The critical semilinear elliptic equation with isolated boundary singularities

TL;DR

This paper analyzes the asymptotic behavior of nonnegative solutions to a critical semilinear elliptic equation with isolated boundary singularities, providing quantitative insights into their boundary behavior.

Contribution

It introduces new quantitative asymptotic estimates for solutions near boundary singularities in critical semilinear elliptic equations.

Findings

Established precise asymptotic behaviors near boundary singularities

Derived bounds for solutions in critical cases

Enhanced understanding of boundary singularity effects

Abstract

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-\Delta u=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.

Equations

$$\{\begin{array}{l}\begin{array}{rlrl}-\Delta u& =n(n-2)u^{\frac{n+2}{n-2}}& & \text{mbox}in{\mathbb{R}}_{+}^n,\\ u& =0& & \text{mbox}on\partial {\mathbb{R}}_{+}^n\setminus \{0\}.\end{array}\end{array}$$

$${\partial }_{tt}^{2}U+{\Delta }_{{\mathbb{S}}^{n-1}}U-\frac{(n-2{)}^2}{4}U+n(n-2)U^{\frac{n+2}{n-2}}=0\text{mbox}on\mathbb{R}\times {\mathbb{S}}_{+}^{n-1},$$

$$U=0\text{mbox}on\mathbb{R}\times \partial {\mathbb{S}}_{+}^{n-1},$$

$$\psi (0)=0,\nabla \psi (0)=0.$$

$$\{\begin{array}{l}\begin{array}{rlrl}-\Delta u& =n(n-2)u^{\frac{n+2}{n-2}}& & \text{mbox}in{Q}_1,\\ u& =0& & \text{mbox}on{\Gamma }_1\setminus \{0\}.\end{array}\end{array}$$

$$u(x)\le C(\gamma )d(x)|x{|}^{-\frac{n}{2}}$$

$$u(x^{\prime },x_n)=\bar{u}(|x^{\prime }|,x_n)(1+O(|x|))\text{mbox}if{x}_n>\frac{1}{\delta }\underset{|y^{\prime }|=|x^{\prime }|}{\max }|\psi (y^{\prime })|,$$

$$u(x)\ge \frac{1}{C}d(x)|x{|}^{-\frac{n}{2}}\text{mbox}forallx\in Q_{1/2}.$$

$$\psi (0)=0,\nabla \psi (0)=0$$

$$\Vert \psi {\Vert }_{C^2({\partial }^{\prime }{B}_1^{+})}\le A_1.$$

$$\{\begin{array}{l}\begin{array}{rlrl}-\Delta u& =a(x)u& & \text{mbox}in{Q}_1,\\ u& =0& & \text{mbox}on{\Gamma }_1,\end{array}\end{array}$$

$$\frac{u(x)}{d(x)}\le C_0\frac{u(y)}{d(y)}\text{mbox}forx,y\in Q_{1/2}$$

$$\left|\frac{u(x)}{d(x)}-\frac{u(y)}{d(y)}\right|\le C_0|x-y{|}^{\alpha }\text{mbox}forx,y\in Q_{1/2},$$

$$\{\begin{array}{l}\begin{array}{rlrl}-\Delta u& \ge 0& & \text{mbox}in{Q}_1,\\ u& =0& & \text{mbox}on{\Gamma }_1\setminus \{0\},\end{array}\end{array}$$

$$u(x)\ge d(x)/c_0\text{mbox}forallx\in Q_{1/2}.$$

$$\Delta (\frac{d(x)}{1-Ad(x)})\ge 0\text{mbox}forx\in Q_{1/2},d(x)<\delta .$$

$$-{\partial }_{\nu }u\ge b_0>0\text{mbox}on{\Gamma }_1\cap \partial B_{1/2}.$$

$$u(x)\ge \frac{1}{C}\frac{d(x)}{1-Ad(x)}\text{mbox}on\partial (Q_{1/2}\cap \{d(x)<\delta \}).$$

$$u(x)\ge \frac{1}{C}\frac{d(x)}{1-Ad(x)}\text{mbox}in{Q}_{1/2}\cap \{d(x)<\delta \}.$$

$$\Delta u+au\le 0\text{mbox}in{Q}_1,u=0\text{mbox}on{\Gamma }_1\setminus \{0\}.$$

$$u\ge 0\text{mbox}in{Q}_1\setminus Q_{\epsilon }.$$

$$-\Delta u\ge 0\text{mbox}in{\mathbb{R}}_{+}^n\setminus B_R^{+},u(x^{\prime },0)=0\text{mbox}for{x}^{\prime }\ne {\bar{x}}^{\prime },$$

$$u(x)\ge \frac{R^n{x}_n}{|x{|}^n}\underset{y\in {\partial }^{\prime \prime }{B}_R^{+}}{\inf }\frac{u(y)}{y_n}\text{mbox}in{\mathbb{R}}_{+}^n\setminus B_R^{+}.$$

$${\varphi }_{\epsilon }(x)=\frac{R^n{x}_n}{|x{|}^n}\underset{y\in {\partial }^{\prime \prime }{B}_R^{+}}{\inf }\frac{u(y)}{y_n}-\epsilon .$$

$$-\Delta u=0\text{mbox}in{B}_1^{+},u(x^{\prime },0)=0\text{mbox}for{x}^{\prime }\ne 0.$$

$$u(x)=a\frac{x_n}{|x{|}^n}+h(x),$$

$$-\Delta h=0\text{mbox}in{B}_1^{+},h(x^{\prime },0)=0\text{mbox}for|x^{\prime }|<1.$$

