数理解析研究所講究録
911 巻 1995 年 52-61

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ASYMPTOTICS FOR SOLUTIONS OF THE TWO DIMENSIONALNONLINEAR ELLIPTIC EQUATIONS WITH CRITICAL GROWTHノ|‘$’\downarrow|\not\in \mathrm{k}$$\nearrow y\backslash ^{\iota}\mathrm{X}$$

TAKAYOSHI OGAWA (NAGOYA UNIVERSITY)1TAKASHI SUZUKI (EHIME UNIVERSITY)\S 1 Introduction and Results.
We study asymptotics for solutions of semilinear elliptic equations which have a
critical growth in two dimensions. Let be a simply connected bounded domain inwith smooth boundary. We consider the following particular equation on .
$\Omega$$\mathrm{R}^{2}$$\partial\Omega$$\Omega$$-\Delta u=\lambda ue^{u^{2}}$

$(E)$

$\{$

,

$x\in\Omega$,,$u>0$ ,

$x\in\Omega$$u=0$ ,

$x\in\partial\Omega$,where is a positive parameter.
The equation (E) arise from a variational problem related to the two-dimensional
Sobolev type inequality. Suppose that, thenwith[25] proved that there are two positive constants andsuch that$\lambda$$u\in H_{0}^{1}(\Omega)$$||\nabla u||_{2}\leq 1$$C_{0}$$\alpha$$(TM)$$r_{\mathrm{R}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}}$$\int_{\Omega}\{\exp(\alpha u2)-1\}d_{X}\leq C0|\Omega|$.

Moser [15] refined this inequality as
is true if
and not true if $\alpha>4\pi$(cf. [17]). The extremal function inwhich maximize the left hand side ofunder the restriction, if it exists, satisfies the equation (E) with
Lagrange Multiplier . In fact, this problem was solved by Carleson-Chang [6] fora radially symmetric case and by Flucher [9] for a general domain case. There are
more existence results to (E). Atkinson-Peletier studied the radially symmetric caseand gave a fine analysis of the behavior of a solution of (E) ([2], [3]). Shaw [23]and Adimurthi [1] considered some variational problems involving (E) and obtained
a solution in some situations. Among these existence results, Adimurthi constructed
a positive and in fact smooth solution to (E) for, whereis the first
eigen value
with zero Dirichlet boundary condition in .
$(\mathrm{T}\mathrm{M})$$\alpha\leq 4\pi$$H_{0}^{1}(\Omega)$$||\nabla u||_{2}\leq 1$$(\mathrm{T}\mathrm{M})$$\lambda$$0<\lambda<\lambda_{0}$$\mathrm{o}\mathrm{f}-\triangle$$\lambda_{0}$$\Omega$

1Current Address: Department of Mathematics, University of California Santa Barbara, CA 93106

USA

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Our main concern is an asymptotic behavior of the solutions (E) as-norm of the solution must blow-up assmooth solution of (E) exists, theIn fact, by multiplying (E) by and integrating by parts, we have$\lambdaarrow 0$$L^{\infty}$. If a$\lambdaarrow 0$

.

$u$

(1)

$|| \nabla u||_{2}^{2}=\lambda\int_{\Omega}u^{2}e^{u^{2}}d_{X}$.

Therefore by Poinc\’are’s inequality,(2)

as$\frac{1}{\lambda}=\frac{1}{||\nabla u||_{2}2}\int_{\Omega}u^{2}e^{u^{2}}d_{X}\leq e^{1}\infty|u||^{2}\frac{||u||_{2}^{2}}{||\nabla u||_{2}2}\leq\frac{1}{\lambda_{0}}e^{||u|}|_{\infty}^{2}arrow\infty$$\lambdaarrow 0$.

We firstly consider the unit disk case. Since by the famous result of Gidas-NiNirenberg [10], a positive smooth solution is necessarily radially symmetric. Then it
has already shown ([18], [19]) that:
Proposition$\mathrm{R}^{2},$$|x|<1\}$ .

$0$

. Let

be any posi tive smooth solu tion of $(E)$ on $\Omega=D\equiv\{x\in$$u$Then we$h\mathrm{a}\mathrm{v}e$aslocally uniformly on(3)$u(x)arrow \mathrm{O}$(4)$\lim_{\lambdaarrow 0}\lambda\int_{D}ue^{u^{2}}dX=0$(5)$\lim_{\lambdaarrow 0}\lambda\int_{D}(e^{u^{2}}-1)d_{X}=0$(6)$\varliminf_{\lambdaarrow 0}\int_{D}|\nabla u|2dx\geq 4\pi$$\lambdaarrow 0$$D\backslash \{0\}$

,

,
,

.

