数理解析研究所講究録
1181 巻 2001 年 52-61

52

Multiple Solutions for Singularly
Perturbed Semilinear Elliptic Problems
in Bounded and Unbounded Domains
Michinori ISHIWATA
石渡通徳
Department of Applied Physics,

School of Science and Engineering,
Waseda University
3-4-1 Okubo, Shinjyuku-ku, Tokyo, 169-8555 Japan.

Main Theorem

1

We are concerned with theturbed semilinear elliptic equations:$\coprod 1\mathrm{U}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{P}^{\mathrm{l}\mathrm{i}_{\mathrm{C}}\mathrm{i}}\mathrm{t}\mathrm{y}$of solutions for the following singularly per-$(\mathrm{P})_{\epsilon}$
where $\in>0,$ $p\in(2,2^{*})(2^{*}$ denotes the critical exponent of the Sobolev embedding) andgiven by $2^{*}=2\mathit{1}\mathrm{V}/(N-2)$ if $N\geq 3,$ $=+\infty$ if$a\in C(\Omega)$ is a function with condition (C) specified below.
As for , we assume that$\mathit{1}\mathrm{V}=1,2$$H^{1}(\Omega)\subset L^{p}(\Omega)$$\Omega$$\bullet$$\Omega$

is a bounded domain in$\bullet\Omega=1\mathrm{R}^{N}(\mathit{1}\mathrm{V}\geq 1)$$\mathbb{R}^{N}(l\mathrm{V}\geq 1)$

, or

.

.
Without loss of generality, we also assume that, the boundary condition should be understood as
When$0\in \mathrm{i}\mathrm{n}\mathrm{t}\Omega$$\Omega=\mathbb{R}^{N}$$u(x)arrow 0$

as $|x|arrow\infty$ .

In order to characterize the topological feature of $a(x)$ . we introduce the following.
condition. for some positive numbers$(\mathrm{C})_{\delta\delta}0$$\delta_{0},$$\delta$

53

(C1):

exist positive constants$a(x)\geq 1+\delta_{0}$ in(C2):$a(x)\geq 1-\delta$$(\mathrm{C})_{\mathit{6}\mathit{5}}0,$:$Ther\epsilon$$r,$$\rho$

which satisfy the followi $ng$ :$B(0, \rho)_{f}$in$\Omega$,, if is a bounded domain,,. $a(X)<1$ and $\lim_{|x|arrow\infty}a(x)=1_{i}$ ifmhere $K$ is some closed subset of which satisfies the following
condition (K):(C3):$\Omega$$\cdot\sup_{x\in(\partial K}-)_{7}a(x)\leq 1$$\iota$$\Omega=\mathbb{R}^{N}$$\sup_{i\in(’K)_{7}},(^{-}l$$\Omega$(K): $B(0, p)\subset K.$$\partial K$$(\partial K),$.

is homotopically equivalent to’$\mathrm{b}^{N-1}$

and.

$=\{x\in \mathbb{R}^{\mathit{1}\mathrm{v}}|\mathrm{d}\mathrm{i}\mathrm{S}\mathrm{t}(x, \partial K)\leq\uparrow\cdot\}\subset\Omega$

Roughly speaking, conditions above imply that $a(x)$ has a “peak” in $K$ (condition”(C1) , the value of $a(x)$ on
is uniforndy less than the level of the(C3) , andis the set which is surrounding the “peak” and is homotopicallyequivalent to $S^{N-1}$ (condition).
Putting $v(x)=u(\in x)$ , we see that (the weak form of)is equivalent to“$\partial K$$)$$\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}$$(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\partial K$$)$$(\mathrm{K})$$(\mathrm{P})_{\epsilon}$(P)\epsilon$-\triangle u+a(\mathrm{C}\prime x)U=u|v|^{p}-2$

and the solution ofin$\Omega,$$v\geq 0,$$v\in H_{0}^{1}(\Omega/\circ)\wedge$,

corresponds to the critical point of the functionalin. Hence it is$I_{\epsilon}(v)=$$(\mathrm{P}’)\underline{\mathrm{c}}$$./\Omega/\epsilon|(\nabla\iota)|^{2}+a(\in x)|\tau’|^{2})$$\mathit{1}M_{p}(\Omega/\in)=\{v\in H_{0}^{1}(\Omega/\Xi);||v||_{L^{p}}(\Omega/\epsilon)=1\}$

