数理解析研究所講究録
第 1779 巻 2012 年 128-139

128

Properties of a least-energy solution to a semilinear
elliptic equation with exponential nonlinearity
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
In this paper we shall study the following problem;(1.1)$\{\begin{array}{ll}\epsilon^{2}\Delta u-u+(e^{\alpha u}-\alpha u-1)=0 in \Omega,u>0 in \Omega,u=0 on \partial\Omega,\end{array}$$\alpha>0$ , andwith the smooth boundary
where
is a bounded domain in.
We are interested in some properties of least-energy solutions to (1.1) as
First we refer to preceding studies of problems similar to (1.1);$\epsilon>0,$$R^{2}$$\Omega$$\partial\Omega$.

$\epsilonarrow 0$(1.2)$\{\begin{array}{ll}\epsilon^{N}\triangle u-u+u^{p}=0 in \Omega,u>0 in \Omega,\frac{\partial u}{\partial n}=0 on \partial\Omega,\end{array}$where
and $N\geq 2$ . Many people have studied the existence of positive solutions. For example Lin,and the behavior of a local maximum point of solutions to (1.2) as
Ni and Takagi [11] proved the existence of positive non-constant solutions for sufficiently, and moreover the solutions are sufficiently close to a zero except for the
small
local maximum value. Later Ni and Takagi [13], [14] showed that the solution found in[11] is a least-energy solution, and the solution has at most one local maximum point, the maximum pointon the boundary. In addition they proved that, as
must be located near the point where the mean curvature attains its maximum. On theother hand, Ni and Wei [15] investigated the problem (1.2) whose boundary condition is
replaced by the Dirichlet boundary condition. They showed that the Dirichlet problem, the maximum point
is situatedalso has a single-peak solution. Moreover, as
near the most distant point from the boundary.
In the cases of both the Neumann boundary condition and the Dirichlet boundary
condition, the proof of the behavior ofis due to asymptotic formulas of a criticalvalue which corresponds to the least-energy solution. In [14] and [15], the uniqueness ofa solution to$\Omega\subset R^{N}$$\epsilonarrow 0$$\epsilon>0$$P_{\epsilon}$$\epsilonarrow 0$$\partial\Omega$$\epsilonarrow 0$$P_{\epsilon}$$P_{\epsilon}$$P_{\epsilon}$$\{\begin{array}{ll}\triangle w-w+w^{p}=0 in R^{N},w>0 in R^{N},w(z)arrow 0 as |z|arrow\infty,w(0)=\max_{z\in R^{N}}w(z), \end{array}$(1.3)

is used to obtain the asymptotic formulas. In contrast, without using the uniqueness ofthe solution to (1.3), del Pino and Felmer [5] proved the same formulas on the behavior

129of the maximum point for both the Dirichlet problem and the Neumann problem. In
addition Byeon [3] also showed the more general result on the behavior of without theuniqueness of the solution to (1.3) for the Neumann problem with $N\geq 3$ . However it isdifficult to know the precise profile of the solution without using the uniqueness of thesolution to (1.3). Thus del Pino and Felmer result or Byeon’s result does not completely
include the Ni and Takagi result.
In the above studies, a single-peak solution is only treated. For a multi-peak solution,e.g., see Gui and Wei [8]. They constructed solutions which has many critical points inthe domain by the Lyapunov-Schmidt reduction method. Similarly Gui, Wei and Winter[9] constructed solutions which have many critical points on the boundary.
Our purpose is to investigate the behavior of the local maximum point of a least-energy
solution and the profile of the least-energy solution to the problem (1.1). Concerning thepreceding studies of the exponential nonlinearity case, Ni and Kabeya [12] investigate
the problem having the exponential nonlinearity with the Neumann boundary condition.
They proved that, for the exponential nonlinearity case, the same result as the power
nonlinearity case holds, that is, least-energy solutions have only one local maximum point$P_{\epsilon}$

$s$

$s$

$P_{\epsilon}\in\partial\Omega$.

First, to define the terminology ”least-energy”, we define the “energy“ functionalcorresponding to (1.1). The functional is as follows:on$J_{\epsilon}(u)= \frac{1}{2}\int_{\Omega}(|\nabla u|^{2}+u^{2})dx-\frac{1}{\alpha}\int_{\Omega}(e^{\alpha u}-\frac{1}{2}(\alpha u)^{2}-\alpha u-1)dx$$H_{0}^{1}(\Omega)$. (1.4)

If there exists a critical pointof , thenis a weak solution to (1.1). In addition
if attains the smallest positive critical value at , thenis said to be a least-energy$J_{\epsilon}$$u_{\epsilon}$$u_{\epsilon}$$J_{\epsilon}$$u_{\epsilon}$$u_{\epsilon}$

solution. In our investigation, we will find a critical value ofMountain Pass Lemma. Namely we set$J_{\epsilon}$by using the well-known$\Gamma=\{h\in C([0,1];H_{0}^{1}(\Omega))|h(0)=0, h(1)=e_{0}\}$

where
value

is a non-negative function incharacterized as$e_{0}\not\equiv 0$$c_{\epsilon}$$H_{0}^{1}(\Omega)$and satisfies,$J_{\epsilon}(e_{0})=0$. Then a critical$c_{\epsilon}= \inf_{h\in\Gamma}\max_{0\leq t\leq 1}J_{\epsilon}(h(t))>0$

