Erratum to : “Multiple Critical Points of Perturbed Symmetric Strongly Indefinite Functionals”

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Erratum

Erratum to: “Multiple critical points of perturbed symmetric strongly indefinite functionals”

[http://dx.doi.org/10.1016/j.anihpc.2008.06.002]

Denis Bonheure

, Miguel Ramos

aUniversité libre de Bruxelles, Département de Mathématique, U.L.B., CP214, Boulevard du Triomphe, 1050 Bruxelles, Belgium bUniversity of Lisbon, CMAF – Faculty of Science, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

Received 21 October 2008; accepted 14 January 2009 Available online 17 March 2009

Abstract

We correct the statement and the proof of Proposition 9 in [D. Bonheure, M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, http://dx.doi.org/10.1016/j.anihpc.2008.06.002].

©2008 Elsevier Masson SAS. All rights reserved.

MSC:35J50; 35J55; 58E05

Keywords:Elliptic system; Strongly indefinite functional; Perturbation from symmetry; Lyapunov–Schmidt reduction; Genericity

The proof of [1, Proposition 9] is incorrect. We weaken the statement of this proposition and present a proof of it. The weaker statement however is enough for the purposes of [1]. We use the notation and assumptions introduced in [1].

LetI^∗:E→Rbe the functional associated to the problem

⎧⎪

⎨

⎪⎩

−u= |v|^q−2v inΩ,

−v= |u|^p−2u inΩ,

u=0 on∂Ω,

v=0 on∂Ω.

(1.1)

Consider the associated reduced functional J^∗(α):=I^∗

α+ψ_α^∗, α−ψ_α^∗

:= max

ψ∈H₀¹(Ω)

I^∗(α+ψ, α−ψ ). (1.2)

DOI of original article: 10.1016/j.anihpc.2008.06.002.

* Corresponding author.

E-mail addresses:denis.bonheure@ulb.ac.be (D. Bonheure), mramos@ptmat.fc.ul.pt (M. Ramos).

1 Research supported by the F.R.S.-FNRS (Belgium).

2 Research partially supported by FCT (Fundação para a Ciência e Tecnologia), program POCI-ISFL-1-209 (Portugal/Feder-EU).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.01.007

Recall that ifαis a critical point ofJ^∗then

−2α=f u^∗_α

+g v_α^∗

, (1.3)

whereu^∗_α:=α+ψ_α^∗,v_α^∗:=α−ψ_α^∗andψ_α^∗is the unique solution of the following equation inH₀¹(Ω):

−2ψ_α^∗=g v^∗_α

−f u^∗_α

. (1.4)

Denote bym^∗(α)theaugmented Morse indexof the critical pointαwith respect toJ^∗, i.e. the number of non-positive eigenvalues of the quadratic form(J^∗)(α).

We will be interested in special critical points constructed via a min-max argument. To that purpose, we introduce the following notations. Let us write

H:=H₀¹(Ω)=E_k⊕E_k^⊥,

where, for eachk∈N0,E_k is spanned by the firstkeigenfunctions of the Laplacian operator inH₀¹(Ω). Arguing as in [1, Lemma 3], we can provide a large constantR_k>0 such thatJ^∗(α) <0 for everyα∈E_k satisfyingα> R_k. Let

G_k:=

σ∈C

B_Rk(0)∩E_k;H σ (−α)= −σ (α)∀α∈H, σ_|∂B

Rk(0)∩E_k=Id and define the minimax levels

b_k:= inf

σ∈Gk

max J^∗

σ (α)

: α∈B_Rk(0)∩E_k

. (1.5)

We next derive a bound onb_k.

Proposition 9.Assume2< pq2^∗. There existC >0andk0∈N0such that for everykk0, kCb⁽¹⁻

1 p−¹_q)^N₂

k .

Proof. For the sake of clarity we divide the proof in several steps.