enough to find critical points of to prove the existence and lIlultiplicity of solutionsof.
It is well-known that for small . there is a relation between the multiplicity ofcritical point of inand the topological feature of. For example. we refer[1. 2. 3. 4].
main theorem reads as follows.
$I_{\epsilon}$$(\mathrm{P})_{\vee}\epsilon$$\in$$I_{\underline{C}}$$a(.\tau)$$\mathit{1}\mathcal{V}I_{p}$$()\iota\iota \mathrm{r}$Main Theorem.
For all. therethe cxiststhat for any. if is a $b_{ou7}1d_{6}ddomai\gamma?$ . th$\delta_{0}>0$$\mathrm{c}=such$$thr\gamma \mathrm{l}$$r\cdot\epsilon$$\mathit{1}Vlor\epsilon O|J\epsilon r$$exi_{-\mathrm{q}}.t.-\delta\backslash ^{\neg}>0.\backslash ^{\neg}uch$that ] $a\in C(\Omega)$ andis Laqtisfi.
at least 2 solutions.
exists anoth er solution $for\in abov\epsilon$ .
$i$$\overline{\mathrm{c}}\in(0, \mathrm{c}=)$$\Omega$$\epsilon re$$6d_{i}$$(\mathrm{C})_{\mathrm{t}5_{0},s}$$(\mathrm{P})_{\xi}$$admit.-\backslash \neg$From now on, we shall only deal with the case of bounded dolnain in the main
theorem. The argulnent for the case of unbounded is almost similar as below.
One can easily prove that if $a(x)\equiv 1$ (in general) thenadmitsat least one solution(ground state solution) for all value
with the aidof Mountain Pass Theorem. In general, one cannot expect the existence of lnultiple
solutions. Indeed, when $a(x)\equiv 1$ and, the uniqueness result for sufficiently
small is known (Dancer [1]). The main theorem above says that as soon as one$\Omega$$a(x)\equiv \mathrm{C}_{0}\mathrm{n}\mathrm{s}\mathrm{t}.$$\overline{\mathrm{c}}\in(0, \infty)$$u_{\mathrm{U}}$$\Omega=\mathrm{b}\mathrm{a}\mathrm{l}\mathrm{l}$$\overline{\mathrm{c}}$$(\mathrm{P})_{\vee}c$54perturb $a(x)$ to have a “peak”, one gets the another (high energy) solutions
even if the perturbation is very small. This “generation of higher energy solution” isa consequence of the change of topology of sollle level set of the functional associatedcaused by the nontrivial shape of $a(x)$ . It is the purpose of this paper totodiscuss this change of topology.
$u_{1},$$u_{2}$$(\mathrm{P})_{\vee}\mathrm{e}$Known Results and Motivations

2

The interest in studying

2.1

arises from several physical and lIlathelnatical contexts.

$(\mathrm{P})_{\mathcal{E}}$Physical Contexts

can be regarded as a (reduced) nonlinear Schr\"odinger
In the physical context.
to the Dirac constant .
equation and sllldll parameterby (this approximation
approximatedwellbe
It is well known that when can), quantum mechanical equation lllay have a
is calledstate, concentrating around asolution corresponding to amechanical equilibrium. It is also well known that classical equilibrium is often
the potential energy. So it is reasonable to expect thatthe point which”
solution concentrating around a point whichhasfor small$a(x)$ . Hence the structure ofpotential
energyoftheattains the minilnuln$\{y\in\Omega|a(y)=\min_{\Omega}o(X)\}$ , the minilnunl set of $a(x),$ lnay play a significant role.
for the existence, multiplicity of solutions of[2] obtain the following result.
Fehner
view,
-Pino
and
of
point
From this$(\mathrm{P})_{\vee}c$$\gamma l$$\mathrm{c}\prime \mathrm{c}(\mathrm{r}1^{\cdot}\mathrm{e}\mathrm{S}1^{)}(11\mathrm{d}\mathrm{s}$