In this case, a microscopic structure for the above asymptotics can be stated under
an assumption of the finite energy.

Theorem 1. Let$\{(u, \lambda)\}$be a solu tions of $(E)$ sa $tis\theta ing$ th finite energy condition:

(7)

$e$

$\varlimsup_{\lambdaarrow 0}\int_{D}|\nabla u|^{2}dx\equiv E0<\infty$

Then there is a su bseq uencesatisfying(8)$\{(u_{m}, \lambda_{m})\}$

and a scaling seq uenceas$u_{m}^{2}( \gamma mx)-u_{m}2(\gamma_{m})arrow 2\log(\frac{2}{1+|_{X|^{2}}})$

locally uniformly on$\mathrm{R}^{2}$.
$\gamma_{m}arrow 0$

as$\lambda_{m}arrow 0$,$\lambda_{m}arrow 0$.

The left hand side of (8) is a scaling sequence of the solution on the scaling param, while the limit function in (8) is a unique explicit solution ofeter$\{\gamma_{m}\}$$-\Delta v=2e^{v}$

(9)

$\{$

$v=0$ ,,

$x\in \mathrm{R}^{2}$,$x\in\partial D$

54(see Chen-Li [7]).
This kind of structure was implicitly suggested by [6]. In [24], Struwe explicitly
pointed out a similar behavior for non-compact maximizing sequence to the left hand
. By using a result by Brezis-Merle [4], we have shown ([20]) that thereside ofis a subsequence of solutions of the same behavior as the above under smallness
condition of the asymptotic energy; $E_{0}<6\pi$ . Theorem 1 removes this restriction.
Next we discuss about general cases. Let
be a bounded domain and simply
connected. Set the blow up set $S$ as$(\mathrm{T}\mathrm{M})$$\Omega$$S=${$x\in\overline{\Omega}|^{\exists}x_{n}arrow x$

We extend Proposition$0$such that $u(x_{n})arrow\infty$ as$\lambdaarrow 0$}.

as in the followings.

Theorem 2. Let be a simply connected boundedsmoothtion of $(E)$ , then we have$dom\mathrm{a}in$$\Omega$. Suppose that

$u$

be a

$sol\mathrm{u}$

as

(10)

$||u||_{\infty}arrow\infty$(11)$\lim_{\lambdaarrow 0}\lambda\int_{\Omega}ue^{u^{2}}dx=0$(12)$\lim_{\lambdaarrow 0}\lambda\int_{\Omega}(e^{u^{2}}-1)dX=0$

Moreover if we$\mathrm{a}ss$$\lambdaarrow 0$

,

,

.

ume
$E_{0}= \varlimsup_{arrow\lambda 0}\int_{\Omega}|\nabla u|^{2}dx<\infty$

then for every $x\in S$ and for any $\delta>0$ such that(13),

$B_{\delta}(x)\subset\Omega$, we have$\varliminf_{\lambdaarrow 0}\int_{B_{\delta}()}x|\nabla u|^{2}d_{X}\geq 4\pi$

and
(14)

as

$\lambdaarrow 0$$u(x)arrow \mathrm{O}$locally uniformly on.

Remark that under the condition $E_{0}<\infty,$(15)$\Omega\backslash S$

is a finite set and$S$$\# S\leq\frac{1}{4\pi}E_{0}$.

We also note that the lower bound like (13) has been proved for the higher dimensional
cases (see e.g., [12]).
We apply the above result to the solution obtained by the variational method. Due
to Nehari’s critical point theory, Adimurthi [1] constructed a solution by finding aminimizer of
(16)

in

$H_{0}^{1}(\Omega)$$J_{\lambda}(v) \equiv\frac{1}{2}||\nabla v||_{2}^{2}-\frac{\lambda}{2}\int_{\Omega}(e^{v^{2}}-1)dX$

under the Nehari constraint (1). That is

55

Proposition 3 (Adimurthi [1]). For
which attainsof$0<\lambda<\lambda_{0}$, there is a minimizer$u\in H_{0}^{1}(\Omega)$$J_{\lambda}(v)$

(17)

.

