数理解析研究所講究録
第 1779 巻 2012 年 11-19

11Stable and unstable solutions to Laplace equations
with nonlinear boundary conditions
Junichi Harada (Waseda University)

1

Introduction

We consider the Laplace equation with a nonlinear boundary condition.
$\triangle u=0$in$\mathbb{R}_{+}^{n}$,$\partial_{\nu}u=u^{q}$on$\partial \mathbb{R}_{+}^{n}$$(u>0)$ ,

(1)

where
and
. The existence and the nonexistence of positive solutions of (1) depends on a exponent $q>1$ . It is known that
if $q\in(1, n/(n-2))$ , there are no positive solutions of (1) ([6]). On the other
hand, there exists a family of positive solutions for $q=n/(n-2)$ ([2], [9]) and for$q>n/(n-2)$ ([3]). These existence and nonexistence theorems for positive solutionsof (1) are completely corresponding to those for positive solutions of$\mathbb{R}_{+}^{n}=\{x\in \mathbb{R}^{n};x_{n}>0\}$$\partial_{\nu}=-\partial/\partial x_{n}$$-\triangle u=u^{p}$$in$$\mathbb{R}^{n}$.

(2)

It is known that (2) has no positive solutions if $p\in(1, (n+2)/(n-2))$ and has afamily of positive radially symmetric solutions for $p\geq(n+2)/(n-2)$ . Moreover
there exists another critical exponent $p_{JL}>p_{S}$ defined by$p_{JL}=\{\begin{array}{ll}\infty if n\leq 10,\frac{n-2\sqrt{n-1}}{n-4-2\sqrt{n-1}} if n\geq 11.\end{array}$This critical exponent
is known to be critical for asymptotic expansions andintersection properties of positive radially symmetric solutions of (2) ([8],[12]). A
goal of this note is to introduce a new critical exponent corresponding toandto study the properties of positive solutions of (1) for $q>n/(n-2)$ .
$p_{JL}$$p_{JL}$

Definition 1.1.
be expressed by

A function$f(x)\in C(\mathbb{R}_{+}^{n})$$f(x)=f(|x^{l}|, x_{n})$is called$x_{n}$.

In this note, we often use a polar coordinate:$r=|x|$ ,$\tan\theta=|x’|/x_{n}$We introduce a singular solution of (1).

.

-axial symmetric if $f(x)$ can12Lemma 1.1 ([11]).

Let $q>(n-1)/(n-2)$ . Then there exists a singular solutionof (1), where $V(\theta)>0$ is a unique solution of$\varphi_{\infty}(x)=V(\theta)r^{-1/(q-1)}$$\partial_{\theta\theta}V+(n-2)(\cot\theta)\partial_{\theta}V=\beta V$

in $(0, \pi/2)$ ,$\partial_{\nu}V=V^{q}$on$\{\pi/2\}$,

where $\beta=m_{q}((n-2)-m_{q})$ and $m_{q}=1/(q-1)$ .
A new critical exponent is defined as follows.

Definition 1.2 (JL-critical exponent).

We set.

$\mu(q)=\inf_{u\in H^{1}(\mathbb{R}_{+}^{n})}\frac{(\int|\nabla u|^{2}dx-\int(q\varphi_{\infty}^{q-1})u^{2}dx’)}{\int_{\partial \mathbb{R}_{+}^{n}}|x|^{-1}u^{2}dx’}$

We call$\mu(q)<0$

JL-supercritical if $\mu(q)>0$ , JL-critical if $\mu(q)=0$ and JL-subcritical if$q$.

Remark 1.1.

From the trace Hardy inequality ([4]):(3),$\int_{\partial \mathbb{R}_{+}^{n}}|x’|^{-1}u^{2}dx’\leq c_{H}\int_{\mathbb{R}_{+}^{n}}|\nabla u|^{2}dx$$\mu(q)$is expressed by $\mu(q)=c_{H}-qV(\pi/2)^{q-1}$ . By using this expression, we can showthat(i) for $n\geq 3$ there exists $q_{0}>n/(n-2)$ such that $\mu(q)<0$ if $q\in(n/(n-2), q_{0})$ ,(ii) there exists

if$q>q_{1}$$n_{0}\in \mathbb{N}$such that for $n>n_{0}$ there exists $q_{1}>q_{0}$ such that $\mu(q)>0$.

