On the uniqueness of solutions of a nonlocal elliptic system

Abstract

We consider the following elliptic system with fractional Laplacian

$$\begin{aligned} -(-\Delta )^su=uv^2,\, \, -(-\Delta )^sv=vu^2,\, \, u,v>0 \ \mathrm{on}\, {\mathbb R}^n, \end{aligned}$$

where $s\in (0,1)$ and $(-\Delta )^s$ is the s-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.

Appendix A: Basic facts about \(L_a\)-subharmonic functions

In this appendix we present several basic facts about $L_a$-subharmonic functions, which are used in this paper.

The first is a mean value inequality for $L_a$-subharmonic function.

Lemma A.1

Let u be a $L_a$-subharmonic function in $B_r\subset {\mathbb R}^{n+1}$ (centered at the origin), then

$$\begin{aligned}u(0)\le C(n,a)r^{-n-1-a}\int _{B_r}y^au.\end{aligned}$$

Here C(n, a) is a constant depending only on n and a.

Proof

Direct calculation gives

$$\begin{aligned} \frac{d}{dr}\left( r^{-n-a}\int _{\partial B_r}y^au\right)= & {} r^{-n-a}\int _{\partial B_r}y^a\frac{\partial u}{\partial r}\\= & {} r^{-n-a}\int _{B_r}\text{ div }\left( y^a\nabla u\right) \\\ge & {} 0. \end{aligned}$$

Thus $r^{-n-a}\int _{\partial B_r}y^au$ is non-decreasing in r. Integrating this in r shows that $r^{-n-1-a}\int _{B_r}y^au$ is also non-decreasing in r. $\square$

By standard Moser’s iteration we also have the following super bound

Lemma A.2

Let u be a $L_a$-subharmonic function in $B_r\subset {\mathbb R}^{n+1}$ (centered at the origin), then

$$\begin{aligned}\sup _{B_{r/2}}u\le C(n,a)\left( r^{-n-1-a}\int _{B_r}y^au^2\right) ^{\frac{1}{2}}.\end{aligned}$$

Here C(n, a) is a constant depending only on n and a.

Lemma A.3

Let $M>0$ be fixed. Any $v\in H^1(B_1^+)\cap C(\overline{B_1^+})$ nonnegative solution to

$$\begin{aligned} \left\{ \begin{array}{l} L_a v\ge 0, \ \text{ in }\ B_1^+, \\ \partial ^a_y v\ge Mv \ \ \text{ on }\ \partial ^0 B_1^+, \end{array} \right. \end{aligned}$$

satisfies

$$\begin{aligned}\sup _{\partial ^0B_{1/2}^+}v\le \frac{C(n)}{M}\int _{B_1^+}y^av.\end{aligned}$$

Proof

This is essentially [23, Lemma 3.5]. We only need to note that, since

$$\begin{aligned}\partial _y^a v\ge 0 \ \ \text{ on }\ \partial ^0 B_1^+,\end{aligned}$$

the even extension of v to $B_1$ is $L_a$-subharmonic (cf. [7, Lemma 4.1]). Then by Lemma A.2,

$$\begin{aligned}\sup _{B_{2/3}^+}v\le C(n)\int _{B_1^+}y^av.\end{aligned}$$

$\square$