$$-\Delta u=f(u)\text{mbox}in\Omega \text{mbox}andu=0\text{mbox}on\partial \Omega \cap \{x_1>0\},$$

$$F(x):=\left(\frac{2x^{\prime }}{|x^{\prime }{|}^2+(1+x_n{)}^2},\frac{|x{|}^2-1}{|x^{\prime }{|}^2+(1+x_n{)}^2}\right).$$

$$v(z):=|J_F{|}^{-\frac{n-2}{2n}}u(x)$$

Full text

1 Introduction

The internal isolated singularity for positive solutions of the semilinear equation $-\Delta u=u^{p}$ has been very well understood, where $\Delta$ is the Laplace operator, $1<p\leq\frac{n+2}{n-2}$ is a parameter and $n\geq 3$ is the dimension. See Lions [21] for $1<p<\frac{n}{n-2}$, Gidas-Spruck [13] for $\frac{n}{n-2}<p<\frac{n+2}{n-2}$, Aviles [1] for $p=\frac{n}{n-2}$, Caffarelli-Gidas-Spruck [9] for $\frac{n}{n-2}\leq p\leq\frac{n+2}{n-2}$ and Korevaar-Mazzeo-Pacard-Schoen [16] for $p=\frac{n+2}{n-2}$. The Sobolev critical exponent $p=\frac{n+2}{n-2}$ case is of particular interest, because the equation connects to the Yamabe problem and the conformal invariance leads to a richer isolated singularity structure. See also Li [18] and Han-Li-Teixeira [14] for conformally invariant fully nonlinear elliptic equations.

The Dirichlet boundary isolated singularity for the same equation has also been studied in many cases. Asymptotic behaviors of singular solutions have been established by Bidaut-Véron-Vivier [5] for $1<p<\frac{n+1}{n-1}$ and Bidaut-Véron-Ponce-Véron [3, 4] for $\frac{n+1}{n-1}\leq p<\frac{n+2}{n-2}$. Existence of singular solutions vanishing on boundaries of bounded domains except finite points has been obtained by del Pino-Musso-Pacard [12] for $p<\frac{n+2}{n-2}$. The exponent $\frac{n+1}{n-1}$ corresponding to $\frac{n}{n-2}$ for the interior singularity was discovered by Brézis-Turner [7]. Under a blow up rate assumption Bidaut-Véron-Ponce-Véron [3, 4] obtain refined asymptotic behaviors for the supercritical case $\frac{n+2}{n-2}<p<\frac{n+1}{n-3}$. We refer to [3] and references therein for related results on boundary singularity.

This paper is concerned with the remaining critical case: $p=\frac{n+2}{n-2}$. The conformal invariance again produces additional complexity and the boundary condition makes the asymptotic analysis of [9] and [16] fail. As said in Bidaut-Véron-Ponce-Véron [4], one can show

Proposition 1.1**.**

Denote $\mathbb{R}^{n}_{+}=\{x=(x^{\prime},x_{n})\in\mathbb{R}^{n}:x_{n}>0\}$. Let $u\in C^{2}(\mathbb{R}^{n}_{+})\cap C(\bar{\mathbb{R}}^{n}_{+}\setminus\{0\})$ be a nonnegative solution of

[TABLE]

Suppose [math] is a non-removable singularity of $u$, then $u$ depends only on $|x^{\prime}|$ and $x_{n}$, and $\partial_{r}u(r,x_{n})<0$ for all $r=|x^{\prime}|>0$.

Note that nothing about the behavior of $u$ at infinity is assumed in Proposition 1.1.

Let $u$ be a solution of (1) and define $U(t,\theta):=|x|^{\frac{n-2}{2}}u(|x|\cdot\theta)$ with $t=-\ln|x|$. Then we have

[TABLE]

[TABLE]

where $\mathbb{S}^{n-1}_{+}=\{\theta=(\theta_{1},\dots,\theta_{n})\in\mathbb{S}^{n-1}:\theta_{n}>0\}$. By Proposition 1.1, $U(t,\theta)=U(t,\theta_{n})$. In contrast to the internal singularity studied by Caffarelli-Gidas-Spruck [9] and Korevaar-Mazzeo-Pacard-Schoen [16], we lose ODE analysis to classify all solutions of equation (2)-(3). del Pino-Musso-Pacard [12] conjectured that there exists a one-parameter family of solutions of (2)-(3). Bidaut-Véron-Ponce-Véron [3, 4] proved that there exists a unique $t$-independent solution. Existence of $t$-dependent solutions and a priori estimates are left open.

Let $\psi$ be a $C^{2}$ function in $\mathbb{R}^{n-1}$ satisfying

[TABLE]

Let $Q_{R}=\{x=(x^{\prime},x_{n}):x_{n}>\psi(x^{\prime})\}\cap B_{R}$ and $\Gamma_{R}=\{x=(x^{\prime},x_{n}):x_{n}=\psi(x^{\prime})\}\cap B_{R}$, where $B_{R}$ is the open ball center at [math] with radius $R$. We consider nonnegative solutions of

[TABLE]

Theorem 1.2**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). Then for each $0<\gamma<1$ there exists a constant $C(\gamma)\geq 1$ such that for all $x\in Q_{1/2}$ with $\frac{d(x)}{|x|}\leq\gamma$,

[TABLE]

and

[TABLE]

where $d(x)=dist(x,\Gamma_{1})$, \bar{u}(x^{\prime},x_{n})=\mathchoice{{\vbox{\hbox{\textstyle-}}\kern-4.86108pt}}{{\vbox{\hbox{\scriptstyle-}}\kern-3.25pt}}{{\vbox{\hbox{\scriptscriptstyle-}}\kern-2.29166pt}}{{\vbox{\hbox{\scriptscriptstyle-}}\kern-1.875pt}}\!\int_{\mathbb{S}^{n-2}}u(|x^{\prime}|\theta,x_{n})\,\mathrm{d}\theta, $O(|x|)\leq C(\gamma)|x|$ as $x\to 0$, and $\delta>0$ depends only on the $C^{2}$ norm of $\psi$.