$0$

$h$

$4\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{S}\mathrm{S}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}\mathrm{i}_{1}11\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}’$‘’$\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{c}\mathrm{a}\iota$$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{S}\mathrm{i}_{\mathrm{C}\mathrm{a}1’}$$1\mathrm{I}\dot{\mathrm{u}}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{i}\mathrm{z}\mathrm{e}$$\mathrm{a}‘ \mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}_{\mathrm{C}\mathrm{a}1}$$(\mathrm{P})_{\Xi}$$\in,$$Cl_{\}\iota i},n=$$(\mathrm{P})_{\mathcal{E}}$$\mathrm{d}\mathrm{e}\mathrm{l}$Proposition 2.1 (Effect of weight function, delin }compactly
Suppo. that A is a bounded.
small
? for
.
trates to a point ? A which attains the minimum of $a(x)$ as$\mathrm{P}\mathrm{i}\mathrm{n}\mathrm{o}- \mathrm{F}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}\}\mathrm{e}\mathrm{r}[2]$$co7ltai\uparrow?ed$$-\backslash ^{\neg}et$$\backslash ^{\neg}e$$\inf_{\Lambda}a(x)$$The\uparrow$$.-\backslash ^{\neg}u\backslash ffi_{Ci_{C}nt}ly$$\in,$$\zeta$$(P)_{\vee}\prime admit.\backslash ^{\neg}a.\backslash ^{\neg}olutio7?u_{\epsilon}$$\mathrm{c}^{-}arrow 0$

$i7$

)

$a \uparrow?d\min_{(9\Lambda}a(\mathrm{J}^{\cdot})>$. mhich concen-.

as thethat there exist at least as lIlany solutions of
Proposition 2.1
if is small enough.
nulllber of connected components of. so in thisectedniay have onl.lj one
In our situation, (that notsavs.
Ourlllaincase Proposition 2.1 provides
but also sollle topological feature
only the nulnb of connected colnponent ofis homotopically equivalent to $S^{N-1}$ ) plays solne role(i.e. the fact thatofon the nmltiplicity of the solutions of$(\mathrm{P})_{\vee}c$$\mathrm{i}_{1}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}$$\mathit{0}_{J},$$(^{\urcorner})\delta_{0}.b\cdot$} $\iota in$$\in$$CO?l\gamma l$$a7\gamma\iota in$$07?lyon\mathrm{r}.\backslash ol\mathrm{t}ltio\gamma?$$0_{1li},,\iota$$(^{\lrcorner}\mathrm{r}$$a_{mjn}$$cornpo\uparrow\prime e7lt$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{I}^{\cdot}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}$$\partial K$$(\mathrm{P})\epsilon.\cdot$

2.2

Mathematical Contexts

can be regarded as an example verifying the folIn the mathematical context,
lowing feature. In many semilinear elliptic problem including small parameters (e.g.
$(\mathrm{P})_{\epsilon}$

55

semilinear elliptic equations with critical or nearly critical exponent $[6, 7]$ , stationary-Hilliard equation [8], Ginzburg-Landau equation [9] , it is commonly observedthat if the parallleter is slIlall enough. then the existence and multiplicity of solutions are controlled by the finite dimensional object. As for singularly perturbed
equations in bounded doma, , the following result holds.
$\mathrm{c}_{\mathrm{a}\mathrm{h}\mathrm{n}}^{1}$$)$$\mathrm{i}\mathrm{n}\mathrm{s}$Proposition 2.2 (Effect of topology of the domain, Benci-Cerami [5])Assume.. that
Const.
thatis bounded andis$\Omega>1$
topologically ontrivial in th. Then for smallof $category_{i}i.e.$ . cat$\Omega+1$
cat
solutions.
. admits at$\Omega\subset \mathbb{R}^{N}$$A_{--}.\backslash .\backslash u\backslash \neg meals^{\neg}O$$a(\alpha\cdot)\equiv$$\Omega$$\epsilon sens\epsilon$$n$$\mathrm{c}_{i}\sim$$l_{6a}St$$(P)_{-}$In this case, the finite dimensional object referred above isquestions naturally arise:$\Omega$

1. Can one replace the (tnontriviality of the topology of the“nontriviality of the shape of the weight function $a(x)$ ” ?

2. What is the finite dimensional object which
plicity of solutions when$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}.\mathrm{t}.\mathrm{h}.\mathrm{e}$. The following$\mathrm{d}_{\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{a}}\mathrm{i}\mathrm{n}$”by theexistence and multi-$a(x)\not\equiv \mathrm{C}(\mathrm{l}\mathrm{n}\mathrm{s}\mathrm{t}.?$Our lnain tlleorel\iota l gives an affirmative (partial) answer for the first questionand suggests that the finite dimensional object asked in question 2 is not
as in
Proposition 2.2 but.
$\Omega$$S^{\Lambda^{\tau}-1}$

Variational and Topological Tools

3

Our lIlain tool relies on the variational approach, which is based on the following
fundamental principle.