To state our results, we prepare notations. Let$e_{i}(\theta)$

be the $i’ th$ eigenfunction of(4)$\{\begin{array}{ll}-\triangle s^{e=\lambda e} in S_{+}^{n-1},\partial_{\nu}e=qV(\pi/2)^{q-l}e on \partial S_{+}^{n-1},\end{array}$where
is the Laplace Beltrami operator on
For simplicity of notations, we set$\triangle s$$S^{n-1}$and set $B_{R}=\{x\in \mathbb{R}_{+}^{n};|x|<R\}$ .

$m_{q}=1/(q-1)$ .

a family of
Theorem 1.1 (JL-supercritical, JL-critical). Theresatisfying the following properties.
symmetric solutions$ex\prime ists$$\{u_{\alpha}(x)\}_{\alpha>0}$(a)$u_{\alpha}(x)=\alpha u_{1}(\alpha^{(q-1)}x)$,$u_{\alpha}(O)=\alpha$,$x_{n}$-axial13(b)$u_{\alpha}(x)<\varphi_{\infty}(x)$(c)$u_{\alpha_{1}}(x)<u_{\alpha_{2}}(x)$(d) there exist,$\lim_{\alphaarrow\infty}u_{\alpha}(x)=\varphi_{\infty}(x)$if,$\alpha_{1}<\alpha_{2\prime}$$\gamma_{2}>\gamma_{1}>m_{qz}c_{\alpha}\neq 0$and$c_{\alpha}’\in \mathbb{R}$such that$u_{\alpha}(x)=\{\begin{array}{ll}V(\theta)r^{-m_{q}}+c_{\alpha}e_{1}(\theta)r^{-\gamma_{1}}+O(r^{-}) if JL- supercritical,V(\theta)r^{-m_{q}}+(c_{\alpha}\log r+c_{\alpha}’)e_{1}(\theta)r^{-\gamma_{1}}+O(r^{-\gamma_{2}}) if JL- critical.\end{array}$Moreover if $u(x),$ $v(x)$ are positive -axial symmetric solutions satisfying $u(O)=$$v(O)$ and $u(x),$, then it holds that $u(x)\equiv v(x)$ .
$x_{n}$$v(x)<\varphi_{\infty}(x)$Theorem 1.2 (JL-subcritical).
tions
satisfying

There exists a family of$x_{n}$-axial symmetric solu-$\{u_{\alpha}(x)\}_{\alpha>0}$$u_{\alpha}(x)\leq c_{\alpha}(1+|x|)^{-1/(q-1)}$.

(5)Moreover one of the following asymptotic expansions holds.
(i) there exist$\gamma_{2}>\gamma_{1}>m_{q},$$A,$ $c_{\alpha}\neq 0$

and$B_{\alpha}\in \mathbb{R}$such that$u_{\alpha}(x)=V(\theta)r^{-m_{q}}+c_{\alpha}e_{1}(\theta)r^{-\gamma_{1}}\sin(A\log r+B_{\alpha})+O(r^{-\gamma_{2}})$(ii) there exist$\gamma_{4}>\gamma_{3}>m_{q}$and$c_{\alpha}\neq 0$,such that$u_{\alpha}(x)=V(\theta)r^{-m_{q}}+c_{\alpha}e_{2}(\theta)r^{-\gamma_{3}}+O(r^{-\gamma_{4}})$.

Moreover if $u(x),$ $v(x)$ are positive -axial symmetric solutions satisfying (5) andfor $|x|>R$ with some $R>0$ , then it holds that $u(x)\equiv v(x)$ .
$x_{n}$$u(x)\geq v(x)$Remark 1.2. A solution $u_{\alpha}(x)(\alpha>0)$ constructed in Theorem 1.1 and the singular solution
do not intersect each other for JL-supercritical case and JLcritical case. On the other hand, $u_{\alpha}(x)(\alpha>0)$ constructed in Theorem 1.2 and thesingular solution
must intersect each other for JL-subcritcal case. Set$\varphi_{\infty}(x)$$\varphi_{\infty}(x)$$Z_{\alpha}=\{x\in \mathbb{R}_{+}^{n};u_{\alpha}(x)=\varphi_{\infty}(x)\}$.

For the case (i) in Theorem 1.2, it holds that for large $R>0$$Z_{\alpha}\backslash B_{R}\sim\{x\in \mathbb{R}_{+}^{n} :A \log|x|+B_{\alpha}=k\pi, k\in N\}$.

