数理解析研究所講究録
第 1838 巻 2013 年 54-63

54

Existence and Nonexistence of solutions to the
Emden-Fowler equation on a geodesic ball in$S^{N}$

大阪府立大学工学研究科

小坂篤志

Atsushi Kosaka
Department of Mathematical Sciences,
Osaka Prefecture University

1

Introduction and preceding studies

In this paper, we consider the following problem(1.1)$\{\begin{array}{ll}\Delta_{S^{N}}u+u^{p}=0 in B_{\theta_{0}},u=0 on \partial B_{\theta_{0}},\end{array}$is the Laplace-Beltrami operator on
where $N\geq 3,$ $S^{N}=\{x\in R^{N+1}||x|=1\},$, and its center isHereis a geodesic ball inwith its geodesic radiuslocated at the North Pole $P_{n}=(x_{1}, x_{2}, \ldots, x_{N+1})=(0,0, \ldots, 1)$ .
The above problem (1.1) is said to be the Emden equation. Usually, the above equation. Namely similar problems to the following
is considered on the Euclidean space
problem are studied by many mathematicians:$S^{N}.$$\triangle_{S^{N}}$$S^{N}$$B_{\theta_{0}}$$\theta_{0}\in(0, \pi)$$R^{N}$(1.2)$\{\begin{array}{ll}\Delta u+u^{p}=0 in\Omega,u=0 on\partial\Omega,\end{array}$where is a bounded domain inwith the smooth boundary. Under $p>1$ $(if N=2)$
or $1<p<p_{*};=(N+2)/(N-2)$ $(if N\geq 3)$ , it is not difficult to prove the existenceof a solution to (1.2), e.g., by using the direct method for variational problems, we canfind a solution
to (1.2). Moreover, by the elliptic regularity theorem and the. On the other
Schauder regularity theorem (e.g. see [4]), we can prove that
hand, we assume $p\geq p_{*}(N\geq 3)$ . Under the assumption, if is a star-shaped domain
in $R^{N}(N\geq 3)$ , by the Pohozaev identity (e.g. see Chapter III- . in [9]). Furthermore
the structure of solutions to (1.2) is investigated in detail (e.g., the number of classicalsolutions or the existence of singular solutions) when $\Omega=B$ $:=\{x\in R^{N}||x|<1\}.$
Concerning this result, e.g., see [5].
On the other hand, for the Emden equation on the sphere , the existence of solutions
is not generally known. In this paper, we focus our attention on the existence of positive
solutions to the Emden equation on caps ofFirst we explain preceding studies of our problem (1.1). Bandle, Brillard and Flucher[1] proved the existence of a positive and radial solution from the viewpoint of the Sobolev$R^{N}$$\Omega$$u\in H_{0}^{1}(\Omega)$$u\in C^{2}(\Omega)$$\Omega$$I$$S^{N}$$S^{N}.$

55
imbedding. Here a solution to (1.1) depending only on the geodesic distance fromissaid to be a radial solution. They proved that if $N\geq 4$ , then there exists a positiveclassical solution to (1.1) for any. On the other hand, if $N=3$ , then there
exists some constantsuch that$P_{n}$$\theta_{0}\in(0, \pi)$$\theta_{c}\in(0, \pi)$(a) a positive classical solution to (1.1) exists if$\theta_{0}\in(\theta_{c}, \pi)$(b) there exist no positive classical solution to (1.1) if;$\theta_{0}\in(0, \theta_{c})$.