is found, and there exists
proposition holds;
Proposition 1.1 Let
two statements hold:$u_{\epsilon}\in H_{0}^{1}(\Omega)$$\epsilon_{0}>0$such that$J_{\epsilon}(u_{\epsilon})=c_{\epsilon}$be sufficiently small. Then, for any(i) There exists a positive solution
it holds that$u_{\epsilon}\in H_{0}^{1}(\Omega)$. Moreover$u_{\epsilon}$. In fact the following$\epsilon\in(0, \epsilon_{0})$, the following

is a classical solution, and(1.5)$J_{\epsilon}(u_{\epsilon})\leq C_{0}\epsilon^{2}$

with some constant $C_{0}>0$ which is independent of . Furthermore
energy solution.
$\epsilon$(ii) The least-energy solution$u_{\epsilon}$

has exactly one local maximum point$P_{\epsilon}$.

$u_{\epsilon}$

is a least-

130
Remark 1.1 The existence of the, least-energy solution to (1.1) is actually proved for. However it is required that is restricted within a small value when weanyshow the inequality (1.5) and Proposition 1.1 (ii).
$u_{\epsilon}$$\epsilon\in(0, \infty)$$\epsilon$

By Proposition 1.1, we see that there exists a least-energy solution, and the solution has
at most one maximum point.
Next we explain the result on the profile and the behavior of the least-energy solution. Concerning the, that is, the number of maximum points and its behavior as
method to investigate those, it seems that we can apply the method used in del Pino andFelmer study [5] for our problem by some modification. However we will investigate themore precise profile of by the method of Ni and Wei, and thus some preliminaries are
required to make use of the method. Hereafter we state those.
First we consider the following whole space problem;$\epsilonarrow 0$$u_{\epsilon}$$s$$u_{\epsilon}$$\{\begin{array}{ll}\Delta w-w+(e^{\alpha w}-\alpha w-1)=0 in R^{2},w>0 in R^{2},w(z)arrow 0 as |z|arrow\infty,w(0)=\max_{z\in R^{2}}w(z).

The solution$w$

to (1.6) seems to be approximate to$u_{\epsilon}$. In fact(1.6)\end{array}$$v_{\epsilon}(y)$$:=u_{\epsilon}(\epsilon y+P_{\epsilon})$satisfies$\{\begin{array}{ll}\Delta v_{\epsilon}-v_{\epsilon}+(e^{\alpha v_{\epsilon}}-\alpha v_{\epsilon}-1)=0 in \Omega_{\epsilon}^{*},v_{\epsilon}>0 in \Omega_{\epsilon}^{*},v_{\epsilon}=0 on \partial\Omega_{\epsilon}^{*},\end{array}$where$\Omega_{\epsilon}^{*}:=\{y\in R^{2}|\epsilon y+P_{\epsilon}\in\Omega\}$

(1.7)

.

. In fact it is proved that uniformly converges, the domain
dilates in. The existence of a solution to (1.6) is guaranteed
to for any compact subset of
by Berestycki, Gallouet alid Kavian [2], and the uniqueness of is proved by Pucci and
Serrin [16], [17]. Moreover, since we can apply the result of Gidas, Ni and Nirenberg [7]for (1.6), it is seen that is radially symmetric and strictly decreasing in $:=|z|$ withthe order of $r^{-1/2}e^{-r}$ . We state these results as Lemma 2.6 again; see Section 2.
In addition, to show our result, we confirm that is nondegenemte. The definition ofis said to be nondegenerate if the linearized
nondegenemte is as follows: the solution
operator$H^{2}(R^{2})arrow L^{2}(R^{2})$$\triangle-1+\alpha(e^{\alpha w}-1)$ :As$R^{2}$$\epsilonarrow 0$$v_{\epsilon}$$\Omega_{\epsilon}^{*}$$R^{2}$

$w$

$w$

$w$

$r$

$w$

$w$

$w$

$:=\{v\in L^{2}(R^{2})|v(z)=v(|z|)\}$ . In fact
has a bounded inverse in the subspace
the solution to (1.6) is nondegenerate, and it is proved in the forthcoming paper [10].
Second we introduce some notations. For the solution to (1.6) and any bounded, let
be a unique solution to the linear problem
smooth domain$L_{rad}^{2}(R^{2})$

$w$

$w$

$U\subset R^{2}$$w_{U}$$\{\begin{array}{ll}\triangle w_{U}-w_{U}+(e^{\alpha w}-\alpha w-1)=0 in U,w_{U}=0 on\partial U.\end{array}$

