In this section, we prove three maximum principles which will be used in the process of moving planes. These maximum principles are on a bounded domain in \({{\mathbb{H}}^{n}}\), on a bounded domain in the left domain of some hyperplane, and on a narrow region.
Lemma 3.1
LetΩbe a bounded domain in\({{\mathbb{H}}^{n}}\). Assume\(u\in {{L}_{sp}}({{\mathbb{H}}^{n}})\cap C_{\mathrm{loc}}^{1,1}({{\mathbb{H}}^{n}})\)is lower semicontinuous onΩ̄and satisfies
$$ \textstyle\begin{cases} (-{{\Delta }_{\mathbb{H}}})_{p}^{s}u(\xi )\ge 0, \quad \xi \in \varOmega , \\ u(\xi )\ge 0, \quad \xi \in \mathbb{H}^{n}\setminus \varOmega , \end{cases} $$
(3.1)
then
$$ u(\xi )\ge 0,\quad \xi \in \varOmega . $$
(3.2)
Furthermore, if\(u=0\)at some point inΩ, then
$$ u(\xi )=0 \quad \textit{almost everywhere in } {{\mathbb{H}}^{n}}. $$
These conclusions also hold on the unbounded regionΩif we further assume that
$$ \mathop{\underline{\lim }}_{{{ \vert \xi \vert }_{\mathbb{H}}}\to \infty } u(\xi )\ge 0. $$
Proof
Suppose that (3.2) is not true, then by the lower semicontinuity of u on Ω̄ there exists \({{\xi }^{0}}\in \bar{\varOmega }\) such that
$$ u \bigl({{\xi }^{0}} \bigr)=\min_{{\bar{\varOmega }}} u< 0. $$
From (3.1), we know that \({{\xi }^{0}}\) is a point in Ω, and
$$\begin{aligned} (-{{\Delta }_{\mathbb{H}}})_{p}^{s}u \bigl({{\xi }^{0}} \bigr)&={{C}_{Q,s,p}}\mathit{PV} \int _{{{\mathbb{H}}^{n}}}{ \frac{{{ \vert u({{\xi }^{0}})-u(\eta ) \vert }^{p-2}}(u({{\xi }^{0}})-u(\eta ))}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\ &\le {{C}_{Q,s,p}} \int _{{{\mathbb{H}}^{n}}\setminus \varOmega }{ \frac{{{ \vert u({{\xi }^{0}})-u(\eta ) \vert }^{p-2}}(u({{\xi }^{0}})-u(\eta ))}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\ &< 0, \end{aligned}$$
which contradicts (3.1). This implies (3.2).
If there exists some point \({{\xi }^{0}}\in \varOmega \) such that \(u({{\xi }^{0}})=0\), then
$$ 0\le (-{{\Delta }_{\mathbb{H}}})_{p}^{s}u \bigl({{\xi }^{0}} \bigr)={{C}_{Q,s,p}} \int _{{{\mathbb{H}}^{n}}}{ \frac{{{ \vert u(\eta ) \vert }^{p-2}}(-u(\eta ))}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta . $$
Using \(u(\xi )\ge 0\), we have \(u(\xi )=0\) almost everywhere in \({{\mathbb{H}}^{n}}\).
