数理解析研究所講究録
第 1896 巻 2014 年 126-136

126

The least energy positive solution for the nonlinear
elliptic three-systems with attractive andrepulsive interaction terms
Yohei Sato
Department of Mathematics, Saitama University

Osaka City University Advanced Mathematical Institute
Zhi-Qiang Wang

Department of Mathematics and Statistics, Utah State University
Chern Institute Mathematics, Nankai University

0. Introduction
In this report, for the bounded domain $\Omega\subset R^{n}(n\leq 3)$ with smooth boundary, we
consider the following nonlinear elliptic 3-system:$- \Delta u_{i}+\lambda_{i}u_{i}=\mu_{i}u_{i}^{3}+\sum_{j=1}^{3}\beta_{i,j}u_{i}u_{j}^{2}$in$\Omega,$$(i=1,2,3)$ ,$(*)$$u_{i}\in H_{0}^{1}(\Omega) (i=1,2,3)$.

$\mu_{i}>0(i=1,2,3)$ and $\beta_{i,j}=\beta_{j,i}(1\leq i<j\leq 3)$ . We consider about a
where. Here weleast energy positive solution of $(*)$ for the case $\beta_{1,2}>0$ andcall a solution $\vec{u}=(u_{1}, u_{2}, u_{3})\in H^{1}(R^{n})^{3}$ is a least energy positive solution of $(*)$ if andonly if aehieves $\inf\{I(\vec{u})|I’(\vec{u})=0, u_{i}>0(i=1,2,3)\}$ where $I(u)$ is a functionalcorresponding to $(*)$ . There are many papers for the existence of non-trivial solutions of-system $(k\geq 3)$ . (cf. [LWel], [LWl], [LW2], [SW2], [S], [TT], [TTVW], [TV] ) To
author’s knowledge, almost existence results for $(*)$ were given under the conditions thatare negative or not large positive.
interaction terms
In this report, we introduce results of our two papers $[SW3]-[SW4]$ but we omit
those proofs. For the proofs, see $[SW3]-[SW4]$ . Roughly speaking our results, whenand $\beta_{1,2}>0$ is sufficiently large, we observe the existence of least energy
positive solution of $(*)$ in Section 1 and the multiple existence of positive solution of $(*)$
in Section 2.
$\lambda_{i},$$\beta_{1,3},$$\beta_{2,3}\leq 0$$\vec{u}$$k$$\cdots$$\beta_{i,j}$$\beta_{1,3},$$\beta_{2,3}\leq 0$

127
1. The Existence of least energy solutions
In this section, whenand $\beta_{1,2}>0$ is sufficiently large, we observe theexistence of least energy positive solution of $(*)$ . Moreover, we observe that, even if
is ball, that solution is not radial symmetric. This is a different property from the single
homogeneous equations. It is well-known in [GNN] that if is ball and $f(u)$ is of classthen any positive solutions in $C^{2}of-\triangle u=f(u)$ in $\Omega u=0$ on
are radial symmetric.
Also, in [LWel], for a -system on $\Omega=R^{n}$ , Lin and Wei showed that, if all interaction
terms are positive, then, the least energy positive solutions must be radially symmetric by
the Schwartz symmetrization.
$\beta_{1,3},$$\beta_{2,3}\leq 0$$\Omega$$\Omega$$C^{1},$$\partial\Omega$$k$

Since we treat
as a parameter which plays an important role, for simplicities, we
often write. We also use the following notations:$\beta_{1,2}$$\beta\equiv\beta_{1,2}$$||u||_{L^{p}(\Omega)}^{p}= \int_{\Omega}|u|^{p}$for$(1\leq p\leq\infty)$$u\in L^{p}(\Omega)$for$|||u|||_{\lambda,\Omega}^{2}=||\nabla u||_{L^{2}(\Omega)}^{2}+\lambda||u||_{L^{2}(\Omega)}^{2}$,$u\in H_{0}^{1}(\Omega)$.

The first theorem is about the existence of a least energy positive solution of $(*)$ .

