Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-H\'enon equations

TL;DR

This paper analyzes the asymptotic behavior and sharp blow-up estimates of positive solutions with isolated singularities to fractional Hardy-Hénon equations, extending classical results to the fractional setting with new methods.

Contribution

It provides a classification of isolated singularities and precise asymptotic behavior for fractional Hardy-Hénon equations, using novel techniques different from the classical ODE approach.

Findings

Classified isolated singularities of positive solutions.

Derived sharp blow-up estimates near singularities.

Described asymptotic behavior at infinity for solutions.

Abstract

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H\'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $\sigma \in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2\sigma < \alpha < 2\sigma$ and $\frac{n+\alpha}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981), but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on $\mathbb{S}^{n}_+$. We also investigate isolated singularities located at infinity of fractional Hardy-H\'enon equations.

Equations

$$(-\Delta {)}^{\sigma }u=|x{|}^{\alpha }{u}^p\text{in}{B}_1\backslash \{0\}$$

$$-\Delta u=|x{|}^{\alpha }{u}^p\text{in}{B}_1\backslash \{0\}$$

$$-2<\alpha <2\text{and}\frac{n+\alpha }{n-2}<p<\frac{n+2}{n-2}.$$

$$\frac{c_1}{|x{|}^{(2+\alpha )/(p-1)}}\le u(x)\le \frac{c_2}{|x{|}^{(2+\alpha )/(p-1)}}\text{near}x=0.$$

$$|x{|}^{(2+\alpha )/(p-1)}u(x)\to C_0\text{as}x\to 0,$$

$$C_0={\left\{\frac{(2+\alpha )(n-2)}{(p-1{)}^2}\left(p-\frac{n+\alpha }{n-2}\right)\right\}}^{1/(p-1)}.$$

$$u(x)=\bar{u}(|x|)(1+O(|x|))\text{as}x\to 0,$$

$$-2<\alpha \le 0\text{and}1<p\le \frac{n+2+\alpha }{n-2}.$$

$$(-\Delta {)}^{\sigma }u=|x{|}^{\alpha }{u}^p\text{in}{B}_1\backslash \{0\}$$

$$(-\Delta {)}^{\sigma }u(x)=c_{n,\sigma }P.V.{\int }_{{\mathbb{R}}^n}\frac{u(x)-u(\xi )}{|x-\xi {|}^{n+2\sigma }}d\xi ,$$

$$u(x)=\bar{u}(|x|)(1+O(|x|))\text{as}x\to 0,$$

$$-2\sigma <\alpha \le 0\text{and}\frac{n+\alpha }{n-2\sigma }<p\le \frac{n+2\sigma +2\alpha }{n-2\sigma }.$$

$$\frac{n+\alpha }{n-2\sigma }<p\le \frac{n+2\sigma +\alpha }{n-2\sigma }\text{and}p\ne \frac{n+2\sigma +2\alpha }{n-2\sigma }.$$

$$u\in {\mathcal{L}}_{\sigma }({\mathbb{R}}^n):=\left\{u\in L_{\text{loc}}^1({\mathbb{R}}^n):{\int }_{{\mathbb{R}}^n}\frac{|u(x)|}{(1+|x|{)}^{n+2\sigma }}dx<+\infty \right\},$$

$$|x{|}^{(2\sigma +\alpha )/(p-1)}u(x)\to C_{p,\sigma ,\alpha }\text{as}x\to 0,$$

$$C_{p,\sigma ,\alpha }={\left\{\Lambda \left(\frac{n-2\sigma }{2}-\frac{2\sigma +\alpha }{p-1}\right)\right\}}^{\frac{1}{p-1}}$$

$$\Lambda (\tau )=2^{2\sigma }\frac{\Gamma (\frac{n+2\sigma +2\tau }{4})\Gamma (\frac{n+2\sigma -2\tau }{4})}{\Gamma (\frac{n-2\sigma -2\tau }{4})\Gamma (\frac{n-2\sigma +2\tau }{4})}.$$

$$u^{\ast }(|x|)\equiv C_{p,\sigma ,\alpha }|x{|}^{-\frac{2\sigma +\alpha }{p-1}}.$$

$$p_S(\alpha ):=\frac{n+2\sigma +2\alpha }{n-2\sigma }.$$

$$p_S(\alpha )<p\le \frac{n+2\sigma +\alpha }{n-2\sigma }.$$

$$-2\sigma <\alpha <2\sigma \text{and}\frac{n+\alpha }{n-2\sigma }<p<\frac{n+2\sigma }{n-2\sigma }.$$

$$\frac{C_1}{|x{|}^{(2\sigma +\alpha )/(p-1)}}\le u(x)\le \frac{C_2}{|x{|}^{(2\sigma +\alpha )/(p-1)}}\text{for }x\in B_{1/2}\backslash \{0\}.$$

$$\underset{|x|\to 0}{\lim }|x{|}^{\frac{2+\alpha }{p-1}}u(x)=0."$$

$$\{\begin{array}{ll}-\text{div}(t^{1-2\sigma }\nabla U)=0& \text{in}{\mathcal{B}}_1^{+},\\ \frac{\partial U}{\partial {\nu }^{\sigma }}(x,0)={\kappa }_{\sigma }|x{|}^{\alpha }{U}^p(x,0)& \text{on}{\partial }^0{\mathcal{B}}_1^{+}\backslash \{0\},\end{array}$$

$${\int }_{{\mathcal{B}}_1^{+}}{t}^{1-2\sigma }\nabla U\cdot \nabla \Phi ={\kappa }_{\sigma }{\int }_{{\partial }^0{\mathcal{B}}_1^{+}}|x{|}^{\alpha }{U}^p\Phi$$

$$-2\sigma <\alpha <2\sigma \text{and}\frac{n+\alpha }{n-2\sigma }<p<\frac{n+2\sigma }{n-2\sigma }.$$

$$\frac{C_1}{|X{|}^{(2\sigma +\alpha )/(p-1)}}\le U(X)\le \frac{C_2}{|X{|}^{(2\sigma +\alpha )/(p-1)}}.$$

$$U(x,t)=C_{p,\sigma,\alpha}\int_{\mathbb{R}^{n}}P_{\sigma}(x-y,t)|y|^{-\frac{2\sigma+\alpha}{p-1}}dy\Big{(}1+o(1)\Big{)}~{}~{}~{}~{}~{}~{}~{}~{}\textmd{as}~{}(x,t)\to 0,$$

$$P_{\sigma }(x,t)=p_{n,\sigma }\frac{t^{2\sigma }}{(|x{|}^2+t^2{)}^{\frac{n+2\sigma }{2}}}$$

$$(-\Delta {)}^{\sigma }u=|x{|}^{\alpha }{u}^p\text{in}\{x\in {\mathbb{R}}^n:|x|>1\}$$

Full text

Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations

**Hui Yang ,   Wenming Zou

** **Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. E-mail address: mahuiyang@ust.hk****Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. W. Zou was supported by NSFC. E-mail address: zou-wm@mail.tsinghua.edu.cn **

Abstract

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation

[TABLE]

with an isolated singularity at the origin, where $\sigma\in(0,1)$ and the punctured unit ball $B_{1}\backslash\{0\}\subset\mathbb{R}^{n}$ with $n\geq 2$. When $-2\sigma<\alpha<2\sigma$ and $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981), but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on $\mathbb{S}^{n}_{+}$. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.

*Keywords: * Isolated singularities; Neumann boundary isolated singularities; Precise asymptotic behavior; Monotonicity formula; Fractional Hardy-Hénon equations

*Mathematics Subject Classification (2010): 35R11; 35J70; 35B09; 35B40 *

1 Introduction and Main Results

In the classical paper [21], Gidas and Spruck studied the asymptotic behavior of positive solutions of the following equation

[TABLE]

with an isolated singularity at the origin, where the punctured unit ball $B_{1}\backslash\{0\}\subset\mathbb{R}^{n}$ with $n\geq 3$. Eq. (1.1) is usually called the Hardy (resp., Lane-Emden, or Hénon) equation for $\alpha<0$ (resp., $\alpha=0$, $\alpha>0$). More specifically, assume

[TABLE]

Let $u\in C^{2}(B_{1}\backslash\{0\})$ be a positive solution of (1.1). Gidas-Spruck [21] proved that either the singularity at $x=0$ is removable, or there exist positive constants $c_{1},c_{2}$ such that

[TABLE]

Further, assume additionally that $p\neq\frac{n+2+2\alpha}{n-2}$, then they used the ODEs method and the sharp blow up estimate (1.2) to derive the precise asymptotic behavior of singular solutions of (1.1)

[TABLE]

where

[TABLE]

When $\alpha=0$, Caffarelli-Gidas-Spruck [8] found that every local positive solution $u$ of (1.1) with $\frac{n}{n-2}\leq p\leq\frac{n+2}{n-2}$ is asymptotically radially symmetric

[TABLE]

where $\bar{u}(|x|)=\frac{1}{|\mathbb{S}^{n-1}|}\int_{\mathbb{S}^{n-1}}u(|x|\omega)d\omega$ is the spherical average of $u$. With the help of this asymptotic radial symmetry, they used the classical ODEs analysis to obtain the precise behavior of positive solutions near the singularity of (1.1) when $\alpha=0$ and $\frac{n}{n-2}\leq p\leq\frac{n+2}{n-2}$.