131
In fact, since the operator: $H^{2}(U)arrow L^{2}(U)$ is bounded and invertible, thesolutionis unique. Moreover we apply the maximum principle for $w-w_{U}$ , and it
follows that$w_{U}(y)<w(y)$for any $y\in U$.
(1.8)
Let be the most distant point from the boundary, that is, is the point with
, wheredenotes the distance between
and the boundaryof . Adding to (1.7), we define$(\triangle-1)$$w_{U}$$Q$$Q$$\max_{P\in\Omega}d(P, \partial\Omega)$$d(Q, \partial\Omega)=$$P\in\Omega$$d(P, \partial\Omega)$$\Omega$$\Omega_{\epsilon}=\{y\in R^{2}|\epsilon y+Q\in\Omega\}$

and we obtainand$w_{\Omega_{\epsilon}}$$w_{\Omega_{\epsilon}^{*}}$,. Moreover we define(1.9)$\{\begin{array}{ll}\varphi_{\epsilon}(y):=w(y)-w_{\Omega_{\epsilon}}(y) y\in\Omega_{\epsilon},\psi_{\epsilon}(x):=-\epsilon\log\varphi_{\epsilon}((x-Q)/\epsilon) x\in\Omega,V_{\epsilon}(y):=e^{\beta\psi_{\epsilon}(Q)}\varphi_{\epsilon}(y) y\in\Omega_{\epsilon},\end{array}$and(1.10)$\{\begin{array}{ll}\tilde{\varphi}_{\epsilon}(y):=w(y)-w_{\Omega_{\epsilon}^{*}}(y) y\in\Omega_{\epsilon}^{*},\tilde{\psi}_{\epsilon}(x):=-\epsilon\log\tilde{\varphi}_{\epsilon}((x-P_{\epsilon})/\epsilon) x\in\Omega,\tilde{V}_{\epsilon}(y):=e^{\beta\overline{\psi}_{\epsilon}(P_{\epsilon})}\tilde{\varphi}_{\epsilon}(y) y\in\Omega_{\epsilon}^{*},\end{array}$where. Since (1.8), the functions and are well-defined. In addition anddefined above have the following properties: for any sequencewithas$karrow\infty$, there exists a subsequencesuch thatuniformly on any compact
subset ofas. Here $u=V_{0}$ is a solution to$\beta=1/\epsilon$$\tilde{\psi}_{\epsilon}$$\psi_{\epsilon}$$V_{\epsilon}$$\tilde{V}_{\epsilon}$$\epsilon_{k}arrow 0$$\{\epsilon_{k}\}$$\{\epsilon_{k_{l}}\}$$R^{2}$$V_{\epsilon_{k_{l}}}arrow V_{0}$$larrow\infty$$\{\begin{array}{ll}\triangle u-u=0 in R^{2},u>0 in R^{2},u(0)=1.

(1.11)\end{array}$

Similarly, there exists a.subsequencesuch thatuniformly on any compact
subset ofas $marrow\infty$ , where is also a solution to (1.11).
We define the functional associated with (1.6) as$\{\epsilon_{k_{m}}\}$$R^{2}$$\tilde{V}_{\epsilon_{k_{m}}}arrow\tilde{V}_{0}$$\tilde{V}_{0}$$I(w):= \frac{1}{2}\int_{R^{2}}(|\nabla w|^{2}+w^{2})dz-\frac{1}{\alpha}\int_{\Omega}(e^{\alpha w}-\frac{1}{2}(\alpha w)^{2}-\alpha w-1)dz$

on$H^{1}(R^{2}),$$(1.12)$

and, from the above preliminaries, we state Theorem 1.1;

Theorem 1.1 Letandbe the least-energy solution to (1.1), the critical value
corresponding toand the maximum point of , respectively. Moreover the uniquesolution to (1.6) is denoted by . Then the following statements hold:(i) The critical value is asymptotically expanded as$u_{\epsilon},$$P_{\epsilon}$$c_{\epsilon}$$u_{\epsilon}$$u_{\epsilon}$$w$$c_{\epsilon}$as$c_{\epsilon}=\epsilon^{2}\{I(w)+\gamma e^{-\beta\tilde{\psi}_{\epsilon}(P_{\epsilon})}+o(e^{-\beta\tilde{\psi}_{\epsilon}(P_{\epsilon})})\}$

where$\beta=1/\epsilon$. Here$\gamma$$V_{*}$,

is a constant defined as$\gamma:=\frac{1}{2}\int_{R^{2}}(e^{\alpha w}-\alpha w-1)V_{*}dz>0$

where$\epsilonarrow 0$,

is a unique, positive and mdially symmetric solution to (1.11).

(1.13)

132(ii) It holds thatas$d(P_{\epsilon}, \partial\Omega)arrow\max_{P\in\Omega}d(P, \partial\Omega)$

Furthermore$\tilde{\psi}_{\epsilon}(P_{\epsilon})arrow 2\max_{P\in\Omega}d(P, \partial\Omega)$(iii) For any sequence$\phi_{\epsilon_{k_{l}}}\in H_{0}^{1}(\Omega_{\epsilon_{k_{l}}}^{*})$

withsuch that$\{\epsilon_{k}\}$$\epsilon_{k}arrow 0$

asas$\epsilonarrow 0$$karrow\infty$$\epsilonarrow 0$.

, there exists a subsequenceas$u_{\epsilon_{k_{l}}}(x)=w_{\Omega_{\epsilon}^{r_{k_{l}}}}(y)+\exp(-\beta_{k_{l}}\tilde{\psi}_{\epsilon_{k_{l}}}(P_{\epsilon_{k_{\downarrow}}}))\phi_{\epsilon_{k_{\downarrow}}}(y)$

where

and

$x=\epsilon_{k_{l}}y+P_{\epsilon_{k_{l}}}$$\beta_{k_{l}}=1/\epsilon_{k_{l}}$$\phi_{0}$$larrow\infty$$\{\epsilon_{k_{l}}\}$and,. Hereas$\Vert e^{-\mu|y|}(\phi_{\epsilon_{k_{l}}}-\phi_{0})\Vert_{L}\infty(\Omega_{\epsilon_{k_{l}}}^{t})arrow 0$

where $0<\mu<1$ . Here.