$\tilde{J}_{\lambda}\equiv\inf\{J_{\lambda}(v)| v\in H_{0}^{1}(\Omega)\backslash \{\mathrm{o}\}, ||\nabla v||_{2}^{2}=\lambda\int_{\Omega}v^{2}e^{v^{2}}d_{X}\}$

The minimizer satisfies solution $(E)$ . Moreover we ave
$h$

(18)

$0<\tilde{J}_{\lambda}<2\pi$

for all$0<\lambda<\lambda_{0}$.

Remark 1. The regularity of the above solution is directly obtained by the similarargument in [4].
.
orRemark 2. There is no positive solution of (E) for
According to Theorem 2, we have the following for the solution in Proposition 3:$\lambda\geq\lambda_{1}$$\lambda\leq 0$

Corollary 4. For th solution
we have$\mathrm{e}$$u$of $(E)$ obtained by a variational formula tion in (14),(19)$\lim||\nabla u||^{2}2=4\pi$$\lambdaarrow 0$

and the blow-up set$S$consists of one interior point of , i.e.,$\Omega$$S=\{^{\exists}x_{0}\in\Omega\}$.

In the similar prob..lem of the higher dimensions, it has Shown that the blow-up
point appearing the singular limit coincides the critical point of the regular part of(see [5], [11], [22] and [26]). Therefore it is expected that
the Green functionin the above theorem would be a maximum point of the regular
the singular point
or in the other word, it might coincide the conformalpart of the Green function
center of the domain (cf. [9]).
In what follows, we shall show the sketch of proofs of Theorems and discuss about
the relation between the solution obtained other variational methods.
$\mathrm{o}\mathrm{f}-\triangle$$x_{0}$$\mathrm{o}\mathrm{f}-\triangle$\S 2 Outline of Proofs.

We show sketch of the proofs of theorems. For the radially symmetric case, wenote the following fact, which plays a crucial role of proof of Theorem 1.
Lemma 5 ([18]). Let
(20)

$u$

be any solution of $(E)$ on $\Omega=D$ . Then we have$ru_{r}(r)arrow 0$

uniformly on $D$ as$\lambdaarrow 0$,

where $r=|x|$ .

This lemma is obtained by a simple use of the Pohozaev identity ([21]) to (E) and
implies (3) $-(5)$ in Proposition .
We now formulate to show theorem 1.
$0$

56

Proof of Theorem 1. For some scaling constant$\gamma>0$, which will be determined later,

we transform the equation (E) by putting(21)$v(r)=u^{2}(\gamma r)-u^{2}(\gamma)$,

into$-\triangle v=2k(r)e-v2\gamma|2\nabla u(\rho\rho)|^{2}$

(22)

,

$0\leq r<\gamma^{-1},$ $\rho=\gamma r$

,

$\{$

$v(1)=0$ ,where.
For each , the scaling parameter$k(r)=\lambda\gamma^{2}e^{u}(\gamma)22u(\gamma r)$

$u$

(23)

$\gamma$are chosen as$u^{2}(\mathrm{O})-u^{2}(\gamma)=2\log 2$

.

Then
(24)

and

$||v||_{L}\infty\leq 2\log 2$$\gammaarrow 0$

by (3) and (23).

Lemma 6. By passing a subsequence, we observe that for some constant $\mu>0$ , we$h\mathrm{a}v\mathrm{e}$

,

(25)

$k(r)arrow\mu$(26)$\gamma^{2}|\nabla_{\rho}u(\rho)|^{2}arrow 0$

locally uniformly in$\mathrm{R}^{2}\backslash \{0\}$.

In fact, (26) is an immediate consequence of Lemma 5. For the convergence of(25), we need to show that(27)

$k(r)|r=1\leq C(E_{0})$,where the constant $C(E_{0})$ is independent of . This bound is obtained by making useof the Pohozaev identity for the equation (22). Using Lemma 5 and (27), we have$\lambda$$\mu\min(r^{-2\eta}, 1)+o(\lambda)\leq k(r)\leq\mu\max(r^{-2\eta}, 1)+o(\lambda)$,

which implies (25). By the apriori bound (24) and the Ascori-Arzela theorem yields
that there exists a limit function
satisfying, for somesubsequence of ,$v_{0}\in C(B)\cap C^{2}(\mathrm{R}^{2}\backslash \{0\})$$v$$v(r)arrow v_{0}(r)$locally uniformly on$(0, \infty)$.

57

Moreover$v_{0}$

satisfies$-\Delta v0=2\mu e^{v\mathrm{o}}$

(28)

,

$x\in \mathrm{R}^{2}\backslash \{0\}$$\{$$v_{0}=0$,$x\in\partial B$

,

.