Proposition 3.1 (Fundamental principle in Morse theory)
Suppo. that $M$ is a Banach-Finsler manifold and $I\in C^{1}(M)$ satisfi es the fol$\backslash ^{\backslash }e$$lo\mathrm{c}\iota’ i?lga.\mathrm{b}^{\neg}.\backslash ^{\neg}umptio\uparrow l$:-condition for all $c\in[a, b]$ .
and $[I\leq b]$ havein topology”.
tha critical(Here $emea\uparrow \mathit{1}\{v\in M|I(u)\leq a\}$ by $[I\leq a]$1.
2.

$I.\backslash ^{\mathrm{B}}c\iota ti.\backslash ^{\backslash }fie.\backslash ^{\backslash }(PS)_{\mathrm{c}}$$[I\leq\zeta \mathit{1}]$$The\gamma l$$\epsilon ree\mathit{1}^{\cdot}i_{-\backslash ^{\neg}t.\backslash ^{\neg}}.-$$\mathrm{t}\ell$$a\prime\prime differe\eta ce$$va\iota_{u}\epsilon\overline{c}\in[a, b]$

)

$.)$

In order to compare the topology of level sets of , various kinds of topologicalinvariants are known. We shall here use the notion of the “category” of sets, definedby:$I$

Definition 3.2 (Notion of$‘\prime \mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}(\Gamma.\mathrm{V}" )$56

Assume $X$ be a topological space andare two closed subsets of X withhen
if and only if is a smallest number among such thatis a closed contractible covening ofin ,$\Omega,\omega$$n=\mathrm{c}\mathrm{a}\mathrm{t}_{\Omega}[\omega]$

$.T$

$n$

$m$

.

$(\omega_{j})_{j=1}^{m}$$\Omega$$\omega$$i.e.$$\omega\subset\Omega$,$\omega=\bigcup_{j1}^{m}=\omega_{j},$$\exists h_{j}\in C([0,1]\mathrm{x}\omega_{j}; x),$$h_{j}(0, X)=x\forall x\in\omega_{j},$$\exists\overline{x}_{/}.\cdot\in Xs.t$$h_{i}(1, x)=\overline{x}_{j}\forall X\in\omega_{j}$.

.

We simply denoteby cat .
In terms of this notion, Lysternik-Schnirehnan theorem (category version) reads
as follows:$\Omega$$\mathrm{c}\mathrm{a}\mathrm{t}_{\Omega}[\Omega]$

Proposition 3.3 (Lysternik-Schnirelman theorem, category version [10])Suppo. thatis a Banach-Finsler $man\dot{\iota}fold_{i}I\in C^{1}(M)$ . a $7ldo,$. Suppose also that for some $b’>b>a$ .
for all$K\cap[I=b]=\phi$ where $K=\{u\in M|(dI)_{v}=0\}$ .
Then $[I\leq b]$ contains at least
critical points.
$\mathrm{M}$$-\backslash ^{\neg}e$$= \inf_{t\backslash ff}I>$$-\infty$$I_{-\backslash ^{\neg}at}.i.,\backslash fie.-\backslash ^{\neg}(PS)_{c}$$c\in[\mathit{0}_{J}, b^{J}]$and$\mathrm{c}\mathrm{a}\mathrm{t}[I\leq b]$4

Sketch of Proof of Main Theorem

4.1

Variational Setting and Notations

We introduce the following notations: for$\omega\subset \mathbb{R}^{N}$$\bullet \mathit{1}\mathrm{t}/l(r\omega)=\{u\in H1(0)\omega|||u|+|_{L^{p}(\omega)}=1\}$,,.
$\bullet I\mathrm{r}rightarrow.\alpha.\mathfrak{U}/(u)=.\int_{\omega}$$(\epsilon:|2\nabla u|22)+\alpha udx$$\bullet S_{p}(_{\mathcal{E}.O}.\omega)=\inf\mu\in M_{\mathcal{P}}1\omega)I\zeta.\circ.\omega(u)$As for$\mathrm{S}_{\rho}’(\in, \mathit{0},\omega)$,.