Remark 1.1 By [7] and [3], any positive classical solution to (1.1) is radially symmetric.
Next, to obtain
exactly, Bandle and Peletier [2] investigated (1.1) with $N=3$ more
precisely. They proved that, and moreover the caseis contained in (b).
We [8] also investigated the structure of solutions. For (1.1), we will treat not only aclassical solution but also a solution having singularity at the North Pole(the solution
is of classon
except for, and the following result.
$\theta_{c}$$\theta_{c}=\pi/2$$\theta_{0}=\pi/2$$P_{n}$$C^{2}$$B_{\theta_{0}}$$\{P_{n}\})$

Theorem A Assume $p=p_{*}$ . If $N\geq 4$ , then, for any, there exists a uniqueclassical solution and a continuum of singular solutions to (1.1). On the other hand, under$\theta_{0}\in(0, \pi)$$N=3,$(a) if $\theta_{0}\in(0, \pi/2]$ , then there does not exist either a classical solution or a singularsolution to (1. 1);(b) if $\theta_{0}\in(\pi/2, \pi)$ , then there exist a unique classical solution and a continuum of
singular solutions to (1.1).

Remark 1.2 Although we only proved Theorem
under $N=3$ in [8], we can prove
Theorem under $N\geq 4$ by using the same idea with some modifications. Furthermore,in [8], (1.1) was treated under more general boundary conditions, that is, we assumed,
instead of the Dirichlet boundaw condition, the Robin boundary condition
$A$

$A$

$u+ \kappa\frac{\partial u}{\partial\nu}=0 \partial B_{\theta_{0}},$

where

$\nu$

is the outer unit normal vector to on$\partial B_{\theta_{0}}$and$\kappa\geq 0.$

The above results are proved in the case of $p=p_{*}$ . On the other hand, our aim in thispaper is to explain results on (1.1) with $p>p_{*}.$

56

Main results

2

In this paper, we investigate (1.1) with a supercritical case $p>p_{*}$ , and we only treat positive radial solutions here. For this purpose, we introduce the polar coordinates. Namely,
let

(2.1)

$\{\begin{array}{l}x_{1}=\sin\theta\sin\varphi_{1}\sin\varphi_{2}\ldots\sin\varphi_{N-1}x_{2}=\sin\theta\sin\varphi_{1}\sin\varphi_{2}\ldots\cos\varphi_{N-1}x_{3}=\sin\theta\sin\varphi_{1}\sin\varphi_{2}\ldots\cos\varphi_{N-2}:.x_{N}=\sin\theta\cos\varphi_{1}x_{N+1}=\cos\theta\end{array}$

with, $\varphi_{i}\in[0, \pi](i=1,2, \ldots, N-2)$ and $\varphi\in[0,2\pi]$ . By (2.1), the Laplace-Beltrami
is expressed byoperator$\theta,$$\triangle_{S^{N}}$$\triangle_{S^{N}}u=\frac{1\partial}{\sin^{N-1}\partial\theta}(\sin^{N-1}\theta\frac{\partial u}{\partial\theta})+\sum_{i=1}^{N-1}\frac{1}{\gamma_{i}}\frac{\partial}{\partial\varphi_{i}}(\sin^{N-i-1}\varphi_{i}\frac{\partial u}{\partial\varphi_{i}})$with$\gamma_{i}=\sin^{2}\theta\sin^{N-i-1}\varphi_{i}\prod_{j=1}^{i-1}\sin^{2}\varphi_{j}.$Hence a positive and radial solution$u$to (1.1) satisfies$\{\begin{array}{ll}\frac{1}{\sin^{N-1}\theta}(u_{\theta}\sin^{N-1}\theta)_{\theta}+u^{p}=0 in (0,\theta_{0}) ,u(\theta)>0 in (0, \theta_{0}) ,u(\theta_{0})=0.

(2.2)\end{array}$Here we define two kinds of solutions to (2.2); a solution to (1.1) is said to be a regular; aconverging to some positive constant asandsolution to (2.2) if
tends
and
solution to (1.1) is said to be a singular solution to (2.2) if. Our main theorems are as follows.
toas
To investigate the structure of solutions to (2.2), we apply results on [6], [10] and [11].
. Namely we defineFirst we transform (2.2) into the Emden-Fowler equation on$u$$C^{2}((0, \theta_{0}))$$\thetaarrow 0$$u(\theta)$$C^{2}((0, \theta_{0}))$$u$$+\infty$$u(\theta)$$\thetaarrow 0$$R^{N}$$\mathcal{T}=g(\theta) :=\int_{\theta}^{\theta_{0}}\frac{d\psi}{\sin^{N-1}\psi}$

