Keeping the notations of the previous sections, we gather the results of Proposition 4 and 5. Assume that
∈ Ω
^{
m
} are given close to
x := (
x_{1}, ...,
x_{
m
}) and satisfy (37). Assume also that
τ := (
τ_{1}, ...,
τ_{
m
}) ∈ [
τ _{

},
τ ^{+}]
^{
m
} ⊂ (0, ∞)
^{
m
} are given (the values of
τ_{} and
τ ^{+} will be fixed shortly). First, we consider some set of boundary data
satisfying (24). We set
According to the result of Proposition 4, we can find
a solution of
in each
that can be decomposed as
where the function
satisfies
Similarly, given some boundary data
satisfying (26), some parameters
satisfying (38), provide
ε ∈ (0,
ε_{
κ
}) and
λ ∈ (0,
λ_{
κ
}), we use the result of Proposition 5, to find a solution
v_{
ext
} of (43) which can be decomposed as
in
where, the function
satisfies
It remains to determine the parameters and the functions in such a way that the function which is equal to
in
and that is equal to
v_{ext} in
is a smooth function. This amounts to find the boundary data and the parameters so that, for each
i = 1 ...,
mon
. Assuming we have already done so, this provides for each
ε and
λ small enough a function
(which is obtained by patching together the functions
and the function
v_{ext}) solution of Δ
v 
λ ∇
v
^{2} =
ρ^{2} (
e^{
v
} +
e^{
γv
}) and elliptic regularity theory implies that this solution is in fact smooth. This will complete the proof of our result since, as
ε and
λ tend to 0, the sequence of solutions we have obtained satisfies the required properties, namely, away from the points
x_{
i
} the sequence
v_{ε,λ}converges to
. Before we proceed, the following remarks are due. First, it will be convenient to observe that the function
can be expanded as
near
. The function
which appear in the expression of
v_{
ext
} can be expanded as
Near
. Here, we have defined
Thus for
x near
, we have
where
.
Next, in (47), all functions are defined on
, but it will be convenient to solve the following equations
on S^{1}. Here, all functions are considered as functions of y ∈ S^{1} and we have simply used the change of variables
to parameterize
.
Since the boundary data, we have chosen satisfy (24) and (26), we can decompose
where
are constant functions on
S^{1},
belong to
and where
are
L^{2}(
S^{1}) orthogonal to
and
. Projecting the equations (
51) over
will yield the system
Let us comment briefly on how these equations are obtained. They simply come from (51) when expansions (48) and (49) are used, together with the expression of
H^{
i
} and
H^{
e
} given in Lemma 2 and Lemma 3, and also the estimates (45) and (46). The system (52) can be readily simplified into
We are now in a position to define
τ _{

} and
τ ^{+} since, according to the above, as
ε and
λ tend to 0 we expect that
will converge to
x_{
i
} and that
τ_{
i
} will converge to
satisfying
and hence, it is enough to choose
τ _{

} and
τ ^{+} in such a way that
We now consider the
L^{2}projection of (51) over
. Given a smooth function
f defined in Ω, we identify its gradient
with the element of
With these notations in mind, we obtain the equations
Finally, we consider the
L^{2}projection onto
L^{2}(
S^{1})
^{⊥}. This yields the system
Thanks to the result of Lemma 4, this last system can be rewritten as
If we define the parameters
t = (
t_{
i
}) ∈ ℝ
^{
m
} by
then, the system we have to solve reads
where as usual, the terms
depend nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and λ) time
, provide ε ∈ (0, ε_{
κ
}) and λ ∈ (0, λ_{
κ
}). Then, the nonlinear mapping which appears on the righthand side of (55) is continuous and compact. In addition, reducing ε_{
κ
} and λ_{
κ
} if necessary, this nonlinear mapping sends the ball of radius
(for the natural product norm) into itself, provided κ is fixed large enough. Applying Schauder's fixed Theorem in the ball of radius
in the product space where the entries live yields the existence of a solution of Eq. (55) and this completes the proof of our Theorem 1. □