Abstract
In this paper, we study the local behaviors of positive solutions of
with an isolated singularity at the origin, where \((-\Delta )^\sigma \) is the fractional Laplacian, \(0<\sigma <1\), \(\tau >-2\sigma \) and \(p>1\). Our first results provide a blowup rate estimate near an isolated singularity, and show that the solution is asymptotically radially symmetric.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The Hardy–Hénon equation
has been studied in many papers, where \(\Delta :=\sum _{i=1}^n\frac{\partial ^2}{\partial x^2_i}\) denotes the Laplacian, \(\tau >-2\), \(p>1\) are parameters, the punctured unit ball \(B_1\setminus \{0\}\subset \mathbb {R}^n\) with \(n\ge 3\).
The blowup rate of solution of (1) has been very well understood. In the special case of \(\tau =0\), there exists a positive constant C such that
See Lions [20] for \(1<p<\frac{n}{n-2}\), Gidas–Spruck [10] for \(\frac{n}{n-2}<p<\frac{n+2}{n-2}\), Korevaar–Mazzeo–Pacard–Schoen [14] for \(p=\frac{n+2}{n-2}\). Aviles [1, 2] treated the case of \(p=\frac{n}{n-2}\) and obtained that
In the case of \(-2<\tau <2\), the upper bound has the following forms near \(x=0\),
Phan–Souplet [22] studied the case of \(-2<\tau \) and \(1<p<\frac{n+2}{n-2}\), and derived that
where \(\nabla u\) denotes the gradient of u. We point out that the first estimate of (5) extends a number of previously results of [10] and [20] regarding them as consequence.
In the classical paper [6], Caffarelli–Gidas–Spruck considered the semilinear elliptic equations
and proved that every solution is asymptotically radially symmetric
Here g(u) is a locally Lipschitz function and is the spherical average of u. A typical example is \(g(u)=u^p\) with \(\frac{n}{n-2}\le p \le \frac{n+2}{n-2}\). Li [15] obtained a weaker asymptotically radially symmetric for more general \(g(x,u)=|x|^{\tau }u^p\), where \(\tau \le 0\), and \(1<p\le \frac{n+2+\tau }{n-2}\).
Inspired by previous work, this paper is aiming at studying the local behaviors of positive solutions of
where \(0<\sigma <1\), \(\tau >-2\sigma \), \(p>1\), the punctured unit ball \(B_1\backslash \{0\}\subset \mathbb {R}^n\), \(n\ge 2\), and \((-\Delta )^\sigma \) is the fractional Laplacian taking the form
here \(\text {P.V.}\) stands for the Cauchy principal value and
with the gamma function \(\Gamma \). The operator \((-\Delta )^\sigma \) is well defined in the Schwartz space of rapidly decaying \(C^\infty \) functions in \(\mathbb {R}^n\).
One can also define the fractional Laplacian acting on spaces of functions with weaker regularity. Considering the space
endowed with the norm
We can verify that if \(u\in C^{2}(B_1\backslash \{0\})\cap L_{\sigma }(\mathbb {R}^n)\), the integral on the right hand side of (8) is well defined in \(B_1\backslash \{0\}\). Moreover, from [24, Proposition 2.4], we have
Problems concerning the fractional Laplacian \((-\Delta )^\sigma \) with an internal isolated singularity have attracted a lot of attention. In particular, Caffarelli–Jin–Sire–Xiong [7] studied the local behaviors of positive solutions of the fractional Yamabe equations
with an isolated singularity at the origin. They obtained sharp blowup rate,
and proved that every local solution of (9) is asymptotically radially symmetric
It is consistent with the result of Korevaar–Mazzeo–Pacard–Schoen [14] work on Laplacian. Jin-de Queiroz–Sire–Xiong [13] further studied Eq. (9) in \(\Omega \setminus \Lambda \), where \(\Omega \) is an open set in \(\mathbb {R}^n\), and \(\Lambda \) is a singular set other than a single point. More work related isolated singular problem see Chen–Quaas [9] for \(1<p<\frac{n+2\sigma }{n-2\sigma }\), Sun–Jin [25] for higher order fractional case.
Our first result provides a blowup rate estimate near an isolated singularity.