By the new variable$\tau$

and a solution$u$.

to (2.2), we define$w:= \frac{u(\theta)}{\tau}.$

(2.3)

57
Then the new function$w$

satisfies the following problem(2.4)$\{\begin{array}{ll}\frac{1}{\tau^{2}}(\tau^{2}w_{\tau})_{\tau}+K(\tau)w_{+}^{p}(\tau)=0 for \tau\in(0, +\infty) ,w(0)=\alpha, w_{\tau}(0)=0, \end{array}$where $w_{+}= \max\{w, 0\}$ and$K(\tau) ;= \tau^{p-1}\sin^{2N-2}\theta$

,

(2.5)

$\alpha ;= -u_{\theta}(\theta_{0})\sin^{2}\theta_{0}.$Thus it suffices to investigate the behavior of solutions to (2.4) instead of (2.2).
Next we explain methods to investigate (2.4). First let(2.6)$\{\begin{array}{ll}\frac{1}{\tau^{2}}(\tau^{2}w_{\tau})_{\tau}+L(\tau)w_{+}^{p}(\tau)=0 for \tau\in(0, +\infty) ,w(0)=\beta, \end{array}$where $w_{+}:= \max\{w, 0\}$ and is a positive constant, and a solution to (2.6) is sometimes
denoted by. Here the function satisfies$\beta$$w(\tau;\beta)$$L$$\{\begin{array}{ll}L(\tau)\in C^{1}((0, +\infty)) , L(\tau)\geq 0 and L(\tau)\not\equiv O on (0, +\infty) ,\tau L(\tau)\in L^{1}(0,1) , \tau^{1-p}L(\tau)\in L^{1}(1, +\infty) .

(L)\end{array}$Hereafter the solution to (2.6) with an initial data is denoted by. From (2.3)
and (2.5), we can easily confirm that $K(\tau)$ satisfies (L). Therefore any result on (2.6) isvalid for (2.4).
Second we define three types of solutions to (2.6) as follows:$w(\tau;\beta)$$\beta$Definition 2.1 (i)solution$w>0$ on $[0, +\infty)$ and
$A$

$w$

$\tau w(\tau)$(ii)

solution
and

$A$

$w$

$A$

solution$w$$\tauarrow+\infty.$to (2.4) is said to be a slowly decaying solution if $w>0$ on $[0, +\infty)$as$\tau w(\tau)arrow+\infty$(iii)

to (2.4) is said to be a rapidly decaying solution if
converges to some positive constant as$\tauarrow+\infty.$to (2.4) is said to be a crossing solution if$w$

has a zero in $(0, +\infty)$ .

Here we remark that a rapidly decaying solution and a slowly decaying solution to (2.6)
are corresponding to a regular solution and a singular solution to (2.2).
Next we refer to some lemmas concerning the structure of solutions to (2.6). We
introduce the following identity$P( \tau;w):=\frac{1}{2}\tau^{2}w_{\tau}\{\tau w_{\tau}+w\}+\frac{\tau^{3}}{p+1}L(\tau)w_{+}^{p+1}$

(2.7)

58
The behavior of (2.7) for sufficiently large depends on a kind of
the next lemma is known.
$\tau$

Lemma 2.1 (Lemma 2.6 in [11]) (a) If$P(\tau;w)>0$ for $\tau\in[z(\beta), +\infty)$ , where(b) Ifthat$w(\tau;\beta)$$w(\tau;\beta)$$z(\beta)$$w$

defined above. Namely

is a crossing solution to (2.6), thenis a zero of$w(\tau;\beta)$.