. it is well known that the following result holds:

Proposition 4.1 (Existence [11] and uniqueness [12] for ground state in)
For any , a. there exists a unique minimizer (up to translation) for
which
positive and radially symmetric with respect to the $origi?l$ .
$\mathbb{R}^{N}$$\overline{\mathrm{c}}$$\in \mathbb{R}^{+}$$\prime 9_{p}(\mathrm{C}\alpha]’,\mathrm{R}^{N})i$$i_{\sim^{\mathrm{S}}}$

In order to discuss the relation between the level set of (in function space) and(in), we define the “truncated barycenter”.
Let
be a cut off function such that $\eta(t)=1$ iff $|t|<R,$ $\eta(\dagger,)=R/t$
iff $|t|\geq R$ . Set
for.
Then it is obvious that for$|\beta_{R}(u)|\leq R$ holds.
Moreover, if
the (intuitive) barycenter of $u\in M_{p}$ is near “infinity”, thenis located near$\partial B_{R}=\{x\in \mathbb{R}^{N}||x|=R\}$ .
Namely,$I$$\partial K/\epsilon$$\mathbb{R}^{N}$$\beta_{R}(u)$$\eta\in C_{0}^{\mathrm{c}\mathrm{u}}(\mathbb{R})$$\beta_{R}(u)=.\int_{\mathrm{R}^{N}}x\eta(|x|)|u|pdX$$\forall u\in M_{p}(]\mathrm{R}^{N})$$\forall u\in M_{p}(\mathbb{R}^{N}),$$\beta_{R}(u)$57

Proposition 4.2 (The range of truncated barycenter)andSupposesatisfies.
asThe$v(x)=\mathrm{c}^{N/p}\prime u(_{\hat{\mathrm{C}}}x)$$(\mathrm{P}^{})_{\epsilon}$, problem

can be rewritten as$(\mathrm{P})_{\epsilon}$$-\triangle u+a(_{\mathrm{C}}\wedge x)U=|v|^{\mathrm{P}^{-2}}v$

It is well known that to solve(V)

as $narrow\infty$ .

$narrow\infty$$7l|\beta_{R}(\overline{u}(\cdot-yn))|arrow R$

Setting$|y_{n}|arrow\infty$$(y_{n})\subset \mathbb{R}^{N}$$\overline{u}\in \mathit{1}M_{p}(1\mathrm{R}^{N})$,$u\geq 0$,$v\in H_{0}^{1}(\Omega/\mathrm{c}’)$.

is equivalent to:$(\mathrm{p}’)_{\underline{-}}$

Find critical points of$I_{1,a_{\epsilon}.\Omega/}\epsilon$on$M_{p}(\Omega/\overline{\mathrm{c}})$.

)asthanks to the Lagrange multiplier rule (in (V) we denote
So hereafter we carry out the progralll (V).
for all , in order to prove lnain
satisfies
Since it is well known that
,
some
for
that
tlleorenl it is enough to verify$a(\circ x’)$$c$$(\mathrm{P}\mathrm{S})_{c}$$I_{1.\mathit{0}_{\epsilon}.\Omega/\epsilon}$$a_{\epsilon}(x).$$b>0=_{\mathrm{k}}9_{p}(1.a_{\epsilon}, \Omega)$$\bullet$$\bullet$cat$[I_{1,f/}(\mathrm{l}\epsilon\cdot‘ \mathcal{E}\leq f)]\geq$

cat$S^{N-1}=\mathit{2}$there exists another critical value

and
$c$

greater thanby virtue of Proposition 3.3.
In order to introduce the “limiting functional”
as follows:
we define$b$$I_{1.b(x).\mathrm{R}^{N}}$associated to$I_{1.a_{6}.\Omega}/\zeta$’$b(\mathrm{n}\cdot)\in C(\mathbb{R}^{N})$$(\mathrm{B})s_{0}.\rho$:$1\leq b(x)\leq 1+\delta_{0}$$b(x)=1$ in $B(0, \rho)^{c}$in.

