\quad x\in\mathbb{R}^{N}\setminus\mathcal{C},$
$W_{\mu}(x)=tV_{\tau_{1}(\alpha)}(x)-\mu V_{\bar{\tau}}(x),\quad
x\in\mathbb{R}^{N}\setminus\mathcal{C}$

where
$\bar{\tau}=\min\\{\tau_{1}(\alpha)p+2\alpha,\frac{1}{2}\tau_{1}(\alpha)\\}<0$.
By (1.12), we have

$\bar{\tau}\in(\tau_{1}(\alpha),0),\
\bar{\tau}-2\alpha<\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}\
\rm{and}\ \bar{\tau}-2\alpha<\tau_{1}(\alpha)p.$ (3.3)

1\. $U$ is Super-solution. We use Proposition 2.2 $(iii)$ to obtain that for
any $x\in A_{\delta_{1}}\setminus\mathcal{C}$,

$\displaystyle(-\Delta)^{\alpha}U(x)+U^{p}(x)\geq-
CtD(x)^{\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}}+t^{p}D(x)^{\tau_{1}(\alpha)p},$

together with
$\tau_{1}(\alpha)p<\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}$,
then there exists $\delta_{2}\in(0,\delta_{1}]$ such that

$(-\Delta)^{\alpha}U(x)+U^{p}(x)\geq 0,\quad x\in
A_{\delta_{2}}\setminus\mathcal{C}.$

2\. $W_{\mu}$ is Sub-solution. We use Proposition 2.2 $(ii)$ and $(iii)$ to
obtain that for $x\in A_{\delta_{1}}\setminus\mathcal{C}$,

$\displaystyle(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)$
$\displaystyle\leq$ $\displaystyle
CtD(x)^{\min\\{\tau_{1}(\alpha),2\tau_{1}(\alpha)-2\alpha+1\\}}$
$\displaystyle-\frac{\mu}{C}D(x)^{\bar{\tau}-2\alpha}+t^{p}D(x)^{\tau_{1}(\alpha)p}.$

Then there exists $\delta_{2}\in(0,\delta_{1}]$ such that for any $\mu\geq 1$,
we have

$(-\Delta)^{\alpha}W_{\mu}(x)+|W_{\mu}|^{p-1}W_{\mu}(x)\leq 0,\ x\in
A_{\delta_{2}}\setminus\mathcal{C}.$

To conclude the proof we use Lemma 3.1 and Proposition 2.2. $\Box$

##  4 Uniqueness and nonexistence

We prove the uniqueness statement by contradiction. Assume that $u$ and $v$
are solutions of problem (1.8) satisfying (1.11). Then there exist $C_{0}\geq
1$ and $\bar{\delta}\in(0,\delta)$ such that

$\frac{1}{C_{0}}\leq v(x)D(x)^{-\tau},\ u(x)D(x)^{-\tau}\leq C_{0},\ \ \forall
x\in A_{\bar{\delta}}\setminus\mathcal{C},$ (4.4)

where $\tau=-\frac{2\alpha}{p-1}$. We denote

$\mathcal{A}=\\{x\in\Omega\setminus\mathcal{C}\ |\ u(x)>v(x)\\}.$ (4.5)

Then $\mathcal{A}$ is open and $\mathcal{A}\subset\Omega$. Then the uniqueness
in Theorem 1.2 $(i)$ and Theorem 1.1 $(i)$ is a consequence of the following
result:

###### Proposition 4.1

Under the hypotheses of Theorem 1.2 $(i)$ and Theorem 1.1 $(i)$, we have

$\mathcal{A}=\O.$

Proof. The procedure of proof is similar as Section§5 in [3], noting that we
need to replace $d(x)$ by $D(x)$ and $\partial\Omega$ by $\mathcal{C}$ .
$\Box$

From Proposition 4.1, we can prove uniqueness part in Theorem 1.1 $(i)$ and
Theorem 1.2 $(i)$ .

The final goal in this note is to consider the nonexistence of solutions of
problem (1.8) under the hypotheses of Theorem 1.1 $(iii)$ and Theorem 1.2
$(ii)$.