is a slowly decaying solution to (2.6), then there exists a sequence
as $jarrow+\infty$ and $P(\hat{\tau}_{j};w)<0$ for any $j.$$\{\hat{\tau}_{j}\}$such$\hat{\tau}_{j}arrow+\infty$

is a rapidly decaying solution to (2.6), then there exists a sequence(c) Ifas $jarrow+\infty.$
andsuch that$w(\tau;\beta)$$\tilde{\tau}_{j}arrow+\infty$$\{\tilde{\tau}_{j}\}$$P(\tilde{\tau}_{j};w)arrow 0$

By using properties of $P(\tau;w)$ , Yanagida and Yotsutani [10], [11] proved a structure
theorem for (2.6). To explain the theorem, we require some preliminaries. First we
introduce the function$G( \tau):=\frac{1}{p+1}\tau^{3}L(\tau)-\frac{1}{2}\int_{0}^{\tau}s^{2}L(s)ds.$

The function$G(\tau)$

is related to $P(\tau;w)$ by the following lemma.

Lemma 2.2 (Lemma 3.2 in [5]) Any solution$w$

to (2.6) satisfies the identity$\frac{d}{d\tau}P(\tau;w)=G_{\tau}(\tau)w_{+}^{p+1}(\tau)$

and its integral form
$P( \tau;w)=G(\tau)w_{+}^{p+1}(\tau)-(p+1)\int_{0}^{\tau}G(s)w_{+}^{p}w_{s}(s)d_{\mathcal{S}}.$

Next we define the following function$H( \tau):=\frac{1}{p+1}\tau^{2-p}L(\tau)-\frac{1}{2}l^{+\infty}s^{1-p}L(s)ds.$

The function$H(\tau)$is corresponding to$G(\tau)$by$G_{\tau}( \tau)=\frac{\tau^{(p+1)/2}}{p+1}(\tau^{-\xi}L)_{\tau}=\tau^{p+1}H_{\tau}(\tau)$with$\xi=\frac{p-5}{2}.$By using$G(\tau)$and$H(\tau)$, we define$\tau_{G}:=\inf\{\tau\in[0, +\infty)|G(\tau)<0\},$$\tau_{H}:=\sup\{\tau\in[0, +\infty)|H(\tau)<0\}.$Here $\tau_{G}=+\infty$ if $G(\tau)\geq 0$ on $(0, +\infty)$ and $\tau_{H}=0$ if $H(\tau)\geq 0$ on $(0, +\infty)$ . From theabove preliminaries, following proposition holds.

59
Lemma 2.3 (Theorem 1 in [10]) Assume (L) andlowing three statements hold.

$G\not\equiv O$

on $(0, +\infty)$ . Then the fol-(i) If $\tau_{G}=+\infty$ , then the structure of solutions to (2.6) is of type $C:w(\tau;\beta)$ is a crossing

solution for any $\beta>0.$(ii) If $\tau_{H}=0$ , then the structure of solutions to (2.6) is of type $S:w(\tau;\beta)$ is a slowly
decaying solution for any $\beta>0.$(iii) If $0<\tau_{H}\leq\tau_{G}<+\infty$ , then the structure of solutions to (2.6) isof type $M$:
there exists a constant $\beta_{*}>0$ such thatis a slowly decaying solution for
is a rapidly decaying solution, andis a crossing solution$w(\tau;\beta)$$\beta\in(0, \beta_{*}),$

for$w(\tau;\beta_{*})$$\beta\in(\beta_{*}, +\infty)$$w(\tau;\beta)$.

By Lemma 2.3, we obtain the next proposition.

Proposition 2.1 Under $p=p_{*}$ , letbe a solution to (2.4). Assume$(N=3)$ or $\theta_{0}\in(0, \pi)(N\geq 4)$ . Then there exists somesuch that$w(\tau;\alpha)$$\theta_{0}\in(\pi/2, \pi)$$\alpha_{*}>0$(i) if$\alpha<\alpha_{*}$, then$w$

is a slowly decaying solution;(ii) if$\alpha=\alpha_{*}$, then$w$

is a rapidly decaying solution;(iii) if$\alpha>\alpha_{*}$, then$w$

is a crossing solution.