In [28], Li proved the asymptotic radial symmetry of positive solutions of (1.1) with

[TABLE]

For other cases of $\alpha$ and $p$, the asymptotic behavior of singular positive solutions of (1.1) has also been understood very well, see Brezis-Lions [6] and Lions [31] for $\alpha=0$ and $1<p<\frac{n}{n-2}$, Zhang-Zhao [36] for $-2<\alpha<2$ and $1<p<\frac{n+\alpha}{n-2}$, Aviles [3] for $-2<\alpha<2$ and $p=\frac{n+\alpha}{n-2}$, Korevaar-Mazzeo-Pacard-Schoen [27] for $\alpha=0$ and $p=\frac{n+2}{n-2}$, and Bidaut-Véron and Véron [4] for $\alpha=0$ and $p>\frac{n+2}{n-2}$.

In recent years, there has been an increasing interest in the study of equations involving a nonlocal diffusion operator, especially, the fractional Laplacian, motivated by models of diverse physical phenomena such as anomalous diffusion and quasi-geostrophic flows [5, 11] and by applications in conformal geometry [12, 22, 23, 24]. Specially, the following type of fractional equation

[TABLE]

has received great interest and has been widely studied in [2, 1, 9, 14, 15, 16, 17, 22, 23, 25, 26, 29, 30, 34, 35] and references therein. Here $\sigma\in(0,1)$ and the fractional Laplacian operator $(-\Delta)^{\sigma}$ is defined as

[TABLE]

where $c_{n,\sigma}$ is a normalization constant depending only on $n$ and $\sigma$ and $P.V.$ stands for the Cauchy principal value. In particular, when $\alpha=0$ and $p=\frac{n+2\sigma}{n-2\sigma}$, the aforementioned asymptotic symmetry result of Caffarelli-Gidas-Spruck has been generalized to the fractional setting by Caffarelli, Jin, Sire and Xiong [9]. More precisely, the authors in [9] classified isolated singularities of positive solutions of (1.3) and showed that every local positive solution $u$ of (1.3) is asymptotically radially symmetric

[TABLE]

where $\bar{u}(|x|)$ is the spherical average of $u$. Li-Bao [29] extended this asymptotic radial symmetry of positive solutions to the equation (1.3) with

[TABLE]

However, since the classical ODEs analysis is a missing ingredient in the fractional setting to further analyze the solutions of (1.3) compared to the case when $\sigma=1$, the precise asymptotic behavior of positive solutions near the singularity to the fractional equation (1.3) remains as an open question. In the recent papers [34, 35], we established a monotonicity formula to classify isolated singularities and prove the precise asymptotic behavior of solutions to the fractional Lane-Emden equation ((1.3) with $\alpha=0$) when $\frac{n}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$. We also refer to Fall-Felli [19] for the precise asymptotic behavior of solutions to fractional elliptic equations with Hardy type potentials.

One of the goals of this paper is to describe the precise asymptotic behavior of positive solutions near the singularity to the problem (1.3) with Hardy weights ($-2\sigma<\alpha<0$) when

[TABLE]

One motivation for studying singular solutions of (1.3) with Hardy weights comes from the study of asymptotic behavior at infinity of solutions of the fractional Lane-Emden equation, which can be reduced to the similar problem for solutions near the origin of (1.3) with Hardy weights via the Kelvin transformation. We assume that $u\in C^{2}(B_{1}\backslash\{0\})$ and

[TABLE]

then $(-\Delta)^{\sigma}u(x)$ is well-defined at every point $x\in B_{1}\backslash\{0\}$. Our first main result is the following precise behavior of singular solutions of (1.3).

Theorem 1.1**.**

Assume $n\geq 2$. Let $u\in C^{2}(B_{1}\backslash\{0\})\cap\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ be a positive solution of (1.3) with $-2\sigma<\alpha\leq 0$, $\frac{n+\alpha}{n-2\sigma}<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$ and $p\neq\frac{n+2\sigma+2\alpha}{n-2\sigma}$. Then either the singularity at $x=0$ is removable, or

[TABLE]

where

[TABLE]

and the function $\Lambda(\tau)$ is defined by

[TABLE]

Remark 1.1**.**

If $\alpha<-2\sigma$, then by Corollary 2.1 Eq. (1.3) has no positive solution in any domain $\Omega$ containing the origin.

Under the assumptions of Theorem 1.1, if $u$ is a positive solution of (1.3) with a non-removable singularity, then Theorem 1.1 tells us that $u$ is asymptotic to a radial solution $u^{*}(|x|)$ to the same equation in $\mathbb{R}^{n}\backslash\{0\}$, where $u^{*}(|x|)$ is

[TABLE]

For when $\sigma=1$, Theorem 1.1 was proved in [21] by Gidas and Spruck. We may also see Caffarelli-Gidas-Spruck [8] for the case $\sigma=$1 and $\alpha=0$. Unlike the proofs of [8, 21] where the ODEs analysis is an important ingredient, our proof of Theorem 1.1 is based on a monotonicity formula, combined with the blow up (down) arguments, the Kelvin transformation and an uniqueness result of solutions of related degenerate equations on $\mathbb{S}^{n}_{+}$. As mentioned earlier, a similar monotonicity formula for the fractional Lane-Emden equation was established and used in our recent papers [34, 35], where Theorem 1.1 was obtained whenever $\alpha=0$. We recall the Hardy-Sobolev exponent

[TABLE]

This exponent plays a critical role in the equation (1.3). When $-2\sigma<\alpha<0$ and $\frac{n+\alpha}{n-2\sigma}<p<p_{S}(\alpha)$, that is for the Hardy-Sobolev subcritical case, our proof of Theorem 1.1 is similar to that in [34, 35]. Remark that, when $-2\sigma<\alpha<0$, Theorem 1.1 also holds in the Hardy-Sobolev supercritical range

[TABLE]

This is essential for applying Theorem 1.1 to study asymptotic behavior at infinity of solutions of the fractional Lane-Emden equation. We emphasize that the proof of Theorem 1.1 in the supercritical case is different from that in subcritical case. One significant difference is that the energy integral (3.1) is non-decreasing in the subcritical case, but it is non-increasing in the supercritical case. The other difference is that it seems difficult to prove that every singular positive solution of (1.3) in $\mathbb{R}^{n}\backslash\{0\}$ is radially symmetric in the supercritical case. These differences lead us to need some new techniques to deal with the supercritical case.

For the Hardy-Sobolev critical case $p=p_{S}(\alpha)$ ($-2<\alpha<0$) and the Hénon’s case $0<\alpha<2\sigma$, we establish the following classification result for isolated singularities, in particular, it implies the sharp blow up estimates of singular solutions.

Theorem 1.2**.**

Assume $n\geq 2$. Let $u\in C^{2}(B_{1}\backslash\{0\})\cap\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ be a positive solution of (1.3). Assume

[TABLE]

Then either the singularity at $x=0$ is removable, or there exist positive constants $C_{1}$ and $C_{2}$ such that

[TABLE]

Remark 1.2**.**

In Theorems 1.1 and 1.2, since we do not use any special structure of the ball, $B_{1}$ can be replaced by any open set containing the origin [math].

The upper bound in (1.8) can be obtained by using a doubling lemma of Poláčik-Quittner-Souplet [33]. To derive the lower bound in (1.8), one main difficulty is to prove Proposition 3.2 in Section 3. In the proof of Theorem 3.3 in [21], Gidas and Spruck proved the lower bound in (1.2) by using the following statement:

”*If $\liminf_{|x|\to 0}|x|^{\frac{2+\alpha}{p-1}}u(x)=0$, then the Harnack inequality (a Harnack inequality similar to (2.21) in this paper) implies that *

[TABLE]

But this seems not obvious and requires more explanation. Aviles also pointed out this point on p.190 in [3]. In this paper, we will make full use of a monotonicity formula (Proposition 3.1) to prove Proposition 3.2. Remark that, our proof also applies to Eq. (1.1) and thus we could give a rigorous proof of the above statement. We believe that the idea used here can be applied in other situations to deal with similar questions. We also mention that Chen-Lin [13] recently proved a similar result as Proposition 3.2 of this paper to a critical elliptic system by applying Pohozaev identity, see Corollary 4.1, Lemma 4.3 and Lemma 4.4 of [13], where a spherical Harnack inequality also holds for $w_{1}+w_{2}$ but the proof is very delicate and complicated. Our poof of Proposition 3.2 is also different from the one in [13].

