CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

28 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA
and
∫
Kξ ⊥ = {φ ∈ L 2 (Ω) :
Ω
e U j
Z ij φ = 0 for i = 1, 2, j = 1, . . . , m}
(see the notations in Step 6 in the proof of Proposition 3.1). Let us remark
that Π ξ φ ∈ C0 k(¯Ω), for any k ≥ 0. Indeed, G(ξ, φ ξ ) = 0 and the linearized
operator:
∂G
∂φ (ξ, φ ξ) = Π ⊥ ξ
[
( )]
Id + p∆ −1 (U ξ + φ ξ ) p−1 Π ⊥ ξ + Π ξ
is invertible for p large. In fact, easily we reduce the invertibility property to
uniquely solve the equation ∂G
∂φ (ξ, φ ξ)[φ] = h in Kξ
⊥ for any h ∈ K⊥ ξ ∩C 0(¯Ω).
By Fredholm’s alternative, we need to show that in Kξ ⊥ ∩C 0(¯Ω) there is only
the trivial solution for the equation ∂G
∂φ (ξ, φ ξ)[φ] = 0, or equivalently for
(
Lφ = p U p−1
ξ
− (U ξ + φ ξ ) p−1) 2∑ m∑
φ + c ij e U j
Z ij ,
i=1 j=1
for any choice of the coefficients c ij , since by elliptic regularity theory φ ∈
C0 2(¯Ω). By Proposition 3.1 and (3.2), we derive that
(
‖φ‖ ∞ ≤ C ′ p‖p U p−1
ξ
− (U ξ + φ ξ ) p−1) φ‖ ∗
≤ C ′ p 2 ‖φ‖ ∞ ‖φ ξ ‖ ∞ ‖p(U ξ + O( 1 p 3 ))p−2 ‖ ∗ < ‖φ‖ ∞
and hence, φ = 0. Similarly, we have also that ξ → φ ξ is a C 1 -function in
H 1 0 (Ω).
5. Variational reduction
After problem (4.1)-(4.4) has been solved, we find a solution of (2.15) (and
hence for (1.1)) if ξ is such that
c ij (ξ) = 0 for all i = 1, 2 , j = 1, . . . , m, (5.1)
where c ij (ξ) are the coefficients in (4.1). Problem (5.1) has a variational
structure. Associated to (1.1), let us consider the energy functional J p given
by
J p (u) = 1 ∫
|∇u| 2 dx − 1 ∫
|u| p+1 dx , u ∈ H0 1 (Ω),
2 Ω
p + 1 Ω
and the finite dimensional restriction
F (ξ) := J p (U ξ + φ ξ ) (5.2)
where φ ξ is the unique solution to problem (4.1)-(4.4) given by Lemma 4.1.
Critical points of F correspond to solutions of (5.1) for large p, as the following
result states:
Lemma 5.1. The functional F (ξ) is of class C 1 . Moreover, for all p sufficiently
large, if D ξ F (ξ) = 0 then ξ satisfies (5.1).