Since
(29)

$||v_{0}||_{L^{\infty}(}B)\leq 2\log 2$,

we conclude$v_{0}=2 \log\frac{2}{1+r^{2}}$and $\mu=1$ . The uniform convergence$v(r)arrow v_{0}(r)$on any compact set$K\subset\subset \mathrm{R}^{2}$

follows immediately. This shows the sketch of proof

Theorem 1.
Next we consider the general case.
Before proving Theorem 2, we need the following inequalities.
Lemma 7. For any positive smooth solution$u$of $(E)$ ,(30)$4 \pi\lambda\int_{\Omega}(e^{u^{2}}-1)d_{X}\leq(\lambda\int_{\Omega}ue^{u^{2}}dx)^{2}$(31)$( \lambda\int_{\Omega}ue^{u^{2}}dx)^{2}\leq\sigma_{\Omega}\lambda\int_{\Omega}(e^{u^{2}}-1)dx$

where$\sigma_{\Omega}$,,

is a constant de termined by the conformal map from $D$ to$\Omega$.

The relation (30) is a consequence of the isoperimetric inequality and simple argument of the distribution functions to ( . Chen-Li [7]). The second inequality (31)is obtained by the Pohozaev identity.
$u$$\mathrm{c}.\mathrm{f}$Proof of Theorem 2. Since (10) has already shown, we show (11) and (12). Rom theinequality (31), it follows for any $t>0$ ,$( \lambda\int_{\Omega}ue^{u^{2}}dx)^{2}\leq\sigma_{\Omega}\lambda\{\int_{u>t}(e^{u^{2}}-1)d_{X+}\int_{u\leq t}(e^{u^{2}}-1)d_{X}\}$$\leq\frac{C_{\Omega}}{t}\lambda\int_{u>t}ue^{u}dX+\lambda\sigma\Omega|\Omega|(e^{t}22-1)$

Then by letting$\lambdaarrow 0$,$\varlimsup_{\lambdaarrow 0}\lambda\int_{\Omega}ue^{u^{2}}dX\leq\frac{C_{\Omega}}{t}$,. This proves (11) and (12) by (30).
which goes to as
To show (13) and (14), we need the following:$0$$tarrow\infty$

58

Lemma 8. For any$K\subset\subset\Omega$(32)

and $1\leq p<\infty$ , we$\int_{K}u^{p}d_{X}arrow 0$

In fact, using the first eigen functionas$h\mathrm{a}ve$$\lambdaarrow 0$$\phi_{1}\mathrm{o}\mathrm{f}-\triangle|_{0}$.

,$\lambda_{1}\int_{\Omega}\phi_{1}udx=\lambda\int_{\Omega}\phi_{1}ue^{u^{2}}dX\leq C\lambda\int_{\Omega}ue^{e^{2}}d_{X}arrow \mathrm{O}$by Theorem 3, which shows. The conclusion follows from $E_{0}<\infty$ and
the Gagliardo-Nirenberg inequality.
Note thatby the boundary condition and Hopf’s lemma (see [8] and [11]).
For $x\in S$ , let. We then assume for the contrary that$\int_{K}udxarrow \mathrm{O}$$S\subset\subset\Omega$$2\delta=d(x, \partial\Omega)$(33)$\int_{B_{2\delta}}|\nabla u|^{2}d_{X<}4\pi$.

Introducing a cut off function, wherewe see by the Schwartz inequality, Lemma 8 and$\phi_{\delta}(\cdot)=\phi(\frac{-x}{\delta})$(34)$\phi\in C_{0}^{\infty}(B_{2}),$$\phi=1$

on$B_{1}$,$|| \nabla(\phi_{\delta}u)||_{2}2\leq(1+\epsilon)\int\phi_{\delta}^{2}|\nabla u|2dx+(1+\frac{1}{\epsilon})\int u^{2}|\nabla\phi_{\delta}|^{2}dX$$\leq(1+\epsilon)\int_{B_{2\delta}}|\nabla u|^{2}+c(1+\frac{1}{\epsilon})\delta^{-2}\int_{B_{2\delta}\backslash B_{\delta}}u^{2}dX$$<4\pi$as$\lambdaarrow 0$. Then consider the localized equation;$-\Delta(\phi_{\delta/}2u)=\lambda\phi_{\delta/2}ue^{u}2-F(u, \phi_{\delta/2})$,$x\in B_{\delta}$