Step 1.According to (1.3) and (1.4), ifαis a critical point ofJ^∗,m^∗(α)is the number of eigenvaluesμ1 of the problem

−2ϕ=μ f

u^∗_α

(ϕ+φ)+g v_α^∗

(ϕ−φ)

, ϕ∈H₀¹(Ω), (1.6)

whereφ∈H₀¹(Ω)solves

−2φ=g v_α^∗

(ϕ−φ)−f u^∗_α

(ϕ+φ). (1.7)

By denotingV =(f(u^∗_α)+g(v_α^∗))/2 andW=(f(u^∗_α)−g(v_α^∗))/2, we can rephrase (1.6)–(1.7) by

−ϕ=μ(V ϕ+W φ) and (−+V )φ= −W ϕ.

Hence,m^∗(α)is the number of eigenvaluesμ1 of the problem

−ϕ=μT ϕ, ϕ∈H₀¹(Ω), whereT is the compact operator

T :=V −W (−+V )⁻¹W.

Since the operatorW (−+V )⁻¹W is positive, we have thatm^∗(α)m^∗_V(α), where the latter quantity denotes the number of eigenvaluesμ1 of the problem

−ϕ=μV (x)ϕ, ϕ∈H₀¹(Ω).

According to a well-known estimate obtained in [2,3,5] (see e.g. [6] for a proof), we have that m^∗_V(α)C

V (x)^N/2 (1.8)

for some universal constantC >0. Going back to the original system−u= |v|^q−2v,−v= |u|^p−2u, we observe that

|u^∗_α|^p=

|v_α^∗|^qand J^∗(α)=I^∗

u^∗_α, v_α^∗

= 1

2− 1 p

u^∗_α ^p+ 1

2−1 q

v^∗_α ^q

=

1−1 p−1

q

u^∗_α ^p

=

1−1 p−1

q

v_α^{∗ q}.

By using this and by applying Hölder inequality in (1.8) we get that m^∗(α)C

J^∗(α)^(p−2)N/2p+J^∗(α)^(q−2)N/q

. (1.9)

We will next refine this estimate by introducing a free parameter.

Step 2.For anyλ >0, define J_λ^∗(α):=I^∗

λα+ψ_α.λ^∗ , α−ψ_α,λ^∗ λ

:=max

ψ∈HI^∗

λα+ψ, α−ψ λ

,

so thatJ₁^∗(α)=J^∗(α). Arguing as in step 1, we can check that ifαis a critical point ofJ_λ^∗then the corresponding Morse indexm^∗_λ(α)is given by the number of eigenvaluesμ1 of the problem

−ϕ=μT_λϕ, ϕ∈H₀¹(Ω), whereT_λis the compact operator

T_λ:=V_λ−W_λ(−+V_λ)⁻¹W_λ,

withV_λ=¹₂(λf(u^∗_α)+g(v_α^∗)/λ)andW_λ=¹₂(f(u^∗_α)−g(v_α^∗)/λ²). Here, of course,u^∗_α:=λα+ψ_α,λ^∗ ,v^∗_α:=α−^ψα,λ_λ^∗ . Then, similarly to (1.8)–(1.9), we get that

m^∗_λ(α)C

λJ_λ^∗(α)^(p−2)/pN/2

+

J_λ^∗(α)^(q−2)/q λ

N/2

. (1.10)

Step 3.As a further preliminary step in our proof, we introduce the map θλ(α):=λ+1

2 α+λ−1 2λ ψ_α,λ^∗ .

We claim thatθ_λ:H→His an odd homeomorphism such that J_λ^∗

θ_λ⁻¹(α)

J₁^∗(α), ∀α∈H. (1.11)

We can already assume thatλ=1, otherwise our statement is obvious. Observe that givenβ∈H it is possible to find an uniqueα∈H such that

ψ_α,λ^∗ = 2λ

λ−1β−λ(λ+1) λ−1 α.

Indeed, using the definition ofψ_α,λ^∗ , this means that we must solve the equation inH:

−2λλ+1

λ−1α= −4λ

λ−1β−g 2λ

λ−1α− 2 λ−1β

+λf

−2λ

λ−1α+ 2λ λ−1β

and, clearly, this problem has a unique solution α∈H for any givenβ∈H. As for (1.11), givenα∈H, letβ = θ_λ⁻¹(α). Thenα=θ_λ(β)and

J_λ^∗(β):=I^∗

λβ+ψ_β,λ, β−ψ_β,λ λ

=I^∗(α+ϕ, α−ϕ)J₁^∗(α),

whereϕ:=λβ+ψ_β,λ−αand we have used the definition ofJ₁^∗in the last inequality.