On the other hand, assume $N=3$ andsolution.

$\theta_{0}\in(0, \pi]$. Then, for any $\alpha>0,$

Proposition 2.1 is equivalent to Theorem A.
We can apply Lemma 2.3 (i) for (2.4) with $p>p_{*}$ whenby direct calculation, we obtain

$p$

$w$

is a crossing

is sufficiently large. In fact,(2.8)$G_{\tau}( \tau)=\frac{1}{p+1}r(\theta,p)\tau^{p+1}\sin^{2N-2}\theta$with$r( \theta,p)=\frac{p+3}{2}-(2N-2)\tau\sin^{N-2}\theta\cos\theta$

.

(2.9)

From (2.3), we see thatis bounded for $\tau\in(0, +\infty)$ . Hence if is sufficiently
large, then $r(\theta,p)>0$ . Therefore, by Lemmas 2.2 and 2.3 (i), the following theorem
holds.
$\tau\sin^{N-2}\theta$$p$

Theorem 2.1 Assume $\theta_{0}\in(\pi/2, \pi)(N=3)$ or $\theta_{0}\in(0, \pi)(N\geq 4)$ . Then there exists
somesuch that, for any $p>p_{c}$ , there does not exist either a regular or a
singular solution to (2.2).
$p_{c}(\theta_{0})>p_{*}$

60
Especially, in the case of $N=3$ , we can obtain$p_{c}( \theta_{0})=4(1+\frac{1}{\sin\theta_{0}})-3.$

On the other hand, it is not easy to investigate the existence of solutions to (2.2) if. To explain the reason, we introduce$p>p_{*}$ is sufficiently near$p_{*}$$C(p)$$:=$$\mathcal{R}(p)$$:=$$S(p)$

$:=$

{
{
{

$\beta>0|w(\tau;\beta)$

is a crossing solution to (2.6)},

is a rapidly decaying solution to (2.6)},$\beta>0|w(\tau;\beta)$ is a slowly decaying solution to (2.6)}.
$\beta>0|w(\tau;\beta)$

For the above sets, the next lemma holds.
Lemma 2.4 (Lemma 2.7 in [6]) The set $C(p)$ is open.
Moreover we introduce the following condition$\{\begin{array}{l}there exists \eta_{1}\in[0, +\infty) such thatG(\tau)\geq 0 for (0, \eta_{1}) and G_{\mathcal{T}}(\tau)\leq 0 for (\eta_{1}, +\infty) .\end{array}$(G)

Under (G), the following lemma holds.
Lemma 2.5 (Lemma 2.6 in [6]) Under (G), $S(p)$ is an open set.

Remark 2.1 Assume $C(p)$ and.
lution between $C(p)$ and$\mathcal{S}(p)$are open. Then there eststs a rapidly decaying so-$\mathcal{S}(p)$If $p=p_{*}$ , then defined by (2.4) satisfies (G), and Lemma 2.5 holds. On the other hand,
if $p>p_{*}$ , then (G) is not satisfied. Hence it seems difficult to investigate the openness of$S(p)$ for $p>p_{*}.$
Moreover the next lenuna implies that the structure of solutions to (2.4) with $p>P*$is qualitatively different from the structure of solutions to (2.4) with $p=p_{*}.$$G$

Lemma 2.6 (Theorem 3 in [11]) If $\lim\inf_{\tauarrow+\infty}G(\tau)>0$ , then there exists $\beta_{c}>0$.
is a crossing solution to (2.6) for any
such that$\beta\in(0, \beta_{c})$$w(\tau;\beta)$is
In fact if $p>p_{*}$ , then $\lim_{\tauarrow+\infty}G(\tau)>0$ . Hence, for a sufficiently small $\beta>0,$
a crossing solution, that is, the structure of solutions to (2.4) with $p>p_{*}$ is different fromthe structure of solutions to (2.4) with $p=p_{*}$ (see Proposition 2.1). Furthermore, from
Lemma 2.6, we expect that there exist at least two rapidly decaying solution to (2.4) if. Hereafter we state this result.
$p>p_{*}$ is sufficiently near
( is sufficiently small),
To investigate the structure of solutions to (2.4) with
we considered a transformed problem from (2.4). Namely let$w(\tau;\beta)$$p_{*}$$p=p_{*}+\epsilon$$t:= \frac{1}{\tau}, v:=\frac{w}{t}$.

