$S^{(\infty)}_{3/2}+\overline{S^{(\infty)}_{3/2}}^{T}$,
$S^{(\infty)}_{5/2}+\overline{S^{(\infty)}_{5/2}}^{T}$. These are positive
definite when $s=0$, hence for all $s$ in an open neighbourhood $V$ of $s=0$.
As $|\lambda_{i,j}|<1$, the original matrices $H_{3/2}$ and $H_{1/2}$ are also
positive definite for all $s\in V$. This completes the proof of Theorem 8.5. ∎

######  Remark 8.7.

We chose the rays with arguments $\scriptstyle\frac{\pi}{2(n+1)}$,
$\pi+\scriptstyle\frac{\pi}{2(n+1)}$ in the above proof for compatibility with
the proof in [10] in the case $n=3$. Examination of the proof given here shows
that in fact any two (collinear) rays could have been used, providing that
they do not coincide with singular directions. ∎

######  Corollary 8.8.

For $s\in V$ we have
$w^{\infty}_{s}|_{(0,\epsilon)}=w^{0}_{m,\hat{c}^{\operatorname{id}}}$.

###### Proof.

For any $x\in(0,\epsilon)$, both $w^{\infty}_{s}|_{(0,\epsilon)}$ and
$w^{0}_{m,\hat{c}^{\operatorname{id}}}$ are solutions to the same Riemann-
Hilbert problem. Hence they coincide. ∎

In the case $n=1$ it is easy to verify that
$V=\mu_{1}(\mathring{\mathcal{A}})$. In [10], in the case $n=3$, the precise
region $V\subseteq\mu_{1}(\mathring{\mathcal{A}})$ of positivity was
calculated, and is a proper subset of $\mu_{1}(\mathring{\mathcal{A}})$. For
the purposes of this article it will be sufficient to know that $V$ is a non-
empty open set, as we shall now appeal to p.d.e. theory to deduce that every
point of $\mu_{1}(\mathring{\mathcal{A}})$ corresponds to a global solution.

######  Theorem 8.9.

There is a bijection between solutions $w$ of the $tt^{*}$-Toda equations
(1.1) on $\mathbb{C}^{\ast}$ such that

$\lim_{t\to 0}\frac{w(|t|)}{\log|t|}=-m,\quad\lim_{t\to\infty}w(|t|)=0$

and points $m\in\mathcal{A}$.

The proof will be given in an appendix (section 10).

Restricting attention to $\mathring{\mathcal{A}}$, it follows from the
construction of section 2 that the solution corresponding to $m$ in Theorem
8.9 must have Stokes data $s=\mu_{1}(m)$, but we do not yet know the
connection matrix data. Theorem 8.5 allows us to obtain this:

######  Corollary 8.10.

Let $m\in\mathring{\mathcal{A}}$, and let $w_{m}^{\text{\em pde}}$ be the
corresponding solution given by Theorem 8.9. Then $w_{m}^{\text{\em pde}}$
must have $e=(1,\dots,1)$.

###### Proof.

For $s\in V$ we know that $w^{0}_{m,\hat{c}^{\operatorname{id}}}$ is a global
solution of the type in Theorem 8.9, where $s=\mu_{1}(m)$. By the
bijectiveness property it coincides with $w_{m}^{\text{pde}}$. Hence
$w_{m}^{\text{pde}}$ has $e=(1,\dots,1)$ for all $m\in V$. But $e$ depends
analytically on $m$, so we must have $e=(1,\dots,1)$ for all
$m\in\mathring{\mathcal{A}}$. ∎

Strictly speaking, the corollary (and its proof) should have been phrased in
terms of the connection matrix $E_{1}$ rather than $e$, because $E_{1}$ exists
for arbitrary solutions of (1.1), whereas we have defined $e$ only for
solutions of the form $w_{m,\hat{c}}$. However the same argument shows that
$E_{1}=E_{1}^{\operatorname{id}}$ for $w_{m}^{\text{pde}}$.

We conclude that the solutions $w_{m}^{\text{pde}}$ with
$m\in\mathring{\mathcal{A}}$ are precisely our solutions
$w^{0}_{m,\hat{c}^{\operatorname{id}}}$, hence the latter are indeed global
solutions.

######  Remark 8.11.

The “boundary conditions” at $t=0$ and $t=\infty$ in Theorem 8.9 are
automatically satisfied if $w$ is a radial solution of (1.1). We shall give a
proof of this fact in section 10.2. ∎

###  8.4. Summary of results

From the point of view of the family of local solutions $w^{0}_{m,\hat{c}}$
constructed in section 2, we have obtained the following characterizations of
global solutions:

######  Corollary 8.12.

Let $m\in\mathring{\mathcal{A}}$. The following conditions are equivalent:

(1) $w^{0}_{m,\hat{c}}$ is a globally smooth solution on $(0,\infty)$,

(2) $\hat{c}=\hat{c}^{\operatorname{id}}$.

(3) The connection matrix $E_{1}$ satisfies $E_{1}=E_{1}^{\operatorname{id}}$,
i.e. $e=(1,\dots,1)$.

(4) The connection matrix $D_{1}$ satisfies $\Omega
D_{1}t^{m}=\Delta(\overline{\Omega D_{1}t^{m}})\Delta$, i.e. $\Omega
D_{1}t^{m}$ belongs to the group $\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$.

###### Proof.

The equivalence of (1),(2),(3) follows immediately from the discussion above.
The equivalence of (3) and (4) is a consequence of formula (5.6) in Theorem
5.2, i.e.

$E_{1}=\tfrac{1}{n+1}D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}\,C.$

Namely, as in the proof of Theorem 6.19, this is equivalent to
$E_{1}=D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}d_{n+1}^{-1}\,E_{1}^{\operatorname{id}}$,
hence (3) is equivalent to

$D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}d_{n+1}^{-1}=I.$

Multiplying by $\Omega$ we obtain $\Omega D_{1}t^{m}\,\Delta=\Omega
d_{n+1}(\overline{D_{1}t^{m}})$. As $\Omega d_{n+1}\bar{\Omega}^{-1}=\Delta$
(cf. the formulae in Appendix A of [10]), we obtain (4). ∎

The significance of $\Omega D_{k}t^{m}$ is that it is the connection matrix
which relates the “undiagonalized form” $\Phi^{(0)}_{k}\Omega^{-1}$ of
$\Phi^{(0)}_{k}$ (see section 4.3) to $\Phi^{(\infty)}$ (see section 4.4),
namely we have the relation

$\Phi^{(\infty)}=(\Phi^{(0)}_{k}\Omega^{-1})(\Omega D_{k}t^{m}).$

The Stokes factors based on $\Phi^{(0)}_{k}\Omega^{-1}$ are $\Omega
Q^{(0)}_{k}\Omega^{-1}$. Using these, a further characterization can be
obtained:

######  Corollary 8.13.

$w^{0}_{m,\hat{c}}$ is a globally smooth solution on $(0,\infty)$ if and only
if all Stokes factors $\Omega Q^{(0)}_{k}\Omega^{-1}$ and all connection
matrices $\Omega D_{k}t^{m}$ lie in the group
$\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$.

###### Proof.

From (4) of Corollary 8.12, the condition $\Omega
D_{1}t^{m}\in\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$ alone implies that
$w^{0}_{m,\hat{c}}$ is a global solution. Conversely, we claim first that
$\Omega Q^{(0)}_{k}\Omega^{-1}\in\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$ for all
$m\in\mathring{\mathcal{A}}$. This means $\Delta\overline{\Omega
Q^{(0)}_{k}\Omega^{-1}}\Delta=\Omega Q^{(0)}_{k}\Omega^{-1}$. This condition
is equivalent to $d_{n+1}\overline{Q^{(0)}_{k}}d_{n+1}^{-1}=Q^{(0)}_{k}$, as
$\Omega^{-1}\Delta\bar{\Omega}=d_{n+1}$ (see the end of the proof of Corollary
8.12). By (4.4), this is equivalent to the reality of the Stokes factors
$\tilde{Q}^{(0)}_{k}$, which we have already established in section 6.2. This
completes the proof of the claim. In view of the formula (5.3), which
expresses all $D_{k}$ in terms of $D_{1}$ and the $Q^{(0)}_{k}$, it follows
that $\Omega D_{k}t^{m}\in\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$ for all $k$.
This completes the proof. ∎

Corollary 8.13 is quite natural, as
$\hat{\alpha}\left|{}_{|\lambda|=1}\right.$ is an
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$-valued connection form.
However, we do not know a direct proof of the above characterization which
exploits this fact.