Theorem 1.1. We suppose thatand
. Then, there exists a$(*)$ has a least energy positive solution
such that, for any
Moreover, there exist a sequence
and $U_{i}\in H_{0}^{1}(\Omega)(i=1,2,3)$ such that$0$$\beta\equiv\beta_{1,2}>0$$\beta_{1,3},$$\beta_{2,3}\leq 0$$\beta>\beta_{*},$$\beta_{*}>$$\vec{u}_{\beta}=(u_{1,\beta}, u_{2,\beta}, u_{3,\beta})$$\beta_{m}arrow\infty$$(\sqrt{\beta_{7n}}u_{1,\beta_{m}}, \sqrt{\beta_{m}}u_{2,\beta_{m}}, u_{3,\beta_{m}})arrow(U_{1}, U_{2}, U_{3})$Here$U_{3}$$H_{0}^{1}(\Omega)^{3}.$is a positive least energy solution of$-\triangle u_{3}+\lambda_{3}u_{3}=\mu_{3}u_{3}^{3}$$u_{3}\in H_{0}^{1}(\Omega)$andstrongly in$(U_{1}, U_{2})$inf$\Omega,$

(1.1)

,

is a positive least energy solution of$-\triangle u_{1}+(\lambda_{1}-\beta_{1,3}U_{3}^{2})u_{1}=u_{1}u_{2}^{2}$$in$$\Omega,$$-\triangle u_{2}+(\lambda_{2}-\beta_{2,3}U_{3}^{2})u_{2}=u_{1}^{2}u_{2}$$in$$\Omega$$u_{1}, u_{2}\in H_{0}^{1}(\Omega)$

In particularin$(U_{1}, U_{2}, U_{3})$

inf

,

(1.2)

.

is a minimizer of the following minimizing problem:$e=$ $u_{3}\in K_{3}(u_{1},u_{2})\in N_{u_{3}}(|||u_{1}|||_{\lambda_{1},\Omega}^{2}+|||u_{2}|||_{\lambda_{2},\Omega}^{2}-\beta_{1,3}||u_{1}u_{3}||_{L^{2}(\Omega)}^{2}-\beta_{2,3}||u_{2}u_{3}||_{L^{2}(\Omega)}^{2})$

,

(1.3)

.

128
where
$K_{3}=${$u\in H_{0}^{1}(\Omega)|u$

is a least energy solution of (1.1)},$N_{u_{3}}=\{(u_{1}, u_{2})\in H_{0}^{1}(\Omega)^{2}|$Remark 1.2. The infimum$e$$|||u_{1}|||_{\lambda_{1},\Omega}^{2}+|||u_{2}|||_{\lambda_{2},\Omega}^{2}-\beta_{1,3}||u_{1}u_{3}||_{L^{2}(\Omega)}^{2}-\beta_{2,3}||u_{2}u_{3}||_{L^{2}(\Omega)}^{2}=2||u_{1}u_{2}||_{L^{2}(\Omega)}^{2}\neq 0\}\cdot$

is also written as$e= \inf_{u_{3}\in K_{3}}\overline{e}(u_{3})$where$\overline{e}(u_{3})=\inf_{(u_{1},u_{2})\in N_{u_{3}}}(|||u_{1}|||_{\lambda_{1},\Omega}^{2}+|||u_{2}|||_{\lambda_{2},\Omega}^{2}-\beta_{1,3}||u_{1}u_{3}||_{L^{2}(\Omega)}^{2}-\beta_{2,3}||u_{2}u_{3}||_{L^{2}(\Omega)}^{2})$For any given
is achieved by a minimizer
non-trivial least energy solution of$u_{3}\in H_{0}^{1}(\Omega),$$\overline{e}(u_{3})$$(u_{1}, u_{2})\in N_{u_{3}}$$-\triangle u_{1}+(\lambda_{1}-\beta_{1,3}u_{3}^{2})u_{1}=u_{1}u_{2}^{2}$in$\Omega,$$-\Delta u_{2}+(\lambda_{2}-\beta_{2,3}u_{3}^{2})u_{2}=u_{1}^{2}u_{2}$in$\Omega$$u_{1}, u_{2}\in H_{0}^{1}(\Omega)$,.

which is a

(1.3)

.

In fact, in [SWl], we showed the existence of a minimizer for
when$(i=1,2)$ are positive constants. Whenare non-negative functions, we can show theexistence of a minimizer by the same way.
$V_{i}(x)\equiv\lambda_{i}-\beta_{i,3}u_{3}^{2}$$\overline{e}(u_{3})$$V_{i}(x)$Remark 1.3. The solution
minimizing problem:$\vec{u}_{\beta}$of Theorem 1.1 was given as a minimizer of the following$c_{\beta}= \inf_{\vec{u}\in M_{\beta}}(|||u_{1}|||_{\lambda_{1},\Omega}^{2}+|||u_{2}|||_{\lambda_{2},\Omega}^{2}+|||u_{3}|||_{\lambda_{3},\Omega}^{2})$,$M_{\beta}=\{\vec{u}\in H_{0}^{1}(\Omega)^{3}| f_{1}(\vec{u})+f_{2}(\vec{u})=0f_{3}(\vec{u})=0, (u_{1},u_{2}).\neq(0,0)u_{3}\neq 0’\},$

where[SW3].