Step 4.We now describe the following min-max construction. By using Fatou’s lemma, we easily see that for any finite dimensional subspaceY ofH,J_λ^∗(α)→ −∞asα → ∞,α∈Y. From now on, we denote byQ^λ_k a large ball Q^λ_k=B_Rλ

k(0)∩Ek and

∂Q^λ_k:=

α∈E_k: α =R_k^λ

, S:=

α∈H_k^⊥₋₁: α =ρ .

The constant ρ =ρ_k is defined in the following way: from now on we restrict ourselves to a fixed interval λ∈ [λ^∗,+∞[with 0< λ^∗<1; then it is possible to fixρ∈ ]0, Rk[in such a way that

inf

I^∗ 2λ

λ+1α, 2λ λ+1α

: α∈S

>0. (1.12)

We stress thatρ does not depend onλ. The positive constantR^λ_k is taken large enough so thatJ_λ^∗(α) <0 for every α∈E_ksuch thatαR_k^λand we chooseR¹_k=R_k. By possibly taking a largerR^λ_k, we also require that

θ_λ⁻¹(α)> ρ, ∀α∈∂Q^λ_k. (1.13)

Finally, we require thatR_k^λR_k¹for everyλ∈ [λ^∗,+∞[.

GivenA⊂H, we say thatAandSlinkif:

(i) Ais compact and symmetric;

(ii) Acontains a subsetB which is odd homeomorphic to∂Q^λ_k and sup_BJ_λ^∗<0;

(iii) γ (A)∩S= ∅for every odd and continuous mapγ:A→Hsuch thatγ|B=Id. Accordingly, we denote Aλ:= {A⊂H: AandSlink}

and

c^∗_λ:= inf

A∈A^λsup

A

J_λ^∗.

The classAλ contains the set Q^λ_k, since ρ < R_k^λ. We also observe thatc^∗_λ>0. Indeed, by definition, we have c^∗_λinfSJ_λ^∗while, using the very definition ofJ_λ^∗, for everyα∈H,

J_λ^∗(α)I^∗

λα+ϕ, α−ϕ λ

=I^∗ 2λ

λ+1α, 2λ λ+1α

,

whereϕ:=^λ(1₁₊⁻_λ^λ)α, and our claim follows from (1.12).

Now, it is standard that there exists a critical pointαλofJ_λ^∗at levelc^∗_λsatisfyingm^∗_λ(αλ)k.

It then follows from (1.9) that kC

λ

c^∗_λ(p−2)/pN/2

+

(c^∗_λ)^(q−2)/q λ

N/2

. (1.14)

This estimate holds uniformly inλand ink, providedλis bounded away from zero. In our final step below we prove that, for everyλ,

c^∗_λb_k. (1.15)

By inserting (1.15) in the inequality (1.14) withλ:=b

1 p−_q¹

k completes then the proof of Proposition 9. We stress that indeedb_k1 for every largek∈N.

Step 5.In order to prove (1.15), givenε >0, letσ ∈Gk be such that sup_{σ (Q}¹

k)J₁^∗bk+ε; we recall thatQ¹_k= B_R1

k(0)∩E_k=B_Rk(0)∩E_k. SinceR¹_kR^λ_k, we can extendσ toQ^λ_k by setting

˜ σ (α):=

σ (α) ifα∈Q¹_k, α ifα∈Q^λ_k\Q¹_k.

LetA:=θ_λ⁻¹(σ (Q˜ ^λ_k)). We claim thatA∈Aλ. Indeed, by lettingB:=θ_λ⁻¹(σ (∂Q˜ ^λ_k))=θ_λ⁻¹(∂Q^λ_k), thanks to (1.11) we see that

sup

B

J_λ^∗ sup

˜ σ (∂Q^λ_k)

J₁^∗=sup

∂Q^λ_k

J₁^∗<0

sinceR_k^λR_k¹. On the other hand, letγ :A→H be any odd and continuous map such thatγ (α)=αfor allα∈ θ_λ⁻¹(∂Q^λ_k). Define

U:=

α∈Q^λ_k: γ θ_λ⁻¹

σ (α)˜ < ρ .