Collecting together the asymptotic results at $t=0$ (Proposition 2.7,
Corollary 6.22) and at $t=\infty$ (Theorem 7.10), we can now give the
asymptotic data of our global solutions:

######  Corollary 8.14.

Let $m\in\mathring{\mathcal{A}}$. Let $w$ denote the corresponding global
solution. Then:

(1) The asymptotic data at $t=0$ of $w$ is given by

$w_{i}=-m_{i}\log|t|-\tfrac{1}{2}\log(\hat{c}^{\operatorname{id}}_{i})^{2}+o(1),$

where $(\hat{c}^{\operatorname{id}}_{i})^{2}=$

$\textstyle{(n+1)}^{m_{n-i}-m_{i}}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+1}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+1}}{n+1}\right)}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+2}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+2}}{n+1}\right)}\cdots\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+n}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+n}}{n+1}\right)}.$

(2) The asymptotic data at $t=\infty$ of $w$ may be expressed as

$\displaystyle w_{0}\sin p\tfrac{\pi}{n+1}+w_{1}$ $\displaystyle\sin
3p\tfrac{\pi}{n+1}+\cdots+w_{\frac{n-1}{2}}\sin np\tfrac{\pi}{n+1}$
$\displaystyle=-\tfrac{n+1}{8}\ s_{p}\ (\pi
L_{p}x)^{-\frac{1}{2}}e^{-2L_{p}x}+O(x^{-\frac{3}{2}}e^{-2L_{p}x})$

where $L_{p}=2\sin\tfrac{p}{n+1}\pi$. Here $p$ ranges over
$1,2,\dots,\frac{n+1}{2}$, so that we have $\frac{n+1}{2}$ linear equations
for $w_{0},\dots,w_{\frac{n-1}{2}}$. ∎

These formulae constitute a solution of the “connection problem”, i.e. they
give a precise relation between the asymptotics at zero and the asymptotics at
infinity.

##  9\. Appendix: modifications when $n+1$ is odd

Our main results are stated in the introduction for arbitrary $n\geq 1$, but
for readability we have taken $n+1$ to be even in most of the calculations in
sections 4-8. Here we give the modifications needed when $n+1$ is odd (the
“odd case”). We write $n+1=2d+1$, with $d\in\mathbb{N}$.

Section 4:

The new features of the case $n+1=2d+1$ begin with the treatment of the Stokes
data. As initial sector at $\zeta=0$ we take

$\Omega^{(0)}_{1}=\\{\zeta\in\mathbb{C}^{\ast}\ |\
-\tfrac{\pi}{2}-\tfrac{\pi}{2(n+1)}<\text{arg}\zeta<\tfrac{\pi}{2}+\tfrac{\pi}{2(n+1)}\\},$

and define

$\Omega^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=e^{-\frac{\pi}{n+1}{\scriptscriptstyle\sqrt{-1}}\,}\Omega^{(0)}_{k},\quad\theta^{(0)}_{k}=-\tfrac{1}{2(n+1)}\pi-(k-1)\pi$

for $k\in\tfrac{1}{n+1}\mathbb{Z}$. As in the even case, $\theta^{(0)}_{k}$
bisects $\Omega^{(0)}_{k}\cap\Omega^{(0)}_{k+\scriptstyle\frac{1}{n+1}}$.

The formal solution $\Psi^{(0)}_{f}$ is defined as in the even case, but we
define the tilde version by
$\tilde{\Psi}^{(0)}_{f}(\zeta)=\Psi^{(0)}_{f}(\zeta)(d_{n+1})^{-d}$. The
symmetries of the connection form $\hat{\alpha}$ are as stated in section 3.1.
The resulting symmetries of $\tilde{\Psi}^{(0)}_{f}$ and
$\tilde{\Psi}^{(0)}_{k}$ are stated in [7]; the symmetries of $\Psi^{(0)}_{f}$
and $\Psi^{(0)}_{k}$ follow from this. At $\zeta=\infty$ we define
$\Omega^{(\infty)}_{k}=\overline{\Omega^{(0)}_{k}},$ as in the even case. We
define
$\tilde{\Psi}^{(\infty)}_{f}(\zeta)=\Psi^{(\infty)}_{f}(\zeta)(d_{n+1})^{d}$.

The Stokes factors in the odd case are defined as in the even case, and they
are indexed by the singular directions $\theta^{(0)}_{k}$. Formulae
(4.1)-(4.9) continue to hold in the odd case, except for (4.4) and (4.8),
which must be replaced by

$\tilde{Q}^{(0)}_{k}=(d_{n+1})^{d}Q^{(0)}_{k}(d_{n+1})^{-d},\quad\tilde{Q}^{(\infty)}_{k}=(d_{n+1})^{-d}Q^{(\infty)}_{k}(d_{n+1})^{d}.$

The cyclic symmetry formula (4.3) implies that
$(M^{(0)})^{n+1}=S^{(0)}_{1}S^{(0)}_{2}$, where
$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$, as in the even
case. However the tilde version is simpler in the odd case: we have
$(\tilde{M}^{(0)})^{n+1}=\tilde{S}^{(0)}_{1}\tilde{S}^{(0)}_{2}$, where
$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi=\omega^{d}(d_{n+1})^{d}M^{(0)}(d_{n+1})^{-d}$.
(Recall that, in the even case,
$(\tilde{M}^{(0)})^{n+1}=-\tilde{S}^{(0)}_{1}\tilde{S}^{(0)}_{2}$, where
$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\hat{\Pi}=\omega^{-\frac{1}{2}}d_{n+1}^{-\frac{1}{2}}M^{(0)}d_{n+1}^{\frac{1}{2}}$.)

For the connection form $\hat{\omega}$, the Stokes sectors and Stokes factors
at $\zeta=0$ are the same as for the connection form $\hat{\alpha}$.

Section 5:

The connection matrices $E_{k},D_{k}$ are defined as in the even case, but
their tilde versions are $\tilde{E}_{k}=(d_{n+1})^{d}E_{k}(d_{n+1})^{d}$,
$\tilde{D}_{k}=(d_{n+1})^{d}D_{k}$. The formulae (5.1)-(5.4) continue to hold
in the odd case.

In the odd case, the first statement of Theorem 5.2 becomes
$E_{k}=\tfrac{1}{n+1}D_{k}t^{m}\,\Delta\,\bar{t}^{-m}\bar{D}_{2-k}^{-1}\,d_{n+1}^{-1}\,C$,
and this gives immediately the following (simpler) version of (5.6):

$E_{1}=\tfrac{1}{n+1}D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}d_{n+1}^{-1}\,C.$

Section 6:

In the even case, the shape of the Stokes factors $\tilde{Q}^{(0)}_{k}$ is
specified by the set $\mathcal{R}^{(0)}_{k}$ in Proposition 6.4 (Proposition
3.4 of [7]). In the odd case, it is specified by the set
$\mathcal{R}^{(0)}_{k}$ in Proposition 3.9 of [7].

Based on this, and the symmetries of the $\tilde{Q}^{(0)}_{k}$, we introduce
the notation

$\tilde{Q}^{(0)}_{k}=I+\sum_{(i,j)\in\mathcal{R}^{(0)}_{k}}s_{i,j}\,e_{i,j},$

where $e_{i,j}=E_{i,j}$ if $0\leq i<j\leq n$ with $j-i$ even, or if $0\leq
j<i\leq n$ with $i-j$ odd. The remaining cases are determined by
$e_{i,j}=-e_{j,i}$. These definitions ensure that Proposition 6.7 continues to
hold.

We define $s_{1},\dots,s_{n}$ as in Definition 6.9, and put $s_{0}=s_{n+1}=1$.
Then the $s_{i}$ are all real and they satisfy $s_{i}=s_{n+1-i}$. The
essential Stokes parameters are $s_{1},\dots,s_{d}$. The analogue of
Proposition 6.10 is that the (monic) characteristic polynomial of
$\tilde{M}^{(0)}$ is

$\textstyle\sum_{i=0}^{n+1}(-1)^{i}s_{i}\mu^{n+1-i}.$

Then $s_{i}$ is the $i$-th symmetric function of
$\omega^{m_{0}+\frac{n}{2}},\omega^{m_{1}-1+\frac{n}{2}},\dots,\omega^{m_{n}-n+\frac{n}{2}}$,
as in the even case.