$f_{i}( \vec{u})=|||u_{i}|||_{\lambda_{i},\Omega}^{2}-\mu_{i}||u_{i}||_{L^{4}(\Omega)}^{4}-\sum_{J\neq i}\beta_{i,j}||u_{i}u_{j}||_{L^{2}(\Omega)}^{2}$. For details, see our paper

Next, we will observe the non-radial symmetry for a least energy positive solution ofeven if is ball. When is ball, it is well-known that least energy solution of (1.1) isa unique positive radial symmetric solution satisfying $U_{3}’(r)<0,$ $r=|x|$ . (The uniqueness
was proved in [K], The radial symmetry was proved in [GNN]. Let
be a least energy solution of $(*)$ . From Theorem 1.1,
converges to the unique positive
radial symmetric solution
and a subsequence of
converges to a least
energy solutionof (1.2) whose potential functions $\lambda_{i}-\beta_{i,3}U_{3}(x)(i=1,2)$ haveminimum on the boundary. Then, we can show thathas a concentrating point. That is,near
is not radial symmetric.
$(*)$$\Omega$$\Omega$$)$$\vec{u}_{\beta}=(u_{1,\beta}, u_{2,\beta}, u_{3,\beta})$$u_{3,\beta}$$U_{3}$$(\sqrt{\beta}u_{1,\beta}, \sqrt{\beta}u_{2,\beta})$$(U_{1}, U_{2})$$\partial\Omega$$\partial\Omega$$(U_{1}, U_{2})$$(U_{1}, U_{2})$129
In fact, in this report, we will observe such concentrating phenomenon for the following
system including (1.2):$-\epsilon^{2}\triangle u_{1}+V_{i}(x)u_{1}=u_{1}u_{2}^{2}$in$\Omega,$$-\epsilon^{2}\triangle u_{2}+V_{2}(x)u_{2}=u_{1}^{2}u_{2}$in$\Omega,$$(1.4)_{\epsilon}$.

$u_{1}, u_{2}\in H_{0}^{1}(\Omega)$whereis a parameter, $V_{i}(x)\in C(\overline{\Omega}, R)(i=1,2)$ are positive functions. Let
be a family of positive least energy solutions of. To state the concentrating point of, we need the following:$\epsilon>0$$(\vec{u}_{\epsilon})_{\epsilon>0}$$(1.4)_{\epsilon}$$(\vec{u}_{\epsilon})_{\epsilon>0}$$b(\lambda_{1}, \lambda_{2})$$b( \lambda_{1}, \lambda_{2})=\inf_{(u_{1},u_{2})\in N_{\lambda_{1},\lambda_{2}}}(|||u_{1}|||_{\lambda_{1},R^{n}}^{2}+|||u_{2}|||_{\lambda_{2},R^{n}}^{2})$

,

(1.5)

$N_{\lambda_{1},\lambda_{2}}=\{(u_{1}, u_{2})\in H^{1}(R^{n})^{2}||||u_{1}|||_{\lambda_{1},R^{n}}^{2}+|||u_{2}|||_{\lambda_{2},R^{n}}^{2}=2||u_{1}u_{2}||_{L^{2}(R^{n})}^{2}\neq 0\}.$We easily see thatenergy solution of$b(\lambda_{1}, \lambda_{2})$is achieved for some $\vec{u}=(u_{1}, u_{2})\in H^{1}(R^{n})^{2}$ and$-\triangle u_{1}+\lambda_{1}u_{1}=u_{1}u_{2}^{2}$in$R^{n},$$-\triangle u_{2}+\lambda_{2}u_{2}=u_{1}^{2}u_{2}$in$R^{n},$$u_{1}, u_{2}\in H^{1}(R^{n})$Also, for$b(\lambda_{1}, \lambda_{2})$$\vec{u}$is a least$(1.6)_{(\lambda_{1},\lambda_{2})}$.

, we have the following:

Lemma 1.4.
(i)$b(\lambda_{1}, \lambda_{2})$: $(0, \infty)^{2}arrow R$ is a continuous function.

(ii)$b(\lambda_{1}, \lambda_{2})$is increasing with respect to $\lambda_{i}(i=1,2)$ .

(iii)$b(\eta\lambda_{1}, \eta\lambda_{2})=\eta^{2}b(\lambda_{1}, \lambda_{2})$We regard
following theorem.

