Proof
Assume
\(u(x)\in \mathcal {L}_{\alpha}\) is a positive solution of the system of the fractional nonlinear PDEs system:
$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u_{i}(x)=x^{\gamma}_{n}u_{1}^{\alpha _{i}}(x)u_{2}^{\beta_{i}}(x), &\mbox{$x \in R^{n}_{+}$},\\ u_{i}(x)=0, &\mbox{$x \notin R^{n}_{+}$}, \end{cases} $$
(2.2)
where
\(i=1,2\),
\(\gamma\geq0\),
\(0<\alpha<2\),
\(\alpha_{i}\),
\(\beta_{i} >0\).
We first show that
$$ \int_{R^{n}_{+}}G(x,y){y^{\gamma}_{n}} u^{\alpha_{i}}_{1}(y)u^{\beta _{i}}_{2}(y)\,dy< \infty. $$
(2.3)
Set
\(P_{R}:=(0, \ldots, 0,R )\in R^{n}_{+} \),
\(B^{+}_{R}(P_{R}):= \{x\in R^{n} : \vert x-P_{R} \vert < R \}\), the ball of radius
R centered at
\(P_{R}\). Let
$$ v_{i}^{R}(x)= \int_{B^{+}_{R}(P_{R})}G_{R}(x,y)y^{\gamma}_{n}u_{1}^{\alpha _{i}}(y)u_{2}^{\beta_{i}}(y) \,dy, $$
(2.4)
where
\(G_{R}(x,y)\), the Green’s function on the ball
\(B^{+}_{R}(P_{R})\), was given in [
10],
$$G_{R}(x,y)=\frac{A_{n,\alpha}}{ \vert x-y\vert ^{n-\alpha}} \biggl[1-\frac{B_{n,\alpha}}{(s_{R}+t_{R})^{\frac{(n-2)}{2}}} { \int_{0}^{\frac{s_{R}}{t_{R}}}\frac{(s_{R}-t_{R}b)^{\frac {(n-2)}{2}}}{b^{\frac{\alpha}{2}}(1+b)}}\,db \biggr],\quad x, y \in B^{+}_{ R }(P_{R}), $$
here
\(s_{R}=\frac{\vert x-y\vert ^{2}}{R^{2}}\),
\(t_{R}=(1-\frac {\vert x-P_{R}\vert ^{2}}{R^{2}})(1-\frac{\vert y-P_{R}\vert ^{2}}{R^{2}})\),
\(A_{n,\alpha} \), and
\(B_{n,\alpha}\) are constants depending on
n and
α.
From the local bounded-ness assumption on
u, one can see that, for each
\(R > 0\),
\(v_{i}^{R}(x)\) is well defined and is continuous. Moreover,
$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}v_{i}^{R}(x)=x^{\gamma}_{n}u_{1}^{\alpha _{i}}(x)u_{2}^{\beta_{i}}(x), &\mbox{$x\in B^{+}_{R}(P_{R})$},\\ v_{i}^{R}(x)=0, &\mbox{$x\notin B^{+}_{R}(P_{R})$}. \end{cases} $$
(2.5)
Let
\(w_{i}^{R}(x)=u_{i}(x)-v_{i}^{R}(x)\), by (
2.2) and (
2.5), we derive
$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}w_{i}^{R}(x)=0, &\mbox{$x\in B^{+}_{R}(P_{R})$},\\ w_{i}^{R}(x)\geq0, &\mbox{$ x\notin B^{+}_{R}(P_{R})$}. \end{cases} $$
(2.6)
Applying the maximum principle (see Proposition
2.1), we derive that
$$ w_{i}^{R}(x) \geq0 ,\quad \forall x\in B^{+}_{R}(P_{R}). $$
(2.7)
It is easy to prove that
$$ v_{i}^{R}(x)\rightarrow v_{i}(x)= \int_{R^{n}_{+}}G(x,y)y^{\gamma }_{n}u_{1}^{\alpha_{i}}(y)u_{2}^{\beta_{i}}(y) \,dy , \quad \mbox{as }R\rightarrow \infty. $$
(2.8)
Obviously,
$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}v_{i}(x)=x^{\gamma}_{n}u_{1}^{\alpha _{i}}(x)u_{2}^{\beta_{i}}(x),&\mbox{$x\in R^{n}_{+}$},\\ v_{i}(x)\equiv0,&\mbox{$x\notin R^{n}_{+}$}. \end{cases} $$
(2.9)
Denote
\(w_{i}(x)=u_{i}(x)-v_{i}(x)\). Using (
2.2), (
2.7), (
2.8), and (
2.9), we have
$$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}w_{i}(x)=0,& w_{i}(x)\geq0,\mbox{$ x\in {R^{n}_{+}}$},\\ w_{i}(x)\equiv0, &\mbox{$ x\notin{R^{n}_{+}} $}. \end{cases} $$
Applying the Liouville theorem (see Theorem
2.1), we deduce that either
$$w_{i}(x)\equiv0 ,\quad x \in{R^{n}} \quad\mbox{or}\quad w_{i} (x)\equiv c_{i}x_{n}^{\frac{\alpha}{2}},\quad \forall x \in{R^{n}_{+}}, i=1,2, $$
for some positive constants
\(c_{i}>0\), then we could write
\(w_{i} = c_{i}x_{n}^{\frac{\alpha}{2}}\),
\(c_{i}\geq0\). That is, the solutions of (
2.2) satisfy
$$ u_{i}(x)=c_{i}x^{\frac{\alpha}{2}}_{n}+\int _{R^{n}_{+}}G(x,y)y^{\gamma}_{n} u_{1}^{\alpha_{i}}(y)u_{2}^{\beta_{i}}(y)\,dy, \quad x \in R^{n}_{+}, i=1,2, $$
(2.10)
where
\(c_{i}\geq0\),
\(G(x,y)\) is defined in (
1.5).