Theorem 1.1
Let \(-2\sigma <\tau \), \(1<p<\frac{n+2\sigma }{n-2\sigma }\). Suppose that \(u\in C^{2}(B_1 \backslash \{0\})\cap L_{\sigma }(\mathbb {R}^n)\) is a positive solution of (7), then there exists a positive constant \(C=C(n,\sigma , \tau , p)\) such that
The result can be understood as an extension of the work (5) and (10). We obtain the blowup rate estimate (12) by using the method of blowing-up and rescaling argument. (For more details and references, see, e.g., [7, 11, 13, 22, 27]). In particular, the doubling property (Proposition 2.1) plays a key role in our proof. The idea, by contradiction, is that if an estimate fails, the violating sequence of solutions \(u_k\) will be increasingly large along a sequence of points \(x_k\) such that each \(x_k\) has a suitable neighborhood, where the relative growth of \(u_k\) remains controlled. After appropriate rescaling, one can blow up the sequence of neighborhoods and pass to the limit to obtain a bounded solution of a limiting problem in the whole of \(\mathbb {R}^n\). At last, by Liouville theorem, we get a contradiction.
With the help of the estimate (12), we are able to show that the solution of (7) is asymptotically radially symmetric.
Theorem 1.2
Let \(-2\sigma <\tau \le 0\), \(\frac{n+\tau }{n-2\sigma }<p\le \frac{n+2\sigma +2\tau }{n-2\sigma }\). Suppose that \(u\in C^{2}(B_1\backslash \{0\})\cap L_{\sigma }(\mathbb {R}^n)\) is a positive solution of (7), then
where is the spherical average of u.
The result can be understood as an extension of the work (6) and (11). Notice that we do not use any special structure of the unite ball \(B_1\), \(B_1\) can be replaced by general open sets containing the origin. We get the result by the method of moving spheres with Kelvin transformation, a variant of the method of moving planes, which has been widely used and has become a powerful and user-friendly tool in the study of nonlinear partial differential equations (see [8, 16,17,18,19]). In addition, by Kelvin transformation, the difference between Eqs. (7) and (9) is that Eq. (9) is a conformally invariant equations but Eq. (7) is not. To deal with the problem, we need the following useful proposition.
Proposition 1.3
For \(x\in \mathbb {R}^n\backslash \{0\}\), \(y\in \mathbb {R}^n\setminus \{x\}\), let \( y_{\lambda }:=x+\frac{\lambda ^2(y-x)}{|y-x|^2},\) we have
if and only if
One consequence of this proposition is the following corollary.
Corollary 1.4
For \(x\in \mathbb {R}^n\backslash \{0\}\), \(0<\lambda <\min \{|x|,|y-x|\}\), we have
The outline of this paper is arranged as follows. In Sect. 2, we will obtain the blowup upper bound (12). Then with the help of the estimate we can devote to asymptotically radially symmetry property of solutions of (7) in Sect. 3. In Appendix, for readers’ convenience, we not only prove some preliminaries, but also collect some basic propositions which will be used in our proof.
2 Upper bound near an isolated singularity
First, we recall the doubling property [23, Lemma 5.1] and denote \(B_{R}(x)\) as the ball in \(\mathbb {R}^{n}\) with radius R and center x. For convenience, we write \(B_R(0)\) as \(B_R\) for short.
Proposition 2.1
Suppose that \(\emptyset \ne D\subset \Sigma \subset \mathbb {R}^n\), \(\Sigma \) is closed and \(\Gamma =\Sigma \setminus D\). Let \(M: D\rightarrow (0,\infty )\) be bounded on compact subset of D. If for a fixed positive constant k, there exists \(y\in D\) satisfying
then there exists \(x\in D\) such that
and for all \(z\in D\cap B_{kM^{-1}(x)}(x)\),
The second one is called the interior Schauder estimates. See [12, Theorem 2.11] for the proof. Readers can see [4] for more regularity issues.
Proposition 2.2
Suppose that \(g\in C^{\gamma }(B_R)\), \(\gamma >0\) and u is a nonnegative solution of
If \(2\sigma +\gamma \le 1\), then \(u\in C^{0,2\sigma +\gamma }(B_{R/2})\). Moreover,
where C is a positive constant depending on n, \(\sigma \), \(\gamma \), R.