$u\in H_{0}^{1}(\Omega)$

for all$\eta,$$\lambda_{1},$$\lambda_{2}>0.$

as $u\in H^{1}(R^{n})$ by setting $u=0$ on

Theorem 1.5. There exist sequences$x_{m}arrow x_{0}$ inwhich is a positive least energy solution of$\epsilon_{m}arrow 0,$$H^{1}(R^{n})^{2}$$R^{n}\backslash \Omega$$\overline{\Omega}$. Now we have theand $\vec{u}_{0}=(u_{1,0}, u_{2,0})\in$
such that$(1.6)_{V_{1}(x_{0}),V_{2}(x_{0})}$$u_{i,\epsilon_{m}}(\epsilon_{m}x-x_{m})arrow u_{i,0}(x)$strongly in $H^{1}(R^{n})$$b(V_{1}(x_{m}), V_{2}(x_{m}))arrow b(V_{1}(x_{0}), V_{2}(x_{0}))=\underline{b}.$Here $\underline{b}=\min_{x\in S}b(V_{1}(x), V_{2}(x))$ .

$(i=1,2)$ ,130Remark 1.6.
of (1.1), setting $V_{i}(x)=\lambda_{i}-$(i) For an unique positive radial symmetric solution$\beta_{i,3}U_{3}(x)(i=1,2)$ , from (ii) of Lemma 1.4, $b(V_{1}(x), V_{2}(x))$ has minimum on theof (1.2) with suitable coeffiboundary. Thus, the least energy solution
cients, has a concentrating point near(ii) When a positive interaction term closes to , Lin-Wei [LWe2] and $Ikoma-Tanaka$$U_{3}$$(U_{1}, U_{2})$$\partial\Omega$$\partial\Omega.$$0$$\beta$[IT] studied a singular perturbation problem for$-\epsilon^{2}\Delta u_{1}+V_{1}(x)u_{1}=\mu_{1}u_{1}^{3}+\beta u_{1}u_{2}^{2}$in$R^{n},$$-\epsilon^{2}\Delta u_{2}+V_{2}(x)u_{2}=\mu_{2}u_{2}^{3}+\beta u_{1}^{2}u_{2}$in$R^{n}$In this case, there are possibilities that least energy positive solution
andhave
always must concentratedifferent concentrating points. But, in our case,anda same point.
$u_{1,\epsilon}$$u_{1,\epsilon}$$u_{2,\epsilon}$$u_{2,\epsilon}$

2. The Existence of -symmetric least energy solutions$G$

In this section, whenis a ball $B=\{x\in R^{N}||x|=1\}$ , we observe the multipleof (1.1)
is a ball, a positive solution
existence of positive solutions of $(*)$ . Whenis unique and radially symmetrically. Thus, for group actions $G\subset O(n)(O(n)$ is theorthogonal group for $n=2,3$ ), by solving minimizing problems on -symmetric function’s
set $H_{0}^{G}(B)=\{u\in H_{0}^{1}(B)|u(gx)=u(x)$ for all $g\in G\}$ , we can expect multiple existenceof positive solutions of $(*)$ . In fact, we can show the following theorem by a similar wayof Theorem 1.1.
$\Omega$$\Omega$$U_{3}$$G$. Let $G\subset O(n)$ be a
and
Theorem 2.1. Assume $\Omega=B,$$\beta^{G}>0$$(*)$ has a-symmetricgroup action. Then, there exists
such that, for any. Moreover, there exist a sequence
positive solution
and $U_{i}\in H_{0}^{1}(B)(i=1,2,3)$ such that$\beta\equiv\beta_{1,2}>0$$\beta_{1,3},$$\beta_{2,3}\leq 0$$\beta>\beta^{G},$$G$$\vec{u}_{\beta}^{G}(x)=(u_{1,\beta}^{G}(x), u_{2,\beta}^{G}(x), u_{3,\beta}^{G}(x))$$\beta_{m}arrow\infty$$(\sqrt{\beta_{m}}u_{1,\beta_{m}}^{G}, \sqrt{\beta_{m}}u_{2,\beta_{m}}^{G}, u_{3,\beta_{m}}^{G})arrow(U_{1}^{G}, U_{2}^{G}, U_{3})$strongly in$H_{0}^{1}(B)^{3}.$Hereis a uniq $ue$ positive radial least energy solution of (1.1) with $\Omega=B$ anda positive least energy -symmetric solution of (1.2) with $\Omega=B.$$U_{3}$$is$$(U_{1}^{G}, U_{2}^{G})$$G$