Next we need to prove \(c_{i}\) must be zero for \(i=1,2\). To this end, we employ a certain type of Kelvin transform and the method of moving planes in integral forms.
For \(z^{0}=\{z^{0}_{1}, \ldots,z^{0}_{n-1} ,0 \}\in\partial {R^{n}_{+}} \), let \(\bar{u}_{i}^{z^{0}}(x)=\bar{u}_{i}(x) =\frac{1}{\vert x-z^{0}\vert ^{n-\alpha}}u_{i}(\frac{x-z^{0}}{\vert x-z^{0}\vert ^{2}}+z^{0})\), be the Kelvin transform of \(u_{i}(x)\) centered at \(z^{0}\).
Through a straightforward calculation by (
2.10), we derive
$$\bar{u}_{i}(x) =\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+\frac {1}{\vert x-z^{0}\vert ^{n-\alpha}} \int_{R^{n}_{+}}G \biggl(\frac{x-z^{0}}{\vert x-z^{0}\vert ^{2}}+z^{0},y \biggr)y^{\gamma }_{n}u_{1}^{\alpha_{i}}(y)u_{2}^{\beta_{i}}(y) \,dy. $$
Let
\(y=\frac{z-z^{0}}{\vert z-z^{0}\vert ^{2}}+z^{0} \), then
\(dy=\frac {1}{\vert z-z^{0}\vert ^{2n}}\,dz\),
$$\begin{aligned} \bar{u}_{i}(x) ={}&\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \frac {1}{\vert x-z^{0}\vert ^{n-\alpha}} \int_{R^{n}_{+}}G(x,z)\bigl\vert x-z^{0}\bigr\vert ^{n-\alpha} \bigl\vert z-z^{0}\bigr\vert ^{n-\alpha} \\ &{}\times\frac{\vert \frac{z_{n}}{\vert z-z^{0}\vert ^{2}}\vert ^{\gamma }u^{\alpha_{i}}_{1}(\frac{z-z^{0}}{\vert z-z^{0}\vert ^{2}}+z^{0})u^{\beta _{i}}_{2}(\frac{z-z^{0}}{\vert z-z^{0}\vert ^{2}}+z^{0})}{\vert z-z^{0}\vert ^{2n}}\,dz \\ ={}&\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int _{R^{n}_{+}}G(x,z)\frac{z_{n}^{\gamma}}{ \vert z-z^{0}\vert ^{n+\alpha+2\gamma}} \frac{u^{\alpha_{i}}_{1}(\frac {z-z^{0}}{\vert z-z^{0}\vert ^{2}}+z^{0})}{\vert z-z^{0}\vert ^{(n-\alpha)\alpha_{i}}} \\ &{}\times\frac{u^{\beta_{i}}_{2}(\frac {z-z^{0}}{\vert z-z^{0}\vert ^{2}}+z^{0})}{\vert z-z^{0}\vert ^{(n-\alpha)\beta_{i}}} \bigl\vert z-z^{0}\bigr\vert ^{(n-\alpha)\alpha_{i}} \bigl\vert z-z^{0}\bigr\vert ^{(n-\alpha)\beta_{i}}\,dz \\ ={}&\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int _{R^{n}_{+}}G(x,z) \frac{z_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(z)\bar{u}^{\beta _{i}}_{2}(z)}{\vert z-z^{0}\vert ^{n+2\gamma+\alpha-(n-\alpha)(\alpha _{i}+\beta_{i})}}\,dz \\ ={}&\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int _{R^{n}_{+}}G(x,y)\frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta_{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}}\,dy, \quad \forall x\in {R^{n}_{+}}\backslash B_{\epsilon}\bigl(z^{0} \bigr), \end{aligned}$$
(2.11)
where
\(\epsilon>0\),
\(\delta=n+\alpha+2\gamma-(n-\alpha)(\alpha _{i}+\beta_{i})\).
Then we have \(\delta=0\)
i.e.
\(\alpha_{i}+\beta_{i}=\frac{n+\alpha+2\gamma}{n-\alpha}\), it is called critical case. When \(\delta>0\), we have \(1<\alpha_{i}+\beta_{i}<\frac{n+\alpha +2\gamma}{n-\alpha} \) and it is called the subcritical case. In this section, we consider these two cases \(1<\alpha_{i}+\beta_{i}\leq \frac{n+\alpha+2\gamma}{n-\alpha} \), then we have \(\delta\geq0\).
Now we introduce some basic notations in the method of moving planes. For a given real number λ, denote \({\Sigma_{\lambda}}=\{x=(x_{1},\ldots,x_{n})\in R^{n}_{+}\mid x_{1}< \lambda\}\), \({T_{\lambda}}=\{x=(x_{1},\ldots,x_{n})\in R^{n}_{+}\mid x_{1}= \lambda\}\). Let \(x^{\lambda}=(2\lambda-x_{1},x_{2},\ldots,x_{n})\) be the reflection of the point \(x=(x_{1},\ldots,x_{n}) \) about the plane \(T_{\lambda}\), and \(\bar{u}_{i}^{\lambda}(x)=\bar{u}_{i}(x^{\lambda})\), \(\bar{w}^{\lambda}_{i}(x)=\bar{u}^{\lambda}_{i}(x)-\bar{u}_{i}(x)\).