We study Eq. (1.3) via the well known extension theorem for the fractional Laplacian $(-\Delta)^{\sigma}$ established by Caffarelli-Silvestre [10], through which one can study the isolated boundary singularities of a degenerate elliptic equation with a nonlinear Neumann boundary condition in the upper half-space $\mathbb{R}^{n+1}_{+}$ (see (2.3) and (2.4) in Section 2). We denote $\mathcal{B}_{R}^{+}$ as the upper half-ball $\mathcal{B}_{R}\cap\mathbb{R}_{+}^{n+1}$, $\partial^{+}\mathcal{B}_{R}^{+}=\partial\mathcal{B}_{R}^{+}\cap\mathbb{R}_{+}^{n+1}$ as the positive part of $\partial\mathcal{B}_{R}^{+}$, and $\partial^{0}\mathcal{B}_{R}^{+}$ as the flat part of $\partial\mathcal{B}_{R}^{+}$ which is the ball $B_{R}$ in $\mathbb{R}^{n}$. More generally, we are concerned with the corresponding degenerate elliptic equation in $\mathcal{B}_{1}^{+}$ with an isolated Neumann boundary singularity

[TABLE]

where $\frac{\partial U}{\partial\nu^{\sigma}}(x,0):=-\lim_{t\rightarrow 0^{+}}t^{1-2\sigma}\partial_{t}U(x,t)$, the constant $\kappa_{\sigma}=\frac{\Gamma(1-\sigma)}{2^{2\sigma-1}\Gamma(\sigma)}$ and $\Gamma$ is the Gamma function. By the extension theorem of Caffarelli and Silvestre, if one knows the behavior of the traces $u(x):=U(x,0)$ of the nonnegative solutions $U(x,t)$ of (1.9) near the singularity, then the behavior of nonnegative solutions of (1.3) follows.

We say that $U$ is a nonnegative weak solution of (1.9) if $U$ is in the weighted Sobolev space $W^{1,2}(t^{1-2\sigma},\mathcal{B}^{+}_{1}\backslash\overline{\mathcal{B}}^{+}_{\epsilon})$ for every $\epsilon>0$, $U\geq 0$, and it satisfies (1.9) in the sense of distributions away from 0, that is,

[TABLE]

for every nonnegative $\Phi\in C_{c}^{\infty}\left((\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}\right)$. It follows from the regularity results in [7, 26] that $U(x,t)$ is locally Hölder continuous in $\overline{\mathcal{B}}^{+}_{3/4}\backslash\{0\}$. We use capital letters, such as $X=(x,t)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, to denote points in $\mathbb{R}_{+}^{n+1}$. Next, we first classify the isolated boundary singularities of the equation (1.9).

Theorem 1.3**.**

Assume $n\geq 2$. Let $U$ be a nonnegative weak solution of (1.9). Assume

[TABLE]

Then either the singularity at $x=0$ is removable, i.e., $U(X)$ can be extended to a continuous solution in $\overline{\mathcal{B}_{1/2}^{+}}$, or there exist two positive constants $C_{1}$ and $C_{2}$ such that

[TABLE]

Furthermore, we can describe the precise asymptotic behavior of singular positive solutions $U(x,t)$ of (1.9) as $(x,t)\to 0$.

Theorem 1.4**.**

Assume $n\geq 2$. Let $U$ be a nonnegative weak solution of (1.9) with $-2\sigma<\alpha\leq 0$, $\frac{n+\alpha}{n-2\sigma}<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$ and $p\neq\frac{n+2\sigma+2\alpha}{n-2\sigma}$. Then either the singularity at $x=0$ is removable, or

[TABLE]

where the constant $C_{p,\sigma,\alpha}$ is given by (1.6)-(1.7), $P_{\sigma}(x,t)$ is the Poisson kernel

[TABLE]

and $p_{n,\sigma}$ is a positive constant chosen such that $\int_{\mathbb{R}^{n}}P_{\sigma}(x,1)dx=1$.

Remark that Theorems 1.3 and 1.4 give us not only the asymptotic behavior of the trace $u(x)$ of a singular solution $U(x,t)$, but also the asymptotic behavior of the solution $U(x,t)$ near the boundary singularity.

The following two theorems treat the isolated singularities located at infinity.

Theorem 1.5**.**

Assume $n\geq 2$. Let $u\in\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ be a nonnegative $C^{2}$ solution of

[TABLE]

with $\alpha>-2\sigma$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$.

• (1)

If $1<p<\frac{n+\alpha}{n-2\sigma}$, then necessarily $u(x)\equiv 0$ in $\{x\in\mathbb{R}^{n}:|x|>1\}$.

• (2)

If $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$, then either the singularity at $\infty$ is removable, i.e., there exists $C>0$ such that

[TABLE]

or there exist positive constants $C_{1}$, $C_{2}$ such that

[TABLE]

Theorem 1.6**.**

Assume $n\geq 2$. Let $u\in\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ be a positive $C^{2}$ solution of (1.14) with $-2\sigma<\alpha\leq 0$, $\frac{n+\alpha}{n-2\sigma}<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$ and $p\neq\frac{n+2\sigma+2\alpha}{n-2\sigma}$. Then either there exists $\beta>0$ such that

[TABLE]

or

[TABLE]

where $C_{p,\sigma,\alpha}$ is given by (1.6).

In particular, we give a complete classification of isolated singularities of positive solutions to the fractional Lane-Emden equation near $\infty$ .

Corollary 1.1**.**

Assume $n\geq 2$. Let $u\in\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ be a positive $C^{2}$ solution of

[TABLE]

with $\frac{n}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$. Then either there exists $\beta>0$ such that

[TABLE]

or

[TABLE]

where $C_{p,\sigma,0}$ is given by (1.6).

Remark 1.3**.**

Our characterization of isolated singularities near $\infty$ of the fractional Lane-Emden equation is complemented by the existence results of fast-decay solutions satisfying (1.16) which have been recently constructed by Ao-Chan-DelaTorre-Fontelos-González-Wei [2, 1]. More precisely, for some exponent $p_{1}=p_{1}(n,\sigma)\in(\frac{n}{n-2\sigma},\frac{n+2\sigma}{n-2\sigma})$, and for every $\beta\in(0,\infty)$, there exists a positive solution of (1.15) satisfying (1.16) which was proved in [2] when $\frac{n}{n-2\sigma}<p<p_{1}$ and in [1] when $p_{1}\leq p<\frac{n+2\sigma}{n-2\sigma}$.

Finally, we establish an uniqueness theorem for global singular solutions.

Theorem 1.7**.**

Assume $n\geq 2$. Let $U\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ be a nonnegative weak solution of

[TABLE]

with $-2\sigma<\alpha\leq 0$, $\frac{n+\alpha}{n-2\sigma}<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$ and $p\neq\frac{n+2\sigma+2\alpha}{n-2\sigma}$. Assume that the two isolated singularities of the trace $u(x)$ of $U(x,t)$ at $x=0$ and $x=\infty$ are non-removable. Then necessarily we have

[TABLE]

where the constant $C_{p,\sigma,\alpha}$ is given by (1.6)-(1.7) and $P_{\sigma}(x,t)$ is the Poisson kernel.

The rest of this paper is organized as follows. In Section 2, we introduce the extension formulation for $(-\Delta)^{\sigma}$ established by Caffarelli-Silvestre [10] and provide some a priori estimates. In Section 3, we establish an important monotonicity formula and prove Theorems 1.2 and 1.3. In Section 4, we show the precise asymptotic behavior of singular solutions stated in Theorems 1.1 and 1.4 and also give the proof of Theorem 1.7. In Section 5, we prove Theorems 1.5 and 1.6.

2 Preliminaries

In this section, we introduce some notations and prove some important estimates which will be used in this paper.