Theorem 2.1 suggests the multiplicity of positive solutions of $(*)$ . However, in orderto get a multiple existence for $(*)$ , we need to show thefor group actions$G\neq G’$ . To observe this, we will discuss an asymptotically behavior of limit equation(1.2) by regarding some coefficients as parameters. That is, for group actions $G\neq G’,$$\vec{u}^{G}\neq\vec{u}^{G’}$131
we show thatandhas different asymptotically behaviors when someparameters go to limits. More precisely, we will show
-peaks near thehasboundary
whereis a number of the minimum orbit for . This argument is similar
to the arguments for the multiplicity of positive solutions $of-\Delta u+u=u^{p}$ on annulus
domain.
We will observe such asymptotically results for more general equations which including(1.2). For radial positive functions $V_{i}(x)\in C(B, R)(i=1,2)$ , we consider the following$(U_{1}^{G’}, U_{2}^{G’})$$(U_{1}^{G}, U_{2}^{G})$$k^{G}$$(U_{1}^{G}, U_{2}^{G})$$\partial B$$k^{G}$$G$

system$-\triangle u_{1}+\eta V_{1}(x)u_{1}=u_{1}u_{2}^{2}$

in $B,$$u_{1}\in H_{0}^{1}(B)$,$-\triangle u_{2}+\eta V_{2}(x)u_{2}=u_{1}^{2}u_{2}$in $B,$$u_{2}\in H_{0}^{1}(B)$.

For a group action $G\subset O(n)(n=2,3)$ , to get
the following minimizing problem on
:

(2.1)

$G$

-symmetric solutions of (2.1), we solve

,

(2.2)

$H_{0}^{G}(B)$$\hat{b}_{\eta}^{G}=\inf_{(u_{1},u_{2})\in\hat{N}_{\eta}^{G}}(|||u_{1}|||_{\etaV_{1},B}^{2}+|||u_{2}|||_{\eta V_{2},B}^{2})$$\hat{N}_{\eta}^{G}=\{(u_{1}, u_{2})\in H_{0}^{G}(B)^{2}||||u_{1}|||_{\eta V_{1},B}^{2}+|||u_{2}|||_{\eta V_{2},B}^{2}=2||u_{1}u_{2}||_{L^{2}(B)}^{2}\neq 0\}.$

By standard ways, we see thatis achieved for somewhich is a Gsymmetric positive solution of (2.1). To discuss an asymptotically behavior of, the functionaswhich was defined in (1.5) also plays important roles.
By using the Schwarz symmetrization, we see that least energy solutions ofisradial symmetry. For $G\subset O(n)$ , let $G[x]=\{gx|g\in G\}$ be an orbit ofand$k^{G}= \min\{\# G[x]|x\in R^{n}\backslash \{0\}\}$ be a element number of the minimum orbit. Now, we
obtain the following theorem which is essential in our arguments.
$(u_{1,\eta}^{G}, u_{2,\eta}^{G})\in\hat{N}_{\eta}^{G}$$\hat{b}_{\eta}^{G}$$(u_{1,\eta}^{G}, u_{2,\eta}^{G})$$\etaarrow\infty$$b(\lambda_{1}, \lambda_{2})$$(1.6)_{(\lambda_{1},\lambda_{2})}$$x\in R^{n}\backslash \{0\}$Theorem 2.2. Assume that $V_{i}(x)\in C(B, R)(i=1,2)$ are positive radial functions anda finite group $G\subset O(n)$ satisfies
where $\underline{b}=\min_{x\in B}b(V_{1}(x), V_{2}(x))$ .
Then we have$k^{G}< \frac{b(V_{1}(0),V_{2}(0))}{\underline{b}}$$\eta^{\frac{n}{2}-2}\hat{b}_{\eta}^{G}arrow k^{G}\underline{b}$as$\etaarrow\infty.$

Moreover, for a family of -symmetric positive solutions
the minimizing problem (2.2), there exist a subsequence
in $B$ with $\# G[x_{m}]=\# G[x_{0}]=k^{G}$ and$G$of (2.1) which achieves
and a sequencesuch that$(u_{1,\eta}^{G}, u_{2,\eta}^{G})$$x_{m}arrow x_{0}$$\eta_{m}arrow\infty$$b(V_{1}(x_{0}), V_{2}(x_{0}))=\underline{b}$$\Vert|u_{i,\eta_{m}}^{G}-\sum_{z\in G[x_{m}]}\sqrt{\eta_{m}}w_{i,0}(\sqrt{\eta_{m}}(\cdot-z))\Vert|_{\eta_{m}V_{l},B}^{2}=o(\eta_{m}^{1-\frac{n}{2}}) (i=1,2)$.