For
\(x,y \in{\Sigma_{\lambda}}\),
\(x\neq y\), by [
5], we have
$$ G(x,y) =G \bigl(x^{\lambda},y^{\lambda} \bigr) > G \bigl(x,y^{\lambda} \bigr)=G \bigl(x^{\lambda},y \bigr). $$
(2.12)
Obviously, we have
$$\begin{aligned} \bar{u}_{i}(x)={}&\frac{c_{i}x_{n}^{\frac{\alpha }{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int_{R^{n}_{+}}G(x,y) \frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}}\,dy \\ ={}&\frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int _{\Sigma_{\lambda}}G(x,y) \frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}}\,dy \\ &{}+ \int_{\Sigma_{\lambda}}G \bigl(x^{\lambda},y \bigr)\frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y^{\lambda}) \bar{u}^{\beta_{i}}_{2}(y^{\lambda})}{\vert y^{\lambda}-z^{0}\vert ^{\delta}} \,dy , \\ \bar{u}^{\lambda}_{i}(x) ={}& \frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x^{\lambda}-z^{0}\vert ^{n-\alpha }}+ \int_{\Sigma_{\lambda}}G \bigl(x^{\lambda},y \bigr) \frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}} \,dy \\ &{}+ \int_{\Sigma_{\lambda }}G(x,y)\frac{y_{n}^{\gamma}\bar{u}^{\alpha_{i}}_{1}(y^{\lambda })\bar{u}^{\beta_{i}}_{2}(y^{\lambda})}{\vert y^{\lambda}-z^{0} \vert ^{\delta}}\,dy . \end{aligned}$$
By an elementary calculation, we derive
$$\begin{aligned} \bar{u}_{i}(x)- \bar{u}^{\lambda}_{i}(x) ={}& \frac{c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x-z^{0}\vert ^{n-\alpha}}- \frac {c_{i}x_{n}^{\frac{\alpha}{2}}}{\vert x^{\lambda}-z^{0}\vert ^{n-\alpha}} \\ &{}+ \int_{\Sigma_{\lambda}} \bigl[G(x,y)-G \bigl(x^{\lambda},y \bigr) \bigr]y_{n}^{\gamma } \biggl[\frac{\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}}-\frac{\bar{u}^{\alpha _{i}}_{1}(y^{\lambda})\bar{u}^{\beta_{i}}_{2}(y^{\lambda })}{\vert y^{\lambda}-z^{0}\vert ^{\delta}} \biggr]\,dy \\ \leq{}& \int_{\Sigma_{\lambda}} \bigl[G(x,y)-G \bigl(x^{\lambda},y \bigr) \bigr]y_{n}^{\gamma } \biggl[\frac{\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}}-\frac{\bar{u}^{\alpha _{i}}_{1}(y^{\lambda})\bar{u}^{\beta_{i}}_{2}(y^{\lambda })}{\vert y^{\lambda}-z^{0}\vert ^{\delta}} \biggr]\,dy \\ \leq{}& \int_{\Sigma_{\lambda}} \bigl[G(x,y)-G \bigl(x^{\lambda},y \bigr) \bigr] \frac {y_{n}^{\gamma}[\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta _{i}}_{2}(y)-\bar{u}^{\alpha_{i}}_{1}(y^{\lambda})\bar{u}^{\beta _{i}}_{2}(y^{\lambda})]}{\vert y-z^{0}\vert ^{\delta}}\,dy. \end{aligned}$$
(2.13)
The proof consists of two steps. In step 1, we will show that, for
λ sufficiently negative,
$$\bar{w}_{i}^{\lambda}(x)= \bar{u}_{i}^{\lambda}(x)- \bar{u}_{i}(x)\geq0 ,\quad \mbox{a.e. } \forall x\in \Sigma_{\lambda} . $$
In step 2, we deduce that
\(T_{\lambda}\) can be moved to the right all the way to
\(z_{1}^{0}\). Furthermore, we obtain
\(\bar{w}_{z^{0}_{1}}\equiv0\),
\(\forall x \in\Sigma_{z_{1}^{0}}\).
Step 1. (Prepare to move the plane from near
\(x_{1}=-\infty\).) In this step, we will show that, for
λ sufficiently negative,
\(\epsilon>0\) sufficiently small
$$ \bar{u}_{i}^{\lambda}(x)\geq\bar{u}_{i}(x) ,\quad \mbox{a.e. } \forall x\in{\Sigma_{\lambda} \backslash{B_{\epsilon } \bigl( \bigl({z^{0}} \bigr)^{\lambda} \bigr)}}, $$
(2.14)
where
\(({z^{0}})^{\lambda}\) is the reflection of
\(z^{0}\) about the plane
\(T_{\lambda}\). Define
\(\Gamma^{\lambda}_{i}=\{x\in\Sigma_{\lambda} \backslash B_{\epsilon}(({z^{0}})^{\lambda}) \mid\bar{u}^{\lambda}_{i}(x)<\bar{u}_{i}(x) \} \), the sets where the inequalities (
2.14) are violated. We will prove that
\(\Gamma^{\lambda}_{i}\) are empty, where
\(i=1,2\).