######  Example 9.1.

In analogy with Example 6.13, we present here the Stokes data for $n+1=3$, the
simplest odd case. From Proposition 3.9 of [7] we have
$\mathcal{R}^{(0)}_{1}=\\{(1,0)\\}$,
$\mathcal{R}^{(0)}_{\frac{4}{3}}=\\{(1,2)\\}$. There is only one Stokes
parameter $s=s_{1}$, and

$\tilde{Q}^{(0)}_{1}=\begin{pmatrix}1&0&0\\\ s&1&0\\\
0&0&1\end{pmatrix},\quad\tilde{Q}^{(0)}_{\scriptstyle\frac{4}{3}}=\begin{pmatrix}1&0&0\\\
0&1&-s\\\ 0&0&1\end{pmatrix}.$

We have

$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{\scriptstyle\frac{4}{3}}\Pi=\begin{pmatrix}0&1&0\\\
-s&s&1\\\ 1&0&0\end{pmatrix},$

with characteristic polynomial is $\mu^{3}-s\mu^{2}+s\mu-1$. Theorem 6.12
gives $s=1+2\cos\frac{2\pi}{3}(m_{0}+1)$. We have
$m_{0}=\frac{k_{1}-k_{0}}{k_{0}+2k_{1}+3}$ (from Definition 2.5). ∎

The computation of the connection matrices $D_{k},E_{k}$ in the odd case can
be carried out as in the even case, but, to obtain the asymptotics of
solutions near zero it is easier to observe that a solution
$w=(w_{0},\dots,w_{2d})$ with $n+1=2d+1$ gives rise to a solution
$w^{\sharp}=(w_{0},\dots,w_{2d},-w_{2d},\dots,-w_{0})$ with $n+1=4d+2$. Then
the asymptotics of $w$ near zero follow immediately from the known asymptotics
of $w^{\sharp}$ (Corollary 8.14).

Section 7:

The Riemann-Hilbert problem can be set up as in the even case, using the rays
$\theta^{(0)}_{k},\theta^{(\infty)}_{k}$ and the Stokes parameters $s_{i}$.
The analogue of formula (7.1) is
$\Psi^{(\infty)}_{k}=\Psi^{(0)}_{2-k}Z_{2-k}$. The analogue of formula (7.5)
is

$Q^{(\infty)}_{k}=\left(\tfrac{1}{n+1}C\right)^{-1}\left(Q^{(0)}_{\scriptstyle\frac{2n+1}{n+1}-k}\right)^{-1}\tfrac{1}{n+1}C.$

Then the analogue of Lemma 9.2, which is proved in the same way, is:

######  Lemma 9.2.

Let $E_{1}=\tfrac{1}{n+1}C$. Then $Z_{k}=\tfrac{1}{n+1}C$ for all $k$.

The analogue of Theorem 7.9, which produces solutions near infinity, is:

######  Theorem 9.3.

Let $s_{1},\dots,s_{d}$ be real numbers, and let the matrices
$\tilde{Q}^{(\infty)}_{k}$ be defined in terms of $s_{1},\dots,s_{d}$ as
above. Then there is a unique solution $w$ of (1.1) on an interval
$(R,\infty)$, where $R$ depends on $s_{1},\dots,s_{d}$, such that the
associated monodromy data is given by the $Q^{(\infty)}_{k}$ (i.e.
$(d_{n+1})^{d}\tilde{Q}^{(\infty)}_{k}(d_{n+1})^{-d}$) and
$E_{1}^{\operatorname{id}}=\tfrac{1}{n+1}C$.

Again, to obtain the asymptotics of solutions near infinity, it is easiest to
observe that a solution $w=(w_{0},\dots,w_{2d})$ with $n+1=2d+1$ gives rise to
a solution $w^{\sharp}=(w_{0},\dots,w_{2d},-w_{2d},\dots,-w_{0})$ with
$n+1=4d+2$. From Theorem 6.12 it follows that the Stokes parameters
$s_{i}^{\sharp}$ for $w^{\sharp}$ are given in terms of the Stokes parameters
$s_{i}$ for $w$ by:

$s_{i}^{\sharp}=\begin{cases}0\quad\ \,\text{if $i$ is odd}\\\
s_{\frac{i}{2}}\quad\text{if $i$ is even}\end{cases}$

(i.e. from the expression for $s^{\sharp}_{i}$ as an elementary symmetric
function, in terms of $m^{\sharp}=(m_{0},\dots,m_{2d},-m_{2d},\dots,-m_{0})$).
Using this, and the known asymptotics of $w^{\sharp}$ near infinity (Corollary
8.14), we obtain the asymptotics of $w$ as stated in the introduction.

Section 8:

As the notation suggests, it is the value $E_{1}=E_{1}^{\operatorname{id}}$
which is taken by global solutions. The arguments concerning global solutions
go through as in the even case.

##  10\. Appendix: p.d.e. theory

In this appendix, for $a,b>0$ and $n\geq 2$, we prove existence and uniqueness
results for solutions of the system

(10.1) $\begin{cases}\Delta u_{1}=e^{au_{1}}-e^{u_{2}-u_{1}}\doteqdot
f_{1}(u_{1},\dots,u_{n}),\\\ \hskip 20.00003pt\cdots\\\ \Delta
u_{i}=e^{u_{i}-u_{i-1}}-e^{u_{i+1}-u_{i}}\doteqdot
f_{i}(u_{1},\dots,u_{n}),\\\ \hskip 20.00003pt\cdots\\\ \Delta
u_{n}=e^{u_{n}-u_{n-1}}-e^{-bu_{n}}\doteqdot
f_{n}(u_{1},\dots,u_{n}),\end{cases}$

where $u_{i}=u_{i}(x)$, $1\leq i\leq n$, and
$x\in\mathbb{R}^{2}\setminus\\{0\\}$. We impose the “boundary conditions”

(10.2) $\lim\limits_{|x|\to 0}\dfrac{u_{i}(x)}{\log|x|}=\gamma_{i}\mbox{ and
}\lim\limits_{|x|\to+\infty}u_{i}(x)=0$

at $0$ and $\infty$, and we assume that $\gamma_{i}$ satisfies

(10.3) $-2\leq a\gamma_{1},\quad-2\leq\gamma_{i+1}-\gamma_{i}\ (1\leq i\leq
n-1),\quad b\,\gamma_{n}\leq 2.$

It is a consequence of uniqueness that $u_{i}$ is necessarily radial.

The tt*-Toda equations — the system (1.1) — are a special case of the system
(10.1). Namely, if $n+1$ in (1.1) is even, then the resulting equations for
$w_{0},\dots,w_{\frac{n-1}{2}}$ have the form of (10.1) with $a=b=2$; if $n+1$
in (1.1) is odd, then the resulting equations for
$w_{0},\dots,w_{\frac{n-2}{2}}$ have the form of (10.1) with $a=2,b=1$. Then
the results from p.d.e. theory needed in section 8 are consequences of the
results proved in this section: Theorem 10.4 below implies Theorem 8.9, and
Theorem 10.5 below justifies the assertion in Remark 8.11 concerning the
boundary conditions. The case $n=2$ of (10.1) was studied in detail by
elementary methods in our previous articles [12],[9],[10]. Here we shall give
analogous results for general $n$, but using methods more appropriate for the
general case.

###  10.1. Existence and uniqueness

The purpose of this subsection is to prove Theorem 10.4 below, i.e. the
existence and uniqueness of solutions of (10.1) subject to (10.2) and (10.3).

######  Remark 10.1.

Suppose $u(x)=(u_{1}(x),\dots,u_{n}(x))$ is a solution of (10.1) with
$\lim\limits_{|x|\to+\infty}u_{i}(x)=0$. Then, near $\infty,$ (10.1) can be
written as

$\begin{cases}\Delta u_{1}=(a+1)u_{1}-u_{2}+O(|u|^{2}),\\\ \hskip
20.00003pt\cdots\\\ \Delta u_{i}=2u_{i}-u_{i-1}-u_{i+1}+O(|u|^{2}),\\\ \hskip
20.00003pt\cdots\\\ \Delta u_{n}=(b+1)u_{n}-u_{n-1}+O(|u|^{2}).\end{cases}$

Since the matrix of coefficients on the right hand side is positive definite,
standard estimates (a straightforward application of the maximum principle)
show that $u_{i}(x)=O(e^{-\epsilon|x|})$ for some $\epsilon>0,$ and then
$|\nabla u_{i}(x)|=O(e^{-\epsilon|x|})$ for $|x|$ large. For the tt*-Toda
equations, the Riemann-Hilbert method allows us to make this more precise
(Theorem 7.10). ∎

First we establish the monotonicity of solutions of (10.1)-(10.2) with respect
to $\gamma=(\gamma_{1},\dots,\gamma_{n})$.