$\mathbb{R}^{N},$$b(x)=1+\delta_{0}$in $B(0, \rho/2)$ ,.
for $\forall\in\in(0,1)$ andNote thatis a consequence
The nontriviality of the topology of the level set
functional.
of the nontriviality of the level set of’ the limiting).
We next investigate the level set of$\forall x\in\Omega/\in$$\mathit{0}_{\text{ノ}\epsilon}.(.\iota\cdot)=a(\in\alpha\cdot)\geq b(x)-\delta$$[I_{1.a\Omega}\in./\xi\leq b]$$I_{\infty}=I_{1.b(x}$$.\mathrm{R}^{N}$$I_{\infty}$

4.2

Linliting$\mathrm{p}_{\Gamma}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\ln$. Let $b(x)$ be a function defined by
Hereafter xve fix positive constants
the last section.
In view of Proposition 4.1 and the fact that ,can verify the following:$\delta_{\mathrm{U}},$$(\mathrm{B})_{\delta 0\cdot \mathit{0}}$$\rho$$5_{p}^{\urcorner}(1.1,1\mathrm{R}^{N})=5_{p}’(1, b(x),$Proposition 4.3 (Inf is not achieved in the lilniting problem.)is $??ot$ achieved.
$, \mathrm{S}_{p}(1.b(X).\mathbb{R}N\mathrm{I}=M\inf_{\mathrm{t}’\in p(\mathrm{R}^{N})}\mathit{1}^{\cdot}\mathrm{R}^{N}(|\nabla u|^{\mathit{2}}+b(.C)|u|^{2}\mathrm{I}^{d}X$$\mathbb{R}N)$

in
we

58

sequence possesses no convergent subseThis result imply that allquence.
, we getthis fact with the colllpactness of elllbedding$1\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{i}_{\mathrm{Z}}\mathrm{i}\mathrm{n}\mathrm{g}$$H^{1}\subset L_{lv\mathrm{C}}^{p}$$\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}$Proposition 4.4 (Behavior of[13]

For an(sequences of the limiting problem$ymi\uparrow?imi\approx i_{7}?g.- equ67\backslash \iota\backslash C\epsilon(u_{n})\subset M_{p}(1\mathrm{R}^{N})$

in
where
, IR ) (see Proposition 4.1).
$H^{1}(\mathbb{R}^{N})$$\overline{v}(\cdot-y_{?l})+o(1)$$S_{p}$$1\mathrm{l}\dot{\mathrm{u}}\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{l}\dot{\mathrm{H}}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}$

$)$

of

.
mini

$I_{\infty i}\exists(y_{n})\subset 1\mathrm{R}^{N}.\backslash ^{\neg}.t$$\overline{\mathrm{t})}(.l\cdot)=\overline{v}_{1_{:}1.\mathrm{B}}N(.I^{\cdot})i.\backslash ^{\neg}a(u7\mathrm{t}iqu\epsilon)$$u_{\iota},(\cdot)=$$7??i_{\sim}^{\sim}6^{\cdot}\mathrm{r}$

of

$N$

$1,$

$\perp$That is, foris very closesuch thatis alnlost concentrated at “infinity”. So by Proposition 4.2,$\mathrm{a}‘ \mathrm{n}\mathrm{y}v\in M_{p}(\mathbb{R}^{N})$to $S_{p}(1, b(X),$$\mathbb{R}^{N}),$$|\beta_{R}(v)|\simeq R.$

Nalnely,$u$$I_{\infty}(v)=I_{1.y(.)},N(|i.\mathrm{R}v)$

Proposition 4.5 (Concentration lelnlna at infinity for the limiting functional),
exists a such that for allFor.
$all\uparrow\tau\in(0, R)$$ther\epsilon$$v\in \mathit{1}\mathrm{t}ff_{p}(\mathbb{R}^{N})$$I_{\infty}(v)\leq S_{p}(1, b(x),$ $]\mathrm{R}^{N})+\alpha\Rightarrow\beta_{R}(v)\not\in B(0, r)$This proposition says thatcan be colnpared with the

4.3

Original$[I_{\infty}\leq S_{p}+\alpha](\subset M_{p}(]\mathrm{R}^{N}))$$B(0, r)^{c}(\subset 1\mathrm{R}^{N})$.

, infinite dimensional object,, finite dimensional object, via$\beta_{R}(\iota))$.

