Parametric surfaces 207
The regularity of F is assured by lemma 6, since u ∈ H
1 g
= u + H
1
and 45
d Fv = d E
H
v + u .
The condition mp
1
is granted by lemma 7. The condition mp
2
follows immediately from lemma 8. Hence, by theorem 13, the functional F admits a Palais-Smale sequence
v
n
⊂ H
1
at a level c 0. By 44 and 45, setting u
n
= v
n
+ u, we obtain a Palais- Smale sequence in H
1 g
for the functional E
H
at level c + E
H
u. Owing to the conformal invariance of the problem, the functional E
H
is not ex- pected to verify the Palais-Smale condition, and a deeper analysis of the Palais-Smale
sequences for E
H
is needed.
6.2. Palais-Smale sequences for E
H
Recalling remark 6, by 41 and 43, a Palais-Smale sequence for the functional E
H
is a sequence u
n
⊂ H
1 g
such that E
H
u
n
→ ¯c 46
1 u
n
= 2H u
n x
∧ u
n y
+ f
n
in D
2
, with f
n
→ 0 in H
−1
47 for some
¯c ∈ R. As a first fact, we have the following result.
L
EMMA
9. Any Palais-Smale sequence u
n
⊂ H
1 g
for E
H
is bounded in H
1
. Proof. Since u
n
⊂ H
1 g
it is enough to prove that sup k∇u
n
k
2
+∞. Setting ϕ
n
= u
n
− u, and keeping into account that d E
H
u = 0, one has
E
H
u
n
= E
H
u +
1 2
d
2
E
H
uϕ
n
, ϕ
n
+ 2V
H
ϕ
n
d E
H
u
n
ϕ
n
= d
2
E
H
uϕ
n
, ϕ
n
+ 6V
H
ϕ
n
. Hence, subtracting, one obtains
3E
H
u
n
= E
H
u +
1 2
d
2
E
H
uϕ
n
, ϕ
n
+ d E
H
u
n
ϕ
n
. Using Lemma 7, one gets
δ k∇ϕ
n
k
2 2
≤ d
2
E
H
uϕ
n
, ϕ
n
= 6E
H
u
n
− E
H
u − 2d E
H
u
n
ϕ
n
≤ C + kd E
H
u
n
k k∇ϕ
n
k
2
. By 46 and 47 one infers that ϕ
n
is bounded in H
1
and then the thesis follows.
208 F. Bethuel - P. Caldiroli - M. Guida
In the case of variable H , it is not clear whether the lemma holds or not. A method to overcome this kind of difficulty can be found in Struwe [42].
From the previous lemma we can deduce that all Palais-Smale sequences for E
H
are relatively weakly compact. The next result states that the weak limit is a solution to D
H
. L
EMMA
10. Let u
n
⊂ H
1 g
be a Palais-Smale sequence for E
H
converging weakly in H
1
to some ¯u ∈ H
1 g
. Then d E
H
¯u = 0, i.e., ¯u is a weak solution to D
H
. Proof. Fix an arbitrary ϕ
∈ C
∞ c
D
2
, R
3
. By 47, one has Z
D
2
∇u
n
· ∇ϕ + 2H Lu
n
, ϕ → 0
where we set Lu, ϕ
= Z
D
2
ϕ · u
x
∧ u
y
. By weak convergence
R
D
2
∇u
n
· ∇ϕ → R
D
2
∇ ¯u · ∇ϕ. Moreover, using the divergence expression 2u
x
∧ u
y
= u ∧ u
y x
+ u
x
∧ u
y
, one has that 2Lu, ϕ
= − Z
D
2
ϕ
x
· u ∧ u
y
+ ϕ
y
· u
x
∧ u . Hence Lu
n
, ϕ → L ¯u, ϕ, since u
n
→ ¯u strongly in L
2
and weakly in H
1
. In conclusion, one gets
Z
D
2
∇ ¯u · ∇ϕ + 2H Z
D
2
ϕ · ¯u
x
∧ ¯u
y
= 0 that is the thesis.
However, the Palais-Smale sequences for E
H
are not necessarily relatively strongly compact in H
1
. In the spirit of Aubin [5] and Sacks-Uhlenbeck [37], and inspired by the concentration-compactness principle by P.-L. Lions [35], Brezis and
Coron in [14] have precisely analyzed the possible defect of strong convergence, as the following theorem states.
T
HEOREM
14. Suppose that u
n
∈ H
1 g
is a Palais-Smale sequence for E
H
. Then there exist
i u ∈ H
1 g
solving 1u = 2H
u
x
∧ u
y
in D
2
, ii a finite number p
∈ N ∪ {0} of nonconstant solutions v
1
, . . . , v
p
to 1u =
2H u
x
∧ u
y
on R
2
, iii p sequences a
1 n
, . . . , a
p n
in D
2
Parametric surfaces 209
iv p sequences ε
1 n
, . . . , ε
p n
in R
+
with lim
n →+∞
ε
n i
= 0 for any i = 1, . . . , p such that, up to a subsequence, we have
u
n
− u −
p
X
i =1
v
i
· − a
i n
ε
i n
H
1
→ 0 Z
D
2
|∇u
n
|
2
= Z
D
2
|∇u|
2
+
p
X
i =1
Z
R
2
|∇v
i
|
2
+ o1 E
H
u
n
= E
H
u +
p
X
i =1
¯ E
H
v
i
+ o1 , where in general ¯
E
H
v =
1 2
R
R
2
|∇v|
2
+
2H 3
R
R
2
v · v
x
∧ v
y
. In case p = 0 any sum
P
p i
=1
is zero and u
n
→ ¯u strongly in H
1
. R
EMARK
7. The conformal invariance is reflected in the concentrated maps v
i ·−a
i n
ε
i n
. This theorem also emphasizes the role of solutions of the H -equation on
whole R
2
, which are completely known see below.
6.3. Characterization of solutions on R