######  Lemma 10.2.

Let $u$ and $v$ be two solutions of (10.1)-(10.2) with
$\gamma^{(1)}=(\gamma_{1}^{(1)},\dots,\gamma_{n}^{(1)})$ and
$\gamma^{(2)}=(\gamma_{1}^{(2)},\dots,\gamma_{n}^{(2)})$. If
$\gamma_{i}^{(1)}<\gamma_{i}^{(2)},1\leq i\leq n,$ then $u(x)\geq v(x)$ for
all $x\in\mathbb{R}^{2}\setminus\\{0\\}$.

###### Proof.

We shall apply the maximum principle. We remark that, in the following
argument, it is sufficient to assume that $u(x)$ is a supersolution, i.e.,
$\Delta u_{i}(x)\leq f_{i}(u)$.

Step 1: $v_{1}\leq u_{1}$. If not, $v_{1}>u_{1}$ at some point in
$\mathbb{R}^{2}\setminus\\{0\\}$. By the assumption
$\gamma^{(2)}_{1}<\gamma_{1}^{(1)},$ the boundary conditions imply that
$(v_{1}-u_{1})(x_{1})=\max_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{1}-u_{1})(x)>0$
for some $x_{1}\in\mathbb{R}^{2}\setminus\\{0\\}$. The maximum principle gives

$e^{av_{1}(x_{1})}-e^{au_{1}(x_{1})}-(e^{(v_{2}-v_{1})(x_{1})}-e^{(u_{2}-u_{1})(x_{1})})\leq
0.$

Since $e^{av_{1}(x_{1})}-e^{au_{1}(x_{1})}>0,$ we obtain
$e^{(v_{2}-v_{1})(x_{1})}-e^{(u_{2}-u_{1})(x_{1})}>0,$ hence

$(v_{2}-u_{2})(x_{1})>(v_{1}-u_{1})(x_{1})>0.$

The assumption $\gamma^{(2)}_{2}<\gamma_{2}^{(1)}$ again implies that

$(v_{2}-u_{2})(x_{2})=\max_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{2}-u_{2})(x)>0$

for some $x_{2}\in\mathbb{R}^{2}\setminus\\{0\\}$. Repeating this process for
$1\leq i\leq n-1,$ we find that there exists
$x_{i}\in\mathbb{R}^{2}\setminus\\{0\\}$ such that

$(v_{i}-u_{i})(x_{i})=\max_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{i}-u_{i})(x)>0$

and applying the maximum principle to the $i$-th equation at $x_{i}$ yields

$(v_{i+1}-u_{i+1})(x_{i})>(v_{i}-u_{i})(x_{i}).$

For the maximum at $x_{i+1}$, we have

$(v_{i+1}-u_{i+1})(x_{i+1})\geq(v_{i+1}-u_{i+1})(x_{i})>(v_{i}-u_{i})(x_{i}).$

At $i=n$, we have, for some $x_{n}\in\mathbb{R}^{2}\setminus\\{0\\}$,

$(v_{n}-u_{n})(x_{n})=\max_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{n}-u_{n})(x)>(v_{n-1}-u_{n-1})(x_{n-1})>0.$

Applying the maximum principle to the n-th equation at $x_{n}$ yields

$e^{(v_{n}-v_{n-1})(x_{n})}-e^{(u_{n}-u_{n-1})(x_{n})}-(e^{-bv_{n}(x_{n})}-e^{-bu_{n}(x_{n})})\leq
0,$

which implies $e^{-bv_{n}(x_{n})}-e^{-bu_{n}(x_{n})}>0$, and then
$v_{n}(x_{n})<u_{n}(x_{n})$. This contradicts $(v_{n}-u_{n})(x_{n})>0$. Thus
we have $v_{1}\geq u_{1}$.

Step 2: Assume $v_{i-1}\leq u_{i-1}$. We want to prove that $v_{i}\leq u_{i}$.
If not, $v_{i}>u_{i}$ at some point in $\mathbb{R}^{2}\setminus\\{0\\}$. By
the assumption $\gamma_{i}^{(2)}>\gamma_{i}^{(1)}$, the boundary conditions
imply that
$(v_{i}-u_{i})(x_{i})=\max\limits_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{i}-u_{i})(x)>0$
for some $x_{i}\in\mathbb{R}^{2}\setminus\\{0\\}$. The maximum principle at
$x_{i}$ yields

$e^{(v_{i}-v_{i-1})(x_{i})}-e^{(u_{i}-u_{i-1})(x_{i})}-(e^{(v_{i+1}-v_{i})(x_{i})}-e^{(u_{i+1}-u_{i})(x_{i})})\leq
0.$

By the assumption $v_{i-1}\leq u_{i-1}$ and $(v_{i}-u_{i})(x_{i})>0$, we have
$(v_{i+1}-v_{i})(x_{i})-(u_{i+1}-u_{i})(x_{i})>0$, i.e.,
$(v_{i+1}-u_{i+1})(x_{i})>(v_{i}-u_{i})(x_{i})$. The assumption
$\gamma_{i+1}^{(2)}>\gamma_{i+1}^{(1)}$ again implies that
$(v_{i+1}-u_{i+1})(x_{i+1})=\max\limits_{x\in\mathbb{R}^{2}\setminus\\{0\\}}(v_{i+1}-u_{i+1})(x)>0$
for some $x_{i+1}\in\mathbb{R}^{2}\setminus\\{0\\}$. Thus repeating the
argument of Step 1 gives a contradiction. Hence we have $v_{i}<u_{i}$. By
induction, we have $v_{i}<u_{i}$ for all $i$. This proves the lemma. ∎

To solve (10.1) we use the variational method. First, we study equation (10.1)
on the ball $B_{R}$ with centre $0$ and radius $R$. In the following, we say
that the $\gamma_{i}$ satisfy (10.3) strictly if

$-2<a\gamma_{1},\quad-2<\gamma_{i+1}-\gamma_{i}\ (1\leq i\leq n-1),\quad
b\,\gamma_{n}<2.$

We fix a large $R>0$ and consider $(\ref{A.1})$ in the domain
$B_{R}\setminus\\{0\\}$ with the boundary conditions

(10.4) $\lim_{r\to 0}\dfrac{u_{i}(r)}{\log r}=\gamma_{i}\text{ and
}u_{i}(x)=0,\ x\in\partial B_{R}.$

######  Lemma 10.3.

Suppose the $\gamma_{i}$ satisfy (10.3) strictly. Then the system (10.1) in
$B_{R}\setminus\\{0\\}$ has a unique solution $u=(u_{1},...,u_{n})$ satisfying
(10.4).

###### Proof.

Let $w_{i}(x)=u_{i}(x)-\gamma_{i}\log\frac{|x|}{R}$. We obtain a system for
$w_{i}$ without singularities at the origin by considering

$\begin{cases}\Delta
w_{1}=e^{a(w_{1}+\gamma_{1}\log\frac{|x|}{R})}-e^{w_{2}-w_{1}+(\gamma_{2}-\gamma_{1})\log\frac{|x|}{R}}\\\
\hskip 20.00003pt\cdots\cdots\\\ \Delta
w_{i}=e^{w_{i}-w_{i-1}+(\gamma_{i}-\gamma_{i-1})\log\frac{|x|}{R}}-e^{w_{i+1}-w_{i}+(\gamma_{i+1}-\gamma_{i})\log\frac{|x|}{R}}\\\
\hskip 20.00003pt\cdots\cdots\\\ \Delta
w_{n}=e^{w_{n}-w_{n-1}+(\gamma_{n}-\gamma_{n-1})\log\frac{|x|}{R}}-e^{-b(w_{n}+\gamma_{n}\log\frac{|x|}{R})}\end{cases}$

with $w|_{\partial B_{R}}=0$. We establish existence by the variational
method. Consider the nonlinear functional