For an unbounded region Ω, the condition \(\underline{\lim }_{{{ \vert \xi \vert }_{\mathbb{H}}}\to \infty } u(\xi )\ge 0\) implies that the negative minimum \({{\xi }^{0}}\) of u cannot be reached at infinity. Then the condition of lower semicontinuity ensures that the proof can go on as above. The proof is ended. □
Let \(T_{\lambda }\) be a hyperplane in \({{\mathbb{H}}^{n}}\) defined by
$$ T_{\lambda }= \bigl\{ \xi \in {{\mathbb{H}}^{n}}\mid t=\lambda , \lambda \in \mathbb{R} \bigr\} . $$
Denote by \(\tilde{\xi }=(y,x,2\lambda -t)\) the H-reflection of \(\xi =(x,y,t)\) about the plane \(T_{\lambda }\) and by
$$ \varSigma _{\lambda }= \bigl\{ \xi \in {{\mathbb{H}}^{n}}\mid t< \lambda \bigr\} $$
the region in the left of the plane \(T_{\lambda }\). Letting
$$ {{u}_{\lambda }}(\xi )={{u}_{\lambda }} \bigl( \bigl\vert (x,y) \bigr\vert ,t \bigr):=u \bigl( \bigl\vert (x,y) \bigr\vert ,2\lambda -t \bigr) $$
and using the H-refection (see [1]), we have
$$ {{u}_{\lambda }}(\xi )=u(y,x,2\lambda -t)=u \bigl({{\xi }^{\lambda }} \bigr). $$
Set
$$ {{w}_{\lambda }}(\xi )={{u}_{\lambda }}(\xi )-u(\xi ). $$
Lemma 3.2
LetΩbe a bounded domain in\({{\varSigma }_{\lambda }}\). Assume that the cylindrical function\(u\in {{L}_{sp}}({{\mathbb{H}}^{n}})\cap C_{\mathrm{loc}}^{1,1}({{\mathbb{H}}^{n}})\)is lower semicontinuous onΩ̄and satisfies
$$ \textstyle\begin{cases} (-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }}(\xi )-(-{{ \Delta }_{\mathbb{H}}})_{p}^{s}u(\xi )\geq 0, \quad \xi \in \varOmega , \\ {{w}_{\lambda }}(\xi )\ge 0, \quad \xi \in {{\varSigma }_{\lambda }} \setminus \varOmega , \\ w_{\lambda }(\xi ^{\lambda })=-w_{\lambda }(\xi ), \quad \xi \in {{\varSigma }_{ \lambda }}, \end{cases} $$
(3.3)
then
$$ w_{\lambda }(\xi )\ge 0,\quad \xi \in \varOmega . $$
(3.4)
Furthermore, if\(w_{\lambda }=0\)at some point inΩ, then
$$ w_{\lambda }(\xi )=0 \quad \textit{almost everywhere in } {{ \mathbb{H}}^{n}}. $$
These conclusions also hold for the unbounded regionΩif we further assume that
$$ \mathop{\underline{\lim }}_{{{ \vert \xi \vert }_{\mathbb{H}}}\to \infty } w_{\lambda }(\xi )\ge 0. $$
Proof
Suppose that (3.4) is incorrect. By the lower semicontinuity of \(w_{\lambda }\) on Ω̄, there exists \({{\xi }^{0}}\in \bar{\varOmega }\) such that
$$ w_{\lambda } \bigl({{\xi }^{0}} \bigr)=\min_{{\bar{\varOmega }}} w_{\lambda }< 0. $$
For simplicity, we denote
$$ G(a)={{ \vert a \vert }^{p-2}}a, \quad a\geq 0. $$
Note that \(G(a)\) is increasing and \({G}'(a)=(p-1){{ \vert a \vert }^{p-2}}\ge 0\). A direct calculation gives
$$\begin{aligned}& (-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-(-{{ \Delta }_{\mathbb{H}}})_{p}^{s}u \bigl({{\xi }^{0}} \bigr) \\& \quad = {{C}_{Q,s,p}}\mathit{PV} \int _{{{\mathbb{H}}^{n}}}{ \frac{G({{u}_{\lambda }}({{\xi }^{0}})-{{u}_{\lambda }}(\eta ))-G(u({{\xi }^{0}})-u(\eta ))}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\& \quad = {{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}}{ \frac{G({{u}_{\lambda }}({{\xi }^{0}})-{{u}_{\lambda }}(\eta ))-G(u({{\xi }^{0}})-u(\eta ))}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\& \quad\quad{} +{{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}}{ \frac{G({{u}_{\lambda }}({{\xi }^{0}})-u(\eta ))-G(u({{\xi }^{0}})-{{u}_{\lambda }}(\eta ))}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\& \quad = {{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}} \biggl( \frac{1}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}- \frac{1}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}} \biggr) \\& \quad\quad{}\times \bigl(G \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}(\eta ) \bigr)-G \bigl(u \bigl({{ \xi }^{0}} \bigr)-u(\eta ) \bigr) \bigr)\,d\eta \\& \quad\quad{} +{{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}} \bigl(G\bigl({{u}_{\lambda }}\bigl({{\xi }^{0}}\bigr)-{{u}_{\lambda }}(\eta )\bigr) -G\bigl(u\bigl({{\xi }^{0}}\bigr)-u_{\lambda }(\eta )\bigr)+ G\bigl({{u}_{\lambda }}\bigl({{\xi }^{0}}\bigr)-{{u}}(\eta )\bigr) \\& \quad\quad{}-G\bigl(u\bigl({{\xi }^{0}}\bigr)-u(\eta )\bigr) \bigr) \bigl( {{ \big\vert {{\bigl({{\eta }^{\lambda }}\bigr)}^{-1}}\circ {{\xi }^{0}} \big\vert }_{\mathbb{H}}}^{Q+ sp}\bigr)^{-1} \,d \eta \\& \quad := {{C}_{Q,s,p}}({{J}_{1}}+{{J}_{2}}). \end{aligned}$$
(3.5)
For \(J_{1}\), we have for any \({{\xi }^{0}},\eta \in {{\varSigma }_{\lambda }}\),
$$ \frac{1}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}- \frac{1}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}>0. $$
By the monotonicity of G and the fact that
$$ \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}( \eta ) \bigr)- \bigl(u \bigl({{\xi }^{0}} \bigr)-u( \eta ) \bigr)={{w}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{w}_{\lambda }}( \eta ) $$
is nonpositive but not identity to 0, we deduce that
$$ G \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}( \eta ) \bigr)-G \bigl(u \bigl({{\xi }^{0}} \bigr)-u( \eta ) \bigr) $$
is also nonpositive but not identity to 0. So we have
For \(J_{2}\), by the mean value theorem,
$$\begin{aligned} {{J}_{2}}&= \int _{{{\varSigma }_{\lambda }}}{ \frac{G({{u}_{\lambda }}({{\xi }^{0}})-{{u}_{\lambda }}(\eta ))-G(u({{\xi }^{0}})-{{u}_{\lambda }}(\eta ))+G({{u}_{\lambda }}({{\xi }^{0}})-u(\eta ))-G(u({{\xi }^{0}})-u(\eta ))}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+ sp}}}\,d \eta \\ &=w_{\lambda } \bigl(\xi ^{0} \bigr) \int _{{{\varSigma }_{\lambda }}}{ \frac{{G}'(g(\eta ))+{G}'(h(\eta ))}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}}\,d \eta \\ &\le 0. \end{aligned}$$
(3.7)
In fact, if \(u_{\lambda }(\eta )\ge u(\eta )\), then we have \(w_{\lambda }(\eta )\ge 0\), i.e., (3.4) holds. If \(u_{\lambda }(\eta )> u(\eta )\), we know G is strictly increasing, then \({G}'(g(\eta ))\ge 0\) and \({G}'(h(\eta ))\ge 0\). Hence, we have (3.7).
Putting (3.6) and (3.7) into (3.5) implies
$$ (-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-(-{{ \Delta }_{\mathbb{H}}})_{p}^{s}u \bigl({{\xi }^{0}} \bigr)< 0. $$
This contradicts (3.3) and we obtain (3.4).
If there exists some point \({{\xi }^{0}}\in \varOmega \) such that \({{w}_{\lambda }}({{\xi }^{0}})=0\), then (3.5) holds and \({{J}_{2}}\ge 0\). Hence from the first inequality in (3.3) we have \({{J}_{1}}\ge 0\), and by the monotonicity of G,
$$ G \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}( \eta ) \bigr)-G \bigl(u \bigl({{\xi }^{0}} \bigr)-u( \eta ) \bigr)\ge 0. $$
We have, for almost all \(\eta \in {{\varSigma }_{\lambda }}\),
$$ \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}( \eta ) \bigr)- \bigl(u \bigl({{\xi }^{0}} \bigr)-u( \eta ) \bigr)={{w}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{w}_{\lambda }}( \eta )=-{{w}_{ \lambda }}(\eta )\ge 0. $$
Using (3.4), we have
$$ {{w}_{\lambda }}(\xi )=0\quad \text{almost everywhere in } {{\varSigma }_{ \lambda }}. $$