Without loss of generality, we consider
\(\bar{u}_{1} \). Denote
\(\Sigma ^{\lambda}_{i}=\{x\in\Sigma_{\lambda} \backslash B_{\epsilon }(({z^{0}})^{\lambda}) \mid\bar{u}_{1}^{\alpha_{i}}(x^{\lambda})\times \bar{u}_{2}^{\beta_{i}}(x^{\lambda})<\bar{u}_{1}^{\alpha_{i}}(x)\bar{u}_{2}^{\beta_{i}}(x) \} \), for
\(y\in\Sigma^{\lambda}_{1}\), we may assume that
\(\bar{u}_{1}(y)>\bar{u}^{\lambda}_{1}(y) \) and
\(\bar{u}_{2}(y)\leq\bar{u}^{\lambda}_{2}(y)\). Define
$$\bar{w}^{\lambda}_{i}(y)=\textstyle\begin{cases} 0,&\mbox{for } \bar{u}_{i}(y)< \bar{u}^{\lambda}_{i}(y) , \\ \bar{u}_{i}(y)-\bar{u}^{\lambda}_{i}(y), &\mbox{for } \bar{u}_{i}(y)>\bar{u}^{\lambda}_{i}(y), \end{cases} $$
and
\(\bar{w}^{\lambda}(y)=(\bar{w}^{\lambda}_{1}(y),\bar{w}^{\lambda}_{2}(y) )\). By the expression of
\(G(x, y)\), it is easy to see
$$ G(x, y)\leq\frac{A_{n,\alpha}}{ \vert x-y\vert ^{n-\alpha}}. $$
(2.15)
Applying the mean value theorem, combining (
2.13) and (
2.15), we have, for
\(x\in\Gamma^{\lambda}_{i}\),
$$\begin{aligned} \bar{u}_{i}(x)-\bar{u}^{\lambda}_{i}(x) \leq{}& \int_{\Sigma_{\lambda}}G(x,y)\frac{y_{n}^{\gamma}[\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta_{i}}_{2}(y)-\bar{u}^{\alpha _{i}}_{1}(y^{\lambda})\bar{u}^{\beta_{i}}_{2}(y^{\lambda })]}{\vert y-z^{0}\vert ^{\delta}}\,dy \\ \leq{}& \int_{\Sigma^{\lambda}_{i}}G(x,y)\frac{y_{n}^{\gamma}[\bar{u}^{\alpha_{i}}_{1}(y)\bar{u}^{\beta_{i}}_{2}(y)-\bar{u}^{\alpha _{i}}_{1}(y^{\lambda})\bar{u}^{\beta_{i}}_{2}(y^{\lambda })]}{\vert y-z^{0}\vert ^{\delta}}\,dy \\ ={}& \int_{\Sigma^{\lambda}_{i}}G(x,y)\frac{y_{n}^{\gamma}\{[\bar{u}^{\alpha_{i}}_{1}(y)-\bar{u}^{\alpha_{i}}_{1}(y^{\lambda})]\bar{u}^{\beta_{i}}_{2}(y)+ \bar{u}^{\alpha_{i}}_{1}(y)[\bar{u}^{\beta_{i}}_{2}(y)-u^{\beta _{i}}_{2}(y^{\lambda})]\} }{\vert y-z^{0}\vert ^{\delta}} \,dy \\ \leq{}&c \int_{\Sigma^{\lambda}_{i}}G(x,y)\frac{y_{n}^{\gamma}[\bar{u}^{\alpha_{i}}_{1}(y)-\bar{u}^{\alpha_{i}}_{1}(y^{\lambda})]\bar{u}^{\beta_{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}} \,dy \\ \leq{}&c \int_{\Sigma^{\lambda}_{i}}G(x,y)\frac{y_{n}^{\gamma}\psi ^{\alpha_{i}-1}_{1}(y)[\bar{u}_{1}(y)-\bar{u}_{1}(y^{\lambda})]\bar{u}^{\beta_{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}} \,dy \\ \leq{}&c \int_{\Sigma^{\lambda}_{i}}\frac{1}{\vert x-y\vert ^{n-\alpha}}\frac {y_{n}^{\gamma}\bar{u}^{\alpha_{i}-1}_{1}(y)[\bar{u}_{1}(y)-\bar{u}_{1}(y^{\lambda})]\bar{u}^{\beta_{i}}_{2}(y)}{\vert y-z^{0}\vert ^{\delta}} \,dy. \end{aligned}$$
(2.16)
Noticing
\(\Sigma_{i}^{\lambda} \subseteq\Gamma^{\lambda}_{j} \) for some
j, applying the Hardy-Littlewood-Sobolev inequality and the Hölder inequality we obtain for any
\(q>\frac{n}{n-\alpha}\),
$$ \Vert \bar{w}_{\lambda} \Vert _{L^{q}({\Gamma_{\lambda}})}\leq c\biggl\Vert \frac{y_{n}^{\gamma} \vert \bar{u}\vert ^{\alpha_{i}+\beta_{i}-1}}{\vert y-z^{0}\vert ^{\delta}} \bar{w}_{\lambda}\biggr\Vert _{L^{\frac{nq}{n+\alpha q}}({\Gamma_{\lambda}}) } \leq c\biggl\Vert \frac{y_{n}^{\gamma} \vert \bar{u}\vert ^{\alpha_{i}+\beta_{i}-1}}{\vert y-z^{0}\vert ^{\delta}}\biggr\Vert _{L^{\frac {n}{\alpha}}({\Gamma_{\lambda}}) }\Vert \bar{w}_{\lambda} \Vert _{L^{q}({ \Gamma_{\lambda}})}, $$
(2.17)
where
\(\Gamma_{\lambda} =\Gamma^{\lambda}_{1}\cup\Gamma ^{\lambda}_{2}\). Since
\(\gamma\geq0\), we can easily see that
\(y^{\gamma}_{n}\) is bounded in each bounded domain
\(\Omega\subset R^{n}_{+}\). Therefore, by our assumption
\(\vert u\vert \in L^{\frac{n(\alpha_{i}+\beta _{i}-1)}{\alpha}}_{\mathrm{loc}}(R^{n}_{+})\),
i.e.
\(\vert u\vert ^{\alpha _{i}+\beta_{i}-1}\in L^{\frac{n}{\alpha}}_{\mathrm{loc}}(R^{n}_{+})\), we derive
$$ y^{\gamma}_{n} \vert u\vert ^{\alpha_{i}+\beta_{i}-1} \in L^{\frac{n}{\alpha }}_{\mathrm{loc}} \bigl(R^{n}_{+} \bigr). $$
(2.18)
Hence, we obtain
$$ \int_{ \hat{\Omega}} \biggl[\frac{y^{\gamma}_{n}\vert \bar{u}\vert ^{\alpha_{i}+\beta _{i}-1}(y)}{{\vert y-z^{0}\vert ^{\delta}}} \biggr]^{\frac{n}{\alpha}} \,dy= \int_{ \Omega} \bigl(y^{\gamma}_{n} \vert u \vert ^{\alpha_{i}+\beta_{i}-1}(y) \bigr)^{\frac{n}{\alpha}}\,dy < \infty $$
(2.19)
for any domain Ω that is a positive distance away from the
\(z^{0}\) and Ω̂ is the image of Ω about the Kelvin transform. By (
2.19), for
λ sufficiently negative and for
\(\epsilon > 0\) sufficiently small,
\(c\Vert \frac{y^{\gamma}_{n}\vert \bar{u}\vert ^{\alpha_{i}+\beta_{i}-1}}{\vert y-z^{0}\vert ^{\delta}} \Vert _{L^{\frac{n}{\alpha }}({\Gamma_{\lambda}}) } \) can be made very small. Combining this with (
2.17), we arrive at
\(\Vert \bar{w}_{\lambda} \Vert _{L^{q}({\Gamma_{\lambda}})}=0 \), and
\(\Gamma^{\lambda}_{i}\) must be of measure zero. Hence (
2.14) holds.