$\displaystyle I(w)=$
$\displaystyle\tfrac{1}{2}\sum_{i=1}^{n}\int_{B_{R}}|\nabla
w_{i}|^{2}+\tfrac{1}{a}\int_{B_{R}}e^{a(w_{1}+\gamma_{1}\log\frac{|x|}{R})}dx$
$\displaystyle\quad+\sum_{i=2}^{n}\int_{B_{R}}e^{w_{i}-w_{i-1}+(\gamma_{i}-\gamma_{i+1})\log\frac{|x|}{R}}dx+\tfrac{1}{b}\int_{B_{R}}e^{-b(w_{n}+\gamma_{n}\log\frac{|x|}{R})}dx$
$\displaystyle\doteqdot\tfrac{1}{2}\sum_{i=1}^{n}\int_{B_{R}}|\nabla
w_{i}|^{2}dx+\sum_{i=1}^{n}\int_{B_{R}}g_{i}(x,w)dx$

on the Sobolev space $H_{0}^{1}(B_{R})$. The functional $I$ is $C^{1}$ on
$H_{0}^{1}(B_{R})$ if $\gamma_{1}>-\frac{2}{a}$, $\gamma_{n}<\frac{2}{b}$ and
$\gamma_{i-1}-\gamma_{i}<2$, $i=2,...,n$ by the Moser-Trudinger theorem. It is
easy to see that any critical point of $I$ is a solution of the above system.
The functional is coercive, i.e.,
$I(w)\geq\frac{1}{2}\sum_{i=1}^{n}\int_{B_{R}}|\nabla w_{i}|^{2}dx$. Suppose
that $w^{(m)}=(w_{1}^{(m)},...,w_{n}^{(m)})$ is a minimizing sequence for $I$,
i.e., $I(w^{m})\to\inf_{H_{0}^{1}(B_{R})}I(w)$. Then $\\{w^{(m)}\\}$ is
bounded on $H_{0}^{1}(B_{R})$. Thus, up to a subsequence, $w^{(m)}\to
w=(w_{1},...,w_{n})$ and $w^{(m)}(x)\to w(x)$, a.e. Thus have

$\int_{B_{R}}|\nabla
w_{i}|^{2}dx\leq\lim\limits_{m\to+\infty}\int_{B_{R}}|\nabla
w_{i}^{(m)}|^{2}dx,$

and by Fatous’s Lemma

$\int_{B_{R}}g_{i}(x,w)dx\leq\lim\limits_{m\to+\infty}\int_{B_{R}}g_{i}(x,w^{(m)})dx.$

Thus $I(w)\leq\lim\limits_{m\to+\inf}I(w^{(m)})$, i.e., $w=(w_{1},...,w_{n})$
is a minimizer for the functional $I$. Let
$u_{i}(x)=w_{i}(x)+\gamma_{i}\log\frac{|x|}{R}$. Thus $u=(u_{1},...,u_{n})$ is
a solution of the system (10.1) on the ball $B_{R}$ with the boundary
conditions (10.4).

Next we shall apply Lemma 10.2 to prove the uniqueness of the solution.
Suppose $u(x)$ is a solution of (10.1) and (10.4). Choose a sequence of strict
parameters $\gamma^{(k)}=(\gamma_{1}^{(k)},\dots,\gamma_{n}^{(k)})$ such that
$\gamma_{i}<\gamma_{i}^{(k+1)}<\gamma^{(k)}_{i},1\leq i\leq n$, and a sequence
of solutions $v_{i}^{(k)}$ on $B_{R}$ satisfying (10.4). The argument of Lemma
10.2 implies that $v_{i}^{(k)}(x)<v_{i}^{(k+1)}(x)<u_{i}(x),1\leq i\leq n,$
for all $x\in B_{R}\setminus\\{0\\}$. Let
$v_{i}(x)=\lim\limits_{k\to\infty}v_{i}^{(k)}(x),x\in B_{R}\setminus\\{0\\}$.
Then it is easy to see that both $v(x)=(v_{1},\dots,v_{n})$ and $u(x)$ are
solutions of (10.1) on $B_{R}$ with the boundary condition (10.4), and
$v(x)\leq u(x)$ holds. Note that $v(x)$ is a minimal solution, that is, if
$u(x)$ is a solution of (10.1) and (10.4) on $B_{R},$ then $v(x)\leq u(x),x\in
B_{R}\setminus\\{0\\}$.

Since $v_{i}(x)\leq u_{i}(x)$ for $x\in B_{R},$ we have $(\nabla
u_{i}\cdot\nu)(x)\leq(\nabla v_{i}\cdot\nu)(x)$ for $x\in\partial B_{R},$
where $\nu$ is the outward normal at $x\in\partial B_{R}$. By integrating
(10.1), we have

(10.5)
$\begin{split}\int_{B_{R}}(e^{av_{1}(x)}-e^{-bv_{n}(x)})dx&=\int_{\partial
B_{R}}\sum_{i=1}^{n}(\nabla
v_{i}\cdot\nu)(x)d\sigma-\sum_{i=1}^{n}2\pi\gamma_{i}\\\ &\geq\int_{\partial
B_{R}}\sum_{i=1}^{n}(\nabla
u_{i}\cdot\nu)(x)d\sigma-\sum_{i=1}^{n}2\pi\gamma_{i}\\\
&=\int_{B_{R}}(e^{au_{1}(x)}-e^{-bu_{n}(x)})dx.\end{split}$

Then we have $u(x)\equiv v(x),$ that is, uniqueness holds. ∎

We denote the solution in Lemma 10.3 by $u^{R}(x)$. From the uniqueness, it
follows that $u^{R}(x)=u^{R}(|x|)$. By studying $u^{R}(x)$ as $R\to+\infty$ we
shall prove our existence theorem for (10.1).

######  Theorem 10.4.

Subject to conditions (10.2) and (10.3), the system (10.1) has a unique
solution on $\mathbb{R}^{2}\setminus\\{0\\}$.

###### Proof.

We first consider $\gamma_{i}$ such that (10.3) holds strictly and also
$\gamma_{i}<0$ for all $i$. We shall show that $u^{R}(x)$ converges for any
sequence $R\to+\infty$. Note that, since $\gamma_{i}<0$, we have
$u_{i}^{R}(x)=u_{i}^{R}(|x|)\geq 0$ by Lemma 10.2, and also that
$u_{i}^{R}(x)$ increases as $R$ increases.

By integrating (10.1), we have

(10.6)
$\begin{split}&R\frac{du_{1}^{R}}{dr}(R)-\gamma_{1}=\int_{0}^{R}\rho(e^{au_{1}^{R}(\rho)}-1)d\rho-\int_{0}^{R}\rho(e^{(u_{2}^{R}-u_{1}^{R})(\rho)}-1)d\rho\\\
&R\frac{du_{i}^{R}}{dr}(R)-\gamma_{i}=\int_{0}^{R}\rho(e^{u_{i}^{R}-u_{i-1}^{R})(\rho)}-1)d\rho-\int_{0}^{R}\rho(e^{(u_{i+1}^{R}-u_{i}^{R})(\rho)}-1)d\rho\\\
&R\frac{du_{n}^{R}}{dr}(R)-\gamma_{n}=\int_{0}^{R}\rho(e^{u_{n}^{R}-u_{n-1}^{R})(\rho)}-1)d\rho-\int_{0}^{R}\rho(e^{-bu_{n}^{R}(\rho)}-1)d\rho.\end{split}$

Summing both sides, we have

(10.7)
$\begin{split}\int_{0}^{R}\rho(e^{au_{1}^{R}(\rho)}-1)d\rho=\int_{0}^{R}\rho(e^{-bu_{n}^{R}(\rho)}-1)d\rho+R\sum_{i=1}^{n}\frac{du_{i}^{R}}{dr}(R)-\sum_{i=1}^{n}\gamma_{i}.\end{split}$

Since $u_{i}^{R}(|x|)\geq 0$, we have $\frac{du_{i}^{R}}{dr}(R)\leq 0$. Thus

$\int_{0}^{R}\rho(e^{au_{1}^{R}(\rho)}-1)d\rho\leq-\sum_{i=1}^{n}\gamma_{i},$

which implies $u_{1}^{R}(x)$ is bounded, hence
$u_{1}(x)=\lim\limits_{R\to+\infty}u_{1}^{R}(x)$ exists. We obtain also

$\int_{0}^{\infty}\rho(e^{au_{1}(\rho)}-1)d\rho\leq\lim\limits_{R\to\infty}\int_{0}^{R}\rho(e^{au_{1}^{R}(\rho)}-1)d\rho\leq-\sum_{i=1}^{n}\gamma_{i}.$

Further, (10.7) implies that

$\int_{0}^{R}\rho(1-e^{-bu_{n}^{R}(\rho)})d\rho+R\sum_{i=1}^{n}\Big{(}-\tfrac{du_{i}^{R}}{dr}(R)\Big{)}$

is bounded as $R\to+\infty$. Since each term of the above is nonnegative,
$R\sum\limits_{i=1}^{n}|\frac{d}{dr}u_{i}^{R}(R)|$ and
$\int_{0}^{R}\rho(1-e^{-bu_{n}^{R}(\rho)})d\rho$ are bounded as $R\to+\infty$.
By (10.6), we can prove by induction on $i$ that $u_{i}^{R}(x)$ is bounded as
$R\to\infty$, and $\lim\limits_{R\to\infty}Ru_{i}^{R}(R)\to 0$.