If the above inequality (5) holds for $\gamma=1$, then either [math] is a removable singularity or there exists a constant $C>0$ such that

[TABLE]

Furthermore, (6) holds for all $x\in Q_{1/2}$.

Without assuming (5) holds up to $\gamma=1$, we are still able to show some sort of asymptotic symmetry for almost all $x\in Q_{1/2}$ close to [math]; see Proposition 5.1.

The second conclusion of Theorem 1.2 is a partial answer of a question of [4] (see Remark 1 of [4]).

The method of proof of Theorem 1.2 can be adapted to study boundary singularity of critical equations with nonlinear Neumann boundary conditions, which we leave to another paper. Motivated by conformal geometry, boundary singularity of linear (degenerate) elliptic equation with a critical nonlinear Neumann boundary condition has been studied in Caffarelli-Jin-Sire-Xiong [10], Jin-de Queiroz-Sire-Xiong [15], and Sun-Xiong [23].

The organization of paper and crucial steps of the proofs are as follows. In section 2, we recall some basic facts of elliptic equations with zero Dirichlet condition, such as boundary Harnack inequality, a special maximum principle and a boundary version of Bôcher theorem. In section 3, we prove the partial upper bound in the main theorem. A classical result of Berestycki-Nirenberg [2] plays an important role. In section 4, we establish the lower bound. The Pohozaev identity and Bôcher theorem are used crucially. In section 5, we prove symmetry results via the moving spheres method developed by Li-Zhu [20]; see also Li-Zhang [19]. In particular, we present a proof of Proposition 1.1 by this method. Here the boundary Harnack inequality is used repeatedly. Our proof of Proposition 1.1 can be adapted to give another proof of a result of Dancer [11].

Acknowledgments: We would like to thank the referee for her/his invaluable suggestions.

2 Preliminaries

Since we will work in neighborhoods of partial boundaries of domains, we let $\psi$ be a $C^{2}$ function in $\mathbb{R}^{n-1}$ satisfying

[TABLE]

and denote $Q_{R}=\{x=(x^{\prime},x_{n}):x_{n}>\psi(x^{\prime})\}\cap B_{R}$, $\Gamma_{R}=\{x=(x^{\prime},x_{n}):x_{n}=\psi(x^{\prime})\}\cap B_{R}$ and $d(x)=dist(x,\Gamma_{1})$, where $B_{R}$ is the open ball center at [math] with radius $R$. We denote $B_{R}^{+}(y)$ for $y=(y^{\prime},0)$ as $\{x\in B_{R}(y):x_{n}>0\}$, $\partial^{\prime}B_{R}^{+}(y):=B_{R}(y)\cap\partial\mathbb{R}^{n}_{+}$ and $\partial^{\prime\prime}B_{R}^{+}(y):=B_{R}(y)\cap\mathbb{R}^{n}_{+}$. We assume

[TABLE]

Lemma 2.1**.**

Let $u\in C^{2}(\bar{Q}_{1})$ be a nonnegative solution of

[TABLE]

where $a(x)\in L^{\infty}(Q_{1})$. Then

[TABLE]

and

[TABLE]

where $C_{0}\geq 1$ and $\alpha\in(0,1)$ are constants depending only on $n$, $A_{1}$, $\|a\|_{L^{\infty}(Q_{1})}$ and $\|u\|_{L^{\infty}(Q_{1})}$.

The above result is a simple version of the boundary Harnack inequality; see Caffarelli-Fabes-Mortola-Salsa [8] or Krylov [17]. If only a differential inequality is assumed, we have

Lemma 2.2**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of

[TABLE]

If $u>0$ somewhere in $Q_{1}$, then there exists a constant $c_{0}>1$ such that

[TABLE]

Proof.

It is easy to check that there exist constants $0<\delta<1/4$ and $A\geq 1$ with $A\delta<1/2$ such that

[TABLE]

It follows from the strong maximum principle that $u>0$ in $Q_{1}$. By Hopf Lemma and the compactness of $\Gamma_{1}\cap\partial B_{1/2}$, there exists a constant $b_{0}>0$ such that

[TABLE]

By the continuity of $\nabla u$ on $Q_{1}\cap\partial B_{1/2}$, one can find a constant $C>0$ such that

[TABLE]

In view of (8), by maximum principle we have

[TABLE]

The lemma follows immediately.

∎

We will also use a maximum principle when the coefficient of zero order term without sign restriction but being small.

Lemma 2.3**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a solution of

[TABLE]

There exists a small constant $\delta=\delta(n)>0$ such that if $|a(x)|\leq\delta|x|^{-2}$, and $u\geq 0$ on $\partial(Q_{1}\setminus Q_{\varepsilon})$ for some $\varepsilon>0$, there holds

[TABLE]

Proof.

Multiplying both sides by $u^{-}=\max\{-u,0\}$ and using the Hardy inequality, the lemma follows immediately.

∎

Lemma 2.4**.**

For $R>0$, let $u\in C^{2}(\mathbb{R}^{n}_{+}\setminus B_{R})$ and be continuous up to the boundary $\partial\mathbb{R}^{n}_{+}\setminus B_{R}$ except a point $\bar{x}=(\bar{x}^{\prime},0)$ with $|\bar{x}^{\prime}|>R$. Suppose that

[TABLE]

and $u\geq 0$. Then

[TABLE]

Proof.