132
Here $(w_{1,0},w_{2,0})$ is a positive least energy solution of (1.5) with
$1arrow 0$. That is, for largeasandfunctions.

$(\lambda_{1}, \lambda_{2})=(V_{1}(x_{0}), V_{2}(x_{0}))$‘$o(\eta_{m^{2}}^{1-g})\eta^{\frac{n}{m^{2}}}$$\eta_{m},$$\eta_{m}arrow\infty$$(u_{1,\eta_{m}}^{G}, u_{2,\eta_{m}}^{G})$is close to$k^{G}$-peak

Remark 2.3.
achieving-symmetric positive solutions(2.2) may be radial symmetric. Thus we can’t look for any more positive solutions byonly Theorem 2.2.
(ii) Theorem 2.1 and 2.2 still hold for -invariant domain . The other corollaries andtheorems below follow from Theorem 2.1 and Theorem 2.2.
(i) For the case$k^{G} \geq\frac{b(V_{1}(0),V_{2}(0))}{\underline{b}},$$G$$(u_{1,\eta}^{G}, u_{2,\eta}^{G})$$G$$\Omega$

From Theorem 2.2, we have the following
be aCorollary 2.4. Suppose that the same assumptions as Theorem 2.2 hold. Let
set of least energy $G$ -symmetric solution of (2.1), $K$ be a set ofleast energy solution of (1.6)$K_{\eta}^{G}$

with $(\lambda_{1}, \lambda_{2})=(V_{1}(x_{0}), V_{2}(x_{0}))andX=\{x\in B|\# G[x]=k^{G}, b(V_{1}(x), V_{2}(x))=b\}and$$X_{\rho}=\{x\in B|\neq G[x]=k^{G}, dist(x, X)<\rho\}$ . Then, for any $\rho>0$ , we have$\sup_{(u_{1},u_{2})\in K_{\eta}^{G}}\inf_{(w_{1},w_{2})\in K,x\in X_{\rho}}\sum_{i=1,2}\Vert|u_{i}-\sum_{z\in G[x]}\sqrt{\eta}w_{i}(\sqrt{\eta}(\cdot-z))\Vert|_{\eta V_{t},B}^{2}=o(\eta^{1-\frac{n}{2}})$Here$o(\eta^{1-\frac{n}{2}})\eta^{g-1}arrow 0$

as

(2.3)

$\etaarrow\infty.$

Proof. Suppose that Corollary 2.4 does not hold. Then there exists $c_{0}>0,$such that$\eta_{m}arrow\infty$and$(u_{1,\eta_{m}}, u_{2,\eta_{m}})\in K_{\eta_{m}}^{G}$$\inf_{(w_{1},w_{2})\in K,x\in X_{\rho}}\sum_{i=1,2}\Vert|u_{i,\eta_{m}}-\sum_{z\in G[x]}\sqrt{\eta_{m}}w_{i}(\sqrt{\eta_{m}}(\cdot-z))\Vert|_{\eta_{m}V_{i},B}^{2}\geq c_{0}\eta^{1-4}2$I

But, this contradicts to Theorem 2.2.

When $n=2$ , for any $k\in N$ , the cyclic group $Z_{k}\subset O(2)$ satisfies $k^{z_{k}}=k$ . Thus, from
Corollary 2.4, we easily find the following multiple existence of positive solutions.
Corollary 2.5. Suppose that $n=2$ and $V_{i}(x)\in C(B, R)(i=1,2)$ are positive radial
where $\underline{b}=\min_{x\in B}b(V_{1}(x), V_{2}(x))$ . Then, there exists
functions and$(2.1)$ has a positive solution close to -peak function in$\eta_{k}>0$ such that, for any$k< \frac{b(V_{1}(0),V_{2}(0))}{\underline{b}}$$\ell$$\eta>\eta_{k},$133
the sense of (2.3) with $G=Z\ell$ and (2.1) has a radial positive solution. That is, (2.1) has
at least $k+1$ positive solutions.

Proof. From Corollary 2.4, for any, there exists $\eta^{Z\ell}>0$ such that,$(2.1)$ has a positive solution close to -peakfor any
solutions in the sense of(2.3). On the other hand, (2.1) always has a radial positive solution. Thus, for$\ell<\frac{b(V_{1}(0),V_{2}(0))}{\underline{b}}$$\eta>\eta^{Z\ell},$$\ell$$\eta\geq\eta_{k}\equiv$$\max\{\eta^{z_{1}}, \cdots, \eta^{z_{k}}\},$$(2.1)$

has at least $k+1$ positive solutions.