Let $u(x)=\lim\limits_{R\to+\infty}u^{R}(x)$. Clearly $u(x)$ is a solution of
(10.1), and the integrals of $e^{au_{1}(x)}-1$, $e^{u_{i+1}(x)-u_{i}(x)}-1$
and $1-e^{-bu_{n}(x)}$ over $\mathbb{R}^{2}$ are all finite. From this, it is
not difficult to see that $\lim\limits_{|x|\to\infty}u_{i}(|x|)=0$, $1\leq
i\leq n$. By integrating (10.1) in a neighbourhood of $0$, we see that, for
any $\epsilon>0$, there is a $\delta>0$ such that if $0<r<\delta$, then

(10.8) $\left|r\tfrac{du_{i}^{R}}{dr}(r)-\gamma_{i}\right|\leq\epsilon$

for any $R>1$. When $R\to\infty$, we obtain $\lim\limits_{r\to
0}\dfrac{u_{i}(x)}{\log|x|}=\gamma_{i}$, $1\leq i\leq n$.

Similar arguments work when $\gamma_{i}>0$ for all $i$, that is, $u^{R}(x)$
converges to $u(x)$ as $R\to+\infty$ and the integrals of $1-e^{au_{1}(x)}$,
$e^{u_{i+1}(x)-u_{i}(x)}-1$ and $e^{-bu_{n}(x)}-1$ over $\mathbb{R}^{2}$ are
all finite. In this case, by (10.7), we have

$\int_{0}^{R}\rho(e^{-bu_{n}^{R}(\rho)}-1)d\rho\leq\sum_{i=1}^{n}\gamma_{i},$

because $u_{n}^{R}(\rho)<0$ and $\frac{du_{i}^{R}}{dR}(R)\geq 0$. Clearly the
limit $u(x)$ satisfies (10.1) and the boundary condition (10.2).

For general $\gamma$ such that (10.3) holds strictly, we can choose
$\gamma^{(1)}$ and $\gamma^{(2)}$ such that
$\gamma_{i}^{(2)}<\gamma_{i}<\gamma_{i}^{(1)}$ and
$\gamma_{i}^{(2)}<0,\gamma_{i}^{(1)}>0$. Let $v_{i}^{R}$, $i=1,2,$ be the
solution of (10.1) corresponding to $\gamma^{(i)}$. Then we have

$v_{1,i}^{R}\leq u_{i}^{R}\leq v_{2,i}^{R}.$

Since $v_{i}^{R}$ exists as $R\to+\infty$, we can prove, by standard estimates
for linear elliptic equations, that $u_{i}^{R}$ converges to $u$ as
$R\to+\infty$ and $v_{1}(x)\leq u(x)\leq v_{2}(x)$. Since both $v_{i}(x)\to 0$
as $R\to+\infty$, we have $u(x)\to 0$ as $R\to+\infty$. The behaviour
$\lim\limits_{x\to 0}\dfrac{u_{i}(x)}{\log|x|}=\gamma_{i}$ can be established
by the standard argument (10.8) as before.

Finally, we consider a $\gamma$ which does not satisfy (10.3) strictly. It is
easy to see there is a sequence of strict parameters
$\gamma^{(k)}=(\gamma_{1}^{(k)},...,\gamma_{n}^{(k)})$ tending to $\gamma$ and
either $\gamma_{i}<\gamma_{i}^{(k+1)}<\gamma_{i}^{(k)}$ for all $k$ or
$\gamma_{i}^{(k+1)}>\gamma_{i}^{k}>\gamma_{i}$. Without loss of generality, we
assume the former case occurs. Then the solution $u^{(k)}$ with
$\gamma=\gamma^{(k)}$ satisfies $u^{(k)}(x)<u^{(k+1)}(x)<\cdots$. By (10.7) as
before, we can prove that $u^{(k)}(x)$ converges to $u(x)$ and this $u(x)$ is
a solution of (10.1) with the boundary condition (10.2).

The uniqueness can be proved by the procedure of Lemma 10.2. As in Lemma 10.3,
let $v_{i}(x)\leq u_{i}(x)$ be two solutions of (10.1) and (10.2), where
$v(x)$ is the minimal solution obtained by the argument of Lemma 10.3. Then,
instead of (10.5), we have

$\begin{split}\int_{\mathbb{R}^{2}}(e^{av_{1}(x)}-e^{-bv_{n}(x)})dx=-\sum_{i=1}^{n}\gamma_{i}=\int_{\mathbb{R}^{2}}(e^{au_{1}(x)}-e^{-bu_{n}(x)})dx\end{split}$

which implies $v_{1}(x)=u_{1}(x)$, and $v_{n}(x)=u_{n}(x)$. By the method of
Lemma 10.2, we can prove $v_{i}(x)=u_{i}(x)$ by induction on $i$.

This completes the proof of Theorem 10.4. ∎

###  10.2. Radial solutions without boundary conditions

The purpose of this subsection is to prove the following theorem (cf. Section
9 of [10], where the same result for the case $n=2$ of (10.1) was proved).

######  Theorem 10.5.

If $u(x)$ is a radial solution of (10.1) in $\mathbb{R}^{2}\setminus\\{0\\}$,
then:

(i) $\lim\limits_{|x|\to 0}\dfrac{u_{i}(x)}{\log|x|}=\gamma_{i}$ exists, and
the $\gamma_{i}$ satisfy (10.3),

(ii) $e^{au_{i}}-1,e^{u_{i+1}-u_{i}}-1$ and $e^{-bu_{n}}-1$ are all in
$L^{1}(\mathbb{R}^{2})$,

(iii) $\lim\limits_{|x|\to\infty}u_{i}(x)=0$.

From (i) and (iii), $u(x)$ satisfies (10.2).

The proof will follow from Lemmas 10.7 and 10.8 below. We begin by
establishing an identity which will be used in the proof of Lemma 10.8.

######  Lemma 10.6.

Suppose $u=(u_{1},\dots,u_{n})$ is a smooth solution of (10.1) in
$\Omega\subseteq\mathbb{R}^{2}$. Then the following Pohozaev identity holds:

$\displaystyle\tfrac{2}{a}\int_{\Omega}(e^{au_{1}}-1)dx+2\sum_{i=2}^{n}\int_{\Omega}(e^{u_{i}-u_{i-1}}-1)dx+\tfrac{2}{b}\int_{\Omega}(e^{-bu_{n}}-1)dx$
$\displaystyle=\tfrac{1}{a}\int_{\partial\Omega}(x\cdot\nu)(e^{au_{1}}-1)ds+\sum_{i=2}^{n}\int_{\partial\Omega}(x\cdot\nu)(e^{u_{i}-u_{i-1}}-1)ds$
$\displaystyle+\tfrac{1}{b}\int_{\partial\Omega}(x\cdot\nu)(e^{-bu_{n}}-1)ds-\int_{\partial\Omega}(x\cdot\nabla
u_{i})\tfrac{\partial u_{i}}{\partial
v}ds+\frac{1}{2}\int_{\partial\Omega}(x\cdot\nu)|\nabla u_{i}|^{2}ds.$

###### Proof.

We multiply (10.1) by $x\cdot\nabla u_{i}$ and then integrate over $\Omega$.
Integration by parts gives the left hand term

$\begin{split}&\int_{\Omega}(x\cdot\nabla u_{i})\Delta u_{i}dx\\\
=&\int_{\partial\Omega}(x\cdot\nabla u_{i})\tfrac{\partial
u_{i}}{\partial\nu}ds-\int_{\Omega}|\nabla
u_{i}|^{2}dx-\tfrac{1}{2}\int_{\Omega}x\cdot\nabla(|\nabla u_{i}|^{2})dx\\\
=&\int_{\partial\Omega}(x\cdot\nabla u_{i})\tfrac{\partial
u_{i}}{\partial\nu}ds-\tfrac{1}{2}\int_{\partial\Omega}(x\cdot\nu)|\nabla
u_{i}|^{2}ds.\end{split}$