For $\varepsilon>0$, let

[TABLE]

By maximum principle we have $u\geq\phi_{\varepsilon}$. Sending $\varepsilon\to 0$, the lemma follows.

∎

To establish the lower bound in Theorem 1.2, we need a well-known boundary Bôcher type theorem. See Marcus-Véron [22] for a nonlinear version.

Lemma 2.5**.**

Let $u\in C^{2}(B_{1}^{+})\cap C^{0}(\bar{B}_{1}^{+}\setminus\{0\})$ be a nonnegative solution of

[TABLE]

Then

[TABLE]

where $a\geq 0$ is a constant and

[TABLE]

Furthermore, if $B_{1}^{+}$ is replaced by $\mathbb{R}^{n}_{+}$, then $h=bx_{n}$ for some nonnegative constant $b$.

3 A partial upper bound

The following lemma is an easy consequence of a classical result of Berestycki-Nirenberg [2].

Lemma 3.1**.**

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, which is convex in the $x_{1}$ direction and symmetric with respect to the hyperplane $\{x_{1}=0\}$. Let $u\in C^{2}(\Omega)$ be a positive solution of

[TABLE]

where $f:[0,\infty)\to[0,\infty)$ is local Lipschitz continuous. Then $\partial_{x_{1}}u(x)<0$ for $x\in\Omega$ with $x_{1}>0$.

In order to use Lemma 3.1, let us define the conformal transform $F:\mathbb{R}^{n}_{+}\to B_{1}$ by

[TABLE]

Then F\Big{|}_{\partial\mathbb{R}^{n}_{+}} is the inverse of stereographic projection. Let $u$ be a solution of (4), then the function

[TABLE]

satisfies

[TABLE]

where $z:=F(x)$ and $|J_{F}|$ is the Jacobian determinant of $F$.

By performing a Kelvin transform with respect to a sphere of small radius below $Q_{1}$ and re-labeling coordinates, we may assume that $\psi$ is convex. Since $\psi(0)=0$ and $\nabla\psi(0)=0$, we have $\psi\geq 0$. For $0<r<1$, denote

[TABLE]

and

[TABLE]

Then

[TABLE]

where $e_{n}=(0^{\prime},1)$. Note that $rx\in Q_{1}$ implies $x\in\{y_{n}>\frac{1}{r}\psi(ry^{\prime})\}$. Clearly,

[TABLE]

and thus

[TABLE]

Now we able to prove a partial upper bound.

Proposition 3.2**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). Then for every $0<\gamma<1$, there exists a constant $C(\gamma)>0$ such that

[TABLE]

Proof.

By (10) and (11), for any constant $0<\delta<1/2$ there exists a constant $r_{0}>0$ such that $\tilde{Q}_{1/r}\cap\{z_{n}<1-\delta\}$ is a convex body for all $0<r<r_{0}$. Together with Lemma 3.1, there exits a constant $c(\delta)>\delta$ with $\lim_{\delta\to 0}c(\delta)=0$, depending only on $n$ and $\delta$, such that $v$ has no critical points in the region $\tilde{Q}_{1/r}\setminus\mathcal{C}_{c(\delta)}$, where $\mathcal{C}_{c(\delta)}$ is the cone generated by the vertex $-e_{n}$ and $B_{c(\delta)}(e_{n})$. Choose $\delta$ small such that

[TABLE]

Since $F$ is a conformal map,

[TABLE]

and $|J_{F}(x)|$ is smooth positive smooth function on $\bar{B}_{1/2}^{+}$, we only need show that for $0<r<r_{0}$

[TABLE]

Suppose the contrary that there exists a sequence $\{z_{j}\}_{j=1}^{\infty}\subset\tilde{Q}_{1/r}\setminus\mathcal{C}_{2c(\delta)}$ such that

[TABLE]

and

[TABLE]

Let $\theta=\arcsin c(\delta)-\arcsin\frac{c(\delta)}{2}>0$ be the cone angle error between $\mathcal{C}_{2c(\delta)}$ and $\mathcal{C}_{c(\delta)}$. It is easy to see that

[TABLE]

Consider

[TABLE]

Let $|\bar{z}_{j}-z_{j}|<\frac{\sin\theta}{2}|z_{j}|$ satisfy

[TABLE]

and let

[TABLE]

Then

[TABLE]

By the definition of $v_{j}$, we have

[TABLE]

Thus, we have

[TABLE]

We also have

[TABLE]

Now, consider

[TABLE]

where

[TABLE]

Then $w_{j}$ satisfies $w(0)=1$ and

[TABLE]

Moreover, it follows from (17) and (18) that

[TABLE]

where

[TABLE]

Since $\psi$ is $C^{2}$ and $\nabla\psi(0)=0$, we have two possibilities:

If $B_{R_{j}}\cap\Omega_{j}\to\mathbb{R}^{n}\cap\{x_{n}>-\rho\}$ for some $\rho>0$, then by the up to boundary estimates for 2nd order linear elliptic equations there exists a subsequence of $\{w_{j}\}$, which is still denoted as $\{w_{j}\}$, satisfying

[TABLE]

for some $w$ satisfying

[TABLE]

By [11], $w=0$. This is impossible since $w(0)=1$.