Step 2. (Move the plane to the limiting position to derive symmetry.)
Inequality (
2.14) provides a starting point to move the plane
\(T_{\lambda}\). Now we start to move the plane
\(T_{\lambda}\) along the
\(x_{1}\) direction as long as (
2.14) holds. Define
$$\lambda_{0} =\operatorname{sup} \bigl\{ \lambda\leq z^{0}_{1} \mid \bar{u}_{i}^{\mu}(x)\geq \bar{u}_{i}(x) , \mbox{ a.e. } \forall x\in{ \Sigma_{\mu}} , \mu\leq\lambda \bigr\} . $$
We prove that
\(\lambda_{0}= z^{0}_{1} \). On the contrary, suppose that
\(\lambda_{0}< z^{0}_{1}\). We will show that
\(\bar{u}(x)\) is symmetric about the plane
\(T_{\lambda _{0}}\), that is,
$$ \bar{u}(x)\equiv\bar{u}^{\lambda_{0}}(x) , \quad\mbox{a.e. } \forall x\in{ \Sigma _{\lambda_{0}}\backslash{B_{\epsilon} \bigl( \bigl(z^{0} \bigr)^{\lambda_{0}} \bigr)}}. $$
(2.20)
Suppose (
2.20) is not true, then, for such
\(\lambda_{0}< z^{0}_{1} \), for all
\(i=1,2\), we have
$$ \bar{u}_{i}^{\lambda_{0}}(x) >\bar{u}_{i}(x)\quad \mbox{a.e. } x \in{\Sigma_{\lambda _{0}}\backslash{B_{\epsilon} \bigl( \bigl(z^{0} \bigr)^{\lambda_{0}} \bigr)}}. $$
(2.21)
In fact, by (
2.14), we have two cases for all
\(i=1,2\); one is
$$ \bar{u}_{i}^{\lambda_{0}}(y)>\bar{u}_{i}(y) \quad\mbox{on a set of positive measure, $i=1$ and 2.} $$
(2.22)
For the other case, without loss of generality, we may assume that
\(\bar{u}_{1}^{\lambda_{0}}(z_{1})>\bar{u}_{1}(z_{1})\) and
$$ \bar{u}_{2}^{\lambda_{0}}(z_{2})=\bar{u}_{2}(z_{2}). $$
(2.23)
For the first case, (
2.21) is proved. For the other case, we have
$$ \bar{u}_{1}^{\alpha_{i}}(y)\bar{u}_{2}^{\beta_{i}}(y)- \bar{u}_{1}^{\alpha_{i}} \bigl(y^{\lambda_{0}} \bigr)\bar{u}_{2}^{\beta_{i}} \bigl(y^{\lambda _{0}} \bigr)< 0. $$
(2.24)
Combining (
2.23) with (
2.24), we obtain
$$0=\bar{u}_{i}(z_{i})-\bar{u}_{i}^{\lambda_{0}}(z_{i}) < \int_{\Sigma_{\lambda_{0}} } \bigl[G(z_{i},y)-G \bigl({z_{i}}^{\lambda},y \bigr) \bigr]\frac {y^{\gamma}_{n} [\bar{u}_{1}^{\alpha_{i}}(y)\bar{u}_{2}^{\beta_{i}}(y)-\bar{u}_{1}^{\alpha_{i}}(y^{\lambda_{0}})\bar{u}_{2}^{\beta_{i}}(y^{\lambda _{0}})]}{\vert y-z^{0}\vert ^{\delta}}\,dy< 0. $$
This is impossible. Hence (
2.21) holds. Next based on (
2.21), we will verify that the plane can be moved further to the right. More precisely, there exists a
\(\zeta> 0\) such that, for all
\(\lambda\in[\lambda_{0}, \lambda_{0} + \zeta)\)
\(\bar{u}_{i}^{\lambda}(x)\geq\bar{u}_{i} (x)\), a.e.
\(\forall x\in\Sigma _{\lambda}\backslash B_{\epsilon}(z^{0})^{\lambda_{0}}\). By inequality (
2.17), we have
$$ \Vert \bar{w}_{\lambda} \Vert _{L^{q}(\Gamma_{\lambda})}\leq \biggl\{ \int_{\Gamma_{\lambda}} \biggl(\frac{y^{\gamma}_{n}\vert \bar{u}\vert ^{\alpha_{i}+\beta_{i}-1}(y)}{\vert y-z^{0}\vert ^{\delta}} \biggr)^{\frac{n}{\alpha}}\,dy \biggr\} ^{\frac{\alpha}{n}} \Vert \bar{w}_{\lambda} \Vert _{L^{q}(\Gamma_{\lambda})}. $$
(2.25)
Equation (
2.19) ensures that one can choose
η sufficiently small so that, for all
λ in
\([\lambda_{0}, \lambda _{0}+\eta) \),
$$ c \biggl\{ \int_{\Gamma_{\lambda}} \biggl(\frac{y^{\gamma}_{n}\vert \bar{u}\vert ^{\alpha_{i}+\beta_{i}-1}(y)}{\vert y-z^{0}\vert ^{\delta}} \biggr)^{\frac{n}{\alpha}}\,dy \biggr\} ^{\frac{\alpha}{n}} \leq\frac{1}{2}. $$
(2.26)
We postpone the proof of this inequality (
2.26) for a moment.