We use capital letters, such as $X=(x,t)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, to denote points in $\mathbb{R}_{+}^{n+1}$. We denote $\mathcal{B}_{R}$ as the ball in $\mathbb{R}^{n+1}$ with radius $R$ and center 0, and $B_{R}$ as the ball in $\mathbb{R}^{n}$ with radius $R$ and center [math]. We also denote $\mathcal{B}_{R}^{+}$ as the upper half-ball $\mathcal{B}_{R}\cap\mathbb{R}_{+}^{n+1}$, $\partial^{+}\mathcal{B}_{R}^{+}=\partial\mathcal{B}_{R}^{+}\cap\mathbb{R}_{+}^{n+1}$ as the positive part of $\partial\mathcal{B}_{R}^{+}$, and $\partial^{0}\mathcal{B}_{R}^{+}$ as the flat part of $\partial\mathcal{B}_{R}^{+}$ which is the ball $B_{R}$ in $\mathbb{R}^{n}$. For a more general domain $\Omega\subset\mathbb{R}^{n+1}_{+}$, we denote $\partial^{0}\Omega$ as the interior of $\overline{\Omega}\cap\partial\mathbb{R}^{n+1}_{+}$ in $\mathbb{R}^{n}$.

As mentioned before, we will study the fractional Hardy-Hénon equation (1.3) via the well known extension theorem for the fractional Laplacian $(-\Delta)^{\sigma}$ established by Caffarelli-Silvestre [10]. Assume $u\in C^{2}(B_{1}\backslash\{0\})\cap\mathcal{L}_{\sigma}(\mathbb{R}^{n})$. For $X=(x,t)\in\mathbb{R}_{+}^{n+1}$, let

[TABLE]

where

[TABLE]

and $p_{n,\sigma}$ is a positive constant chosen such that $\int_{\mathbb{R}^{n}}P_{\sigma}(x,1)dx=1$. Then $U\in C^{2}(\mathbb{R}_{+}^{n+1})\cap C\left((\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}\right)$, $t^{1-2\sigma}\partial_{t}U\in C\left((\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}\right)$ and

[TABLE]

By the extension formulation in [10], we have

[TABLE]

where $\frac{\partial U}{\partial\nu^{\sigma}}(x,0):=-\lim_{t\rightarrow 0^{+}}t^{1-2\sigma}\partial_{t}U(x,t)$, the constant $\kappa_{\sigma}=\frac{\Gamma(1-\sigma)}{2^{2\sigma-1}\Gamma(\sigma)}$ and $\Gamma$ is the Gamma function.

Instead of Eq. (1.3) we may study the following degenerate elliptic equation with an isolated Neumann boundary singularity

[TABLE]

By (2.3) and (2.4), the asymptotic behavior of solutions near the singularity of (1.3) can be obtained from that of the traces $u(x):=U(x,0)$ of the solutions $U(x,t)$ of (2.5).

We recall that $U$ is a nonnegative weak solution of (2.5) if $U$ is in the weighted Sobolev space $W^{1,2}(t^{1-2\sigma},\mathcal{B}^{+}_{1}\backslash\overline{\mathcal{B}}^{+}_{\epsilon})$ for every $\epsilon>0$, $U\geq 0$, and it satisfies (2.5) in the sense of distributions away from 0, that is,

[TABLE]

for every nonnegative $\Phi\in C_{c}^{\infty}\left((\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}\right)$.

We say that the origin [math] is a removable singularity of solution $U$ of (2.5) if $U(x,t)$ can be extended as a continuous function near the origin, otherwise we say that the origin [math] is a non-removable singularity.

We say $U\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}})$ if $U\in W^{1,2}(t^{1-2\sigma},\mathcal{B}_{R}^{+})$ for all $R>0$, and we say $U\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ if $U\in W^{1,2}(t^{1-2\sigma},\mathcal{B}_{R}^{+}\backslash\overline{\mathcal{B}}_{\epsilon}^{+})$ for all $R>\epsilon>0$.

We now establish the basic singularity and decay estimates. In the case $\sigma=1$, that is for the Laplacian, the corresponding results were proved in [21, 32].

Proposition 2.1**.**

Let $n\geq 2$, $\alpha\in\mathbb{R}$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$.

• (1)

Suppose that $U$ is a nonnegative weak solution of (2.5). Then there exists a constant $C=C(n,p,\alpha,\sigma)$ such that

[TABLE]

• (2)

Suppose that $U$ is a nonnegative weak solution of

[TABLE]

where $B_{1}^{c}:=\{x\in\mathbb{R}^{n}:|x|>1\}$. Then there exists a constant $C=C(n,p,\alpha,\sigma)$ such that

[TABLE]

To prove Proposition 2.1, we need the following lemma.

Lemma 2.1**.**

Let $n\geq 2$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$. Let $K\in C^{1}(\overline{B_{1}})$ satisfy

[TABLE]

for some constants $C_{1},C_{2}>0$. Suppose that $U$ is a nonnegative weak solution of

[TABLE]

Then there exists a constant $C$, depending only on $n,\sigma,\gamma,p,C_{1},C_{2}$, such that

[TABLE]

Proof.

Suppose by contradiction that there exists a sequence of solutions $U_{i}$ of (2.11) and a sequence of points $x_{i}\in B_{1}$ such that

[TABLE]

where the functions $M_{i}$ are defined by

[TABLE]

By the doubling lemma of Poláčik-Quittner-Souplet [33], there exists another sequence $y_{i}\in B_{1}$ such that

[TABLE]

and

[TABLE]

where $\lambda_{i}:=M_{i}(y_{i})^{-1}$. Note that $\lambda_{i}\to 0$ as $i\to\infty$. We now define

[TABLE]

with

[TABLE]

Then $\bar{U}_{i}$ satisfies $\bar{U}_{i}(0)=1$ and

[TABLE]

where $\bar{K}_{i}(x)=K(y_{i}+\lambda_{i}x)$ for $x\in\partial^{0}\Omega_{i}$. Moreover, by (2.13) we have

[TABLE]

On the other hand, by (2.10) we know that $C_{2}\leq\bar{K}_{i}(x)\leq C_{1}$ and, for each $R>0$ and $i\geq i_{0}(R)$ large enough,

[TABLE]

and

[TABLE]

Therefore, by Arzela-Ascoli’s theorem, there exists $\bar{K}\in C(\mathbb{R}^{n})$ such that, after extracting a subsequence, $\bar{K}_{i}\to\bar{K}$ in $C_{loc}(\mathbb{R}^{n})$. Moreover, from (2.16) we have for any $y,z\in\mathbb{R}^{n}$ that

[TABLE]

and hence the function $\bar{K}$ is actually a constant $K_{0}>0$.

It follows from Corollary 2.10 and Theorem 2.15 of Jin-Li-Xiong [26] that there exists $\gamma\in(0,1)$ such that for every $R>1$,

[TABLE]

where $C(R)$ is independent of $i$. Thus, there is a subsequence of $i\rightarrow\infty$, still denoted by itself, and a function $\bar{U}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}})\cap C_{loc}^{\gamma}(\overline{\mathbb{R}^{n+1}_{+}})$ such that as $i\rightarrow\infty$,

[TABLE]

Moreover, $\bar{U}$ is a nonnegative solution of

[TABLE]

and $\bar{U}(0)=1$. Since $p<\frac{n+2\sigma}{n-2\sigma}$, this contradicts the Liouville type theorem in [26] (See Theorem 1.8 and Remark 1.9 in [26] ). ∎

*Proof of Proposition 2.1. * Suppose either $\Omega=\{x\in\mathbb{R}^{n}:0<|x|<1\}$ and $0<|x_{0}|<\frac{1}{2}$, or $\Omega=\{x\in\mathbb{R}^{n}:|x|>1\}$ and $|x_{0}|>2$. Take

[TABLE]

Then, for any $y\in B_{1}$, we have $\frac{|x_{0}|}{2}<|x_{0}+\lambda y|<\frac{3|x_{0}|}{2}$. Hence $x_{0}+\lambda y\in\Omega$ in either case. Define

[TABLE]

Then $W$ is a nonnegative solution of

[TABLE]

where $K(y)=|y+\frac{x_{0}}{\lambda}|^{\alpha}$ for $y\in B_{1}$. Clearly

[TABLE]

Therefore, $\|K\|_{C^{1}(\overline{B_{1}})}\leq C_{1}(\alpha)$ and $K(y)\geq C_{2}(\alpha)$ in $\overline{B_{1}}$ for some constants $C_{1}(\alpha),C_{2}(\alpha)>0$. By Lemma 2.1 we obtain $W(0)\leq C$. This implies that

[TABLE]

The desired conclusion follows. $\square$

Corollary 2.1**.**

Let $n\geq 2$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$. Suppose that $U$ is a nonnegative weak solution of (2.5). If $\alpha<-2\sigma$, then $U(x)\equiv 0$ in $(\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}$.