From the antisymmetry of \({{w}_{\lambda }}\),
$$ {{w}_{\lambda }}(\xi )=0\quad \text{almost everywhere in } {{ \mathbb{H}}^{n}}. $$
□
Lemma 3.3
LetΩbe a bounded narrow domain in\({{\varSigma }_{\lambda }}\)and locate in\(\{\xi\mid \lambda -l< t<\lambda \}\)for small l. Assume that the cylindrical function\(u\in {{L}_{sp}}({{\mathbb{H}}^{n}})\cap C_{\mathrm{loc}}^{1,1}({{\mathbb{H}}^{n}})\)is lower semicontinuous onΩ̄. If\(c(x)\)is bounded from below inΩandusatisfies
$$ \textstyle\begin{cases} (-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }}(\xi )-(-{{ \Delta }_{\mathbb{H}}})_{p}^{s}u(\xi )+c(\xi ){{w}_{\lambda }}(\xi ) \ge 0, \quad \xi \in \varOmega , \\ {{w}_{\lambda }}(\xi )\ge 0, \quad \xi \in {{\varSigma }_{\lambda }} \setminus \varOmega , \\ {{w}_{\lambda }}({{\xi }^{\lambda }})=-{{w}_{\lambda }}(\xi ),\quad \xi \in {{\varSigma }_{\lambda }}, \end{cases} $$
(3.8)
then
$$ w_{\lambda }(\xi )\ge 0,\quad \xi \in \varOmega . $$
(3.9)
Furthermore, if\(w_{\lambda }=0\)at some point inΩ, then
$$ w_{\lambda }(\xi )=0 \quad \textit{almost everywhere in } {{ \mathbb{H}}^{n}}. $$
These conclusions also hold for the unbounded regionΩif we further assume that
$$ \mathop{\underline{\lim }}_{{{ \vert \xi \vert }_{\mathbb{H}}}\to \infty } w_{\lambda }(\xi )\ge 0. $$
Proof
By the proof of Lemma 3.2, we have
$$\begin{aligned} & (-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-(-{{ \Delta }_{\mathbb{H}}})_{p}^{s}u \bigl({{\xi }^{0}} \bigr) \\ &\quad ={{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}} \biggl( \frac{1}{{{ \vert {{\eta }^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}}- \frac{1}{{{ \vert {{({{\eta }^{\lambda }})}^{-1}}\circ {{\xi }^{0}} \vert }_{\mathbb{H}}}^{Q+sp}} \biggr) \\ &\quad\quad{}\times \bigl(G \bigl({{u}_{\lambda }} \bigl({{\xi }^{0}} \bigr)-{{u}_{\lambda }}(\eta ) \bigr)-G \bigl(u \bigl({{ \xi }^{0}} \bigr)-u(\eta ) \bigr) \bigr)\,d\eta \\ &\quad\quad{} +{{C}_{Q,s,p}}\mathit{PV} \int _{{{\varSigma }_{\lambda }}} \bigl(G\bigl({{u}_{\lambda }}\bigl({{\xi }^{0}}\bigr)-{{u}_{\lambda }}(\eta )\bigr) -G\bigl(u\bigl({{\xi }^{0}}\bigr)-u(\eta )\bigr)+ G\bigl({{u}_{\lambda }}\bigl({{\xi }^{0}}\bigr)-{{u}}(\eta )\bigr) \\ &\quad\quad{} -G\bigl(u\bigl({{\xi }^{0}}\bigr)-u_{\lambda }(\eta )\bigr) \bigr) \bigl({{ \big\vert {{\bigl({{\eta }^{\lambda }}\bigr)}^{-1}}\circ {{\xi }^{0}} \big\vert }_{\mathbb{H}}}^{Q+ sp} \bigr)^{-1} \,d \eta \\ &\quad :={{C}_{Q,s,p}}({{I}_{1}}+{{I}_{2}}). \end{aligned}$$
(3.10)
Obviously,
$$ {{I}_{2}}\le 0. $$
(3.11)
Similar to (3.6), we know
Denote
$$ {{\delta }_{{{\xi }^{0}}}}=\operatorname{dist} \bigl({{\xi }^{0}}, \partial {{\varSigma }_{ \lambda }} \bigr)= \bigl\vert \lambda -{{t}^{0}} \bigr\vert . $$
Combining (3.10), (3.11), and \(I_{1}<0\), we have
$$ \frac{(-{{\Delta }_{\mathbb{H}}})_{p}^{s}{{u}_{\lambda }}({{\xi }^{0}})-(-{{\Delta }_{\mathbb{H}}})_{p}^{s}u({{\xi }^{0}})}{{{\delta }_{{{\xi }^{0}}}}}< 0. $$
(3.12)
Noting that \({{\xi }^{0}}\) is a negative minimum of \({{w}_{\lambda }}\), we infer \(\nabla {{w}_{{{\lambda }_{0}}}}({{\xi }^{0}})=0\), and so
$$ \frac{\partial {{w}_{\lambda }}}{\partial t} \bigl({{\xi }^{0}} \bigr)= \lim _{{{\delta }_{k}}\to 0} \frac{{{w}_{\lambda }}({{\xi }^{0}})}{{{\delta }_{{{\xi }^{0}}}}}=0, $$
i.e.,
$$ \frac{c({{\xi }^{0}}){{w}_{\lambda }}({{\xi }^{0}})}{{{\delta }_{{{\xi }^{0}}}}} \le o(1). $$
(3.13)
Now (3.12) and (3.13) contradict (3.8), and then (3.9) is proved. □