If \(2\sigma +\gamma >1\), then \(u\in C^{1,2\sigma +\gamma -1}(B_{R/2})\). Moreover,
where C is a positive constant depending on n, \(\sigma \), \(\gamma \), R.
Next, in order to prove Theorem 1.1, we start with the following lemma.
Lemma 2.3
Let \(1< p <\frac{n+2\sigma }{n-2\sigma }\), \(0<\alpha \le 1\) and \(c(x)\in C^{2,\alpha }(\overline{B_1})\) satisfy
for some positive constants \(C_1\), \(C_2\). Suppose that \(u\in C^{2}(B_1)\cap L_{\sigma }(\mathbb {R}^n)\) is a nonnegative solution of
then there exists a positive constant C depending only on n, \(\sigma \), p, \(C_1\), \(C_2\) such that
Proof
Arguing by contradiction, for \(k=1,2,\cdots \), we assume that there exist nonnegative functions \(u_k\) satisfying (17) and points \(y_k\in B_{1}\) such that
Define
Via Proposition 2.1, for \(D=B_1\), \(\Gamma =\partial B_1\), there exists \(x_k\in B_{1}\) such that
and for any \(z\in B_1\) and \(|z-x_k|\le kM_k^{-1}(x_k)\),
It follows from (19) that
Consider
Combining (22), we obtain that for any \(y\in B_k\),
that is,
Therefore, \(w_k\) is well defined in \(B_k\) and
From (20), we find that for all \(y\in B_k\),
That is,
Moreover, \(w_k\) satisfies
and
where \(c_k(y):=c( x_k+\lambda _{k}y)\).
By condition (16), we obtain that \(\{c_k\}\) is uniformly bounded in \(\mathbb {R}^n\). For each \(R>0\), and for all y, \(z\in B_R\), we have
for k is large enough. Therefore, by Arzela–Ascoli’s Theorem, there exists a function \(c\in C^2(\mathbb {R}^n)\), after extracting a subsequence, \(c_k\rightarrow c\) in \(C^2_{loc}(\mathbb {R}^n)\). Moreover, by (21), we obtain
This implies that the function c actually is a constant C. By (16) again, \(c_k \ge C_2>0\), we conclude that C is a positive constant.
On the other hand, applying Proposition 2.2 a finite number of times to (23) and (24), there exists some positive \(\gamma \in (0,1)\) such that for every \(R\in (1,k)\),
where C(R) is a positive constant independent of k. Thus, after passing to a subsequence, we have, for some nonnegative function \(w\in C^2_{\mathrm{loc}}(\mathbb {R}^n)\),
Moreover, w satisfies
and
Since \(p<\frac{n+2\sigma }{n-2\sigma }\), this contradicts the Liouville-type result [12, Remark 1.9] that the only nonnegative entire solution of (26) is \(w=0\). Then we conclude the lemma. \(\square \)
We now turn to prove Theorem 1.1.
Proof
For \(x_0\in B_{1/2}\backslash \{0\}\), we denote \(R:=\frac{1}{2}|x_0|\). Then for any \(y\in B_1\), we have \(\frac{|x_0|}{2}<|x_0+Ry|<\frac{3|x_0|}{2}\), and deduce that \(x_0+Ry\in B_1\backslash \{0\}\). Define
Therefore, we obtain that
where \(c(y):=|y+\frac{x_0}{R}|^{\tau }\). Notice that
Moreover,
Applying Lemma 2.3, we obtain that
That is,
Hence,
Since \(x_0\in B_{1/2}\setminus \{0\}\) is arbitrary, Theorem 1.1 is proved. \(\square \)
3 Asymptotical radial symmetry
3.1 Proof of Theorem 1.2
Proof
Assume that there exists some positive constant \(\varepsilon \in (0,1)\) such that for all \(0<\lambda <|x|\le \varepsilon \), \(y\in B_{3/4}\backslash B_\lambda (x)\) and \(y\ne 0\),
where
Let \(r>0\) and \(x_1\), \(x_2\in \partial B_r\) be such that
and define
Then
Via some direct computations and \(|x_1|^2=|x_2|^2=r^2\), we find that
which follows from this and (28) that \(\lambda<|x_3| <\varepsilon \) by choosing \(r<\frac{3\varepsilon }{4}\).