If $B_{R_{j}}\cap\Omega_{j}\to\mathbb{R}^{n}$, then by the interior estimates for 2nd order linear elliptic equations there exists a subsequence of $\{w_{j}\}$, which is still denoted as $\{w_{j}\}$, satisfying

[TABLE]

for some $w$ satisfying

[TABLE]

By [9] we have

[TABLE]

for some point $\bar{y}\in\mathbb{R}^{n}$ and $1\leq\lambda\leq 2^{\frac{n-2}{2}}$.

Since $w_{j}\to w$ in $C^{2}_{loc}(\mathbb{R}^{n})$ and $\nabla w(\bar{y})=0$ and $\nabla^{2}w(\bar{x})$ is negative definite, for large $j$ there exists $y_{j}\in B_{1}(\bar{y})$ such that $\nabla w_{j}(y_{j})=0$. By the definition of $w_{j}$, we have

[TABLE]

This is impossible, since $\bar{z}_{j}+\frac{y_{j}}{u(\bar{z}_{j})^{\frac{2}{n-2}}}\in\tilde{Q}_{1/r}\setminus\mathcal{C}_{c(\delta)}$ where $v$ does not have any critical point. Therefore, (13) holds. Scaling back to $u$ and using (12), we proved Proposition 3.2.

∎

Corollary 3.3**.**

Assume the assumptions in Proposition 3.2. Let $0<\gamma<1$. Then for $x\in Q_{1/4}$ with $\frac{d(x)}{|x|}\leq\gamma$,

[TABLE]

[TABLE]

and for any $0<s<\frac{1}{4}$, $x,y\in Q_{2s}\setminus Q_{s/2}\mbox{ with }\frac{d(x)}{|x|},\frac{d(y)}{|y|}\leq\gamma,$

[TABLE]

where $r=|x|$, $\theta=\frac{x}{|x|}$, $C(\gamma)>0$ depends on $\gamma$ but not $s$.

Proof.

For $0<s<\frac{1}{4}$, let

[TABLE]

Denote $\tilde{Q}_{r}:=\{x:sx\in Q_{rs}\}$, $\tilde{\Gamma}_{r}:=\{x:sx\in\Gamma_{rs}\}$ for any $r>0$, and $\tilde{d}(x)=dist(x,\tilde{\Gamma}_{1})$. Then

[TABLE]

It follows from Proposition 3.2 that $v_{s}(x)\leq C(\gamma^{\prime})$ for $x\in\tilde{Q}_{3}\setminus\tilde{Q}_{1/4}$ with $\frac{\tilde{d}(x)}{|x|}<\gamma^{\prime}$, where $0<\gamma<\gamma^{\prime}<1$. By Lemma 2.1 and the standard linear elliptic equations theory, for $x,y\in\tilde{Q}_{2}\setminus Q_{1/2}$ with $\frac{\tilde{d}(x)}{|x|}\leq\gamma$ and $\frac{\tilde{d}(y)}{|y|}\leq\gamma$ we have

[TABLE]

[TABLE]

[TABLE]

where $C$ depends only on $n,C(\gamma^{\prime})$. Scaling back to $u$, the above three inequalities yield (23), (24) and (25), respectively.

Therefore, we complete the proof.

∎

Remark 3.4**.**

If $u(x)\leq C|x|^{\frac{2-n}{2}}$ for all $x\in Q_{1/4}$, Corollary 3.3 holds for $\gamma=1$.

4 A lower bound and removability

Lemma 4.1** (Pohozaev identity).**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). Then for all $0<r<1$ there holds

[TABLE]

where $c_{0}$ is constant independent of $r$.

The proof of Lemma 4.1 is standard by now. $P(u,r)$ is called Pohozaev integral sometimes in the literature.

Proposition 4.2**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). If

[TABLE]

and

[TABLE]

then

[TABLE]

Proof.

Suppose the contrary that

[TABLE]

Since $\liminf_{Q_{1}\ni x\to 0}d(x)^{-1}|x|^{\frac{n}{2}}u(x)=0$, by the Harnack inequality (25) in annulus we can find sequences $x_{j}=(0^{\prime},(x_{j})_{n})\to 0$ and $y_{j}=(0^{\prime},(y_{j})_{n})\to 0$ as $j\to\infty$ satisfying

[TABLE]

where $0<\bar{C}_{0}^{*}\leq C_{0}$. In view of this oscillation picture, without loss of generality we assume $(x_{j})_{n}$ are local minimum of $x_{n}^{-1}|x|^{\frac{n}{2}}u(x)$ restricted to the line $(0^{\prime},x_{n})$. It follows that

[TABLE]

Let $r_{j}=|x_{j}|=(x_{j})_{n}>0$, and

[TABLE]

Denote $\tilde{Q}_{j}:=\{x:r_{j}x\in Q_{1}\}$, $\tilde{\Gamma}_{j}:=\{x:r_{j}x\in\Gamma_{1}\}$, and $\tilde{d}_{j}(x)=dist(x,\tilde{\Gamma}_{j})$. It follows from (25) that for any $R>1$ there exists $C(R)>0$ such that

[TABLE]

Furthermore, $w_{j}$ satisfies

[TABLE]

[TABLE]

By the up to boundary estimates for linear elliptic equation, after passing to a subsequence, as $j\to\infty$,

[TABLE]

and $0<w\in C^{2}_{loc}(\bar{R}^{n}_{+}\setminus\{0\})$ satisfies

[TABLE]

[TABLE]

By Lemma 2.5,

[TABLE]

where $a,b\geq 0$ are constants. By (27) and $w_{j}(e_{n})=1$, we have

[TABLE]

and $a+b=1$. Thus $a=b=\frac{1}{2}$.