Now combining (
2.25) and (
2.26), we have
\(\Vert \bar{w}_{\lambda} \Vert _{L^{q}(\Gamma_{\lambda})}=0 \), and
\(\Gamma_{\lambda} \) must be of measure zero. Hence, for these values of
\(\lambda> \lambda_{0}\), we have
\(\bar{w}_{\lambda}(x)\geq0\), a.e.
\(\forall x\in\Sigma_{\lambda _{0}}\backslash B_{\epsilon}((z^{0})^{\lambda})\),
\(\epsilon>0 \). This contradicts the definition of
\(\lambda_{0}\). Therefore (
2.20) must hold. That is, if
\(\lambda_{0}< z^{0}_{1} \), then we must have
$$ \bar{u}_{i}(x)\equiv\bar{u}_{i}^{\lambda}(x),\quad \mbox{a.e. } \forall x \in\Sigma _{\lambda_{0}}\backslash B_{\epsilon}\bigl( \bigl(z^{0} \bigr)^{\lambda_{0}} \bigr). $$
(2.27)
Recall that, by our assumption,
\(c_{i_{0}}>0\) and
$$ \bar{u}_{i_{0}}(x)=\frac{c_{i_{0}}x^{\frac{\alpha }{2}}_{n}}{\vert x-z^{0}\vert ^{n-\alpha}}+ \int_{R^{n}_{+}}G(x,y)\frac{y^{\gamma}_{n}\bar{u}_{1}^{\alpha _{1}}(y)\bar{u}_{2}^{\beta_{1}}(y)}{\vert y-z^{0}\vert ^{\delta}}\,dy. $$
(2.28)
It follows that
\(\bar{u}_{i_{0}}\) is singular at
\(z^{0}\), hence by (
2.27),
\(\bar{u}_{i_{0}}\) must also be singular at
\((z^{0})^{\lambda}\). This is impossible, because
\(z^{0}\) is the only singularity of
ū. Hence, we must have
\(\lambda_{0}=z^{0}_{1} \). Since
ϵ is an arbitrary positive number, we have actually derived that
$$\bar{u}_{i}^{\lambda_{0}}(x)\geq\bar{u}_{i}(x), \quad\mbox{a.e. } \forall x\in\Sigma _{\lambda_{0}}, \lambda_{0}=z^{0}_{1} . $$
Entirely similarly, we can move the plane from near
\(x_{1} = +\infty\) to the left and obtain
\(\bar{u}_{i}^{\lambda_{0}}(x)\leq\bar{u}_{i}(x)\), a.e.
\(\forall x\in\Sigma_{\lambda_{0}}\),
\(\lambda_{0}= z^{0}_{1} \). Therefore we have
\(\bar{w}_{\lambda_{0}} (x)\equiv0\), a.e.
\(\forall x\in\Sigma_{\lambda _{0}}\),
\(\lambda_{0}=z_{1}^{0}\).
Now we prove inequality (
2.26). For any small
\(\eta>0\),
\(\forall\varepsilon>0\), we can choose
R sufficiently large so that
$$ \biggl( \int_{(R^{n}_{+} \backslash B_{\varepsilon}(z^{0}))\backslash B_{R}} \biggl[\frac{y_{n}^{\gamma} \vert \bar{u}\vert ^{\alpha _{i}+\beta_{i}-1}(y)}{\vert y-z^{0}\vert ^{\delta}}\,dy \biggr]^{\frac{n}{\alpha}} \biggr)^{\frac {\alpha}{n}} \leq\eta. $$
(2.29)
For any
\(\tau>0\), define
\(E_{i}^{\tau}=\{x \in({\Sigma_{\lambda_{0}}}\backslash {B_{\varepsilon}((z^{0})^{\lambda_{0}})}) \cap{B_{R}(0)} \mid\bar{u}_{i}^{\lambda_{0}}(x)- \bar{u}_{i}(x)>\tau \} \), and
\(F_{i}^{\tau}=\{ ({\Sigma_{\lambda_{0}}}\backslash {B_{\varepsilon}((z^{0})^{\lambda_{0}})}) \cap{B_{R}(0)} \} \backslash E_{i}^{\tau}\). Obviously,
\({\lim_{\tau\rightarrow0}}\mu(F_{i}^{\tau})=0 \).
For \(\lambda> {\lambda_{0}}\), let \(D_{\lambda}=\{ ({\Sigma_{\lambda}}\backslash{B_{\varepsilon }((z^{0})^{\lambda})}) \backslash({\Sigma_{\lambda_{0}}}\backslash {B_{\varepsilon}((z^{0})^{\lambda_{0}})})\cap{B_{R}(0)} \} \).
It is easy to see that
$$ \bigl\{ {\Gamma^{\lambda}_{i}} \cap{B_{R}(0)} \bigr\} \subset \bigl({\Gamma ^{\lambda}_{i}} \cap E_{i}^{\tau} \bigr)\cup F_{i}^{\tau} \cup D_{\lambda}. $$
(2.30)
For
λ sufficiently close to
\(\lambda_{0}\),
\(\mu (D_{\lambda}) \) is very small. We will show that
\(\mu(\Gamma^{\lambda}_{i} \cap E_{i}^{\tau} ) \) is sufficiently small as
λ close to
\(\lambda_{0}\).