For the case (ii) in Theorem 1.2, it holds that for large $R>0$$Z_{\alpha}\backslash B_{R}\sim\{x\in \mathbb{R}_{+}^{n}:e_{2}(\theta)=0\}$.

Unfortunately, we do not know which case (i) or (ii) actually occurs.

14

2

Proof

Step l-(i) Existence. (JL-supercritical case, JL-critical case)
For JL-supercritical case and JL-critical case, we construct solutions of (1) satisfying(b) by a different way from [3]. For simplicity, we set$D_{R}=\{x\in\partial \mathbb{R}_{+}^{n};|x|<R\}$,$S_{R}=\{x\in \mathbb{R}_{+}^{n};|x|=R\}$.

First we construct suitable super-solutions.
Lemma 2.1. There exists
insuch that$\delta_{0}>0$$C^{2}(\overline{B}_{1+\delta_{0}})$$\overline{u}\equiv\varphi_{\infty}$

We fix$\delta\in(0, \delta_{0})$$B_{1+\delta_{0}}$,$x_{n}$-axial symmetric function
and$\overline{u}\in$$B_{1+\delta_{0}}\backslash \overline{B}_{1}$on$\partial_{\nu}\overline{u}=\overline{u}^{q}$$D_{1+\delta_{0}}$.

. Here we consider approximation problems.
in$\triangle u=0$in

$B_{1},\overline{u}<\varphi_{\infty}$in$-\triangle\overline{u}\geq 0$

and a positive$B_{1+\delta}$,

on$\partial_{\nu}u=u^{q}$$D_{1+\delta}$,$u=\overline{u}$on$S_{1+\delta}$,(6)

where is a function given in Lemma 2.1. Here we call $u(x)$ a weak solution of (6)if $u\in\{u\in H^{1}(B_{1+\delta});u-\overline{u}=0 on S_{1+\delta}\}$ satisfies$\overline{u}$$\int_{B_{1+\delta}}\nabla u\cdot\nabla\psi=\int_{D_{1+\delta}}u^{q}\psi$for any. To construct solutions of (6), we construct a monotoneand define $u_{i+1}(x)$ inductively by. We set$\psi\in C_{c}^{\infty}(\overline{B}_{1+\delta}\backslash \overline{S}_{1+\delta})$sequence$\{u_{i}(x)\}_{i\in N}$$\triangle u_{i+1}=0$$u_{0}(x)=\overline{u}(x)$in$B_{1+\delta}$

Thenfollowing lemma.

$u_{\delta}(x)= \lim_{iarrow\infty}u_{i}(x)$Lemma 2.2. Let
2.1. Then for$q$,$\partial_{\nu}u_{i+1}=u_{i}^{q}$on,$u_{i+1}=\overline{u}$on$S_{1+\delta}$.

gives a solution of (6). More precisely, we obtain thebe give in Lemmabe JL-supercritical or JL-critical andsymmetric
positive-axialweak solution
a
exits
there
$\delta_{0}>0$$\delta\in(0, \delta_{0})$$u_{\delta}\in H^{1}(B_{1+\delta})$$D_{1+\delta}$of (6) such that$x_{n}$$u_{\delta}(x)\leq\overline{u}(x)$Next we show a boundedness of$u_{\delta}(x)$.

near the origin.

Lemma 2.3. Let be JL-supercritical or JL-critical andof (6) constructed in Lemma 2.2. Then it holds that$q$$u_{\delta}(x)$be a weak solution$u_{\delta}\in L^{\infty}(B_{1+\delta})$.

-estimates.
To show Lemma 2.3, we use a technique similar to [10] with local
The following local-estimates are easily derived from the argument of Theorem
8.17 in [5] with Lemma 2.1 in [7].
$L^{\infty}$$L^{\infty}$

15
Lemma 2.4.

Let $u\in H^{1}(B_{1})$ be a weak solution of$-\Delta u+b(x)\cdot\nabla u+c(x)u=0$in$B_{1}$,

on$\partial_{\nu}u=K(x’)u$$D_{1}$

with $K\in L^{\gamma}(D_{1})$ for some $\gamma>n-1$ . Then there exists $c>0$ depending on
andsuch that$\Vert K\Vert_{L\gamma(D_{1})},$$\Vert b\Vert_{L}\infty(B_{1})$$\Vert c\Vert_{L^{\infty}(B_{1})}$$\Vert u\Vert_{L}\infty(B_{1/2})\leq c\Vert u\Vert_{L^{2}(B_{1})}$.