$larrow\infty$(1.14),

is a solution toin$\Delta\phi_{0}-\phi_{0}+\alpha(e^{\alpha w}-1)(\phi_{0}-\tilde{V}_{0})=0$$R^{2}$.

Remark 1.2 Although the uniqueness of a solution to (1.11) is not guaranteed, the fol(it isand the mdially symmetric solution
lowing fact is known: for any solution(1.11),
itfunction)
that
holds
to
explicitly solved by using the modified Bessel$V$$V_{*}$$\frac{1}{2}\int_{R^{2}}(e^{\alpha w}-\alpha w-1)Vdz=\frac{1}{2}\int_{R^{2}}(e^{\alpha w}-\alpha w-1)V_{*}dz$,

where is the solution to (1.6). This fact is used in Theorem 1.1, and, conceming theproof of this fact, see Caffarelli and Littman [4].
$w$

To prove Theorem 1.1 (iii), we require the uniqueness and nondegeneracy of the solution
is approximate to the solution
to (1.6). The asymptotic equality implies that, and the remainder term is exponentially small.
to (1.11) with$w$$u_{\epsilon}$$w_{\Omega_{e_{k_{l}}}^{*}}$$U=\Omega_{\epsilon_{k_{l}}}^{*}$

2

Proof of Proposition 1.1

In this section we show Proposition 1.1. Before beginning the proof, we state the next
two lemmas.

Lemma 2.1 Let
inequality holds:$\Omega$satisfy the cone condition. For$u\in H_{0}^{1}(\Omega)$$\Vert u\Vert_{Lp(\Omega)}\leq K_{0}p^{\frac{1}{2}}\Vert u\Vert_{H_{0}^{1}(\Omega)}$and $p>1$ , the following,

where the constantis dependent on the cone property of but is independent ofMoreover,.
andfor $\xi\in[1, \infty)$ , let $\Omega_{\xi}^{d};=\{y|\xi y\in\Omega\}$ . Then there also holds$\Omega$$K_{0}$$|\Omega|$$\Vert u\Vert_{L^{p}(\Omega_{\xi}^{d})}\leq K_{0}p^{\frac{1}{2}}\Vert u\Vert_{H_{0}^{1}(\Omega_{\xi}^{d})}$

.

$p$

133
Concerning the cone condition, e.g. see Adams [1]. Lemma 2.1 implies that the imbedding
constant
is not required to be replaced when the domain is dilated. This lemma isshown by the inequality$K_{1}$$\Omega$$\Vert u\Vert_{Lp(\Omega)}\leq K_{1}p^{\frac{1}{2}}|\Omega|^{\frac{1}{p}}\Vert u\Vert_{H_{0}^{1}(\Omega)}$with some constant $K_{1}>0$ (see Lemma 2.1 in [18]) and the strategy in which we use
a family of subsets ofwhich have finite intersection property (see Lemma 5.14 andCorollary 5.16 in [1] .
$R^{2}$$)$

Lemma 2.2 (Proposition 4.3 in [19]) Let $M$ be a bounded smooth domain, and set$\{u_{j}\}\subset H^{1}(M)$ .
strongly in
If $u_{j}arrow u$ weakly in $H^{1}(M)$ as $jarrow\infty$ , then$L^{1}(M)$ aswith any.
By using the above lemmas, we state the proof of Proposition 1.1 (i).
$e^{lu_{j}}arrow e^{lu}$$jarrow\infty$$l\in \mathbb{R}$Proof of Proposition 1.1 (i). If the solution

is found, then, from Lemma 2.2
and the Schauder regularity theorem, it is proved that is a classical solution. Moreover,by the maximum principle, it follows thatin . Hence it suffices that we show theexistence of.
In order to apply the Mountain Pass Lemma for (1.4), it suffices that the following
three conditions are satisfied:(a) The functional
satisfies the Palais-Smale condition.
$u_{\epsilon}\in H_{0}^{1}(\Omega)$$u_{\epsilon}$$u_{\epsilon}>0$$\Omega$$u_{\epsilon}\in H_{0}^{1}(\Omega)$$J_{\epsilon}$(b) There exists some $t_{*}>0$ such that $J_{\epsilon}(u)\geq c>0$ for
where some constant $c>0$ .
(c) For sufficiently small
and$J_{\epsilon}(e_{0})=0$$\epsilon>0$$u\in H_{0}^{1}(\Omega)$

with$\Vert u\Vert_{H_{0}^{1}(\Omega)}=t_{*}$, there exists a nonnegative function
for $t\in[0,1]$ , where a constant $C_{0}>0$ .