In fact,
\(\bar{w}_{i}^{\lambda}(x)=\bar{u}_{i}^{\lambda}(x)-\bar{u}_{i}(x) = \bar{u}_{i}^{\lambda}(x)-\bar{u}_{i}^{\lambda_{0}}(x)+\bar{u}_{i}^{\lambda _{0}}(x) -\bar{u}_{i}(x) < 0\),
\(\forall x \in(\Gamma^{\lambda}_{i} \cap E_{i}^{\tau} )\). Therefore,
\(\bar{u}_{i}^{\lambda_{0}}(x)-\bar{u}_{i}^{\lambda}(x)>\bar{u}_{i}^{\lambda }(x)-\bar{u}_{i}^{\lambda}(x) >\tau\),
\(\forall x \in(\Gamma^{\lambda }_{i} \cap E_{i}^{\tau} )\). It follows that
$$ \bigl(\Gamma^{\lambda}_{i} \cap E_{i}^{\tau} \bigr) \subset H_{i}^{\tau} = \bigl\{ x \in B_{R}(0) \mid\bar{u}_{i}^{\lambda_{0}}(x)- \bar{u}_{i}^{\lambda}(x) >\tau \bigr\} . $$
(2.31)
By the well-known Chebyshev inequality, for fixed
τ, as
λ is close to
\(\lambda_{0} \),
\(\mu (E_{i}^{\tau}) \) can be sufficiently small. By (
2.30) and (
2.31), we derive that
\(\mu (\Gamma^{\lambda}_{i} \cap B_{R}(0) ) \) can be made as small as we wish. Combining this with (
2.29), we deduce that (
2.26) holds.
Since we can choose any direction that is perpendicular to the \(x_{n}\)-axis as the \(x_{1}\) direction, we have actually shown that the Kelvin transform of the solution \(\bar{u}(x)\) is rotationally symmetric about the line parallel to the \(x_{n}\)-axis and passing through \(z^{0}\). Now we take any two points \(X^{1}\) and \(X^{2}\), with \(X^{l}=(x^{\prime l},x_{n})\in R^{n-1}\times[0,\infty)\), \(l=1,2\). Let \(z^{0}\) be the projection of \(\bar{X}= \frac{X^{1}+X^{2}}{2}\) on \(\partial R^{n}_{+}\). Set \(Y^{l}= \frac{X^{l}-z^{0}}{\vert X^{l}-z^{0}\vert ^{2}}+z^{0}\), \(l=1,2\). From the above arguments, it is easy to see \(\bar{u}(Y ^{1}) = \bar{u}(Y^{2})\), hence \(u(X^{1}) = u(X^{2})\). This implies that \(u_{i}\) is independent of \(x'=(x_{1},\ldots,x_{n-1}) \). That is, \(u_{i}=u_{i}(x_{n})\), and we will show that this will contradict the finiteness of the integral \(\int_{R^{n}_{+}}G(x,y)y^{\gamma}_{n} u^{\alpha_{i}}_{1}(y)u^{\beta _{i}}_{2}(y)\,dy\). To continue, we need the following lemma.
Set
\(x=(x',x_{n})\),
\(y=(y',y_{n})\in R^{n-1}\times(0,+\infty)\),
\(r^{2}=\vert x'-y'\vert ^{2} \) and
\(a^{2}=\vert x_{n}-y_{n}\vert ^{2} \). If
\(u_{i}=u_{i}(x_{n}) \) is a solution of
$$ u_{i}(x)= \int_{R^{n}_{+}}G(x,y)y^{\gamma}_{n} u^{\alpha_{i}}_{1}(y)u^{\beta _{i}}_{2} (y)\,dy, $$
(2.32)
then, for each fixed
\(x\in R^{n}_{+} \), letting
R be large enough, by elementary calculations, we have
$$\begin{aligned} +\infty>u_{i}(x_{n})={}& \int^{\infty}_{0}y^{\gamma}_{n} u^{\alpha _{i}}_{1}(y_{n})u^{\beta_{i}}_{2}(y_{n}) \int_{R^{n-1}} G(x,y)\,dy'\,dy_{n} \\ \geq{}&C \int^{\infty}_{R}y^{\gamma}_{n} u^{\alpha_{i}}_{1}(y_{n})u^{\beta _{i}}_{2}(y_{n})y_{n}^{\frac{\alpha}{2}} \int_{R^{n-1}\backslash B_{R}(0)} \frac{1}{\vert x-y\vert ^{n}}\,dy' \,dy_{n} \\ \geq{}&C \int^{\infty}_{R}y^{\gamma}_{n} u^{\alpha_{i}}_{1}(y_{n})u^{\beta _{i}}_{2}(y_{n})y_{n}^{\frac{\alpha}{n-2}}{ \int^{\infty}_{R}} \frac {r^{2}}{(r^{2}+a^{2})^{\frac{n}{2}}}\,dr \,dy_{n} \\ \geq{}&C \int^{\infty}_{R}y^{\gamma}_{n} u^{\alpha_{i}}_{1}(y_{n})u^{\beta _{i}}_{2}(y_{n})y_{n}^{\frac{\alpha}{2}} \frac{1}{\vert x_{n}-y_{n}\vert }{ \int ^{\infty}_{\frac{R}{a}}} \frac{\tau^{n-2}}{(\tau^{2}+1)^{\frac {n}{2}}}\,d\tau \,dy_{n} \\ \geq{}&C \int^{\infty}_{R} u^{\alpha_{i}}_{1}(y_{n})u^{\beta _{i}}_{2}(y_{n})y_{n}^{\gamma+\frac{\alpha}{2}-1} \,dy_{n}. \end{aligned}$$
(2.33)
Equation (
2.33) implies that there exists a sequence
\(\{ y^{k}_{n}\}\rightarrow\infty \) as
\(k\rightarrow\infty \), such that