,

$\{$

$\phi_{\delta/2}u=0$

where

,

$x\in\partial B_{\delta}$,$F(u, \psi)=2\nabla u\cdot\nabla\psi_{\delta}/2+u\triangle\phi_{\delta/2}\in L^{2}(B_{\delta})$$L^{1+\eta’}(B_{2\delta})$. Since by, we see for $\eta>0$ ,$\lambda\phi_{\delta/2}ue^{u^{2}}\in L^{1+\eta}(B_{\delta})$

The standard elliptic estimate implies$||u||L\infty(B_{\delta}/2)\leq C$,

which impossible since $x\in S$ . Hence we have(35)$\int_{B_{2\delta}}|\nabla u|^{2}dx\geq 4\pi$.

.

$(\mathrm{T}\mathrm{M})$and (34)$e^{\phi_{\delta}^{2}u^{2}}\in$59

For any compact set(E),$K\subset\subset\Omega\backslash S$, we have$||u||_{L(}\infty K$$||\triangle u||L\infty(K)\leq C$

By Lemma 8, this implies$||u||_{L(}\infty K$)$arrow 0$

as)$\leq C$and hence by the equation.

$\lambdaarrow 0$. This proves (14).

Proof of Corollary 4. Since (12), (13) and (18) we see.

$4\pi\leq\varliminf_{\lambdaarrow 0}||\nabla u||_{2}^{2}\leq\varlimsup_{\lambdaarrow 0}||\nabla u||2\varlimsup 2=J2\lambdaarrow 0\lambda(u)\leq 4\pi$This shows (19) and by (15),$S=\{x\mathrm{o}\}$.

\S 3 Relations of Variational Solutions.

Finally we remark the relation between the solution in Proposition 3 and the solution obtained by the variational problem by Shaw. In [23], Shaw considered a different
kind of variational solution to (E).

Proposition 9 (Shaw). For$u\in H_{0}^{1}(\Omega)$with$||\nabla u||_{2}2<4\pi$$I(u)= \int_{\Omega}(e^{u^{2}}-1)dx=\mu$

there exists a minimizer$of||\nabla u||^{2}2$

with

,

.

The ”dual” of Shaw’s formulation is the natural variational problem associate to$(\mathrm{T}\mathrm{M})$.

Proposition 10 ([6], [9]). There exists a maxim izer of$I(u)= \int_{\Omega}(e^{u^{2}}-1)dx$in$H_{0}^{1}(\Omega)$

$mul$

with

$||\nabla u||_{2}2\leq 4\pi$. The maximizer solves $(E)$ with a certain Lagrange

tiplier .
$\lambda$

The solution obtained in Proposition 10 is in fact a minimizer of$J_{\lambda}(u)$.

Proposition 11. Let be a solution obtained in Proposition 10 with some . Then
it is a solution obtain $ed$ in Proposition 3, i.e.,$\lambda$$u$$J(u)_{\lambda\lambda}=\tilde{J}$be a solution which maximize $I(u)$ with, we can choose some $t_{0}>0$ such that

Proof of Proposition 11. Suppose$4\pi$. For any$v\in H_{0}^{1}(\Omega)$.

$u$$|| \nabla v||_{2}2=\lambda\int_{\Omega}v^{2}e^{t^{22}}\mathrm{O}d_{X}v$.

$||\nabla u||_{2}^{2}\leq$60

Note that for all $t>0,$, we have$J_{\lambda}(tv)\leq J_{\lambda}(t0v)$

since$\frac{\partial}{\partial t}J_{\lambda}(tv)|_{tt_{0}}==0$. By setting $s^{2}=$$||\nabla u||_{2}2/||\nabla v||_{2}2$$J_{\lambda(u)||u|}= \frac{1}{2}\nabla|_{2}^{2}-\frac{\lambda}{2}\int_{\Omega}(e^{u^{2}}-1)dX$$\leq\frac{1}{2}||\nabla(Sv)||_{2^{-\frac{\lambda}{2}}}^{2}\int_{\Omega}(e^{(u)^{2}}-S1)d_{X=}J_{\lambda}(_{S}v)\leq J_{\lambda}(t_{0}v)$

Therefore$J_{\lambda}(u)= \inf_{v\in V}J_{\lambda}(v)$, where.

$V= \{v\in H_{0}^{1}(\Omega)\backslash \{\mathrm{o}\}, ||\nabla v||_{2}^{2}=\lambda\int_{\Omega}v^{2}e^{v}d_{X}\}2$.

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