$e_{0}\in H_{0}^{1}(\Omega)$,such that$J_{\epsilon}(tv)\leq C_{0}\epsilon^{2}$

The condition (a) is shown by using Lemma 2.2 which implies that the mapping
is compact for any $l>0$ . Thus the compactness ofisguaranteed, and therefore
satisfies the Palais-Smale condition.
Next we confirm the condition (b). By using Lemma 2.1, we obtain$H_{0}^{1}(\Omega)\ni u\mapsto e^{lu}\in L^{1}(\Omega)$$J_{\epsilon}$$J_{\epsilon}$$J_{\epsilon}(u) \geq\{\frac{\epsilon^{2}}{2}-\sum_{k=3}^{\infty}\frac{(\alpha k^{\frac{1}{2}}K_{0})^{k}}{k!}\Vert u\Vert_{H_{0}^{1}(\Omega)}^{k-2}\}\Vert u\Vert_{H_{0}^{1}(\Omega)}^{2}$

Hence it suffices that we take sufficiently small.
Finally the condition (c) is confirmed. We take.

$t_{*}$$\epsilon>0$

sufficiently small, and define$v(x)=\{\begin{array}{ll}\epsilon^{2}(1-\frac{|x|}{\epsilon}) (|x|<\epsilon),0 (|x|\geq\epsilon).\end{array}$

If is sufficiently large, then
. In contrast, since condition (b),when
is sufficiently small, and hence there exists such that
. It suffices that wetake $e_{0}:=t_{0}v$ . In addition, for the function , we obtain$t$$J_{\epsilon}(tv)<0$$J_{\epsilon}(tv)>0$$t$$t_{0}$$J_{\epsilon}(t_{0}v)=0$$v$$\{\begin{array}{ll}\int_{\Omega}|\nabla v|^{2}dx=D_{0}\epsilon^{-2} with D_{0}=\pi,\int_{\Omega}v^{s}dx=D_{s}\epsilon^{1-s} with D_{s}=2\pi\int_{0}^{1}(1-r)^{s}rdr and s>0.\end{array}$

(2.1)

134
The functional$t\in[0, t_{0}]$is estimated by using (2.1), and hence it follows that

$J_{\epsilon}$.

$J_{\epsilon}(tv)\leq C_{0}\epsilon^{2}$forSince the conditions $(a)-(c)$ are satisfied, we can apply the Mountain Pass Lemma andfind a solution to (1.1). Furthermore, from$u_{\epsilon}$$c_{\epsilon}= \inf_{h\in\Gamma}\max_{0\leq t\leq 1}J_{\epsilon}(h(t))>0$,$\Gamma=\{h\in C([0,1];H_{0}^{1}(\Omega))|h(0)=0, h(1)=e_{0}\}$, we see thatand
(1.5) is obtained.
Finally we define$e_{0}\in\Gamma$$J_{\epsilon}(u_{\epsilon})=c_{\epsilon} \leq\max_{0\leq t\leq 1}J_{\epsilon}(e_{0})\leq C_{0}\epsilon^{2}$for$M[v]$ $:= \sup_{t\geq 0}J_{\epsilon}(tv)$and let$c_{*}$,. Therefore the inequality$v\in H_{0}^{1}(\Omega)$,be$c_{*}:= \inf\{M[v]|v\in H_{0}^{1}(\Omega),$ $v\not\equiv O$, and therefore
Then we can show that
3.1 in [13] . The proof is finished.
$c_{\epsilon}=c_{*}$$u_{\epsilon}$

and $v\geq 0$ in$\Omega\}$.

is a least-energy solution (see Lemma$\blacksquare$$)$

From Proposition 1.1 (i) and Lemma 2.1, the following results is obtained:

Lemma 2.3 Let be the least-energy solution to (1.1). Then, for $p\in[1, \infty)$ and suffi, the following inequalities hold:ciently small$u_{\epsilon}$$\epsilon>0$$\int_{\Omega}(\epsilon^{2}|\nabla u_{\epsilon}|^{2}+|u_{\epsilon}|^{2})dx=\int_{\Omega}u_{\epsilon}f(u_{\epsilon})dx\leq C_{1}\epsilon^{2}$$su_{\frac{p}{\Omega}}u_{\epsilon}\leq C_{1}x\in$,$\int_{\Omega}u_{\epsilon}^{p}dx\leq M_{p}\epsilon^{2}$

where

$C$

and

$M_{p}$

,

,

(2.2)

are independent of .
$\epsilon$

By using the above lemma, Proposition 1.1 (ii) is proved. Hereafter
andare theleast-energy solution and the corresponding critical value of (1.4), respectively.
Next we show Proposition 1.1 (ii), and we refer to the following two lemmas before
beginning the proof. Hereafter let be a local maximum point of the least-energy solution
found in Proposition 1.1 (i), is the unique solution to (1.6), and is the functional
defined in (1.12).
$u_{\epsilon}$$c_{\epsilon}$$P_{\epsilon}$

$I$

$w$

$u_{\epsilon}$

Lemma 2.4 It holds that$c_{\epsilon}\leq\epsilon^{2}(I(w)+o(1))$

as$\epsilonarrow 0$.