###### Proposition 4.2

Under the hypotheses of Theorem 1.1 $(iii)$ and Theorem 1.2 $(ii)$, we assume
that $U_{1}$ and $U_{2}$ are both sub-solutions (or both super-solutions) of
(1.8) satisfying that $U_{1}=U_{2}=0$ in $\Omega^{c}$ and

$\displaystyle 0$ $\displaystyle<$
$\displaystyle\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}$ $\displaystyle<$
$\displaystyle\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}\leq\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}<+\infty,$

for $\tau\in(-1,0)$. For the case $\tau p>\tau-2\alpha$, we further assume
that
$(i)$ if $U_{1},U_{2}$ are sub-solutions, there exist $C>0$ and
$\tilde{\delta}>0$,

$(-\Delta)^{\alpha}U_{2}(x)\leq-CD(x)^{\tau-2\alpha},\quad x\in
A_{\tilde{\delta}}\setminus\mathcal{C};$ (4.6)

or
$(ii)$ if $U_{1},U_{2}$ are super-solutions, there exist $C>0$ and
$\tilde{\delta}>0$,

$(-\Delta)^{\alpha}U_{1}(x)\geq CD(x)^{\tau-2\alpha},\quad x\in
A_{\tilde{\delta}}\setminus\mathcal{C}.$ (4.7)

Then there doesn’t exist any solution $u$ of (1.8) such that

$\limsup_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}\frac{U_{1}(x)}{u(x)}<1<\liminf_{x\in\Omega\setminus\mathcal{C},\
x\to\mathcal{C}}\frac{U_{2}(x)}{u(x)}.$ (4.8)

Proof. The proof is similar as Proposition 6.1 in [3], noting again that we
need to replace $d(x)$ by $D(x)$ and $\partial\Omega$ by $\mathcal{C}$ .
$\Box$

With the help of Proposition 2.2, for given $t_{1}>t_{2}>0$, we construct two
sub-solutions (or both super-solutions) $U_{1}$ and $U_{2}$ of (1.8) such that

$\lim_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}U_{1}(x)D(x)^{-\tau}=t_{1},\
\lim_{x\in\Omega\setminus\mathcal{C},x\to\mathcal{C}}U_{2}(x)D(x)^{-\tau}=t_{2}.$

So what we have to do is to prove that for any $t>0$, we can construct super-
solution (sub-solution) of problem (1.8).

Proof of Theorem 1.1 $(iii)$ and Theorem 1.2 $(ii)$. We divide our proof of
the nonexistence results into several cases under the assumption $p>1$.

Zone 1: We consider $\tau\in(\tau_{1}(\alpha),0)$ and
$\alpha\in(0,\alpha_{0}).$ By Proposition 2.2 $(ii)$, there exists
$\delta_{1}>0$ such that

$(-\Delta)^{\alpha}V_{\tau}(x)\geq\frac{1}{C}D(x)^{\tau-2\alpha},\ \ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$ (4.9)

Since $V_{\tau}$ is $C^{2}$ in $\Omega\setminus\mathcal{C}$, then there exists
$C>0$ such that

$|(-\Delta)^{\alpha}V_{\tau}(x)|\leq C,\ \ x\in\Omega\setminus
A_{\delta_{1}}.$ (4.10)

Let $\bar{U}:=V_{\tau}+C\bar{V}\quad\rm{in}\ \
\mathbb{R}^{N}\setminus\mathcal{C}$, then we have $\bar{U}>0$ in
$\Omega\setminus\mathcal{C}$,

$(-\Delta)^{\alpha}\bar{U}\geq 0\ \ {\rm{in}}\ \
\Omega\setminus\mathcal{C}\quad{\rm{and}}\ \
(-\Delta)^{\alpha}\bar{U}(x)\geq\frac{1}{C}D(x)^{\tau-2\alpha},\ \ x\in
A_{\delta_{1}}\setminus\mathcal{C}.$

Then, we have that $t\bar{U}$ is super-solution of (1.8) for any $t>0$. Using
Proposition 4.2, we see that there is no solution of (1.8) satisfying (1.14).
Zone 2: We consider $\tau-2\alpha<\tau p$ and

$\tau\in\left\\{\begin{array}[]{lll}(-1,0),&\alpha\in[\alpha_{0},1),\\\\[5.69054pt]
(-1,\tau_{1}(\alpha)),&\alpha\in(0,\alpha_{0}).\end{array}\right.$

Let us define

$W_{\mu,t}=tV_{\tau}-\mu\bar{V}\quad{\rm{in}}\ \
\mathbb{R}^{N}\setminus\mathcal{C},$

where $t,\mu>0$. By Proposition 2.2 $(i)$, for $x\in
A_{\delta_{1}}\setminus\mathcal{C}$,

$\displaystyle(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq-\frac{t}{C}D(x)^{\tau-2\alpha}+t^{p}D(x)^{\tau
p}.$

For any fixed $t>0$, there exists $\delta_{2}\in(0,\delta_{1}]$, for all
$\mu\geq 0$,