Proof.

Since $\alpha<-2\sigma$, by (2.7) we know

[TABLE]

Assume by contradiction that there exists $X_{0}\in(\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\backslash\{0\}$ such that $U(X_{0})>0$. Then the maximum principle implies that

[TABLE]

By Proposition 3.1 in [26], we have

[TABLE]

a contradiction with (2.18). ∎

Now we recall a Harnack inequality. For its proof, see [7, 26].

Lemma 2.2**.**

Let $U\in W^{1,2}(t^{1-2\sigma},\mathcal{B}^{+}_{1})$ be a nonnegative weak solution of

[TABLE]

If $a\in L^{q}(B_{1})$ for some $q>\frac{n}{2\sigma}$, then we have

[TABLE]

where $C$ depends only on $n,\sigma$ and $\|a\|_{L^{q}(B_{1})}$.

One very useful consequence of Proposition 2.1 is the following Harnack inequality.

Lemma 2.3**.**

Let $n\geq 2$, $\alpha\in\mathbb{R}$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$.

• (1)

Suppose that $U$ is a nonnegative weak solution of (2.5). Then there exists a constant $C=C(n,p,\alpha,\sigma)$ such that for all $0<r<\frac{1}{8}$, we have

[TABLE]

• (2)

Suppose that $U$ is a nonnegative weak solution of (2.8). Then there exists a constant $C=C(n,p,\alpha,\sigma)$ such that for all $r>8$, we have

[TABLE]

Proof.

Let

[TABLE]

for $X\in\mathcal{B}_{4}^{+}\backslash\overline{\mathcal{B}_{1/4}^{+}}$. Then $V_{r}$ satisfies

[TABLE]

where $v_{r}(x)=V_{r}(x,0)$ and $a_{r}(x)=r^{2\sigma+\alpha}|x|^{\alpha}\left(u(rx)\right)^{p-1}$. By Proposition 2.1,

[TABLE]

where $C$ is a positive constant independent of $r$ and $U$. By Harnack inequality in Lemma 2.2 and the standard Harnack inequality for uniformly elliptic equations, we have

[TABLE]

where $C$ is another positive constant independent of $r$ and $U$. We complete the proof by rescaling back to $U$. ∎

3 Classification of Isolated Singularities at $x=0$

In this section, we classify the isolated singularities of positive solutions of (2.5) near the origin. To this end, we need to establish a monotonicity formula for the nonnegative solutions $U$ of (2.5) (resp. of (2.8)). Let $U$ be a nonnegative solution of (2.5) (resp. of (2.8)), we define

[TABLE]

We recall that the Hardy-Sobolev critical exponent is defined by

[TABLE]

Then, we have the following monotonicity formula.

Proposition 3.1**.**

Let $n\geq 2$, $\alpha\in\mathbb{R}$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$.

• (1)

Suppose that $p\leq p_{S}(\alpha)$ and $U$ is a nonnegative weak solution of (2.5) (resp. of (2.8)). Then $E(r;U)$ is non-decreasing in $r\in(0,1)$ (resp. in $r\in(1,\infty)$). Moreover,

[TABLE]

where $J_{1}=\frac{n-2\sigma}{p-1}\left(\frac{n+2\sigma+2\alpha}{n-2\sigma}-p\right)\geq 0$.

• (2)

Suppose that $p>p_{S}(\alpha)$ and $U$ is a nonnegative weak solution of (2.5) (resp. of (2.8)). Then $E(r;U)$ is non-increasing in $r\in(0,1)$ (resp. in $r\in(1,\infty)$). Moreover,

[TABLE]

where $J_{1}=\frac{n-2\sigma}{p-1}\left(\frac{n+2\sigma+2\alpha}{n-2\sigma}-p\right)<0$.

Proof.

We shall take the standard polar coordinates in $\mathbb{R}^{n+1}_{+}$: $X=(x,t)=r\theta$, where $r=|X|$ and $\theta=\frac{X}{|X|}$. Let $\theta_{1}=\frac{t}{|X|}$ denote the component of $\theta$ in the $t$ direction and

[TABLE]

denote the upper unit half-sphere.

Let $U$ be a nonnegative weak solution of (2.5). Using the classical change of variable in Fowler [20],

[TABLE]

Direct calculations show that $V$ satisfies

[TABLE]

where

[TABLE]

Multiplying (3.2) by $V_{s}$ and integrating, we have

[TABLE]

For any $s\in(-\infty,0)$, we define

[TABLE]

Then, by (3.3) we get

[TABLE]

Note that

[TABLE]

Hence, $\widetilde{E}(s)$ is non-decreasing in $s\in(-\infty,0)$ if $p\leq p_{S}(\alpha)$ and $\widetilde{E}(s)$ is non-increasing in $s\in(-\infty,0)$ if $p>p_{S}(\alpha)$.

Now, rescaling back to $U$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Substituting these into (3.4) and noting that $s=\ln r$ is increasing in $r$, we easily obtain that $E(r;U)$ is non-decreasing in $r\in(0,1)$ if $p\leq p_{S}(\alpha)$ and it is non-increasing in $r\in(0,1)$ if $p>p_{S}(\alpha)$.

If $U$ is a nonnegative solution of (2.8), we just need to replace $s\in(-\infty,0)$ in the above proof with $s\in(0,\infty)$. The proof is finished. ∎

By using the monotonicity of $E(r;U)$, we prove the following proposition, which will play an essential role in deriving the lower bound of singular positive solutions.

Proposition 3.2**.**

Let $U$ be a nonnegative weak solution of (2.5) with $-2\sigma<\alpha<2\sigma$ and $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$. If

[TABLE]

then

[TABLE]

Proof.

We consider separately the case $p\leq p_{S}(\alpha)$ and the case $p>p_{S}(\alpha)$.

Case 1: $p\leq p_{S}(\alpha)$. Suppose by contradiction that

[TABLE]

Then there exist two sequences of points $\{x_{i}\}$ and $\{y_{i}\}$ satisfying

[TABLE]

such that

[TABLE]

Let $g(r)=r^{\frac{2\sigma+\alpha}{p-1}}\bar{u}(r)$, where $\bar{u}(r)=\frac{1}{|\partial B_{r}|}\int_{\partial B_{r}}u$ denotes the spherical average of $u$ over $\partial B_{r}$. By the Harnack inequality (2.21), we have

[TABLE]

Hence, there exists a sequence of local minimum points $r_{i}$ of $g(r)$ such that

[TABLE]

Define

[TABLE]

where $e_{1}=(1,0,\cdots,0)\in\mathbb{R}^{n+1}$. It follows from Harnack inequality (2.21) that $V_{i}$ is locally uniformly bounded away from the origin and satisfies

[TABLE]

Note that by the Harnack inequality (2.21), $r_{i}^{\frac{2\sigma+\alpha}{p-1}}U(r_{i}e_{1})\rightarrow 0$ as $i\rightarrow\infty$. By Corollary 2.10 and Theorem 2.15 in [26] there exists $\gamma\in(0,1)$ such that for every $R>1>r>0$,

[TABLE]

where $v_{i}(x)=V_{i}(x,0)$ and $C(R,r)$ is independent of $i$. Then after passing to a subsequence, $\{V_{i}\}$ converges to a nonnegative function $V\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying

[TABLE]

By a Bôcher type theorem in [26], we have

[TABLE]

where $a,b$ are nonnegative constants. Recall that $r_{i}$ is a local minimum point of $g(r)$ for every $i$ and note that

[TABLE]

Hence, we have

[TABLE]

Let $v(x)=V(x,0)$. Then $v_{i}(x)\rightarrow v(x)$ in $C_{loc}^{2}(\mathbb{R}^{n}\backslash\{0\})$. By (3.7) we obtain

[TABLE]

which implies that

[TABLE]

On the other hand, $V(e_{1})=1$ implies

[TABLE]

Combining (3.8) with (3.9), we get

[TABLE]

Since $-2\sigma<\alpha<2\sigma$ and $\frac{n+\alpha}{n-2\sigma}<p$, we have $0<a,b<1$. Next we compute $E(r;U)$.