For the right hand terms, we obtain

$\begin{split}&\int_{\Omega}(x\cdot\nabla
u_{1})(e^{au_{1}}-e^{u_{2}-u_{1}})dx+\sum_{i=2}^{n-1}\int_{B_{r,1}}(x\cdot\nabla
u_{i})(e^{u_{i}-u_{i-1}}-e^{u_{i+1}-u_{i}})dx\\\ &+\int_{\Omega}(x\cdot\nabla
u_{n})(e^{u_{n}-u_{n-1}}-e^{-bu_{n}})dx\\\ =&\int_{\Omega}x\cdot\nabla
u_{1}e^{au_{1}}dx+\sum_{i=2}^{n}\int_{\Omega}x\cdot(\nabla u_{i}-\nabla
u_{i-1})e^{u_{i}-u_{i-1}}dx-\int_{\Omega}x\cdot\nabla u_{n}e^{-bu_{n}}dx\\\
=&\tfrac{1}{a}\int_{\partial\Omega}(x\cdot\nu)(e^{au_{1}}-1)ds+\sum_{i=2}^{n}\int_{\partial\Omega}(x\cdot\nu)(e^{u_{i}-u_{i-1}}-1)ds\\\
&+\tfrac{1}{b}\int_{\partial\Omega}(x\cdot\nu)(e^{-bu_{n}}-1)ds-\tfrac{2}{a}\int_{\Omega}(e^{au_{1}}-1)dx\\\
&-2\sum_{i=2}^{n}\int_{\Omega}(e^{u_{i}-u_{i-1}}-1)dx-\tfrac{2}{b}\int_{\Omega}(e^{-bu_{n}}-1)dx.\end{split}$

Equating these gives the identity stated in the lemma. ∎

######  Lemma 10.7.

Assume that $u(x)$ is a smooth radial solution of (10.1) on
$B_{R}\setminus\\{0\\}$. Then there exist $\beta_{i}>0$ such that
$|u_{i}(x)|\leq-\beta_{i}\log|x|$ for small $|x|$, $i=1,\dots,n$. Moreover,
the limits $\lim\limits_{x\to 0}\dfrac{u_{i}(x)}{\log|x|}$ exist,
$i=1,\dots,n$, and $e^{au_{1}},e^{u_{i}-u_{i-1}},e^{-bu_{n}}\in
L^{1}(B_{R}),i=2,\dots,n$. It follows that the $\gamma_{i}$ satisfy (10.3).

###### Proof.

Step 1: We shall prove that there exist $\beta_{i}>0$ such that
$|u_{i}(x)|\leq-\beta_{i}\log|x|$ for small $|x|$. Without loss of generality
we take $R=1$. For any $0<|x_{0}|<\frac{1}{4},$ we choose
$r_{0}=\frac{1}{2}|x_{0}|$ and $|y_{0}-x_{0}|=\frac{1}{4}r_{0}$. Then
$x_{0}\in B_{r_{0}}(y_{0})\subset B_{1}$ and $0\not\in B_{r_{0}}(y_{0})$. We
consider the system (10.1) on the ball $B_{r_{0}}(y_{0})$. With $r=|x-y_{0}|,$
the function $w(r)=-\log r(r_{0}-r)$ is a smooth function on
$B_{r_{0}}(y_{0})\setminus\\{0\\},$ and satisfies $w(r)\to+\infty\mbox{ as
}r\to 0\mbox{ or }r\to r_{0},$ and

$w^{\prime\prime}+\dfrac{1}{r}w^{\prime}=\frac{r_{0}}{r(r_{0}-r)^{2}},\
0<r<r_{0}.$

For $i=1,\dots,n,$ let $w_{i}(x)=\beta_{i}w(r),$ where $\beta_{i}$ are
positive constants to be chosen later. For $x\in B_{r_{0}}(y_{0}),$ if we
choose $r_{0}$ small enough, then

$\Delta
w_{i}=\dfrac{\beta_{i}r_{0}}{r(r_{0}-r)^{2}}\leq\dfrac{1}{(r(r_{0}-r))^{2}}.$

On the other hand, for $i=2,\dots,n-1,$ we have

$\begin{split}&e^{aw_{1}}-e^{w_{2}-w_{1}}=\dfrac{1}{(r(r_{0}-r))^{a\beta_{1}}}-\dfrac{1}{(r(r_{0}-r))^{\beta_{2}-\beta_{1}}},\\\
&e^{w_{i}-w_{i-1}}-e^{w_{i+1}-w_{i}}=\dfrac{1}{(r(r_{0}-r))^{\beta_{i}-\beta_{i-1}}}-\dfrac{1}{(r(r_{0}-r))^{\beta_{i+1}-\beta_{i}}},\\\
&e^{w_{n}-w_{n-1}}-e^{-bw_{n}}=\dfrac{1}{(r(r_{0}-r))^{\beta_{n}-\beta_{n-1}}}-\dfrac{1}{(r(r_{0}-r))^{-b\beta_{n}}}.\end{split}$

If we choose $a\beta_{1}>\beta_{2}-\beta_{1}>\cdots>\beta_{n}-\beta_{n-1}\geq
2,$ and $r_{0}$ small enough, then on
$B_{r_{0}}(y_{0}),w(x)=(w_{1}(x),\dots,w_{n}(x))$ is a supersolution, that is,

$\begin{split}&\Delta w_{1}\leq e^{aw_{1}}-e^{w_{2}-w_{1}}\\\ &\Delta
w_{i}\leq e^{w_{i}-w_{i-1}}-e^{w_{i+1}-w_{i}},i=2,\dots,n-1,\\\ &\Delta
w_{n}\leq e^{w_{n}-w_{n-1}}-e^{-bw_{n}}.\end{split}$

Since $w_{i}(r)=+\infty,$ if $r=0$ or $r=r_{0},$ the supersolution property of
$w$ implies that $u_{i}(x)\leq w_{i}(x)$ for all $x\in B_{r_{0}}(y_{0})$. The
proof of this inequality can be carried out as in Lemma 10.2 by applying the
maximum principle.

At $x=x_{0},$ we have $u_{i}(x_{0})\leq-\beta_{i}\log|x_{0}|$ for some
$\beta_{i}$. If we let $\hat{u}_{i}=-u_{n+1-i},\ 1\leq i\leq n,$ then
$\hat{u}_{i}$ satisfies (10.1) with the pair $(\hat{a},\hat{b})=(b,a)$. Then
$\hat{u}_{i}(x_{0})\leq-\hat{\beta}_{i}\log|x_{0}|$ for some
$\hat{\beta}_{i}$. This completes the proof of Step 1.

Step 2: $e^{au_{1}},e^{u_{i}-u_{i-1}},e^{-bu_{n}}\in
L^{1}(B_{1}),i=2,\dots,n$. As in the proof of Lemma 10.6, we have

$\begin{split}&\sum_{i=1}^{n}\int_{\partial B_{r,1}}(x\cdot\nabla
u_{i})\tfrac{\partial
u_{i}}{\partial\nu}ds-\tfrac{1}{2}\sum_{i=1}^{n}\int_{\partial
B_{r,1}}(x\cdot\nu)|\nabla u_{i}|^{2}ds\\\ =&\tfrac{1}{a}\int_{\partial
B_{r,1}}(x\cdot\nu)e^{au_{1}}ds+\sum_{i=2}^{n}\int_{\partial
B_{r,1}}(x\cdot\nu)e^{u_{i}-u_{i-1}}ds+\tfrac{1}{b}\int_{\partial
B_{r,1}}(x\cdot\nu)e^{-bu_{n}}ds\\\
&-\tfrac{2}{a}\int_{B_{r,1}}e^{au_{1}}dx-2\sum_{i=2}^{n}\int_{B_{r,1}}e^{u_{i}-u_{i-1}}dx-\tfrac{2}{b}\int_{B_{r,1}}e^{-bu_{n}}dx,\end{split}$

where $B_{r,1}=\\{x\ |\ r<|x|<1\\}$.

Noting that $x\cdot\nu=|x|$ on the ball, we may rewrite this as follows:

$\begin{split}&\tfrac{2}{a}\int_{B_{r,1}}e^{au_{1}}dx+2\sum_{i=2}^{n}\int_{B_{r,1}}e^{u_{i}-u_{i-1}}dx+\tfrac{2}{b}\int_{B_{r,1}}e^{-bu_{n}}dx\\\
&+\tfrac{1}{a}\int_{\partial
B_{r,1}}re^{au_{1}}ds+\sum_{i=2}^{n}\int_{\partial
B_{r,1}}re^{u_{i}-u_{i-1}}ds+\tfrac{1}{b}\int_{\partial
B_{r,1}}re^{-bu_{n}}ds\\\ =&\tfrac{1}{a}\int_{\partial
B_{1}}e^{au_{1}}ds+\sum_{i=2}^{n}\int_{\partial
B_{1}}e^{u_{i}-u_{i-1}}ds+\tfrac{1}{b}\int_{\partial
B_{1}}e^{-bu_{n}}ds-\sum_{i=1}^{n}\int_{\partial B_{1}}(\tfrac{\partial
u_{i}}{\partial\nu})^{2}ds\\\ &+\sum_{i=1}^{n}\int_{\partial
B_{r}}r(\tfrac{\partial
u_{i}}{\partial\nu})^{2}ds+\tfrac{1}{2}\sum_{i=1}^{n}\int_{\partial
B_{1}}|\nabla u_{i}|^{2}ds-\tfrac{1}{2}\sum_{i=1}^{n}\int_{\partial
B_{r}}r|\nabla u_{i}|^{2}ds\\\ \leq&\tfrac{1}{2}\sum_{i=1}^{n}\int_{\partial
B_{r}}r|\nabla u_{i}|^{2}ds+O(1).\end{split}$

By Step 1, we may choose $r_{m}\to 0$ such that
$|\tfrac{du_{i}}{dr}(r_{m})|\leq r_{m}^{-1}$. Since each term on the left hand
side is positive,
$\lim\limits_{m\to+\infty}\int_{B_{r_{m},1}}e^{u_{i}-u_{i-1}}dx$
$(i=2,\dots,n)$ and $\lim\limits_{m\to+\infty}\int_{B_{r_{m},1}}e^{-bu_{n}}dx$
exist. Hence $e^{au_{1}},e^{u_{i}-u_{i-1}},e^{-bu_{n}}\in
L^{1}(B_{1}),i=2,\dots,n$.