$(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq 0,\quad
A_{\delta_{2}}\setminus\mathcal{C}.$ (4.11)

To consider $x\in\Omega\setminus A_{\delta_{2}}$, in fact, there exists
$C_{1}>0$ such that

$t|(-\Delta)^{\alpha}V_{\tau}(x)|+t^{p}V_{\tau}^{p}(x)\leq C_{1},\quad
x\in\Omega\setminus A_{\delta_{2}}$

and

$\displaystyle(-\Delta)^{\alpha}W_{\mu,t}(x)+|W_{\mu,t}|^{p-1}W_{\mu,t}(x)\leq
C_{1}t-\mu,\quad x\in\Omega\setminus A_{\delta_{2}}$

For given $t>0$, there exists $\mu(t)>0$ such that

$(-\Delta)^{\alpha}W_{\mu(t),t}(x)+|W_{\mu,t}|^{p-1}W_{\mu(t),t}(x)\leq 0,\ \
x\in\Omega\setminus A_{\delta_{2}}.$ (4.12)

Therefore, together with (4.11) and (4.12), for any given $t>0$, there sub-
solutions $W_{\mu(t),t}$ of problem (1.8) and by Proposition 4.2, we see that
there is no solution $u$ of (1.8) satisfying (1.14).
Zone 3: We consider $\tau-2\alpha>\tau p$ and

$\tau\in\left\\{\begin{array}[]{lll}(-1,0),&\alpha\in[\alpha_{0},1),\\\\[5.69054pt]
(-1,\tau_{1}(\alpha)),&\alpha\in(0,\alpha_{0}).\end{array}\right.$

We denote that

$U_{\mu,t}=tV_{\tau}+\mu\bar{V}\quad\rm{in}\ \
\mathbb{R}^{N}\setminus\mathcal{C},$

where $t,\mu>0$. Here $U_{\mu,t}>0$ in $\Omega\setminus\mathcal{C}$. By
Proposition 2.2 $(i)$,

$\displaystyle(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq-
CtD(x)^{\tau-2\alpha}+t^{p}D(x)^{\tau p},\quad x\in
A_{\delta_{1}}\setminus\mathcal{C}.$

For any fixed $t>0$, there exists $\delta_{2}\in(0,\delta_{1}]$, for all
$\mu\geq 0$,

$(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq 0,\quad x\in
A_{\delta_{2}}\setminus\mathcal{C}.$ (4.13)

For $x\in\Omega\setminus A_{\delta_{2}}$, we see that
$(-\Delta)^{\alpha}V_{\tau}$ is bounded and

$\displaystyle(-\Delta)^{\alpha}U_{\mu,t}(x)+U^{p}_{\mu,t}(x)\geq-Ct+\mu.$

For given $t>0$, there exists $\mu(t)>0$ such that

$(-\Delta)^{\alpha}U_{\mu(t),t}(x)+U^{p}_{\mu(t),t}(x)\geq 0,\ \
x\in\Omega\setminus A_{\delta_{2}}.$ (4.14)

Combining with (4.13) and (4.14), we have that for any $t>0$, there exists
$\mu(t)>0$ such that

$(-\Delta)^{\alpha}U_{\mu(t),t}(x)+U^{p}_{\mu(t),t}(x)\geq 0,\ \ \
x\in\Omega\setminus\mathcal{C}.$

Therefore, for any given $t>0$, there is a super-solution $U_{\mu(t),t}$ of
problem (1.8) and by Proposition 4.2, we see that there is no solution of
(1.8) satisfying (1.14).

We see that Zones 1 and 2 cover Theorem 1.1 part $(iii)$ a) since
$\tau>-2\alpha/(p-1).$ From Zones 1, 2 and 3 we cover Theorem 1.1 part $(iii)$
b) since $\tau_{1}(\alpha)>2\alpha/(p-1)$. Moreover, from Zone 1 to Zone 3, we
cover the parameters in part $(iii)$ c) of Theorem 1.1, since
$\tau_{1}(\alpha)<2\alpha/(p-1)$. Finally Theorem 1.2 part ii) can be obtained
from Zone 2 and Zone 3. This complete the proof. $\Box$

Acknowledgements. The authors thanks Peter Bates for proposing the problem.
H.C. was partially supported by Conicyt Ph.D. scholarship. P.F. was partially
supported by Fondecyt # 1110291 and Programa BASAL-CMM U. de Chile. A.Q. was
partially supported by Fondecyt # 1110210 and Programa BASAL-CMM U. de Chile.

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