$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\iota \mathrm{n}$Now we turn to the original problelIl. We regard the original functional. We firstas a perturbed functional of$I_{\infty}$$I_{\epsilon}(v)=./\Omega/\vee \mathrm{e}(|\nabla v|^{2}+a(\in x)|v|^{\mathit{2}})d_{X}$, the relation between the level set of functional and $B(0,$note that under
described in Proposition 4.5 still holds for the perturbed functional .
$(\mathrm{A})_{\mathit{5}_{0}.\mathrm{b}.\rho}$$\uparrow\cdot \mathrm{I}^{1^{-}}$$I_{\epsilon,\vee}$

Proposition 4.6 (Concentration lemlIla at infinity for original functional)
satisfying
such that for a
For all $7’\in(0, R)$ . there exist
following holds:
there existsthat forand for all $v\in M_{p}(\Omega/\in)$ ,$\delta,$$7?y$$\eta$$\mathrm{c}-=.\backslash ^{\neg}uCh$$(A)_{\delta_{0}.b.\rho}th\epsilon$$a$$all\in\in(0,\overline{\in}]$$I_{\epsilon}(v)\leq S_{p}(1, b(x),$ $\Omega/\in)+\eta\Rightarrow\beta_{R}(v)\not\in B(0, r)$.

So we can construct the mapping:. Next we construct.
the mapping:
For anyset
where
is a cut off function such that is radiallywith respect to the origin,$\varphi(x)=1$ iff $|x|<1/2,0\leq\varphi(x)\leq 1$ iff $1/2\leq|a\cdot|<1,$
iff $|x|\geq 1$ . Let us
define.
. Then the following holds.
$\beta_{R}$$\Phi_{\epsilon}$$[I_{\vee}, \leq S_{p}+\eta]arrow B(0, r)^{\mathrm{c}}$$\partial K/\overline{\mathrm{c}}arrow[I_{\epsilon}\leq S_{p}+\eta]$$y_{\epsilon}\in\partial K/\in$$v_{\epsilon.y_{\epsilon}}(x)=\varphi(\in(x-y\xi)/r)\overline{v}_{1_{1}1.\mathrm{R}}N(X-y\epsilon)$$\mathrm{s}\mathrm{y}_{11\mathrm{u}\mathrm{n}\mathrm{e}}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}$$\varphi$$\varphi(\mathrm{L}\iota\cdot)=0$$\Phi_{\xi}(y_{\vee}G)=\mathrm{t}_{\sim.y_{\mathcal{E}}\vee}^{)}$$\forall y\vee-\in\partial K/\in$$\varphi\in C_{0}^{\infty}(]\mathrm{R}^{n})$59Proposition 4.7 (Construction of an embedding mapping fromtion space)$I_{\overline{\vee}} \mathrm{o}\Phi_{\epsilon}(y_{\epsilon})arrow c\leq S_{p}(1, \sup_{(\partial\kappa})ra(X),$$i\uparrow?y\in\partial Ku’ herey_{\epsilon}=y/\mathrm{c}\wedge$$1\mathrm{R}^{N})\leq S_{p}(1, b(X),$$\mathbb{R}^{N}$

to the func-$as\inarrow 0$$\mathbb{R}^{N})$uniformly.

satisfies (PS). For in Proposition 4.6, we can choose
It is well known thatby virtue ofso that $I\epsilon 0\Phi\epsilon(y\epsilon)\leq b<b’=S_{p}(1, b(X),$
Proposition 4.7. It is also obvious that without loss of generality we can choose
so as not to be a critical value of , since otherwise we already get infinitely many
critical values. These facts combined with Proposition 3.3 imply that
$I_{\epsilon}$$\eta$$\mathbb{R}^{N})+3\eta/4\forall y_{\epsilon}\in\partial K/\in$$b$$\overline{\mathrm{c}},$$b$$I_{\epsilon}$

the number of critical points in$[I_{\Xi}\leq b]\geq \mathrm{c}\mathrm{a}\mathrm{t}[I_{\epsilon}\leq b]$.

from below.
So in order to prove lnain theorem, we have to estimate
out this with the aid of comparison theorem of category (Proposition
We can4.8) and the fact$\mathrm{c}\mathrm{a}\mathrm{t}[I\in\leq b]$$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{r}_{d}\mathrm{v}$$\beta_{R}$:$[I_{\epsilon}\leq b]arrow B(0, r)^{c}$and$\Phi_{\epsilon}$:$\partial K/\inarrow[I_{\epsilon}\leq b]$

in view of Proposition 4.6 and 4.7.