It follows from (27) that
Since
then
and
Hence,
On the other hand,
then
for some \(C=C(\varepsilon )\). That is,
Hence for any \(x\in \partial B_r\),
In conclusion, we have
It follows that
Therefore, in order to complete the proof of Theorem 1.2, it suffices to prove (27). \(\square \)
3.2 Proof of (27)
Since the operator \((-\Delta )^\sigma \) is nonlocal, the traditional methods on local differential operators, such as on Laplacian, may not work on this nonlocal operator. To circumvent this difficulty, Caffarelli and Silvestre [5] introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions with the connormal derivative boundary condition.
In order to describe the method in a more precise way, let us give some notations. We use capital letters, such as \(X=(x,t)\) to denote points in \(\mathbb {R}^{n+1}_{+}\). We denote \(\mathcal {B}_R(X)\) as the ball in \(\mathbb {R}^{n+1}\) with radius R and center X, and \(\mathcal {B}^+_R(X)\) as \(\mathcal {B}_R(X)\cap \mathbb {R}^{n+1}_+\). We also write \(\mathcal {B}_R(0)\), \(\mathcal {B}^+_R(0)\) as \(\mathcal {B}_R\), \(\mathcal {B}_R^+\) for short respectively. For a domain \(D\subset \mathbb {R}^{n+1}_+\) with boundary \(\partial D\), we denote \(\partial ' D:=\partial D\cap \partial \mathbb {R}^{n+1}_+\) and \(\partial ''D:=\partial D \cap \mathbb {R}^{n+1}_+\). In particular, \(\partial ' \mathcal {B}_R^+(X):=\partial \mathcal {B}^+_R(X)\cap \partial \mathbb {R}^{n+1}_+\) and \(\partial '' \mathcal {B}_{R}^+(X):=\partial \mathcal {B}^+_{R}(X)\cap \mathbb {R}^{n+1}_+\).
More precisely, for \(u\in C^{2}(B_1 \backslash \{0\})\cap L_{\sigma }(\mathbb {R}^n)\), define
where
with a constant \(\beta (n,\sigma )\) such that \(\int _{\mathbb {R}^n}\mathcal {P}_{\sigma }(x,1)\hbox {d}x=1\). Then
satisfying
In order to study the behaviors of the solution u of (7), we just need to study the behaviors of U defined by (29). In addition, by works of Caffarelli and Silvestre [5], it is known that up to a constant,
where the connormal derivative
From this and (7), we have
For all \(0<|x|<\frac{1}{4}\), \(X=(x,0)\) and \(\lambda >0\), define the Kelvin transformation of U as
The aim is to show that there exists some positive constant \(\varepsilon \in (0,1)\) such that for \(0<\lambda <|x|\le \varepsilon \),
In particular, choose \(\xi =(y,0)\), \(y\in \mathbb {R}^n\setminus \{0\}\), then for \(0<\lambda <|x|\le \varepsilon \),
that is (27).
3.3 Proof of (33)
To prove (33), for fixed \(x\in B_{1/4}\backslash \{0\}\), we first define
and then show \(\bar{\lambda }(x)=|x|\).
For sake of clarity, the proof of (33) is divided into three steps. For the first step, we need the following Claim 1 to make sure that \(\bar{\lambda }(x)\) is well defined.
Claim 1: There exists \(\lambda _0(x)<|x|\) such that for all \(\lambda \in (0,\lambda _0(x))\),
Second, we give that
Claim 2: There exists a positive constant \(\varepsilon \in (0,1)\) sufficiently small such that for all \(0<\lambda <|x|\le \varepsilon \),
Last, we are going to prove that
Claim 3:
Proof of Claim 1
First of all, we are going to show that there exist \(\mu \) and \(\lambda _0(x)\) satisfying \(0<\lambda _0(x)<\mu <|x|\) such that for all \(\lambda \in (0,\lambda _0(x))\),
Then we will prove that for all \(\lambda \in (0,\lambda _0(x))\),
For every \(0<\lambda<\mu <\frac{1}{2}|x|\), \(\xi \in \partial '' \mathcal {B}^+_{\mu }(X)\), we have \(X+\frac{\lambda ^2(\xi -X)}{|\xi -X|^2}\in \mathcal {B}^+_{\mu }(X)\). Thus we can choose
such that for every \(0<\lambda<\lambda _0(x)<\mu \),
The above inequality, together with
implies that for all \( \lambda \in (0,\lambda _0(x))\),
We will make use of the narrow domain technique of Berestycki and Nirenberg from [3], and show that, for sufficiently small \(\mu \), that for \(\lambda \in (0,\lambda _0(x))\),
It is a straightforward computation to show that
which yield
where \(y_{\lambda }:=x+\frac{\lambda ^2(y-x)}{|y-x|^2}\), \(p^*:=n+2\sigma -p(n-2\sigma )\).