Lemma 2.5 Let
be sufficiently small and $R_{0}>0$ satisfy
there exists a positive constant $\eta>0$ , independent of , such that$\epsilon_{1}>0$$\epsilon$$u_{\epsilon}(x)\geq\eta$with any$\epsilon\in(0, \epsilon_{1})$.

for$x\in B_{\epsilon R_{0}}(P_{\epsilon})$(2.3)$B_{2\epsilon_{1}R_{0}}(P_{\epsilon})\subset\Omega$. Then135
The inequality (2.3) is a rougher estimate than (1.13), and it is obtained from the resulton the behavior of $w(r)$ for sufficiently large and the fact thatis the least-energy
solution. Lemma 2.5 follows from the Harnack inequality.
$r$$u_{\epsilon}$

Proof of Proposition 1.1 (ii). Define$\rho_{\epsilon}:=\frac{d(P_{\epsilon},\partial\Omega)}{\epsilon}$

and we show that

as

,

.
First suppose that there exist some constant $C>0$ and some sequencesuch thatas $jarrow\infty$ . Then, since, there exists somesuch thatas $jarrow\infty$ by passing a subsequence if necessary (the subsequence is still denotedby. We introduce a diffeomorphism straightening the boundary. By translation androtation, we can set the coordinates such thatis at the origin, -axis is tangent to, and -axis points to the interior of . Then there exists a smooth functionat, which is defined for sufficiently small, such that$\rho_{\epsilon}arrow\infty$$\epsilonarrow 0$$\{\epsilon_{j}\}$$\rho_{\epsilon_{j}}\leq C$$P_{0}\in\partial\Omega$$d(P_{\epsilon}j’\partial\Omega)\leq C\epsilon_{j}$$P_{\epsilon_{j}}arrow P_{0}$$\{\epsilon_{j}\})$$P_{0}$$\partial\Omega$$P_{0}$$\zeta_{P_{0}}(x_{1})$

(i)
(ii)

$|x_{1}|$$\zeta_{P_{0}}(0)=0$, and$\zeta_{P_{0}}’(0)=0$.

$\partial\Omega\cap \mathcal{N}=\{(x_{1}, x_{2})|x_{2}=\zeta_{P_{0}}(x_{1})\}$

a neighborhood ofFor$y\in \mathbb{R}^{2}$$x_{1}$$\Omega$$x_{2}$$P_{0}$

and$\Omega\cap \mathcal{N}=\{(x_{1}, x_{2})|x_{2}>\zeta_{P_{0}}(x_{1})\}$.

near , we define the map$0$$x=\Phi_{P_{0}}(y)=(\Phi_{P_{0},1}(y), \Phi_{P_{0},2}(y))$, where$\mathcal{N}$isas$\{\begin{array}{l}\Phi_{P_{0},1}(y)=y_{1}-y_{2}\zeta_{P_{0}}’(y_{1}),\Phi_{P_{0},2}(y)=y_{2}+\zeta_{P_{0}}(y_{1}).\end{array}$Since, the differential map
satisfies $D\Phi_{P_{0}}=I$ (the identity map). Hence,by the inverse function theorem,
has the inverse map $y=\Psi_{P_{0}}(x)(=(\Phi_{P_{0}})^{-1}(x))$ in
the small neighborhood of $x=P_{0}$ . We assume thatis defined in the closed ball(in the rest of this paper, $B_{R}(P)$ denotes $\{x\in R^{2}||x-P_{\epsilon}|<R\}$ ), where$\kappa>0$
is a sufficiently small constant.
Set$\zeta_{P_{0}}^{l}(0)=0$$D\Phi_{P_{0}}$$\Phi_{P_{0}}$$\Psi_{P_{0}}$$\overline{B_{\kappa}(0)}\in \mathbb{R}^{2}$$v_{\epsilon}(y):=u_{\epsilon}(\Phi_{P_{O}}(y))$,

and hence, from (2.2), it follows that$\int_{B_{\kappa/\epsilon}(0)}v_{\epsilon}^{p}dx\leq M_{p}$for $p\geq 1$ .

Thus, by the Sobolev imbedding theorem and the regularity theorem for elliptic equations,
there exists some function $v_{0}\in C^{2}(R^{2})$ such that$v_{\epsilon}arrow v_{0}$Here the function$v_{0}$

in$C_{loc}^{2}(R^{2})$as$\epsilonarrow 0$.

satisfies$\{\begin{array}{ll}\triangle v_{0}-v_{0}+(e^{\alpha vo}-\alpha v_{0}-1)=0 in R_{+}^{2},v_{0}>0 in R_{+}^{2},v_{0}=0 on \partial R_{+}^{2},\end{array}$

(2.4)

136
is the upper half-plane. IFhrom Theorem 1.1 in [6], the problem (2.4) only has.
in , and hence. It is inconsistent withasthe solution
has two
Next we prove thathas only one local maximum point. Suppose that
local maximum points
and . Set

where

$R_{+}^{2}$$u_{\epsilon}>0$$v_{0}\equiv 0$$\epsilonarrow 0$$\Omega$$\rho_{\epsilon}arrow\infty$$u_{\epsilon}$$u_{\epsilon}$$P_{\epsilon}$$P_{\epsilon}’$,$w_{\epsilon}(y):=u_{\epsilon}(\epsilon y+P_{\epsilon})$and hence, from (2.2) it follows that$)$for $p\geq 1$$\int_{B_{\rho_{e}}(0)}w_{\epsilon}^{p}dx\leq M_{p}$Hence, by the Sobolev imbedding theorem and the regularity theorem for elliptic equations, there exists some function $w_{0}\in C^{2}(R^{2})$ such thatin$w_{\epsilon}arrow w_{0}$as$C_{loc}^{2}(R^{2})$$\epsilonarrow 0$

The function
is a solution to (1.6). Moreover it holds that
following lemma;$w_{0}$

Lemma 2.6 The problem (1.6) has a unique solution
metric, strictly decreasing and $w(r)=O(r^{-1/2}e^{-r})$ as$w$.