Proof of existence of solutions of (1). Let
be a function given in Lemma
2.2 and set. Then Lemma 2.3 implies that
. First we
claim that
. On the contrary, suppose that there exist $M_{0}>0$ anda sequencesuch thatand. Then there exits a subsequence, which is denoted by the same symbol such thatconverges to somefunctioninsatisfying. It is easily verified
that the function
is a positive bounded solution of$u_{\delta}(x)$$M_{\delta}<\infty$$M_{\delta}= \sup_{B_{1+\delta}}u_{\delta}(x)$$\lim_{\deltaarrow 0}M_{\delta}=\infty$$\{\delta_{i}\}_{i\in N}$$\delta_{i}arrow 0$$M_{\delta_{i}}\leq M_{0}$$\{\delta_{i}\}_{i\in N}$$u_{\delta_{i}}(x)$$u_{\infty}(x)$$H^{1}(B_{1})$$u_{\infty}(x)\leq\varphi_{\infty}(x)\leq\overline{u}(x)$$u_{\infty}(x)$$\triangle u_{\infty}=0$

Hence fromin$B_{1}$,$\partial_{\nu}u_{\infty}=u_{\infty}^{q}$$u_{\infty}(x)\leq\varphi_{\infty}(x)(u_{\infty}\not\equiv\varphi_{\infty})$on$D_{1}$,

on$u_{\infty}=\varphi_{\infty}$$S_{1}$.

, we obtain$\int_{B_{1}}|\nabla(u_{\infty}-\varphi_{\infty})|^{2}<qV_{B}^{q-1}\int_{D_{1}}r^{-1}(u_{\infty}-\varphi_{\infty})^{2}$.

By the trace Hardy inequality (3) and $qV_{B}^{q-1}\leq c_{H}$ (see Remark 1.1), it holds
that. However this contradicts to. Hence the claim
is proved. We set$u_{\infty}\in L^{\infty}(B_{1})$$u_{\infty}\equiv\varphi_{\infty}$$\lim_{\deltaarrow 0}M_{\delta}=\infty$$\overline{u}_{\delta}(x)=M_{\delta}^{-1}u_{\delta}(M_{\delta}^{-(q-1)}x)$Then$\overline{u}_{\delta}(x)$is a solution of$\triangle\overline{u}_{\delta}=0$

in

and

$B_{M_{\delta}^{q-1}}$.

$\partial_{\nu}\overline{u}_{\delta}=\overline{u}_{\delta}^{q}$on$D_{M^{q-1}}$. Since$M^{-1}\varphi_{\infty}(M^{-(q-1)}x)=\varphi_{\infty}(x)$for any $M>0$ , it is verified that
. Hence byin, there exists $R>0$ such thatfor small $\delta>0$ . Thus taking, we can obtain a positive -axial symmetric solution $u(x)$ of (1) satisfying
and $u(x)<\varphi_{\infty}(x)$ . Finally we put$\overline{u}_{\delta}(x)\leq\varphi_{\infty}(x)$$0$$\lim_{|x|arrow\infty}\varphi_{\infty}(x)=$$B_{M_{\delta}^{q-1}}$$\max_{B_{R}}\overline{u}_{\delta}(x)=1$$\deltaarrow 0$$x_{n}$$\max_{x\in \mathbb{R}_{+}^{n}}u(x)=1$$u_{\alpha}(x)=\alpha u(\alpha^{q-1}x)$Thenproof.

$u_{\alpha}(x)$is a solution of (1) and satisfies$(\alpha>0)$.

$u_{\alpha}(x)<\varphi_{\infty}(x)$, which completes the$\square$16
isStep l-(ii) Existence. (JL-subcritical case) Since the singular solution
not stable in a sense of Definition 1.2, arguments in Step l-(i) can not be applicable.
To show the existence of solutions of (1) satisfying (5), we need another technique.
Here we omit the detail.
$\varphi_{\infty}(x)$Step 2 Asymptotic expansion I.
In this step, we obtain the first asymptotic expansion:$\lim_{|x|arrow\infty}|x|^{1/(q-1)}u(x)=V(\theta)$

in $C([0, \pi/2])$ .

