CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

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and by Remark 3.2 we deduce that
‖φ ξ ‖ H 1
0 (Ω) = O (‖φ ξ‖ ∞ + ‖N(φ ξ )‖ ∗ + ‖R‖ ∗ ) ≤ C p 3 .
The proof is now complete since φ ξ solves (4.1)-(4.4): in order to show
the validity of (4.2), let us remark that p|φ ξ | → 0 in C(¯Ω) and by elliptic
regularity theory p|φ ξ | → 0 in C 1 (¯Ω \ ∪ m j=1 B(ξ j, ɛ)) and so, we can proceed
as in Remark 2.1 to show that U ξ + φ ξ > 0 in Ω.
□
Let ξ 1 , ξ 2 ∈ O ε . Since
∆(φ ξ1 − φ ξ2 ) + pU p−1
ξ 1
(φ ξ1 − φ ξ2 ) = ((U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p )
(
+ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1
)
ξ 1
(φ ξ2 − φ ξ1 ) + ∆(U ξ2 − U ξ1 )
2∑ m∑
+ (c ij (ξ 1 ) − c ij (ξ 2 )) e U j
(ξ 1 )Z ij (ξ 1 )
+
i=1 j=1
2∑ m∑
i=1 j=1
and by (3.2)
(
)
c ij (ξ 2 ) e U j
(ξ 1 )Z ij (ξ 1 ) − e U j
(ξ 2 )Z ij (ξ 2 )
(
)
‖ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1
ξ 1
(φ ξ2 − φ ξ1 ) ‖ ∗
≤ C p 2 ‖φ ξ 1
− φ ξ2 ‖ ∞ ‖p(U ξ1 + O( 1 p 3 ))p−2 ‖ ∗
= o( 1 p ‖φ ξ 1
− φ ξ2 ‖ ∞ )
uniformly in O ε , by Proposition 3.1 and (4.6) we get
‖φ ξ1 − φ ξ2 ‖ ∞ ≤ Cp‖(U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p ‖ ∗
+ C 2∑ m∑
p 3 ‖e U j
(ξ 1 )Z ij (ξ 1 ) − e U j
(ξ 2 )Z ij (ξ 2 )‖ ∗
i=1 j=1
+ Cp‖∆(U ξ2 − U ξ1 )‖ ∗ ,
for any p ≥ p 0 and ξ 1 , ξ 2 ∈ O ε (here, ‖ · ‖ ∗ is considered with respect to ξ 1 ).
Hence, for fixed p ≥ p 0 , the map ξ → φ ξ is continuous in C 0 (¯Ω) and, in view
of Remark 3.2, in H0 1(Ω). Further, this map is a C1 -function in C 0 (¯Ω) as it
follows by the Implicit Function Theorem applied to the equation:
[
( ) p ]
G(ξ, φ) := Π ⊥ ξ U ξ + Π ⊥ ξ φ + ∆ −1 U ξ + Π ⊥ ξ φ + Π ξ φ = 0, φ ∈ C 0 (¯Ω),
where Π ξ , Π ⊥ ξ
are the maps from L2 (Ω) respectively onto
2∑ m∑
K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m}
i=1 j=1