$$ u^{\alpha_{i}}_{1} \bigl(y^{k}_{n} \bigr)u^{\beta_{i}}_{2} \bigl(y^{k}_{n} \bigr) \bigl(y^{k}_{n} \bigr)^{\gamma +\frac{\alpha}{2}}\rightarrow0. $$
(2.34)
Similarly to (
2.33), for any
\(x=(0,x_{n})\in R^{n}_{+} \), we derive that
$$ +\infty>u_{i}(x_{n})\geq C_{0}{ \int_{0}^{\infty}} y_{n}^{\gamma}u^{\alpha _{i}}_{1}(y_{n})u^{\beta_{i}}_{2}(y_{n})y_{n}^{\frac{ \alpha}{2}}\frac{1}{\vert x_{n}-y_{n}\vert } \,dy_{n} x_{n}^{\frac{ \alpha}{2}}. $$
(2.35)
Let
\(x_{n} = 2R\) be sufficiently large. By (
2.35), we deduce that
$$\begin{aligned} +\infty>u_{i}(x_{n})\geq{}& C_{0}{ \int_{0}^{1}} y_{n}^{\gamma}u^{\alpha _{i}}_{1}(y_{n})u^{\beta_{i}}_{2}(y_{n})y_{n}^{\frac{ \alpha}{2}}\frac{1}{\vert x_{n}-y_{n}\vert } \,dy_{n} x_{n}^{\frac{ \alpha}{2}} \\ \geq{}&\frac{C_{0}}{2R}(2R)^{\frac{\alpha}{2}} { \int_{0}^{1}} y_{n}^{\gamma }u^{\alpha_{i}}_{1}(y_{n})u^{\beta_{i}}_{2}(y_{n})y_{n}^{\frac{ \alpha }{2}}\,dy_{n} \geq C_{1}(2R)^{\frac{\alpha}{2}-1} =C_{1}x_{n}^{\frac{\alpha}{2}-1}. \end{aligned}$$
(2.36)
Then by (
2.35) and (
2.36), for
\(x_{n} = 2R\) sufficiently large, we also obtain
$$\begin{aligned} u_{i}(x_{n})\geq{}&C_{0}{ \int_{\frac{R}{2}^{R}}}y_{n}^{\gamma}u^{\alpha _{i}}_{1}(y_{n})u^{\beta_{i}}_{2}(y_{n})y_{n}^{\frac{ \alpha}{2}} \frac{1}{\vert x_{n}-y_{n}\vert }\,dy_{n} x_{n}^{\frac{ \alpha}{2}} \\ \geq{}&C_{0}{ \int_{\frac{R}{2}^{R}}}y_{n}^{\gamma}C_{1}^{\alpha_{i}+\beta _{i}}y_{n}^{(\alpha_{i}+\beta_{i})(\frac{\alpha}{2}-1)}y_{n}^{\frac{ \alpha}{2}} \frac{1}{\vert x_{n}-y_{n}\vert }\,dy_{n} x_{n}^{\frac{ \alpha}{2}} \\ \geq{}&C_{0} C_{1}^{\alpha_{i}+\beta_{i}}R^{(\alpha_{i}+\beta_{i})(\frac{\alpha }{2}-1)+\gamma}(2R)^{\frac{ \alpha}{2}}\frac{2}{3R} { \int_{\frac{R}{2}^{R}}}y_{n}^{\frac{ \alpha}{2}}\,dy_{n} \\ :={}&AR^{(\alpha_{i}+\beta_{i})(\frac{\alpha}{2}-1)+\gamma+\alpha} :=A_{1}x_{n}^{(\alpha_{i}+\beta_{i})(\frac{\alpha}{2}-1)+\gamma+\alpha}. \end{aligned}$$
Continuing this way
m times, for
\(x_{n} = 2R\), we have
$$ u_{i}(x_{n})\geq A(m,\alpha_{i}+ \beta_{i},\alpha,\gamma)x_{n}^{(\alpha _{i}+\beta_{i})^{m}(\frac{\alpha}{2}-1)+\frac{(\alpha_{i}+\beta _{i})^{m}-1}{\alpha_{i}+\beta_{i}-1}(\gamma+\alpha)}. $$
(2.37)
For any fixed
α and
γ in their respective domain, we choose
m to be an integer greater than
\(\frac{-\alpha^{2}-\alpha \gamma+\gamma+3}{\alpha+\gamma}\) and 1. That is,
$$ m\geq \operatorname{max} \biggl\{ \biggl\lceil \frac{-\alpha^{2}- \alpha\gamma+\gamma+3}{\alpha +\gamma} \biggr\rfloor +1,1 \biggr\} , $$
(2.38)
where
\(\lceil a\rfloor\) is the integer part of
a.
We claim that, for such a choice of m, we have
$$ \tau(\alpha_{i}+\beta_{i}):= \biggl[({ \alpha_{i}+\beta_{i}})^{m} \biggl( \frac{\alpha }{2}-1 \biggr)+\frac{({\alpha_{i}+\beta_{i}})^{m}-1}{{\alpha_{i}+\beta _{i}}-1}(\alpha+\gamma) \biggr]({ \alpha_{i}+\beta_{i}})+\frac{\alpha }{2}+\gamma\geq0. $$
(2.39)
We postpone the proof of (
2.39) for a moment. Now by (
2.37) and (
2.39), we derive that
$$u^{\alpha_{i}+\beta_{i}}_{i}(x_{n})x^{\frac{\alpha}{2}+\gamma}_{n} \geq A(m,{\alpha_{i}+\beta_{i}},\alpha, \gamma)x^{\tau({\alpha_{i}+\beta _{i}})}_{n} \geq A(m,{\alpha_{i}+ \beta_{i}},\alpha,\gamma)>0, $$
for all
\(x_{n}\) sufficiently large. This contradicts (
2.34). So there is no positive solution of (
2.32). This implies that
\(u(x)\) must be constant. By our positive assumption on u, we have
\(u_{i}(x) = b_{i} > 0\),
\(i = 1, 2\). Taking
\(u_{i}\) into (
2.2), we have
\(0=(-\Delta)^{\frac{\alpha }{2}}u_{i}(x)=x^{\gamma}_{n}u_{1}^{\alpha_{i}}(x)u_{2}^{\beta_{i}}(x)>0\). This is impossible. Hence, in (
2.10),
\(c_{i}\) must be zero,
\(i = 1, 2\).