(7)

Let $u(x)$ be a solution of (1) given in Theorem 1.1 or Theorem 1.2 with $u(O)=1$ .
To investigate the asymptotic expansion of -axial symmetric solutions of (1), we
introduce$v(t, \theta)=r^{1/(q-1)}u(r, \theta)$ ,$r=e^{t}(t\in \mathbb{R})$ .
$x_{n}$

Then$v(t, \theta)$

is a solution of(8)$\{\begin{array}{ll}v_{u}+\alpha v_{t}-\beta v+\Delta_{S}v=0 in \mathbb{R}\cross(0, \pi/2),\partial_{\theta}v=v^{q} on \mathbb{R}\cross\{\pi/2\},\end{array}$where$\alpha=(n-2)-2m_{q}$ ,It is easily seen thatassociated with (8).

$\alpha,$$\beta>0$$\beta=m_{q}((n-2)-m_{q})$.

if $q>n/(n-2)$ . Define the energy function $E(t)$.

$E(t)= \frac{1}{2}\Vert\partial_{t}v(t)\Vert_{2}^{2}-\frac{\beta}{2}\Vert v(t)\Vert_{2}^{2}-\frac{1}{2}\Vert\partial_{\theta}v(t)\Vert_{2}^{2}+\frac{1}{q+1}v(t, \pi/2)^{q+1}$

Then it is easily verified that$\partial_{t}E(t)=-\alpha\Vert v_{t}(t)\Vert^{2}\leq 0$.

For JL-supercritical case and JL-critical case, since(b) in Theorem 1.1,
is uniformly bounded oncase, from (5),is also uniformly bounded on
regularity theory,
are uniformly bounded on
it follows that$\varphi_{\infty}(x)\leq|V|_{\infty}r^{-1/(q-1)},$$\mathbb{R}\cross(0, \pi/2)$$v(t, \theta)$$\mathbb{R}\cross(0, \pi/2)$$v(t, \theta)$$v_{t}(t, \theta),$ $v_{\theta}(t, \theta)$. For JL-subcritical. Hence by a elliptic$\mathbb{R}\cross(0, \pi/2)$$\alpha\int_{-\infty}^{\infty}\int_{0}^{\pi/2}|v_{t}|^{2}d\sigma dt=\lim_{tarrow-\infty}E(t)-\lim_{tarrow\infty}E(t)<\infty$

We set$v_{\infty}( \theta)=\lim_{tarrow\infty}v(t, \theta)$

Since $\lim_{tarrow-\infty}E(t)=0$ andof$\Delta_{S}v=\beta v$$\partial_{t}E(t)<0,$

in $(0, \pi/2)$ ,$v_{\infty}(\theta)$$hom$. Therefore.

.

is a nontrivial nonnegative solution$\partial_{\theta}v=v^{q}$on$\{\pi/2\}$.

17
Lemma 1.1, it follows that$\mathbb{R}om$$v_{\infty}(\theta)\equiv V(\theta)$. Thus (7) is derived.

Step 3 Asymptotic expansion II.
Finally we derive more precise asymptotic behavior than Step 2 and obtain (d) in
Theorem 1.1 and (i), (ii) in Theorem 1.2. To obtain a higher expansion of,
we study the asymptotic behavior of$v(t, \theta)$$w(t, \theta)=V(\theta)-v(t, \theta)$Then$w(t, \theta)$.

is a solution of$\{\begin{array}{ll}w_{u}+\alpha w_{t}-\beta w+\triangle_{S}w=0 in\mathbb{R}\cross(0, \pi/2),\partial_{\theta}w=qV_{B}^{q-1}w+f(w) on \mathbb{R}\cross\{\pi/2\},\end{array}$where $f(w)=(V_{B}^{q}-(V_{B}-w)^{q}-qV_{B}^{q-1}w)=O(w^{2})$ and $V_{B}=V(\pi/2)$ . From Step
2, it follows that
in $C([0, \pi/2])$ .
$\lim_{tarrow\infty}w(t, \theta)=0$

By using eigenfunctions of (4), we expand$w(t, \theta)$by$w(t, \theta)=\sum_{i=1}^{\infty}y_{i}(t)e_{i}(\theta)$

The coefficient$y_{i}(t)$.

satisfies$y_{i}’’+\alpha y_{i}’-(\beta+\lambda_{i})y_{i}=x_{i}$

,

(9)

where
$x_{i}(t)=f(w(t, \pi/2))e_{i}(\pi/2)$.

The characteristic equation of (9) is given by$\gamma_{i}^{2}+\alpha\gamma_{i}-(\beta+\lambda_{i})\gamma_{i}=0$

.