Step 3: We prove that $\lim\limits_{x\to 0}\frac{u_{i}(x)}{\log|x|}$ exists.
Recall that if $h(x)$ is a harmonic function on $B_{R}\setminus\\{0\\}$ and
satisfies $|h(x)|\leq\beta\log\frac{1}{|x|}$ for small $x$, then
$h(x)=c\log\frac{1}{|x|}+$ a smooth harmonic function on $B_{R}$.

Let $w(x)$ be a solution of $\Delta w+g(x)=0,x\in B_{R}\setminus\\{0\\}$ and
$|w(x)|\leq\beta\log\frac{1}{|x|}$ for small $x$. Then
$w(x)=\int_{B_{R}}G(x-y)g(y)dy+h(x),$ where $h$ is harmonic on
$B_{R}\setminus\\{0\\}$ with $|h(x)|\leq\beta\log\frac{1}{|x|}$, and $G$ is
the Green’s function. Then it is not difficult to show that $\lim\limits_{x\to
0}\frac{w(x)}{\log|x|}=\lim\limits_{x\to 0}\frac{h(x)}{\log|x|}$. ∎

######  Lemma 10.8.

Suppose $u(x)$ is a smooth radial solution of (10.1) on
$\mathbb{R}^{2}\setminus\\{0\\}$. Then $\lim\limits_{|x|\to+\infty}u(x)=0$.

###### Proof.

Step 1: We prove that $u(x)$ is bounded as $|x|\to+\infty$. We construct a
supersolution as in Lemma 10.7. Choose any
$a\beta_{1}>\beta_{2}-\beta_{1}>...>\beta_{n}-\beta_{n-1}\geq 2$, and consider

$v(x)=-\log(r-s_{1})(s_{2}-r),r=|x|$

where $s_{2}>s_{1}>0$ and $\frac{s_{2}-s_{1}}{2}=\alpha<1$. Then

$\begin{split}\Delta
v(x)&=\dfrac{s_{1}}{r(r-s_{1})^{2}}+\dfrac{s_{2}}{r(s_{2}-r)^{2}}\\\
&=\big{(}\dfrac{s_{1}}{r}(s_{2}-r)^{2}+\dfrac{s_{2}}{r}(r-s_{1})^{2}\big{)}\big{(}(r-s_{1})(s_{2}-r)\big{)}^{-2}.\end{split}$

By noting that
$(r-s_{1})(s_{2}-r)\leq(\frac{s_{2}-s_{1}}{2})^{2}=\alpha^{2}<1$ for
$s_{1}\leq r\leq s_{2}$, we can find $s_{1}$ large enough that
$w=(w_{1},...,w_{n})$, $w_{i}(x)=\beta_{i}v(x)$ is a supersolution of (10.1)
for $s_{1}<r<s_{2}$, i.e.,

$\begin{split}&\Delta w_{1}\leq e^{aw_{1}}-e^{w_{2}-w_{1}},\\\ &\Delta
w_{i}\leq e^{w_{i}-w_{i-1}}-e^{w_{i+1}-w_{i}},i=2,...,n-1\\\ &\Delta w_{n}\leq
e^{w_{n}-w_{n-1}}-e^{-bw_{i}}\end{split}$

where $s_{i}$ depends only on $\beta_{i}$ and $\alpha$. Thus the maximum
principle implies

$u_{i}(x)\leq\beta_{i}v(x),s_{1}\leq|x|\leq s_{2}.$

In particular, if $s_{1}$ and $s_{2}$ are chosen so that
$|x|=\frac{s_{1}+s_{2}}{2}$, we have proved that $u_{i}(x)$ is bounded above.
Similarly, we let $\hat{u}_{i}=-u_{n+1-i}$ and obtain that $u_{i}(x)$ is
bounded below.

Step 2: We prove that $\lim\limits_{|x|\to+\infty}u_{i}(x)=0$. Set
$\overline{l}=\max\limits_{1\leq i\leq
n}\\{\overline{\lim}_{|x|\to+\infty}u_{i}(x)\\}.$ Suppose
$\overline{\lim}_{|x|\to+\infty}u_{i_{0}}(x)=\overline{l}$. Take
$x_{m}\in\mathbb{R}^{2}$ such that $|x_{m}|\to+\infty$ and
$\lim\limits_{m\to+\infty}u_{i_{0}}(x_{m})=\overline{l}$. Then we define

$v_{i}^{(m)}(x)=u_{i}(x_{m}+x),1\leq i\leq n.$

Since $u_{i}(x)$ is bounded, standard p.d.e. estimates show that there is a
subsequence (still denoted by $v^{(m)}$) of $v^{(m)}$ such that $v^{(m)}\to v$
as $m\to+\infty$ and such that $v=(v_{1},...,v_{n})$ is a smooth solution of
(10.1) in $\mathbb{R}^{2}$. Further, $v$ is bounded and

$v_{i}(x)\leq v_{i_{0}}(0)=\overline{l},\forall x\in\mathbb{R}^{2}\mbox{\text{
and }}1\leq i\leq n.$

If $i_{0}=n$, by applying the maximum principle to $v_{n}$ we obtain

$-bv_{n}(0)\geq v_{n}(0)-v_{n-1}(0).$

Thus $\overline{l}\geq v_{n-1}(0)\geq(1+b)v_{n}(0)=(1+b)\overline{l},$ which
implies $\overline{l}\leq 0$.

If $1<i_{0}<n$, by applying the maximum principle to $v_{i_{0}}$ we obtain

$v_{i_{0}+1}(0)-v_{i_{0}}(0)\geq v_{i_{0}}(0)-v_{i_{0}-1}(0)\geq 0.$

Thus $v_{i_{0}+1}(0)\geq v_{i_{0}}(0)$. Hence $v_{i_{0}+1}(0)=v_{i_{0}}(0)$.
Repeating this argument, we have $v_{i_{0}}(0)=\cdots=v_{n}(0)=\overline{l}$,
which implies $\overline{l}\leq 0$.

If $i_{0}=1$, then again the maximum principle yields

$\overline{l}\geq v_{2}(0)\geq(1+a)v_{1}(0)=(1+a)\overline{l},$

and then we have $\overline{l}\leq 0$ as well.

Applying this argument to $\hat{u}_{i}(x)=-u_{n+1-i}(x)$, we obtain

$-\underline{\lim}_{|x|\to+\infty}u_{i}(x)=\overline{\lim}_{|x|\to+\infty}\hat{u}_{i}(x)\leq
0.$

Thus $\underline{\lim}_{|x|\to+\infty}u_{n+1-i}(x)\geq 0$ for all $i$. This
implies $\lim\limits_{|x|\to+\infty}u_{i}(x)=0$. This proves Lemma 10.8. ∎

Lemma 10.7 gives (i) and (ii) of Theorem 10.5, and Lemma 10.8 gives (iii).
Thus we have completed the proof of Theorem 10.5.

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Department of Mathematics
Faculty of Science and Engineering
Waseda University
3-4-1 Okubo, Shinjuku, Tokyo 169-8555
JAPAN,

Department of Mathematical Sciences
Indiana University-Purdue University, Indianapolis
402 N. Blackford St.
Indianapolis, IN 46202-3267
USA

and

St. Petersburg University
7/9 Universitetskaya nab.
199034 St. Petersburg
RUSSIA,

Taida Institute for Mathematical Sciences
Center for Advanced Study in Theoretical Sciences
National Taiwan University
Taipei 10617
TAIWAN