By (23) and (24), we have

[TABLE]

It follows from Lemma 4.1 that

[TABLE]

On the other hand,

[TABLE]

Therefore, as $j\to\infty$

[TABLE]

We obtain a contradiction. The proposition is proved.

∎

Proposition 4.3**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). If

[TABLE]

then [math] is a removable singular point of $u$.

Proposition 4.3 is included in Theorem 7.1 of [3]. We provide

Another proof of Proposition 4.3.

Since the critical equation is conformally invariant, we may assume $Q_{\sigma}$ is convex for some small $\sigma>0$, otherwise one may preform a kelvin transform centered in $Q_{1}$. For any $0<\mu\leq n$, we have

[TABLE]

For any $\varepsilon>0$, let

[TABLE]

where $\alpha>0$ is a constant to be fixed. Hence,

[TABLE]

By (28), for any $\delta>0$ there exists $\tau>0$ such that

[TABLE]

Therefore, we have

[TABLE]

Choose $\delta<n-1$ small and thus the assumptions in Lemma 2.3 are satisfied. Thinks to Lemma 2.1, one can choose $\alpha$ such that $\alpha\frac{x_{n}}{|x|}\geq u$ on $\partial B_{\tau}^{+}\cap Q_{\tau}$. Since $Q_{\tau}$ is convex, $\phi_{\varepsilon}\geq 0=u$ on the bottom boundary of $Q_{\tau}$. By (28), we have

[TABLE]

It follows from Lemma 2.3 that $u\leq\phi_{\varepsilon}$ in $Q_{\tau}$ for all $\varepsilon>0$. Sending $\varepsilon\to 0$, we have

[TABLE]

It follows that [math] is a removable singularity. We complete the proof.

∎

5 Asymptotic symmetry and proof of Theorem 1.2

Proposition 5.1**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). Suppose that $\psi$ is concave and [math] is a non-removable singularity. Then there exists $\varepsilon>0$ such that for every $x=(x^{\prime},x_{n})\in\Gamma_{1}$ with $|x^{\prime}|<\varepsilon$ there holds

[TABLE]

where

[TABLE]

Proof.

To present our idea more clear, let us assume $\psi=0$ at the moment. The proposition is proved as long as the three steps have been through:

• (a).

There exists $0<\varepsilon<1/10$ such that for every $x=(x^{\prime},0)$ with $|x^{\prime}|<\varepsilon$

[TABLE]

• (b).

There exists $0<\lambda_{1}<|x|$ such that

[TABLE]

• (c).

Let

[TABLE]

Then $\bar{\lambda}_{x}=|x|$.

Proof of (a). By Lemma 2.2, there exists a constant $c_{0}>0$ such that

[TABLE]

For $0<\lambda<|x|<\varepsilon,~{}y\in\partial^{\prime\prime}B_{3/4}$, we have

[TABLE]

and

[TABLE]

Thus

[TABLE]

It follows from Corollary 3.3 with $\gamma=\frac{3}{22}$ that

[TABLE]

Hence

[TABLE]

provided

[TABLE]

Proof of (b). For any fixed $0<|x|<\varepsilon$, we claim there exist $0<\lambda_{3}<\lambda_{2}<|x|$ such that

[TABLE]

Indeed, for every $0<\lambda_{2}<|x|$ and every $y\in B_{3/4}^{+}\setminus B_{\lambda_{2}}^{+}(x)$, we have $x+\frac{\lambda^{2}(y-x)}{|y-x|^{2}}\in B_{\lambda_{2}}^{+}(x)$. Hence,

[TABLE]

where $c_{0}$ is the constant in (29),

[TABLE]

and Lemma 2.1 has been used. Therefore, (30) is confirmed.

We are going to use the narrow domain technique to conclude that the remaining case: $u_{x,\lambda}\leq u$ in $B_{\lambda_{2}}^{+}(x)\setminus B_{\lambda}^{+}(x)$.

By (30), $u_{x,\lambda}\leq u$ on \partial\big{(}B_{\lambda_{2}}^{+}(x)\setminus B_{\lambda}^{+}(x)\big{)} for all $0<\lambda<\lambda_{3}$. Multiplying both sides of the equation

[TABLE]

by $(u_{x,\lambda}-u)^{+}$ and integrating by parts, where we used mean value theorem and $0\leq\tau=\tau(x)\leq 1$, we have, using Hölder inequality,

[TABLE]

where $C(n)=n(n-2)2^{\frac{4}{n-2}}$. Since $(u_{x,\lambda}-u)^{+}\in H_{0}^{1}(B_{\lambda_{2}}^{+}(x)\setminus B_{\lambda}^{+}(x))$, by Sobolev inequality we have

[TABLE]

where $S(n)>0$ depends only dimension $n$. Choosing $\lambda_{2}$ small to ensure

[TABLE]

we obtain

[TABLE]

which implies $u_{x,\lambda}\leq u$ in $B_{\lambda_{2}}^{+}(x)\setminus B_{\lambda}^{+}(x)$ because $(u_{x,\lambda}-u)^{+}$ is continuous in $B_{\lambda_{2}}^{+}(x)\setminus B_{\lambda}^{+}(x)$. Let $\lambda_{1}=\lambda_{3}$ and we complete the proof.