$I$

When $n=3$ , the subgroups $G=Z_{2},$$P_{12}\subset O(3)$ satisfy $k^{G}=2,4,8$ ,or 12,
respectively. Hereis the -regular polyhedron group. Thus, from Corollary 2.4, we alsofind the following corollary.
$P_{4},$ $P_{8},$$P_{q}$$q$

Corollary 2.6. Suppose that $n=3$ and $V_{i}(x)\in C(B, R)(i=1,2)$ are positive radialfunctions and 2 (or 4, 8, 12, respectively)
where $\underline{b}=\min_{x\in B}b(V_{1}(x), V_{2}(x))$ .
Then, there exists $\eta_{0}>0$ such that, for any$(2.1)$ has a positive solution close to 2$(or 4, 8, 12,$ respectively) $-peak$ functions in the sense of (2.3) with $G=Z_{2}$ (orrespectively) and (2.1) has a radial positive solution.
$< \frac{b(V_{1}(0),V_{2}(0))}{\underline{b}}$$\eta>\eta_{0},$$P_{4},$ $P_{8},$ $P_{12},$Here, we return to the our original equation $(*)$ and limit equation (1.2). For $\eta>0,$$\lambda_{i}’>0$

and $\beta_{i,3}’<0(i=1,2)$ , we set$\lambda_{i}=\eta\lambda_{i}’, \beta_{i,3}=\eta\beta_{i,3}’ (i=1,2)$.

For (1.2), since $U_{3}(x)$ is radial and decreasing with respect to $r=|x|$ , from (bl) $-(b2),$$b(\lambda_{1}-$: $(0, \infty)^{2}arrow R$ has maximum at $x=0$ and minimum on$\beta_{1,3}U_{3}(x)^{2},$ $\lambda_{2}-\beta_{2,3}U_{3}(x)^{2})$$\partial B.$

We remark that, from (b3),. From Corollary 2.5 and Corollary 2.6, we have the following multiple existence
$(*)$
for
.
$b(\lambda_{1}-\beta_{1,3}U_{3}(x)^{2}, \lambda_{2}-\beta_{2,3}U_{3}(x)^{2})=\eta^{2}b(\lambda_{1}’-\beta_{1,3}’U_{3}(x)^{2},$$\lambda_{2}’-$$\beta_{2,3}’U_{3}(x)^{2})$Theorem 2.7. Assume that $\Omega=B.$ For $\eta>0$ , we assume that $\beta\equiv\beta_{1,2}>0,$$\beta_{i,3}=\eta\beta_{i,3}’<0(i=1,2)$ .
Let
be the maximum integer satisfying $k<$$\lambda_{i}=$$\eta\lambda_{i}’>0,$$k$$\frac{b(\lambda_{1}’-\beta_{1,3}’U_{3}(0)^{2},\lambda_{2}’-\beta_{23}’U_{3}(0)^{2})}{b(\lambda_{1},\lambda_{2})}.$(i) When $n=2$ , for any

exists$\beta_{k}(\eta)>0$, there exists $\eta_{k}>0$ such that, for any
such that, for any
and $\ell=1,$, we have$\epsilon,$$\rho>0$$\beta>\beta_{k}(\eta)$$\cdots,$$\eta>\eta_{k}$, there$k$

inf$\sup$L_{\beta}^{Z\ell(w_{1},w_{2})\in K,x\in x_{\rho}^{\sum_{i=1,2}}}\Vert|\sqrt{\beta}u_{i}-\sum_{z\in Z\ell[x]}\sqrt{\eta}w_{i}(\sqrt{\eta}(\cdot-z))\Vert|_{V_{l},B}^{2}\leq\eta^{1-\frac{n}{2}}\epsilon.$$\vec{u}\in