It follows from Proposition 2.19 in [26] that $|\nabla_{x}V_{i}|$ and $|t^{1-2\sigma}\partial_{t}V_{i}|$ are locally uniformly bounded in $C_{loc}^{\beta}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ for some $\beta>0$. Hence, there exists a constant $C>0$ such that

[TABLE]

and

[TABLE]

By the Harnack inequality (2.21), we also have

[TABLE]

Thus, we estimate

[TABLE]

[TABLE]

and

[TABLE]

where the constant $C$ is independent of $i$. By the definition of $E(r;U)$, we have

[TABLE]

Since $E(r;U)$ is non-decreasing in $r\in(0,1)$ for this case, we obtain

[TABLE]

On the other hand, by the scaling invariance of $E(r;U)$, we have for every $i$ that

[TABLE]

Hence, we have

[TABLE]

Letting $i\rightarrow\infty$, we obtain

[TABLE]

Here we have used the facts $2\sigma+\alpha>0$ and $\frac{2\sigma+\alpha}{p-1}-(n-2\sigma)<0$ in the last inequality. We get a contradiction. This completes the proof of Case 1.

Case 2: $p>p_{S}(\alpha)$. In this case, it follows from Proposition 3.1 (2) that $E(r;U)$ is non-increasing in $r\in(0,1)$. If we proceed as in the proof of Case 1, then we obtain $E(r;U)\leq 0$ for $r\in(0,1)$ in (3.10), and so we cannot get a contradiction in the final proof. Thus, a new method is needed to deal with this supercritical case. In fact, the following method is available for all $p\neq p_{S}(\alpha)$.

Step 1. If $\liminf_{|x|\to 0}|x|^{\frac{2\sigma+\alpha}{p-1}}u(x)=0$, then

[TABLE]

Since $\liminf_{|x|\to 0}|x|^{\frac{2\sigma+\alpha}{p-1}}u(x)=0$, there exists a sequence of points $\{x_{i}\}$ such that

[TABLE]

Let $r_{i}:=|x_{i}|$. By the Harnack inequality (2.21),

[TABLE]

where $e_{1}=(1,0,\cdots,0)\in\mathbb{R}^{n+1}$. Define

[TABLE]

It follows from Proposition 2.1 and Harnack inequality (2.21) that $W_{i}$ is locally uniformly bounded away from the origin. Moreover, $W_{i}$ satisfies

[TABLE]

and

[TABLE]

By Corollary 2.10, Theorem 2.15 and Proposition 2.19 in [26] there exists $\gamma\in(0,1)$ such that for every $R>1>r>0$

[TABLE]

where $C(R,r)$ is independent of $i$. Then after passing to a subsequence, $\{W_{i}\}$ converges to a nonnegative function $W\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying

[TABLE]

By (3.13) we have $W(e_{1})=0$. This together with Lemma 2.2 implies that $W\equiv 0$ in $\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\}$. Since $E(r;U)$ is invariant under the scaling,

[TABLE]

By the monotonicity of $E(r;U)$ (Proposition 3.1), we obtain

[TABLE]

Step 2. Let $W$ be a nonnegative solution of (3.14) in $\mathbb{R}^{n+1}_{+}$. If $E(r;W)\equiv 0$ for $r\in(0,\infty)$, then

[TABLE]

Since $p\neq p_{S}(\alpha)$, we have $J_{1}=\frac{n-2\sigma}{p-1}\left(\frac{n+2\sigma+2\alpha}{n-2\sigma}-p\right)\neq 0$. By Proposition 3.1 we get

[TABLE]

This implies that $W$ is homogeneous of degree $-\frac{2\sigma+\alpha}{p-1}$. That is, there exists $\varphi\in C^{2}(\mathbb{S}^{n}_{+})$ such that

[TABLE]

where $X=(x,t)=r\theta$ with $r=|X|$ and $\theta=\frac{X}{|X|}$. Let $\theta_{1}=\frac{t}{|X|}$ denote the component of $\theta$ in the $t$ direction. A calculation similar to the proof of Proposition 3.1 shows that $\varphi$ satisfies

[TABLE]

where

[TABLE]

Multiplying (3.15) by $\varphi$ and integrating on $\mathbb{S}^{n}_{+}$, we obtain

[TABLE]

On the other hand, by the proof of Proposition 3.1, $E(r;W)\equiv 0$ gives

[TABLE]

Combining (3.16) with (3.17), we easily get

[TABLE]

and so $\varphi\equiv 0$ on $\partial\mathbb{S}^{n}_{+}$. By (3.16) and $J_{2}>0$, we obtain $\varphi=0$ on $\mathbb{S}^{n}_{+}$. Hence $W\equiv 0$ in $\mathbb{R}^{n+1}_{+}$.

Step 3. End of Proof. For $\lambda>0$ small, define

[TABLE]

Then $U^{\lambda}$ is also a nonnegative solution of (2.5) in $\mathcal{B}_{1/\lambda}^{+}$. It follows from Proposition 2.1 and Harnack inequality (2.21) that $U^{\lambda}$ is locally uniformly bounded away from the origin. By Corollary 2.10, Theorem 2.15 and Proposition 2.19 in [26] there exists $\gamma\in(0,1)$ such that for every $R>1>r>0$

[TABLE]

where $C(R,r)$ is independent of $\lambda$. Hence, there is a subsequence $\lambda_{i}$ of $\lambda\to 0$ such that $\{U^{\lambda_{i}}\}$ converges to a nonnegative function $U^{0}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying

[TABLE]

Moreover, by the scaling invairance of $E$ and Step 1, we have for any $r>0$ that

[TABLE]

The conclusion of Step 2 gives $U^{0}\equiv 0$ in $\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\}$. Since the limiting function $U^{0}$ is unique for any subsequence of $\lambda\to 0$, we obtain

[TABLE]

In particular,

[TABLE]

which immediately implies $\lim_{|x|\to 0}|x|^{\frac{2\sigma+\alpha}{p-1}}u(x)=0$. ∎

Proposition 3.3**.**

Let $U$ be a nonnegative weak solution of (2.5) with $-2\sigma<\alpha<2\sigma$ and $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$. If

[TABLE]

then the singularity at $x=0$ is removable, i.e., $U(x,t)$ can be extended to a continuous function near the origin [math].

Proof.

By the Harnack inequality (2.21), we have

[TABLE]

For any $0<\mu<n-2\sigma$ and $0<\delta<\frac{1}{2}$, as in [9], we define

[TABLE]

where $X=(x,t)\neq 0$. Then $\Psi_{\mu}$ satisfies

[TABLE]

Let $\mu_{0}=\frac{2\sigma+\alpha}{p-1}$ and $\tau\in(0,\frac{2\sigma+\alpha}{p-1})$ be fixed. Note that $0<\frac{2\sigma+\alpha}{p-1}<n-2\sigma$ due to $-2\sigma<\alpha$ and $\frac{n+\alpha}{n-2\sigma}<p$. Let

[TABLE]

where $\epsilon,C$ are positive constants. Then we can choose small $\delta=\delta(\tau,\alpha,\sigma,p,n)\in(0,\frac{1}{2})$ such that

[TABLE]

Let $a(x):=\kappa_{\sigma}|x|^{\alpha}u^{p-1}(x)$. By the assumption we have $\lim_{|x|\to 0}a(x)|x|^{2\sigma}=0$. Hence, there exists $r_{0}\in(0,1)$ such that

[TABLE]

Thus, we have

[TABLE]

Furthermore, we note that

[TABLE]

Hence, for any $\epsilon>0$, by (3.18) there exists $r_{\epsilon}>0$ small such that

[TABLE]

On the other hand, we can choose $C=C(\tau,r_{0},U)$ sufficiently large so that

[TABLE]

The maximum principle gives that

[TABLE]

Letting $\epsilon\to 0$, we have

[TABLE]

By standard rescaling arguments and Proposition 2.19 in [26], we obtain

[TABLE]

and

[TABLE]

Since $\tau\in(0,\frac{2\sigma+\alpha}{p-1})$ is arbitrary, it is not difficult to verify that $U\in W^{1,2}(t^{1-2\sigma},\mathcal{B}_{1}^{+})$. Next we will prove that $U$ is a nonnegative weak solution of

[TABLE]

In fact, for $\epsilon>0$ small, let $\eta_{\epsilon}\in C^{\infty}(\mathbb{R}^{n+1})$ be a cut-off function satisfying

[TABLE]

and

[TABLE]

For any $\psi\in C_{c}^{\infty}\left((\mathcal{B}_{1}^{+}\cup\partial^{0}\mathcal{B}_{1}^{+})\right)$, using $\psi\eta_{\epsilon}$ as a test function in (2.6) gives

[TABLE]

But

[TABLE]

By (3.19) and $\alpha>-2\sigma$, we have $|\cdot|^{\alpha}u^{p}\in L_{loc}^{1}(B_{1})$. Letting $\epsilon\to 0$ in (3.23), we get

[TABLE]

Hence $U$ is a nonnegative weak solution of (3.22). Again, by (3.19) and $\alpha>-2\sigma$, we obtain

[TABLE]

for some $q>\frac{n}{2\sigma}$. It follows from Proposition 2.6 in [26] that $U$ is Hölder continuous in $\overline{\mathcal{B}_{1/2}^{+}}$. ∎

Proof of Theorem 1.3. The proof of Theorem 1.3 is now just a combination of Harnack inequality in Lemma 2.3, Propositions 2.1, 3.2 and 3.3. $\square$

Proof of Theorem 1.2. It follows from the extension theorem of Caffarelli-Silvestre [10] and Theorem 1.3. $\square$

4 Precise Asymptotic Behavior

In this section, we prove Theorem 1.4, Theorem 1.1 and Theorem 1.7. We begin by showing the boundedness of the energy integral $E(r;U)$ defined in (3.1).