Proof of (c). By the previous step, we see that $\bar{\lambda}_{x}$ is well defined. If $\bar{\lambda}_{x}<|x|$, we have $u-u_{x,\bar{\lambda}_{x}}\geq 0$ in $B_{3/4}^{+}\setminus B_{\bar{\lambda}_{x}}(x)$. By item (a) and strong maximum principle $u-u_{x,\bar{\lambda}_{x}}>0$ in $B_{3/4}^{+}\setminus\bar{B}_{\bar{\lambda}_{x}}(x)$. It follows from Lemma 2.2 that for every $r>\bar{\lambda}_{x}$ there exists $c_{r}>0$ such that

[TABLE]

For $y\in B_{3/4}$ with $|y-x|\geq r$ and $\bar{\lambda}_{x}<\lambda<r\leq\frac{|x|+\bar{\lambda}_{x}}{2}$, making use of Lemma 2.1 we have

[TABLE]

where $\alpha\in(0,1)$ and $C\geq 1$ depend only on $n$ and $\|u\|_{L^{\infty}(B_{(|x|+\bar{\lambda}_{x})/2}(x))}$, and we have used

[TABLE]

and

[TABLE]

Let $\delta>0$ satisfy

[TABLE]

Then by (33) we have for all $\bar{\lambda}_{x}\leq\lambda\leq\bar{\lambda}_{x}+\delta$

[TABLE]

This implies that $u-u_{x,\lambda}\geq 0$ on $\partial(B_{r}^{+}(x)\setminus B_{\lambda}^{+}(x))$. Using narrow domain technique as before, we immediately obtain

[TABLE]

whenever $r$ is chosen such that

[TABLE]

In conclusion,

[TABLE]

This contradicts to the definition of $\bar{\lambda}_{x}$. Hence, $\bar{\lambda}_{x}=|x|$.

Therefore, we proved Proposition 5.1 when $\psi=0$.

If $\psi\neq 0$ is concave, we note that for each $y\in\Gamma_{1}\cap B_{\lambda}(x)$, where $x\in\Gamma_{1}$, $|x^{\prime}|<\varepsilon$ and $\lambda<|x^{\prime}|$, whenever $|y^{\prime}+\frac{\lambda^{2}(y^{\prime}-x^{\prime})}{|y-x|^{2}}|\leq\frac{3}{4}$ then $y+\frac{\lambda^{2}(y-x)}{|y-x|^{2}}\in Q_{3/4}$. Hence, with a little modification of the above proof for case $\psi=0$, we complete the proof of Proposition 5.1.

∎

Proof of Proposition 1.1.

We are going to show that for all $x\in\partial\mathbb{R}^{n}_{+}$, $x\neq 0$ there holds

[TABLE]

The idea is the same as that of the proof of Proposition 5.1.

Step 1. We prove that (38) holds for all $0<\lambda<\lambda_{1}$ with $\lambda_{1}>0$ small.

Corresponding to the step (a) of the proof of Proposition 5.1, by Lemma 2.4 and Lemma 2.1 we have for $0<\lambda_{3}<\lambda_{2}<|x|$

[TABLE]

and

[TABLE]

It follows from Lemma 2.1 that

[TABLE]

In view of (39), by setting $\lambda_{3}\leq\lambda_{2}/C^{1/n}$ we showed that

[TABLE]

As in the step (b) of the proof of Proposition 5.1, by narrow domain technique we can prove easily that

[TABLE]

where $\lambda_{2}$ is selected to ensure (32).

Step 2. Define

[TABLE]

By the previous step, $\bar{\lambda}_{x}>0$ is well defined. We shall prove $\bar{\lambda}_{x}=|x|$. If not, i.e., $\bar{\lambda}_{x}<|x|$, we want to show that there exists $0<\delta<\frac{|x|-\bar{\lambda}_{x}}{2}$ such that (38) holds for all $0<\lambda<\bar{\lambda}_{x}+\delta$. This obviously contradicts to the definition of $\bar{\lambda}_{x}$.

By the definition of $\bar{\lambda}_{x}$, we have $u-u_{x,\bar{\lambda}_{x}}\geq 0$ in $\mathbb{R}^{n}_{+}\setminus B_{\bar{\lambda}_{x}}(x)$ and thus

[TABLE]

Since [math] is a non-removable singularity of $u$, we have $\limsup_{y\to 0}(u-u_{x,\bar{\lambda}_{x}})(y)=\infty$. By strong maximum principle, we have

[TABLE]

By Lemma 2.4 and Lemma 2.1,

[TABLE]

and

[TABLE]

for every $r>\bar{\lambda}_{x}$. $r$ will be fixed to ensure (37) when using the narrow domain technique. Choosing $0<\delta<\frac{|x|-\bar{\lambda}_{x}}{2}$ sufficiently small ensures that

[TABLE]

Indeed, notice that $x+\frac{\lambda^{2}(y-x)}{|y-x|^{2}}\in B_{\frac{|x|+\bar{\lambda}_{x}}{2}}^{+}(x)$. By Lemma 2.1 and computing as in deriving (34) we have

[TABLE]

where $\alpha\in(0,1)$ and $C\geq 1$ depend only on $n$ and $\|u\|_{L^{\infty}(B_{(|x|+\bar{\lambda}_{x})/2})}$. Hence, (41) holds by setting

[TABLE]

Hence,

[TABLE]

By narrow domain technique, the above inequality holds for all $y\in\mathbb{R}^{n}_{+}$ with $|y|\geq\lambda$. Therefore, step 2 is finished.

Let $e=(e^{\prime},0)\in\mathbb{R}^{n}$ be an arbitrary unit vector, $a>0$ constant, and $y\in\mathbb{R}^{n}_{+}$ satisfying $ye-a<0$, (38) holds for $x=Re$ and $\lambda=R-a$:

[TABLE]

Sending $R\to\infty$, we have

[TABLE]

Proposition 1.1 follows immediately.