4.4

Topological$\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{u}\iota \mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}$The following comparison theorem can be proved by a standard argument:

Proposition 4.8 (Comparison theorem for category): closed subsets. Suppose also
: topological spaces, $a\subset A,$$Barrow
A_{j}$$aarrowb,$
is homotopicallycontinuous mappings such that::equivalent to $i\uparrow ljectionaarrow A$ . Then $cat_{B}[b]\geq cat_{A}[a]$ .
$s_{up\mathrm{P}^{oS\epsilon}}A,$$\exists\Phi$$b\subset B$$B$$\beta 0\Phi$$\exists\beta$is, we can verify
In our case, with solIle technicalfor sniall .
equivalent to injection from
$B=b=[I_{\epsilon}\leq b]$ , we
$A=B(0,\uparrow’)C,$
Applying Proposition 4.8 with. Thus we have establishedfind that cat
the existence of at least two critical points of with the level below .
To get another critical value of greater than , we follow the following standard$\beta_{R}0\Phi_{\xi}$$\arg_{\mathrm{U}\mathrm{l}11}\mathrm{e}11\mathrm{t}$$\partial K/\in \mathrm{t}oB(0, \uparrow\cdot)c$$\mathrm{h}_{01\mathrm{I}\mathrm{l}\mathrm{O}}\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$\mathrm{C}-$$a=\partial K/\mathrm{c}\wedge,$$[I_{\epsilon}\leq b]\geq \mathrm{c}\mathrm{a}\mathrm{t}B(0.r)\mathrm{c}[\partial K/\overline{\mathrm{c}}]=\mathrm{c}\mathrm{a}\mathrm{t}S^{N-1}=2$$b’$$I_{\epsilon}$$b’$$I_{\Xi}$argument.
It is easy to find$v\in M_{p}(\Omega/\overline{\mathrm{c}})$such that$tu+(1-t)\Phi(\in y_{\epsilon})\neq 0$

Let us define$\forall t\in[0,1],$$\eta(t, y_{\epsilon})\in C([0,1]\cross\partial K/\in;\mathrm{i}lff_{p}(\Omega/\in))$$\forall y_{\xi}\in\partial K/\in$.

by$\eta(t, y_{\epsilon})=(tv+(1-t)\Phi_{\epsilon}(y_{\Xi}))/||tv+(1-t)\Phi\in(y\xi)||p$.

60. We shall show that
Then it is easy to see that
there exists at least one critical value in $[b’, c]$ .
Suppose on the contrary there is no critical value in $[b’, c]$ . Then by the wellknown deformation lelnlna, there exists $f\in C’(M;pM_{p})$ such that$b’<c \equiv\max_{t\in[1].\partial K}0.y\epsilon\in/\epsilon I\mathrm{o}\eta(\xi t, y\epsilon)$$f=\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{y}$

Then

on

$[I_{\xi}\leq b]\mathrm{a}\mathrm{n}\mathrm{d}.t([I_{\epsilon}\leq c])\subset[I_{\Xi}\leq b’]$.

is contralctible in $B(0, r)^{c}$ by the contraction
defined by. Sinceand$S^{N-1}$
both homotopically equivalent to, we have the contradiction.
In summary, we find that there exists at least$\bullet$$\bullet$$g(t, y_{\epsilon})\in C([0,1]\mathrm{x}$$\partial K/\mathrm{c}\wedge$$\partial K/\overline{\circ};B(0, r)c)$

two critical points in$g(t, y_{\xi})=\beta_{R}0.t\mathrm{o}\eta(t, y_{\xi})$$[I_{\xi}\leq b’]$one critical value greater than$\partial K/\in$$B(0, ?^{\tau})\mathrm{c}$

is

and
$b’$

.

Thus the main theorem is proved.
Remark. 1. The same type of multiplicity result also holds for (P). for thecase where
and $a(x)$ has a peal , or is a,n exterior
with bounded$a(x)$
complement andhas a “creek” around the “hole” of . These results together
with detailed argument of the proof of facts described above will be the subject of
the forthcoming paper $[14, 15]$ .
2. Another type of multiplicity result for $-\triangle u+u=a(x)u+.f\cdot(x)$ inisdiscussed in Adachi-K. Tanaka [16].
$\Omega=\mathbb{R}^{N}$$<$$\Omega$$\mathrm{d}_{\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{a}}\mathrm{i}\mathrm{n}$$\Omega$$1\mathrm{R}^{N}$

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