$w_{0}\equiv w$. Furthermore.

$w$

in $R^{2}hom$ theis mdially sym-

$rarrow\infty$First we assume that there exists some constant such that
. Then,from the above argument,(we can takehas two maximum points insuchthat $R>C$ ). The function is approximate to , this function has only one maximum
point at the origin and strictly decreases for $r>0$ . It is a contradiction to the assumption
that
has two local maximum point.
. Here we obtain the lower estimate
Second suppose that
as$C$$|P_{\epsilon}-P_{\epsilon}’|/\epsilon\leq C$$R$$B_{R}(P_{\epsilon})$$u_{\epsilon}$$w$$u_{\epsilon}$$u_{\epsilon}$$|P_{\epsilon}-P_{\epsilon}’|/\epsilonarrow\infty$$\epsilonarrow 0$

as$c_{\epsilon}\geq\epsilon^{2}(I(w)+c+o(1))$

with some constant $c>0$ .
In fact we divide$c_{\epsilon}=J_{\epsilon}(u_{\epsilon})$(2.5)$\epsilonarrow 0$

into two part;$I_{1}= \int_{B_{eR}(P_{e})}\{\frac{1}{2}(|\nabla u|^{2}+u^{2})-(e^{\alpha u}-\frac{1}{2}(\alpha u)^{2}-\alpha u-1)\}dx$$I_{2}=J_{\epsilon}(u_{\epsilon})-I_{1}$where $R_{0}>0$ satisfies,,$B_{\epsilon Ro}(P_{\epsilon})\subset\Omega$. By direct calculation it follows that

$I_{1}\geq\epsilon^{2}(I(w)-c_{1}R_{0}^{2}+o(1))$

where $c_{1}>0$ is independent of and$\epsilon$$R_{4}$

a

$s$

(2.6)

$\epsilonarrow 0$. In addition, from Lemma 2.5, we can obtain$I_{2}\geq c_{2}\epsilon^{2}$

,

(2.7)

where $c_{2}>0$ is independent of and . In Lemma 2.5, we can takearbitrarily large
if is taken sufficiently small. Therefore, from (2.6) and (2.7), the inequality (2.5) holds
ifsatisfies $c_{2}-c_{1}R_{0}^{2}>0$ . Since (2.5) is inconsistent with Lemma 2.4, the number ofmaximum points is exactly one. Thus Proposition 1.1 is proved.
$\epsilon$$R_{0}$$R_{O}$$\epsilon_{1}$$R_{C}$$\blacksquare$

137

3

Proof of Theorem 1.1

In this section Theorem 1.1 is shown. Hereafter the least-energy solution to (1.1)
is denoted by , and the corresponding critical value tois written by . Point
denotes the maximum point of . In addition let be the unique solution to (1.6). The
functionsandare defined in (1.9) and (1.10), respectively.
Before beginning the proof, two lemmas are stated. Recall that is the most distantpoint from the boundary, that is,. Next define$u_{\epsilon}$$u_{\epsilon}$$P_{\epsilon}$$c_{\epsilon}$$w$$u_{\epsilon}$$\tilde{\psi}_{\epsilon}$$\psi_{\epsilon}$$Q$$d(Q, \partial\Omega)=\max_{P\in\Omega}d(P, \partial\Omega)$$\partial\Omega$$\phi_{\epsilon}(y)=e^{\beta\tilde{\psi}_{\epsilon}(P_{\epsilon})}(w_{\epsilon}(y)-w_{\Omega_{\epsilon}^{*}}(y)))$where$w_{\epsilon}(y)=u_{\epsilon}(\epsilon y+P_{\epsilon})$, and two lemmas are as follows:

Lemma 3.1 (i) For the critical value corresponding to the least-energy solution
the next equality and inequality hold:$c_{\epsilon}$$c_{\epsilon}=\epsilon^{2}\{I(w)+\gamma e^{-\beta\tilde{\psi}_{\epsilon}(P_{\epsilon})}+o(e^{-\beta\overline{\psi}_{\epsilon}(P_{\epsilon})})\}$$\epsilonarrow 0$(ii) As
small. Here the constant$\epsilonarrow 0,$$\gamma$. Moreover, for any
such that, for any$\sigma_{0}>0$

Lemma 3.2 For sufficiently small$\epsilon>0$(i) For $s>2$ , there exists a constant$\{\epsilon_{k}\}$and$C_{s}$

assuch that$\Vert e^{-\mu|y|}\phi_{\epsilon}\Vert_{W^{2,s}(\Omega_{\epsilon}^{*})}\leq C_{s}$$\epsilon_{k}arrow 0$$karrow\infty$$\{\epsilon_{k_{l}}\}$$larrow\infty$$\triangle\phi_{0}-\phi_{0}-\alpha(e^{\alpha w}-1)\tilde{V}_{0}=0$Here$\tilde{V}_{0}$.

, there exists a subsequencesuch, whereasis sufficiently small,is a classical solution towith$\phi_{0}$.