Proposition 4.1**.**

Let $n\geq 2$, $-2\sigma<\alpha<2\sigma$ and $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$. Assume that $U$ is a nonnegative weak solution of (2.5) (resp. of (2.8)). Then $E(r;U)$ is uniformly bounded in $r\in(0,\frac{1}{8})$ (resp. in $r\in(8,\infty)$). Further, the limit

[TABLE]

exists and it is finite.

Proof.

Suppose $U$ is a nonnegative weak solution of (2.5). For any $r\in(0,\frac{1}{8})$, define

[TABLE]

Then $V$ satisfies

[TABLE]

where $v(x)=V(x,0)$. It follows from Proposition 2.1 and Lemma 2.3 that

[TABLE]

where $C$ is a positive constant depending only on $n,p,\sigma$ and $\alpha$. By Proposition 2.19 in [26], we have

[TABLE]

Hence, there exists $C>0$ depending only on $n,p,\sigma$ and $\alpha$ such that

[TABLE]

and

[TABLE]

Thus, a direct computation gives

[TABLE]

[TABLE]

[TABLE]

where $C$ is a positive constant depending only on $n,p,\sigma$ and $\alpha$. Now we easily conclude that $E(r;U)$ is uniformly bounded in $r\in(0,\frac{1}{8})$. By the monotonicity of $E(r;U)$, we obtain that the limit

[TABLE]

exists and is finite.

Similarly, let $U$ be a nonnegative weak solution of (2.8), we can prove that $E(r;U)$ is uniformly bounded in $r\in(8,\infty)$, and then the limit $\lim_{r\rightarrow+\infty}E(r;U)$ exists and is finite. ∎

Next, we show an uniqueness result for a degenerate elliptic equation on $\mathbb{S}^{n}_{+}$.

Proposition 4.2**.**

Assume $n\geq 2$, $\sigma\in(0,1)$, $-2\sigma<\alpha<2\sigma$ and $p>\frac{n+\alpha}{n-2\sigma}$. Let $\varphi\in C^{2}(\mathbb{S}^{n}_{+})\cap C(\overline{\mathbb{S}^{n}_{+}})$ be a solution of

[TABLE]

where

[TABLE]

If $\varphi\equiv c$ on $\partial\mathbb{S}^{n}_{+}$ for some positive constant $c$, then necessarily $c=C_{p,\sigma,\alpha}$ and

[TABLE]

where $C_{p,\sigma,\alpha}$ is given by (1.6), $\omega_{\alpha}\in C^{2}(\mathbb{S}^{n}_{+})\cap C(\overline{\mathbb{S}^{n}_{+}})$ is the restriction of $V_{\alpha}$ to $\overline{\mathbb{S}^{n}_{+}}$, and $V_{\alpha}$ is the Caffarelli-Silvestre extension of the function $v_{\alpha}=C_{p,\sigma,\alpha}|x|^{-\frac{2\sigma+\alpha}{p-1}}$, as defined in (2.1).

Proof.

For $x\in\mathbb{R}^{n}\backslash\{0\}$, let $v_{\alpha}(x)=C_{p,\sigma,\alpha}|x|^{-\frac{2\sigma+\alpha}{p-1}}$ where $C_{p,\sigma,\alpha}$ is given by (1.6). Then by Lemma 3.1 in Fall [18], we know that

[TABLE]

Let $V_{\alpha}$ be the Caffarelli-Silvestre extension of $v_{\alpha}$, that is,

[TABLE]

Then we have

[TABLE]

It is easy to check that $V_{\alpha}$ is a homogeneous function. Setting \omega_{\alpha}=V_{\alpha}\Big{|}_{\overline{\mathbb{S}^{n}_{+}}}. Then, $\omega_{\alpha}\equiv C_{p,\sigma,\alpha}$ on $\partial\mathbb{S}^{n}_{+}$ and for $X=(x,t)=r\theta\in\mathbb{R}^{n+1}_{+}\backslash\{0\}$,

[TABLE]

A direct calculation shows that $\omega_{\alpha}$ also satisfies (4.1). Define $\omega=\varphi-\left(\frac{c}{C_{p,\sigma,\alpha}}\right)^{p}\omega_{\alpha}$. Then $\omega$ satisfies

[TABLE]

Multiplying (4.2) by $\omega$ and integrating, we obtain

[TABLE]

Note that $J_{2}>0$ because of $\alpha>-2\sigma$ and $p>\frac{n+\alpha}{n-2\sigma}$. Thus, the above equality leads to $\omega\equiv 0$ on $\mathbb{S}^{n}_{+}$, and hence $\omega\equiv 0$ on $\overline{\mathbb{S}^{n}_{+}}$. The proposition follows immediately. ∎

Proof of Theorem 1.4. Suppose that $U$ is a nonnegative weak solution of (1.9) and the origin [math] is a non-removable singularity, we only need to establish (1.12). We consider separately the subcritical case $p<p_{S}(\alpha)$ and the supercritical case $p>p_{S}(\alpha)$.

Case 1: $p<p_{S}(\alpha)$. We define the scaling

[TABLE]

Then $U^{\lambda}$ satisfies

[TABLE]

Since [math] is a non-removable singularity, by Theorem 1.3 there exist $C_{1},C_{2}>0$ such that

[TABLE]

Thus, $U^{\lambda}$ is locally uniformly bounded away from the origin. It follows from Corollary 2.10 and Theorem 2.15 in [26] that there exists $\gamma>0$ such that for every $R>1>r>0$

[TABLE]

where $u^{\lambda}(x)=U^{\lambda}(x,0)$ and $C(R,r)$ is independent of $\lambda$. Then there is a subsequence $\lambda_{k}$ of $\lambda\rightarrow 0$ such that $\{U^{\lambda_{k}}\}$ converges to a nonnegative function $U^{0}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying

[TABLE]

By (4.3) we have

[TABLE]

Moreover, by the scaling invariance of $E(r;U)$ and Proposition 4.1, we have for any $r>0$ that

[TABLE]

That is, $E(r;U^{0})$ is a constant. It follows from Proposition 3.1 that $U^{0}$ is homogeneous of degree $-\frac{2\sigma+\alpha}{p-1}$. Hence, there exists $\varphi^{0}\in C^{2}(\mathbb{S}^{n}_{+})\cap C(\overline{\mathbb{S}^{n}_{+}})$ such that

[TABLE]

where $X=(x,t)=r\theta$ with $r=|X|$ and $\theta=\frac{X}{|X|}$. A calculation similar to the proof of Proposition 3.1 shows that $\varphi^{0}$ satisfies

[TABLE]

where $\theta_{1}=\frac{t}{|X|}$ denotes the component of $\theta$ in the $t$ direction and

[TABLE]

By (4.4), $\varphi^{0}$ also satisfies

[TABLE]

On the other hand, since $p<p_{S}(\alpha)$, from Theorem 1.1 in [30] we know that $U^{0}(x,t)$ is cylindrically symmetric about the origin. In particular, the trace $u^{0}(x):=U^{0}(x,0)$ is radially symmetric about the origin. Hence, $\varphi^{0}$ is a positive constant on $\partial\mathbb{S}^{n}_{+}$. By Proposition 4.2 we have

[TABLE]

where $\omega_{\alpha}$ is defined as in Proposition 4.2. It follows from the form (4.5) of $U^{0}$ and the proof of Proposition 4.2 that

[TABLE]

where $C_{p,\sigma,\alpha}$ is given by (1.6). Since the limiting function $U^{0}(x,t)$ is unique, we conclude that $U^{\lambda}(x,t)\rightarrow U^{0}(x,t)$ for any sequence $\lambda\rightarrow 0$ in $C^{\gamma/2}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$. In particular,

[TABLE]

uniformly for $X\in\overline{\mathbb{S}^{n}_{+}}$. This immediately implies that (1.12) holds.