(10)

Then it is verified that(A) for the case $i\geq 2,$ (10) admits two real roots satisfying,(B) for the case $i=1$ (JL-supercritcal), (10) admits two real roots satisfying$\gamma_{i}^{-}<0,$$\gamma_{1}^{+}<0$$\gamma_{i}^{+}>0$$\gamma_{1}^{-}<$,(C) for the case $i=1$ (JL-critcal), (10) admits one real root satisfying(D) for the case $i=1$ (JL-subcritcal), (10) does not admit real roots.

$\gamma_{1}<0$,Then it is verified that for the case (A)$y_{i}(t)=y_{i}(0)e^{\gamma_{i}^{-}t}- \frac{e^{\gamma_{i}^{-}t}}{\sqrt{\alpha^{2}+4(\beta+\lambda_{i})}}\int_{0}^{t}(e^{-\gamma_{i}^{-}s}-e^{-\gamma_{i}^{+}s})x_{i}(s)ds$,$- \frac{e^{\gamma_{i}^{+}t}-e^{\gamma_{i}^{-}t}}{\sqrt{\alpha^{2}+4(\beta+\lambda_{i})}}\int^{\infty}e^{-\gamma_{i}^{+}s}x_{i}(s)ds$

18
for the case (B)$y_{1}(t)=y_{1}(0)e^{\gamma_{1}^{-}t}+ \frac{y_{1}’(0)-\gamma_{1}^{-}y_{1}(0)}{\sqrt{\alpha^{2}+4(\beta+\lambda_{1})}}(e^{\gamma_{1}^{+}t}-e^{\gamma_{1}^{-}t})$$+ \int_{0}^{t}\frac{1-e^{(2\gamma_{1}^{-}+\alpha)(t-s)}}{\sqrt{\alpha^{2}+4(\beta+\lambda_{1})}}e^{\gamma_{1}^{+}(t-s)}x_{1}(s)ds$,

for the case (C)$y_{1}(t)=y_{1}(0)e^{\gamma_{1}t}+(y_{1}’(0)- \gamma_{1}y_{1}(0))te^{\gamma_{1}t}+\int_{0}^{t}(t-s)e^{\gamma_{1}(t-s)}x_{1}(s)ds$

and for the case (D)$y_{1}(t)= \frac{1}{A}(\frac{\alpha}{2}y_{1}(0)+y_{1}’(0))(\sin At)e^{-(\alpha t)/2}+y_{1}(0)(\cos At)e^{-(\alpha t)/2}$$+ \frac{1}{A}\int_{0}^{t}$$((\sin At)(\cos As)-(\sin As)(\cos At))$ $e^{-\alpha(t-s)/2}x_{1}(s)ds$ ,where $A=\sqrt{|\alpha^{2}+4(\beta+\lambda_{1})|}/2$ . Since$\gamma_{1}^{-}<\gamma_{1}^{+}<0$$\gamma_{1}<0$,,$\gamma_{i+1}^{-}<\gamma_{i}^{-}(i\geq 1)$$\gamma_{i+1}^{-}<\gamma_{i}^{-}(i\geq 1)$$\gamma_{i+1}^{-}<\gamma_{i}^{-}<-\alpha/2(i\geq 2)$

,

,

,

$\gamma_{i}^{+}>0(i\geq 2)$

if JL-supercritical,$\gamma_{i}^{+}>0(i\geq 2)$

if JL-critical,$\gamma_{i}^{+}>0(i\geq 2)$

if JL-subcritical,

we obtain$w(t, \theta)\sim y_{1}(t)e_{1}(\theta)=\{\begin{array}{ll}ce^{\gamma_{1}^{+}}{}^{t}e_{1}(\theta)+o(e^{\gamma_{1}^{+}t}) if JL- supercritical,(ct+c’)e^{\gamma_{1}^{t}}e_{1}(\theta)+o(e^{\gamma_{1}t}) if JL- critical,c\sin(At+B)e^{-\alpha t/2}+o(e^{-\alpha t/2}) if JL- subcritical\end{array}$. For JL-supercritical and JL-critical case, by using the samefor some
technique as [1], we can assure $c\neq 0$ . Hence these asymptotic formula give (d) in
Theorem 1.1. However for JL-subcritical case, we do not know $c\neq 0$ . Hence (i) in
Theorem 1.2 holds if $c\neq 0$ , on the other hand, (ii) in Theorem 1.2 holds if $c=0$ .
$c,$$c^{l},$$B\in \mathbb{R}$

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