Erratum to: Persistence of Coron’s solution in nearly critical problems

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. VIII (2009), 207-209

Erratum to:

Persistence of Coron’s solution in nearly critical problems

MONICAMUSSO ANDANGELAPISTOIA

Formula (3.17) in our paper [1] is wrong. This erratum is devoted to give the right formula and to list the main changes that need to be made.

Results in Section 2 change as follows. In (2.1) we assume

ξ :=µτ, τ ∈

R

Lemma 1.1. Let

R_ε,µ(x):=P_εU_µ,ξ(x)−U_µ,ξ(x)+αNµ^N−22 H(x, ξ) +αN

1

µ^N2−2(1+ |τ|²)^N−22 ϕ_ωx ε

.

Then there exists a positive constant c such that for any x ∈\εω R_ε,µ(x)≤cε^N−24

ε^N2−1

|x|^N−2 +ε

if N ≥4, (1.2) R_ε,µ(x)≤cε¹⁴

ε

|x| +√ ε

if N =3. (1.3) Proof. The functionR:=R_ε,µsolves− R=0 in\εωwith

R(x)=αN

− µ^N−22

(µ²+ |x−ξ|²)^N−22 + µ^N−22

|x−ξ|^N−2

+ 1

µ^N−22 (1+ |τ|²)^N−22 ϕω

x

ε , x∈∂, R(x)=αN

− µ^N−22

(µ²+ |x−ξ|²)^N−22 +µ^N−22 H(x, ξ)

+ 1

µ^N−22 (1+ |τ|²)^N−22 , x ∈∂εω.

Received January 8, 2009.

208 MONICAMUSSO ANDANGELAPISTOIA

Therefore (1.2) and (1.3) follow, because ε^−N4−2R(x)=O

ε+ε^N2−2

, x∈∂

and

ε^−N−24 R(x)=O

ε^−N−32

, x∈∂εω.

In Section 3 the functiondefined in (3.17) becomes

(τ,d):= −F(τ) 1

d^N−2 −a₃H(0,0)d^N−2+b₁+b₂logd, (1.4) where

F(τ):=α_N^p+1c_ω 1 (1+ |τ|²)^N2−2

R^N

1

|y+τ|^N−2 1

(1+ |y²)^N2+2d y. (1.5)

The rest of the section remains unchanged.

In Section 4 the proof of Lemma (4.1) changes as follows. Estimate (4.7) becomes

\εω

U_µ,ξ^p+1=α_N^p+1

\εω

µ^N

(µ²+ |x−ξ|²)^Nd x =α_N^p+1

\εωµ

1

(1+ |y−τ|²)^Nd y

=α_N^p+1

IR^N

1

(1+ |y|²)^Nd y+O ε

µ N

+µ^N

.

Estimate (4.11) becomes

\εω

P_εU_µ,ξ −U_µ,ξ

U_µ,ξ^p d x =

\εω

R_ε,µU_µ,ξ^p d x

−α_N^p+1

\εω

µ^N−22 H(x, ξ)+ 1

µ^N−22 (1+|τ|²)^N2−2ϕ_ωx ε

µ^N2+2

(µ²+|x−ξ|²)^N+22 d x.

ERRATUM TO: PERSISTENCE OFCORON’S SOLUTION 209

Estimate (4.13) becomes 1

µ^N2−2(1+ |τ|²)^N−22

\εω

ϕ_ωx ε

µ^N+22

(µ²+ |x−ξ|²)^N+22 d x

= 1

(1+ |τ|²)^N−22

\εω−ξ µ

ϕ_ωµ

ε(y+τ) 1

(1+ |y|²)^N+22 d y

= ε

µ _N−2

1 (1+ |τ|²)^N−22

\εω−ξ µ

f_ε(y) 1

|y+τ|^N−2

1

(1+ |y|²)^N+22 d y

= ε

µ N−2



c_ω 1 (1+ |τ|²)^N2−2

IR^N

1

|y+τ|^N−2

1

(1+ |y|²)^N2+2d y+o(1)



.

The rest of the proof remains unchanged.

References

[1] M. MUSSO and A. PISTOIA,Persistence of Coron’s solution in nearly critical problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5)6(2007), 331–357.

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and

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