Case 2: $p_{S}(\alpha)<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$. We consider the Kelvin transform

[TABLE]

for $|Y|>1$. Then $\widetilde{U}$ satisfies

[TABLE]

where $\vartheta:=p(n-2\sigma)-(n+2\sigma+\alpha)$ and $B_{1}^{c}=\{x\in\mathbb{R}^{n}:|x|>1\}$. Using Theorem 1.3 we have

[TABLE]

Note that

[TABLE]

due to $\frac{n+\alpha}{n-2\sigma}<p$ and $p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$. Moreover,

[TABLE]

Therefore, after performing the Kelvin transform, the new exponent $\vartheta$ satisfies

[TABLE]

For any $\lambda>0$, define

[TABLE]

Then $\widetilde{U}^{\lambda}$ satisfies

[TABLE]

By (4.8),

[TABLE]

It follows from Corollary 2.10 and Theorem 2.15 in [26] that there exists $\gamma>0$ such that for every $R>1>r>0$,

[TABLE]

where $\widetilde{u}^{\lambda}(y):=\widetilde{U}^{\lambda}(y,0)$ and $C(R,r)$ is independent of $\lambda$. Then there is a subsequence $\lambda_{k}$ of $\lambda\rightarrow+\infty$ such that $\{U^{\lambda_{k}}\}$ converges to a nonnegative function $\widetilde{U}^{\infty}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying

[TABLE]

By (4.9) we have

[TABLE]

Moreover, by the scaling invariance of $E(r;\widetilde{U})$ and Proposition 4.1, we have for any $r>0$ that

[TABLE]

That is, $E(r;\widetilde{U}^{\infty})$ is a constant. It follows from Proposition 3.1 that $\widetilde{U}^{\infty}$ is homogeneous of degree $-\frac{2\sigma+\vartheta}{p-1}$. Notice that we have $p<p_{S}(\vartheta)$, the same argument as in Case 1 gives that $\widetilde{U}^{\infty}$ has the form

[TABLE]

where $C_{p,\sigma,\vartheta}$ is given by (1.6). By the uniqueness of the limiting function $\widetilde{U}^{\infty}$, we conclude that $\widetilde{U}^{\lambda}(y,t)\rightarrow\widetilde{U}^{\infty}(y,t)$ for any sequence $\lambda\rightarrow+\infty$ in $C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$. In particular,

[TABLE]

uniformly for $X\in\overline{\mathbb{S}^{n}_{+}}$. Hence we have

[TABLE]

From (1.7) we have that $\Lambda(\tau)=\Lambda(-\tau)$. Therefore

[TABLE]

By the definition of Kelvin transform, we now easily get that (1.12) holds. This completes the proof of Theorem 1.4. $\square$

Proof of Theorem 1.1. It follows from the extension theorem of Caffarelli-Silvestre [10] and Theorem 1.4. $\square$

Now we give the proof of the uniqueness of global singular solutions in Theorem 1.7, which is similar to that of Theorem 1.4. But it is very different from the proof of the uniqueness theorem of Gidas-Spruck [21] (See Theorem 1.4 in [21]).

Proof of Theorem 1.7. Suppose that $U$ is a nonnegative weak solution of (1.18) and the two singularities [math] and $\infty$ of $u(x)$ are non-removable. By Theorem 1.3 there exist two positive constants $C_{1}$ and $C_{2}$ such that

[TABLE]

Since the singularity at $\infty$ is non-removable, by using the Kelvin transformation and Theorem 1.3, it is not difficult to prove that (4.11) also holds near $X=\infty$. Hence we have

[TABLE]

Now we consider separately the subcritical case $p<p_{S}(\alpha)$ and the supercritical case $p>p_{S}(\alpha)$.

Case 1: $p<p_{S}(\alpha)$. For any $\lambda>0$, we define the scaling

[TABLE]

Then $U^{\lambda}$ also satisfies (1.18) and (4.12). The same arguments as in the proof of Theorem 1.4 imply that there is a subsequence $\lambda_{k}$ of $\lambda\rightarrow 0$ such that $\{U^{\lambda_{k}}\}$ converges to a nonnegative function $U^{0}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying (1.18) and (4.12). By the scaling invariance of $E(r;U)$ and Proposition 4.1, we have for any $r>0$ that

[TABLE]

Similar to the proof of Theorem 1.4, it follows from Proposition 3.1 and Proposition 4.2 that $U^{0}$ has the explicit expression

[TABLE]

where $C_{p,\sigma,\alpha}$ is given by (1.6). On the other hand, let $\lambda\to+\infty$, there is another subsequence $\lambda_{k}$ of $\lambda\rightarrow+\infty$ such that $\{U^{\lambda_{k}}\}$ converges to a nonnegative function $U^{\infty}\in W_{loc}^{1,2}(t^{1-2\sigma},\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})\cap C^{\gamma}_{loc}(\overline{\mathbb{R}^{n+1}_{+}}\backslash\{0\})$ satisfying (1.18) and (4.12). By the scaling invariance of $E(r;U)$ and Proposition 4.1, we have for any $r>0$ that

[TABLE]

Similar to the above proof, we obtain that $U^{\infty}$ also has the explicit expression (4.14). By (4.13) and (4.15) we have

[TABLE]

This together with the monotonicity of $E(r;U)$ on $(0,\infty)$ implies that $E(r;U)$ is a constant. By a very similar argument as in the proof of Theorem 1.4, it follows from Proposition 3.1 and Proposition 4.2 that $U(x,t)$ has the form (1.19).

Case 2: $p_{S}(\alpha)<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$. We consider the Kelvin transform

[TABLE]

for $Y\in\mathbb{R}^{n+1}_{+}\backslash\{0\}$. Denote $\vartheta:=p(n-2\sigma)-(n+2\sigma+\alpha)$. Then the same arguments as in Case 1 give that $\widetilde{U}$ has the form

[TABLE]

where $C_{p,\sigma,\vartheta}$ is given by (1.6). A simple calculation shows that $U(x,t)$ satisfies (1.19). The proof of Theorem 1.7 is completed. $\square$

5 Isolated Singularities at Infinity

In this section, we prove Theorems 1.5 and 1.6.

Proof of Theorem 1.5. Let $u(x)$ be a nonnegative solution of (1.14) with $\alpha>-2\sigma$ and $1<p<\frac{n+2\sigma}{n-2\sigma}$. We consider the Kelvin transform

[TABLE]

Then $\widetilde{u}\in C^{2}(B_{1}\backslash\{0\})\cap\mathcal{L}_{\sigma}(\mathbb{R}^{n})$ and $\widetilde{u}$ satisfies

[TABLE]

where $\varrho:=p(n-2\sigma)-(n+2\sigma+\alpha)$.

(1) If $1<p<\frac{n+\alpha}{n-2\sigma}$, then $\varrho<-2\sigma$. By Corollary 2.1 we have $\widetilde{u}(y)\equiv 0$ in $B_{1}\backslash\{0\}$, this implies that $u(x)\equiv 0$ for $|x|>1$.

(2) If $\frac{n+\alpha}{n-2\sigma}<p<\frac{n+2\sigma}{n-2\sigma}$, then $-2\sigma<\varrho<-\alpha<2\sigma$ and

[TABLE]

It follows from Theorem 1.2 that either the singularity at $y=0$ is removable, or there exist $c_{1},c_{2}>0$ such that

[TABLE]

If the singularity at $y=0$ is removable, then

[TABLE]

If (5.2) holds, then

[TABLE]

This completes the proof. $\square$

Proof of Theorem 1.6. Let $u(x)$ be a positive solution of (1.14) with $-2\sigma<\alpha\leq 0$, $\frac{n+\alpha}{n-2\sigma}<p\leq\frac{n+2\sigma+\alpha}{n-2\sigma}$ and $p\neq\frac{n+2\sigma+2\alpha}{n-2\sigma}$. We define the Kelvin transform $\widetilde{u}(y)$ of $u(x)$ as in (5.1). Then $\widetilde{u}(y)$ satisfies

[TABLE]

where $\varrho:=p(n-2\sigma)-(n+2\sigma+\alpha)$. Note that

[TABLE]

Hence, under the assumptions of Theorem 1.6, we have

[TABLE]

Thus, Theorem 1.1 could be applied to the equation (5.3) and we obtain that either the singularity near $y=0$ is removable, or

[TABLE]

If the singularity near $y=0$ is removable, then $\widetilde{u}(y)$ can be extended to a continuous function near the origin 0. Hence, there exists $\beta>0$ such that

[TABLE]

If (5.4) holds, then

[TABLE]

This completes the proof. $\square$