Let \((U_{X,\lambda }-U)^+:=\max (0,U_{X,\lambda }-U)\) which equals 0 on \(\partial ''(\mathcal {B}^+_{\mu }(X)\backslash \mathcal {B}^+_{\lambda }(X))\). Multiplying the equation in (37) by \((U_{X,\lambda }-U)^+\) and integrating by parts in \(\mathcal {B}_{\mu }^+(X)\backslash \overline{\mathcal {B}_{\lambda }^+(X)}\), we have
Combining Corollary 1.4 with \(\lambda ^2=|x-y_{\lambda }||x-y|\), we have
which implies that
due to \(-2\tau \le p^*\) and \(\frac{\lambda }{|x-y|}\le 1\). Therefore,
and
For any \( y\in B_{\mu }(x)\backslash B_{\lambda }(x)\),
where \(y_{\lambda }:=x+\frac{\lambda ^2(y-x)}{|y-x|^2}\). Combining \(y_{\lambda }\in B_{\lambda }(x)\subset \overline{B}_{\frac{|x|}{2}}(x)\) with \(u\in C^{2}(B_{1} \setminus \{0\})\), we deduce that there exists a positive constant C depending on x, such that for any \( y\in B_{\mu }(x)\backslash B_{\lambda }(x)\),
With the help of mean value theorem and \(\lambda <\frac{1}{2}|x|\), we obtain
where the trace inequality (Proposition 4.1) is used in the last inequality and \(C(n,p,\sigma ,\tau , |x|)\) is a positive constant.
We can fix \(\mu \) sufficiently small such that
Then
Since (36), we deduce that
Therefore,
After that, we are going to prove (35). Let
which satisfies
and
In addition, combining Proposition 4.2, we can choose \(\mu \) small enough such that
and
By the standard maximum principle (Proposition 4.3), we have
Then for all \(\xi \in \mathcal {B}^{+}_{3/4}\backslash \mathcal {B}^{+}_{\mu }(X)\) and \(\lambda \in (0,\lambda _0)\subset (0,\mu )\), we have
where (39) is used in the last inequality. Then Claim 1 is proved. \(\square \)
Proof of Claim 2
For \(y\in B_1\), \(\frac{3}{8}\le |y|\le \frac{7}{8}\) and \(0<\lambda<|x|<\frac{1}{8}\), we have
Hence
and
It follows from Theorem 1 that
Thus, for \(0<\lambda<|x|<\frac{1}{8},\ \frac{3}{8}\le |y|\le \frac{7}{8}\), we conclude that
Since \(\frac{n+\tau }{n-2\sigma }<p\le \frac{n+2\sigma +2\tau }{n-2\sigma }\), we have \(\frac{p(n-2\sigma )-n-\tau }{p-1}>0\). By Harnack inequality (Proposition 4.4), \(\varepsilon >0\) can be chosen sufficiently small to guarantee that for all \(0<\lambda <|x|\le \varepsilon \) and \(|\xi |=\frac{3}{4}\),
\(\square \)
Proof of Claim 3
We prove Claim 3 by contradiction. Assume \(\bar{\lambda }(x)<|x|\le \varepsilon \) for some \(x\ne 0\). We want to show that there exists a positive constant \(\widetilde{\varepsilon }\in (0,\frac{|x|-\bar{\lambda }(x)}{2})\) such that for \(\lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\widetilde{\varepsilon })\),
which contradicts the definition of \(\bar{\lambda }(x)\), then we obtain \(\bar{\lambda }(x)=|x|\).