$\epsilon$(2.10)The transformation (2.10) is said to be the Kelvin transformation. By (2.10), we obtainthe next lemma.

61Lemma 2.7 Assume$w$

is a rapidly decaying solution to (2.4). Thensatisfies$v$

defined in (2.10)$\{\begin{array}{ll}\frac{1}{t^{2}}(t^{2}v_{t})_{t}+\tilde{K}(t)v_{+}^{p}=0 for t\in(0, +\infty) ,v(0)=\eta>0, v_{t}(0)=0, \end{array}$(2.11)

where $v+= \max\{v, 0\}$ and$\tilde{K}(t) :=t^{-4}\sin^{2N-2}\theta$

Similarly, for a rapidly decaying solution
solution to (2.4).

$v$.

(2.12)

to (2.11), $w(\tau)=v(t)/\tau$ is a rapidly decaying

Lemma 2.7 imphes that the number of rapidly decaying solutions to (2.4) is the same as
the number of rapidly decaying solutions to (2.11).
Next, for (2.11), we define$\tilde{P}(t;v) :=\frac{1}{2}t^{2}v_{t}\{tv_{t}+v\}+\frac{t^{3}}{p+1}\tilde{K}(t)v_{+}^{p+1},$$\tilde{G}(t) :=\frac{1}{p+1}t^{3}\tilde{K}(t)-\frac{1}{2}\int_{0}^{t}s^{2}\tilde{K}(s)ds$(2.13)$\tilde{H}(t) :=\frac{1}{p+1}t^{2-p}\tilde{K}(t)-\frac{1}{2}l^{+\infty}\mathcal{S}^{1-p}\tilde{K}(s)d_{\mathcal{S}},$and$\tilde{C}(p)$$:=$$\tilde{\mathcal{R}}(p)$$:=$$\tilde{\mathcal{S}}(p)$$:=${ $\beta>0|w(\tau;\beta)$ is a crossing solution to (2.11)},{ $\beta>0|w(\tau;\beta)$ is a rapidly decaying solution to (2.11)},{ $\beta>0|w(\tau;\beta)$ is a slowly decaying solution to (2.11)}.

Then, from (2.5), (2.10) and (2.12), it follows that
$P(\tau;w)=\tilde{P}(t;v)$,$G(\tau)=\tilde{H}(t)$$H(\tau)=\tilde{G}(t)$,(2.14).

From the above properties, we can obtain the following corollary.
Corollary 2.1 Under $p=p_{*}$ , the structure of solutions to (2.11) is of type $M:$ , that is,
the structure of solutions to (2.11) is the same as the structure of solutions to (2.4).

Therefore, in the case of $p=p_{*},$ $(2.11)$ has a unique rapidly decaying solution. Moreover,for (2.11), it is not so difficult to prove the openness ofunder $p>p_{*}$ . Namely thefollowing two lemmas hold.
$\tilde{\mathcal{S}}(p)$62Lemma 2.8 For $p>p_{*},\tilde{S}(p)$ is an open set.
In fact, from (2.14), we see that
Lemma 2.2, we can prove thatfor $p=p_{*}.$
if
Now we investigate structures of$\tilde{G}_{t}(t)<0$

for sufficiently large . By this properties and( is sufficiently small)for$t$$\lim_{tarrow+\infty}\tilde{P}(t;v)<0$$p=p_{*}+\epsilon$$\epsilon$$\lim_{tarrow+\infty}\tilde{P}(t;v)<0$$P\uparrow(\eta)${$\tilde{C}(p)$$:= \sup$ $p’>p_{*}|v(t;\eta)$. Let$v(t;\eta)$

be a solution to (2.11), and we define

is a crossing solution for any $p\in(p_{*},p’)$ }.