This is a bounded, symmetric neighborhood of the origin inE_k. According to the Borsuk–Ulam theorem, there exists α∈∂U such thatγ (β)∈E_k^⊥₋₁, withβ:=θ_λ⁻¹(σ (α)). Of course we have˜ γ (β)ρ. Now, ifα∈∂Q^λ_kthenγ (β)= θ_λ⁻¹(α)and soγ (β)> ρ, see (1.13). Thusα /∈∂Q^λ_k and thereforeγ (β) =ρ. In conclusion,γ (β)∈γ (A)∩S.

This proves the required linking property and shows thatA∈Aλ. As a consequence, c^∗_λsup

A

J_λ^∗.

But, again by (1.11) and the fact thatR^λ_kR¹_k we have that sup

A

J_λ^∗= sup

θ_λ⁻¹(σ (Q˜ ^λ_k))

J_λ^∗ sup

˜ σ (Q^λ_k)

J₁^∗= sup

(Q^λ_k\Q¹_k)∪σ (Q¹_k)

J₁^∗= sup

σ (Q¹_k)

J₁^∗.

In conclusion,

c^∗_λbk+ε, ∀ε >0, and this establishes (1.15). 2

The proof of [1, Theorem 1] follows easily from Proposition 9. [1, Claim 2 of Theorem 1] has to be adapted to the new statement of Proposition 9. We include the details for completeness.

We denote again byE_k the eigenspace associated to the firstkeigenfunctions of the Laplacian operator inH₀¹(Ω) and we fix a large constantR˜k>0 such thatJ (α) <˜ 0 for everyα∈Eksatisfyingα>R˜k. Let

Gk:=

σ ∈C B_R˜

k(0)∩Ek;H₀¹(Ω) σ (−α)= −σ (α), σ|∂B_R˜

k(0)∩Ek=Id , and define the minimax levels

b˜_k:= inf

σ∈Gk

maxJ˜ σ (α)

: α∈B_R˜

k(0)∩E_k

. (1.16)

Proof of [1, Theorem 1]. Assume by contradiction thatJ˜does not admit an unbounded sequence of critical values.

Let(b˜_k)_kbe the sequence of minimax levels ofJ˜defined by (1.16).

Claim 1.There existC, k₀>0such that for allkk₀,

b˜_kCk^p/(p−1). (1.17)

The claim follows exactly as in [4, Prop. 10.46].

Claim 2.There existC>0andk₀ >0such that for allkk₀,

b˜kCk^{2pq/N (pq−p−q)}. (1.18)

Let us fix a smallc >0 in such a way that the functional I (u, v)ˆ = ∇u,∇v −cF (u)−cG(v)

is such thatI˜− ˆI is bounded from below inH₀¹(Ω)×H₀¹(Ω). We also consider the associated reduced functionalJˆ defined by

J (α)ˆ := ˆI (α+ ˆψ_α, α− ˆψ_α):= max

ψ∈H₀¹(Ω)

I (αˆ +ψ, α−ψ ) and the corresponding minimax numbers

bˆ_k:= inf

σ∈G_kmaxJˆ σ (α)

: α∈B_Rˆ

k(0)∩E_k ,

where takingRˆ_k= ˜R_klarger if necessary, we can assume thatJ (α) <ˆ 0 for everyα∈E_ksatisfyingα>Rˆ_k. Clearly, the sequenceb˜_k− ˆb_k is bounded from below. According to Proposition 9, we have thatk^{2pq/N (pq−p−q)}Cbˆ_k,so that the claim follows.

The conclusion easily follows. 2 References

[1] D. Bonheure, M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, http://dx.doi.org/10.1016/

j.anihpc.2008.06.002.

[2] M. Cwickel, Weak type estimates and the number of bound states of Schrödinger operators, Ann. Math. 106 (1977) 93–102.

[3] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976) 751–753.

[4] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf., vol. 65, Amer. Math.

Soc., Providence, RI, 1986.

[5] G. Rosenbljum, The distribution of the discrete spectrum for singular differential operators, Soviet Math. Dokl. 13 (1972) 245–249.

[6] B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.