Divide the region \(\mathcal {B}^+_{3/4}\backslash \mathcal {B}^+_{\lambda }(X)\) into three parts,
where \(\delta _1\), \(\delta _2\) will be fixed later. To obtain (41) it suffices to prove that it established on \(K_1\), \(K_2\), \(K_3\). And then we are going to prove it respectively. By Claim 1, we have
With the help of Claim 2 and the strong maximum principle, we deduce that
Choose \(R\in (0,\frac{|x|-\bar{\lambda }(x)}{2})\), then
where \(y_{\bar{\lambda }(x)}:=x+\frac{\bar{\lambda }(x)^2(y-x)}{|y-x|^2}\).
By (42) and Proposition 4.2, we get
Thus, there exists a positive constant \(\delta _1\in (0,\frac{|x|-\bar{\lambda }(x)}{2})\) and a positive constant \(C_1\) such that
By the uniform continuity of U on compact sets, there exists a positive constant \(\varepsilon _1\) small enough such that for \(\lambda \in (\bar{\lambda }(x), \bar{\lambda }(x)+\varepsilon _1\)),
From the above argument, we conclude that for \( \lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\varepsilon _1)\),
Next, choose \(\delta _2\) to be small, the aim is to show that there exist positive constants \(C_2\) and \(\varepsilon _2\), such that for \( \lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\varepsilon _2)\),
From (42) and \(K_{2}\) is compact, there exist a positive constant \(C_2\) such that
By the uniform continuity of U on compact sets, there exists a positive constant \(\varepsilon _2\) sufficiently small such that for all \( \lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\varepsilon _2)\),
Hence for all \( \lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\varepsilon _2)\), we have
Then we obtain (44).
Last, let us focus on the region \(K_3\). We can choose a positive constant \(\widetilde{\varepsilon }\) as small as we want (less then \(\varepsilon _1\) and \(\varepsilon _2\)) such that for \( \lambda \in (\bar{\lambda }(x),\bar{\lambda }(x)+\widetilde{\varepsilon })\),
Using the narrow domain technique as that the proof of (34) in Claim 1, we can choose \(\delta _2\) to be small such that
Together with (43), (44) and (45), we can see that the moving sphere procedure may continue beyond \(\bar{\lambda }(x)\) where we reach a contradiction. \(\square \)
At last, we give a proof for Proposition 1.3 and Corollary 1.4, which have been used in our proof.
Proof of Proposition 1.3
Via a straightforward calculation, (13) is rewritten as
that is,
Let
If \(|x|^2-2x\cdot y=0\), it is easy to see that f(s) is a affine function, and
Therefore,
If \(|x|^2-2x\cdot y\ne 0\), it follows that f(s) is a quadratic polynomial, and always has two roots,
Now, let us divide into the following three cases to consider.
Case 1: For \(|x|^2-2x\cdot y<0\), then \(s_2<0<s_1\), which implies that
Case 2: For \(|x|^2-2x\cdot y>0\) and \(x\cdot y\ge 0\), then \(0<s_1\le s_2\), it is obviously to obtain that
Case 3: For \(|x|^2-2x\cdot y>0\) and \(x\cdot y<0\), then \(0<s_2\le s_1\). As before, we have
Combining (46), (47), (48) and (49), we finish the proof of this proposition. \(\square \)
Proof of Corollary 1.4
If \(x\cdot y\ge 0\), it is easy to see that
It follows that \(\lambda \) satisfies (14) whether \(x\cdot y\) is bigger or smaller than \(|x|^2/2\).
If \(x\cdot y<0\), by a direct calculation, we have
This also implies that \(\lambda \) satisfies (14). Therefore, (13) has been established. \(\square \)
4 Appendix
The first one is called the trace inequality.
Proposition 4.1
[12, Proposition 2.1] If \(U\in C^2_c(\mathbb {R}^{n+1}_{+})\), then there exists a positive constant C depending only on n and \(\sigma \) such that
The next one is on a maximum principle for positive supersolutions with an isolated singularity.
Proposition 4.2
[12, Proposition 3.1] Suppose that \(U\in C^2(\mathcal {B}_R^+ \cup \partial '\mathcal {B}_R^+\setminus \{0\})\) and \(U>0\) in \(\mathcal {B}_R^+ \cup \partial '\mathcal {B}_R^+\setminus \{0\} \) is a solution of
then
We also recall the standard maximum principle.