∎

Proposition 5.2**.**

Let $u\in C^{2}(\bar{Q}_{1}\setminus\{0\})$ be a nonnegative solution of (4). Then there exists a constant $\bar{\delta}>0$ depends on the $C^{2}$ norm of $\psi$ such that for $x\in Q_{1/2}$ with $\frac{d(x)}{|x|}<\gamma<1$ and $x_{n}>\frac{1}{\delta}\max_{|y^{\prime}|=|x^{\prime}|}|\psi(y^{\prime})|$ we have

[TABLE]

where \bar{u}(x^{\prime},x_{n})=\mathchoice{{\vbox{\hbox{\textstyle-}}\kern-4.86108pt}}{{\vbox{\hbox{\scriptstyle-}}\kern-3.25pt}}{{\vbox{\hbox{\scriptscriptstyle-}}\kern-2.29166pt}}{{\vbox{\hbox{\scriptscriptstyle-}}\kern-1.875pt}}\!\int_{\mathbb{S}^{n-2}}u(|x^{\prime}|\theta,x_{n})\,\mathrm{d}\theta and $O(|x|)\leq C(\gamma)|x|$ as $x\to 0$.

Proof.

Suppose first that $\psi$ is concave. For $r>0$, let $x_{\alpha}=(x_{\alpha}^{\prime},x_{n})$ and $x_{\beta}=(x^{\prime}_{\beta},x_{n})$ be two points in $Q_{\varepsilon}$ with $x_{n}>\max_{|y^{\prime}|=r}\psi(y^{\prime})$ such that

[TABLE]

where $\varepsilon$ is the one in Proposition 5.1. Let

[TABLE]

We want to find $\tilde{x}_{\beta}=(x_{\beta}^{\prime},t)$ and $\lambda>0$ such that

[TABLE]

It follows that

[TABLE]

and

[TABLE]

It is easy to check that there exist positive constants $\bar{r}$ and $\bar{C}(\varepsilon)$, depending only on $\varepsilon$ and $\psi(x_{\gamma}^{\prime})$, such that that if $|x_{n}|\leq\bar{C}(\varepsilon)\sqrt{r}$ and $r<\bar{r}$ then $\lambda^{2}<|x_{\gamma}|^{2}$. By Proposition 5.1, we have

[TABLE]

Let $s=\sqrt{r^{2}+x_{n}^{2}}$ and $u_{s}(x)=s^{\frac{n-2\sigma}{2}}u(sx)$. Then

[TABLE]

where $V(x):=n(n-2)u_{s}^{\frac{4}{n-2}}$ and $\tilde{Q}_{R}$ and $\tilde{\Gamma}_{R}$ are the scalings of $Q_{sR}$ and $\Gamma(sR)$. Let $\Omega_{\gamma}=\{x\in\tilde{Q}_{3/2}\setminus\tilde{Q}_{3/4}:\frac{d_{sx}}{s|x|}<\gamma\}$. By elliptic estimates to the boundary and Lemma 2.1, we have

[TABLE]

for every $\theta\in(0,1)$, where $C$ depends only on $n,\psi$ and the constant $C(\frac{\gamma+1}{2})$ in Proposition 3.2. Since $|\tilde{x}_{\beta}-x_{\beta}|=\frac{4(x_{n}-\psi(x_{\gamma}^{\prime}))|x_{\alpha}^{\prime}-x_{\beta}^{\prime}|}{\varepsilon}$, by mean value theorem we have

[TABLE]

Hence,

[TABLE]

Therefore, the proposition is proved if $\psi$ is concave.

If $\psi$ is not concave, let $B_{\rho}(\rho e_{n})$ be an inner tangential ball of $Q_{1}$ contacting $Q_{1}$ at [math], where $\rho>0$ for $\Gamma_{1}$ is of $C^{2}$. Let

[TABLE]

$D=\phi^{-1}Q_{1}$ and $\Lambda=\phi^{-1}\Gamma_{1}$. Then $0\in\Lambda$, $D$ is concave at [math] and

[TABLE]

Note that $|x|\leq C(\rho)|y|$ for $|y|\leq\frac{\rho}{100}$, and

[TABLE]

where $x=\frac{\rho^{2}(y-\rho e_{n})}{|y-\rho e_{n}|}+\rho e_{n}$. By what we proved for concave $\psi$, the proposition follows immediately.

∎

Proposition 5.3**.**

Suppose that $u$ is a solution of (4) and $u(x)\leq Cd(x)|x|^{-\frac{n}{2}}$. Then there exists a constant $\bar{\delta}>0$ depends on the $C^{2}$ norm of $\psi$ such that

[TABLE]

Proof.

By the proof of Proposition 5.2, we only consider concave $\psi$ and $x_{n}\geq\frac{1}{\tilde{C}}\sqrt{|x^{\prime}|}$ for some $\tilde{C}>0$.

For $r>0$ and $x_{n}\geq\frac{1}{\tilde{C}}\sqrt{r}$, let $x_{\alpha}=(x_{\alpha}^{\prime},x_{n})$ and $x_{\beta}=(x^{\prime}_{\beta},x_{n})$ be two points such that

[TABLE]

Let $u_{s}$ satisfy (43) with $s=\sqrt{r^{2}+|x_{n}|^{2}}$. By mean value theorem, Harnack inequality and interior estimates, we have

[TABLE]

where we used $x_{n}\geq\frac{1}{\tilde{C}}\sqrt{r}$. It follows that

[TABLE]

We complete the proof.

∎

Proof of Theorem 1.2.

The first part of the theorem follows from Proposition 3.2 and Proposition 5.2. The second part follows from Proposition 4.2, Proposition 4.3 and Proposition 5.3. ∎