Hereis a set of least energy -symmetric solutions of $(*)$ and $K$ is a set ofleast energy solutions of (1.6) and $X_{\rho}=\{1-\rho<|x|\leq 1\}$ . In particular, $(*)$ has$L_{\beta}^{Z\ell}$$Z\ell$134
isof $(*)(1\leq\ell\leq k+1)$ . Hereat least $k+1$ positive solutions
is close to -peak functions which peak’s locations are near-symmetry and$\partial B(1\leq\ell\leq k)$ andis radial symmetry.
holds, for any $\rho>0$ , there exists(ii) When $n=3$ , if 2 (or 4, 8, 12, respectively)
such that, for any, there exists$\eta_{k}>0$ such that, for any
and $G=Z_{2}$ $(or P_{4}, P_{8}, P_{12},$ respectively) , we have$\vec{u}^{\ell}$$\vec{u}^{\ell}=(u_{1}^{\ell}, u_{2}^{\ell}, u_{3}^{\ell})$$I$$(u_{1}^{\ell}, u_{2}^{\ell})$$Z\ell$$\vec{u}^{k+1}$$\leq k$$\epsilon,$$\beta>\beta_{k}(\eta)$$\beta_{k}(\eta)>0$$\eta>\eta_{k}$$\sup_{\vec{u}\in L_{\beta}^{G}}\inf_{(w_{1},w_{2})\in K,x\in X_{\rho}}\sum_{i=1,2}\Vert|\sqrt{\beta}u_{i}-\sum_{z\in G[x]}\sqrt{\eta}w_{i}(\sqrt{\eta}(\cdot-z))\Vert|_{V_{i},B}^{2}\leq\eta^{1-\frac{n}{2}}\epsilon.$

is a set of least energy -symmetric solutions of $(*)$ and $K$ is a set of leastHere
energy solutions of (1.6) and $X_{\rho}=\{1-\rho<|x|\leq 1\}.$$G$$L_{\beta}^{G}$

Proof. From Corollary 2.5 and Corollary 2.6, there exists $\eta_{k}>0$ such that, for all$\eta>\eta_{k},$

.

(2.4)

$\sup_{(U_{1},U_{2})\in K_{\eta}^{G}}\inf_{(w_{1},w_{2})\in K,x\in X_{\rho}}\sum_{i=1,2}\Vert|U_{i}-\sum_{z\in G[x]}\sqrt{\eta}w_{i}(\sqrt{\eta}(\cdot-z))\Vert|_{\eta V_{i},B}^{2}\leq\eta^{1-\frac{n}{2}}\epsilon$

is a set of least energy -symmetric solution of (2.1). Next, from Theorem 2.2,
Here
, we havethere exists $\beta(\eta)>0$ such that, for all$G$$K_{\eta}^{G}$$\beta>\beta(\eta)$$\sup_{\vec{u}\in L_{\eta}^{G}}\inf_{(U_{1},U_{2})\in K_{\eta}^{G}}\sum_{\iota=1,2}\Vert|\sqrt{\beta}u_{i}-U_{i}\Vert|_{\eta V_{\ell},B}^{2}\leq\eta^{1-g}\epsilon$

.

(2.5)

1

From $(2.4)-(2.5)$ , we obtain Theorem 2.7.
Moreover, since $U_{3}(0)arrow\infty$ as$\lambda_{3}arrow\infty$, from (b3), we observe that$\frac{b(\lambda_{1}’-\beta_{1,3}’U_{3}(0)^{2},\lambda_{2}’-\beta_{2,3}’U_{3}(0)^{2})}{b(\lambda_{1},\lambda_{2})}arrow\infty$a$s$$\lambda_{3}arrow\infty$

.

(2.6)

Therefore, from Theorem 2.7 and (2.6), we easily lead the following theorem.

Theorem 2.8. Assume that $\Omega=B$ and $n=2$ . Then, for any $k\in N$ , there exists, there, such that, for any, there exists
such that, for any$(*)$ has at least $k+1$ positive, such that, for any
existsis close. Here is -symmetry andsolutions$\lambda_{k}>0,$$\lambda_{3}>\lambda_{k}$$\beta_{k}(\eta, \lambda_{3})>0$$\eta>\eta_{k}(\lambda_{3})$$\eta_{k}(\lambda_{3})>0$$\beta\equiv\beta_{1,2}>\beta_{k}(\eta, \lambda_{3}),$$\vec{u}^{\ell}=(u_{1}^{\ell}, u_{2}^{\ell}, u_{3}^{\ell})(1\leq\ell\leq k+1)$$\vec{u}^{\ell}$$Z\ell$$(u_{1}^{\ell}, u_{2}^{\ell})$to -peak functions which peak’s locations are near $\partial B(1\leq l\leq k)$ and$\ell$$\vec{u}^{k+1}$

is radial

symmetry.

Acknowledgments. This work is partially supported by the JSPS Strategic Young
Researcher Overseas Visits Program for Accelerating Brain Circulation “Deepening and135
Evolution of Mathematics and Physics, Building of International Network Hub based on
OCAMI”.

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