, the following two statements hold:$\Vert e^{-\mu|y|}(\phi_{\epsilon_{k_{l}}}-\phi_{0})\Vert_{L^{\infty}(B_{(1-\delta_{2})\rho_{\epsilon_{k_{l}}}}(0))}arrow 0$$\rho_{\epsilon}=d(P_{\epsilon}, \partial\Omega)/\epsilon$, there exists a sufficiently$\epsilon<\epsilon_{1z}$$\tilde{\psi}_{\epsilon}(P_{\epsilon})\leq(2+\sigma_{0})d(P_{\epsilon}, \partial\Omega)$(ii) For any sequence
that

(3.1)

is the same as that defined in Theorem 1.1 (i).

$\psi_{\epsilon}(Q)arrow 2d(Q, \partial\Omega)$$\epsilon_{1}>0$,,$c_{\epsilon}\leq\epsilon^{2}\{I(w)+\gamma e^{-\beta\psi_{\epsilon}(Q)}+o(e^{-\beta\psi_{\epsilon}(Q)})\}$

as,$u_{\epsilon}$

is a solution to (1.11). Furthermore$\delta_{2}$in$R^{2}$.

$e^{-\mu|y|}\phi_{0}\in W^{2,s}(\mathbb{R}^{2})$for $s>1$ .

Lemma 3.1 is proved by a very similar argument in Sections 5 and 6 in [15], and the proof
of Lemma 3.2 is almost identical to Proposition 6.1 in [15]. From the above lemmas,
Theorem 1.1 is shown.

Proof of Theorem 1.1. First Theorem 1.1 (i) and (ii) follow from Lemma 3.1. In fact
(1.13) is the same as (3.1). Moreover, by comparing the equality and the inequality inLemma 3.1, we obtainas$\tilde{\psi}_{\epsilon}(P_{\epsilon})\geq\psi_{\epsilon}(Q)+o(1)$From Lemma 3.1 (ii) it holds that, for any$\sigma_{0}>0$$\epsilonarrow 0$.

,$(2+\sigma_{0})d(P_{\epsilon}, \partial\Omega)\geq\tilde{\psi}_{\epsilon}(P_{\epsilon})\geq 2d(Q, \partial\Omega)+o(1)$as$\epsilonarrow 0$

.

(3.2)

138
On the contrary, by the definition of , it holds that$Q$

for any$d(P_{\epsilon}, \partial\Omega)\leq d(Q, \partial\Omega)$$\epsilon>0$

.

(3.3)

Thus, from (3.2), (3.3) and the arbitrariness of , Theorem 1.1 (ii) is shown.
Thus it suffices that Theorem 1.1 (iii) is proved. From Lemma 3.2, (1.14) holds in. In addition, by using the Green function $of-\triangle+1$ in ;$\sigma_{0}$$R^{2}$$B_{(1-\delta_{2})\rho_{\epsilon_{k_{\downarrow}}}}(0)$$(\phi_{\epsilon}-\phi_{0})(y)=$$\int_{R^{2}}G(|z-y|)\{f’(w(z))(\phi_{\epsilon}-\phi_{0})(z)+H_{\epsilon}(\phi_{\epsilon})(z)+f’(w(z))\tilde{V}_{0}(z)\}dz$

Here $G$ is the Green function $of-\triangle+1$ in$f(t):=e^{\alpha t}-$$R^{2}$

.

, and

at–l,

$H_{\epsilon}(\phi_{\epsilon}):=e^{\beta\overline{\psi}_{\epsilon}(P_{\epsilon})}\{f(\tilde{v}_{\epsilon})-f(w)-e^{-\beta\overline{\psi}_{\epsilon}(P_{\epsilon})}f’(w)\phi_{\epsilon}\}$.

For $G$ , it holds thatas$G(r)\leq C\frac{e^{-r}}{(1+r)^{\frac{1}{2}}}$

,

(3.4)

as $rarrow 0$ ,$G(r)=C(|\log r|+1)$

where $C$ is a positive constant. In addition, for
are obtained:$rarrow\infty$$w_{\epsilon}$and$\tilde{V}_{0}$(3.5), the following two inequalities$w_{\epsilon_{k_{l}}}(z)\leq C_{\delta}e^{-(1-\delta)|z|}$

,

(3.6)

$\tilde{V}_{0}(z)\leq Ce^{(1+^{\sigma_{2}})|z|}\lrcorner$

,

(3.7)

(which depends on ) andwhere $z\in R^{2},$ $\delta>0$ is taken arbitrary, and$(3.4)-(3.7)$, the estimate oftive constants. IFhrom Lemma 3.2 (i) andis directly calculated, and therefore$\delta$$C_{\delta}$$C$are posi-$e^{-\mu|y|}(\phi_{\epsilon_{k_{l}}}-\phi_{0})$in$\Omega_{\epsilon_{k_{l}}}^{*}\backslash B_{(1-\delta_{2})\rho_{\epsilon}k_{l}}(0)$$\Vert e^{-\mu|y|}(\phi_{\epsilon_{k_{l}}}-\phi_{0})\Vert_{L}\infty(\Omega_{\epsilon_{k_{l}}}^{r}\backslash B_{(1-\delta_{2})\rho_{\epsilon}k_{l}}(0))arrow 0$

as$larrow\infty$.

Therefore Theorem 1.1 (iii) follows $hom$ Lemma 3.2. The proof of Theorem 1.1 is completely finished.
$\blacksquare$

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