Proposition 4.3
[12, Lemma 2.5] Suppose that \(U\in C^{2}(D)\cap C^{1}(\overline{D})\) is a solution of
where \(D\subset \mathbb {R}^{n+1}_{+}\) is an open domain. If \(U\ge 0\) on \(\partial ''D\), then \(U\ge 0\) in D.
The last one is the Harnack inequality, and Tan-Xiong [26] provide more details for the Harnack inequality.
Proposition 4.4
[12, Proposition 2.6] Suppose that \(U \in C^{2}(\mathcal {B}_{2R}^+)\cap C^{1}(\overline{\mathcal {B}_{2R}^+})\) is a nonnegative solution of
If \(a\in L^q(B_{2R})\) for some \(q>\frac{n}{2\sigma }\), then we have
where the positive constant C depends only on n, \(\sigma \), R and \(\Vert a\Vert _{L^q(B_{2R})}\).
References
Aviles, P.: On isolated singularities in some nonlinear partial differential equations. Indiana Univ. Math. J. 32, 773–791 (1983)
Aviles, P.: Local behavior of solutions of some elliptic equations. Commun. Math. Phys. 108, 177–192 (1987)
Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bull. Braz. Math. Soc. 22, 1–37 (1991)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincar Anal. Non Linaire 31, 23–53 (2014)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)
Caffarelli, L., Jin, T., Sire, Y., Xiong, J.: Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities. Arch. Ration. Mech. Anal. 213, 245–268 (2014)
Cao, D., Li, Y.Y.: Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator. Methods Appl. Anal. 15, 81–95 (2008)
Chen, H., Quaas, A.: Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, arXiv:1509.05836
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)
Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equ. 6, 883–901 (1981)
Jin, T., Li, Y.Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. 16, 1111–1171 (2014)
Jin, T., de Queiroz, O.S., Sire, Y., Xiong, J.: On local behavior of singular positive solutions to nonlocal elliptic equations. Calc. Var. Partial Differ. Equ. 56(9), 25 (2017)
Korevaar, N., Mazzeo, R., Pacard, F., Schoen, R.: Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135, 233–272 (1999)
Li, C.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math. 123, 221–231 (1996)
Li, Y.Y.: Conformally invariant fully nonlinear elliptic equations and isolated singularities. J. Funct. Anal. 233, 380–425 (2006)
Li, Y.Y., Lin, C.S.: A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents. Arch. Ration. Mech. Anal. 203, 943–968 (2012)
Li, Y.Y., Zhang, L.: Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations. J. Anal. Math. 90, 27–87 (2003)
Li, Y.Y., Zhu, M.: Uniqueness theorems through the method of moving spheres. Duke Math. J. 80, 383–418 (1995)
Lions, P.L.: Isolated singularities in semilinear problems. J. Differ. Equ. 38, 441–450 (1980)
Ni, W.M.: Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations. Proc. Symp. Pure Math. 39, 379–399 (1986)
Phan, Q.H.: Souplet, Ph: Liouville-type theorems and bounds of solutions of Hardy-Hénon equations. J. Differ. Equ. 252, 2544–2562 (2012)
Polácik, P., Quittner, P.: Souplet, Ph: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I: elliptic equations and systems. Duke Math. J. 139(3), 555–579 (2007)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)
Sun, L., Xiong, J.: Classification theorems for solutions of higher order boundary conformally invariant problems. I. J. Funct. Anal. 271, 3727–3764 (2016)
Tan, J., Xiong, J.: A Harnack inequality for fractional Laplace equations with lower order terms. Discrete Contin. Dyn. Syst. 31, 5–983 (2011)
Xiong, J.: The critical semilinear elliptic equation with isolated boundary singularities. J. Differ. Equ. 263, 1907–1930 (2017)
Singular solutions of semilinear elliptic and parabolic equations: Zhang, Qi S., Zhao, Z. Math. Ann. 310, 777–794 (1998)
Acknowledgements
We would like to express our deep thanks to Professor Jingang Xiong for useful discussions on the subject of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are supported in part by the National Natural Science Foundation of China (11631002).
Rights and permissions
About this article
Cite this article
Li, Y., Bao, J. Local behavior of solutions to fractional Hardy–Hénon equations with isolated singularity. Annali di Matematica 198, 41–59 (2019). https://doi.org/10.1007/s10231-018-0761-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-018-0761-9