.The
Here if a solution $v(t;\eta)$ is not a crossing solution for any $p>p*$ , thenis bounded. In fact by the same argument as in Theorem 2.1, the following
hmctionproposition is proved.
$p_{\dagger}(\eta)=p_{*}$$p_{\dagger}(\eta)$

Proposition 2.2 Assume $\theta_{0}\in(\pi/2, \pi)(N=3)$ or $\theta_{0}\in(0, \pi)(N\geq 4)$ . Then there
such that, for any $p>p_{c}’$ and $\eta>0$ , a solution $v(t;\eta)$ to (2.11) isexists somea slowly decaying solution.
$p_{c}’(\theta_{0})>p_{*}$

is bounded.
, we can prove some properties. First, we refer to the following lemma.

Proposition 2.2 implies thatFor$p_{\dagger}(\eta)$$p_{\dagger}(\eta)$

Lemma 2.9 (Theorem 2 in [11]) If $\lim\sup_{\tauarrow+\infty}G(\tau)<0$ , then there exists $\beta_{s}>0$.
such thatis a slowly decaying solution to (2.6) for any$\beta\in(0, \beta_{\delta})$$w(\tau;\beta)$

If $p>p_{*}$ , then we can apply Lemma 2.9 for (2.11). Thus, for sufficiently small $\eta>0$ , it.
for $\eta>0$ . In fact let
holds thatOn the other hand,
Then, by the continuity of solutions to (2.11) conceming parameters, it holds that
. Therefore, from Remark 2.1, we see( is sufficiently small), that is,that there exists at least one rapidly decaying solution to (2.11).
Next we state our main result, that is, there exists at least two rapidly decaying
solutions to (2.11) when $p>p_{*}$ is sufficiently near . The following lemma is essential
in the proof of our main result.
$\eta_{0}\in\tilde{C}(p_{*})$$p_{\dagger}(\eta)\not\equiv p_{*}$$p_{\dagger}(\eta)=P*\cdot$$\eta_{0}\in$$\tilde{C}(p_{*}+\epsilon)$$p_{\dagger}(\eta_{0})>p_{*}$$\epsilon$$p_{*}$

Lemma 2.10 As$\etaarrow+\infty,$$p_{\dagger}(\eta)arrow p_{*}.$

is sufficiently
By Lemma 2.10, sufficiently large $\eta>0$ is not contained insmall). Hence, for sufficiently large $\eta>0,$ $v(t;\eta)$ is a rapidly decaying solution or a
is a slowly decaying solution for someslowly decaying solution. In addition if$\eta_{0}>\eta_{M};=\max\{\eta>0|\eta\in\tilde{C}(p)\}$ , then, from Lemma 2.8, there exists
is a rapidly decaying solution. Thus we see that there exists at least twosuch that
rapidly decaying solutions to (2.11). Therefore, by Lemma 2.7, the following theorem isobtained.
$\tilde{C}(p_{*}+\epsilon)(\epsilon$$v(t;\eta_{0})$$\eta_{1}\in(\eta_{M}, \eta_{0})$$v(t;\eta_{1})$Theorem 2.2 Assume $\theta_{0}\in(\pi/2, \pi)(N=3)$ or $\theta_{0}\in(0, \pi)(N\geq 4)$ . Then there exists, there eststs at least two regular solutions tosuch that, for any
some(2.2) with $p=p_{*}+\epsilon.$$\epsilon_{0}(\theta_{0})>0$$\epsilon\in(0, \epsilon_{0})$63References[1] C. Bandle, A. Brillard and M. Flucher, Green’s function, harmonic transplantation,and best Sobolev constant in spaces of constant curvature, Trans. Amer. Math. Soc.
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