On the shape of solutions of an asymptotically linear problem

On the shape of solutions of an asymptotically linear problem

MASSIMOGROSSI

Abstract. In this paper we study the problem −u= |u|u in

u=0 on∂ (0.1)

where is a smooth bounded domain ofR^N, N ≥ 1, > 0. We will show that, under some assumptions, the solutions to (0.1) are close to suitable linear combinations of eigenfunctions of the problem

−u=λu in

u=0 on∂ .

Mathematics Subject Classification (2000):35J60.

1.

Introduction and statement of the main results

Let us consider the following problem

−u= |u|^p−1u in

u=0 on∂ (1.1)

where p > 1 and is a smooth bounded domain of

R

^{N, N ≥} 1. Problem (1.1) has been extensively studied in the last years; there is a wide literature on existence, multiplicity and qualitative properties of solutions to (1.1) as pandvary. Let us recall a classical result (see [7] for a proof).

Theorem 1.1. Let1< p< ^N_N⁺₋²₂. Then there exist infinitely many pairs of solution to(1.1).

In this paper we study (1.1) whenpis close to 1,i.e. p=1+withpositive small enough and we try to characterize its set of solutions. Settingp=1+(1.1)

becomes

−u= |u|u in

u=0 on∂. (1.2)

Supported by M.I.U.R., project “Variational methods and nonlinear differential equations”.

Received May 29, 2008; accepted in revised form October 23, 2008.

To our knowledge in this case there are not many results. One of them is due to C. S. Lin (see [5]):

Theorem 1.2. Let be a convex domain and p sufficiently close to1. Then the positive solution to(1.2)is unique.

In [5] a crucial tool is the scaling invariance of the problem. Theorem 1.2 was extended by Dancer (see [3]) without assuming the convexity of the domain.

Another result for a similar problem can be found in [4].

In this paper we will use a different approach and we study the set of solutions to (1.2) as→0. Roughly speaking, we show that, forsmall enough, the problem (1.2) is “close” to the eigenvalue problem

−u=λu in

u=0 on∂ .

Let us denote byσ (−)the set of the eigenvalues of−with Dirichlet boundary condition and, for anyλ ∈ σ(−), bym(λ)the multiplicity of the eigenvalueλ. Finally, for any λ ∈ σ (−), let W^(λ) = span

e^(λ)₁ , ..,e_m^(λ)_(λ)

be the eigenspace associated to the eigenvalueλwith

e^(λ)_i e^(λ)_j =δ_i^j. We have the following result:

Theorem 1.3. Letλ¯ ∈σ (−). Then there exists0 =0(λ,¯ N, ) > 0such that for any0 < < 0and for any λ ≤ ¯λ, λ ∈ σ(−), we have that(1.2)has the solutions

u_j,= ±λ¹ _m(λ)

i=1

γ_i^(j,λ)e^(λ)_i +φ

j =1, ..,m(λ) (1.3) whereφ →0in C²()and

γ₁^(j,λ), .., γ_m^(j_(λ)^,λ) is a critical point of the functional J :

R

^m(λ)→

R

^,

J

1, .., m(λ)

=

_m(λ)

i=1

ie^(λ)_i

2 2 log

m(λ) i=1

ie_i^(λ) −1

(1.4) for any j =1, ..,m(λ).

Hence we have that any eigenspace W^(λ) associated to λgenerates at least m(λ) solutions to (1.2) provided that is small enough. The proof of Theorem 1.3 uses a finite dimensional reduction of Liapounov-Schmidt type, a tool widely used in perturbation problem. Note that we do not require any “non-degeneracy”

assumption on the critical pointsγ₁^(1,λ),..,γ₁^(m(λ),λ).

Our aim is now to show that, under an a priori estimate on the solutions, the result in Theorem 1.2 is sharp. We have the following:

Theorem 1.4. Let u_nbe a sequence of nontrivial solutions to −u= |u|ⁿu in

u=0 on∂ (1.5)

with n → 0as n → +∞and suppose that there exist M ≥ λ1 (λ1 is the first eigenvalue inσ (−)) and C>0such that

|un|²≤C Mⁿ². (1.6)

Then there existsλ∈σ(−)such that, up to a subsequence again denoted by u_n,

|u_n|² _n

→λ∈σ(−), as n→ +∞, (1.7) and

u_n =λⁿ¹ _m(λ)

i=1

γie_i^(λ)+φn

, (1.8)

with φn → 0in C²()as n → +∞ and

γ1, .., γm(λ)

is a critical point of the functional(1.4).

At this stage a question arises: Does each W^(λ) generate exactly m(λ)solu- tions? The answer is affirmative under a non-degeneracy assumption on the critical points of the functional (1.4).

Theorem 1.5. Let us suppose that for any λ ≤ ¯λthe function J defined in(1.4) has exactly m(λ) nondegenerate critical points γ⁽¹⁾, .., γ^(m(λ)) ∈

R

^{m(λ). Then,} for small enough,(1.2)admits exactly_λ¯

i=1m(λi)pairs of nontrivial solutions

satisfying

|u|²≤C M², (1.9)

for some M ≥λ1and C >0.

Note that if the eigenvalue is simple (i.e. m(λ)=1) it is possible to prove that any critical point of the functional J is nondegenerate (see proof of the Corollary 1.6). Then we have the following:

Corollary 1.6. Suppose that the any eigenvalue inσ(−)is si mple. Then for any λk ∈σ(−)there exists¯0 = ¯0(λ,¯ N, ) >0such that for any0< < 0(1.2) has exactly k pairs of nontrivial solutions satisfying(1.9).

The last corollary can be applied to many situations; indeed, from a Michelet- ti’s result (see [6]) we have that all eigenvalues of − are simple for “generic”

domains close to(see also [8]).

After this result was completed we have learned that some ideas of the paper can be found in [1] where the authors proved some qualitative properties of the solutions to (1.1).

The paper is organized as follows: in Sections 2-3 we give the proof of Theo- rem 1.3. In Section 4 we prove Theorem 1.4 and in Section 5 we prove Theorem 1.5 and its corollary.

2.

The Lyapunov-Schmidt reduction

Let us denote byσ (−)the set of the eigenvalues of−with zero Dirichlet bound- ary condition and, for anyλ∈σ(−),m(λ)be the multiplicity of the eigenvalueλ. Finally, for anyλ∈σ (−), denote byW_k^(λ)=span

e^(λ)₁ , ..,e^(λ)_m(λ)

the eigenspace associated to the eigenvalueλ.

In this section we fix λ ∈ σ(−) and set k = m(λ), e^(λ)_i = e_i, for i = 1, ..,m(λ)andW_k^(λ) =W_k. Forγ =(γ1, .., γk)∈

R

^kwe look for solutions to (1.2) as

u(x)=λ¹ _k

i=1

γie_i(x)+φ(x)

withφ∈ H²()∩H₀¹() (2.1) where H²()andH₀¹()denote the usual Sobolev spaces.

Set byu = u_H2()∩H₀¹()andu2= u_L²₍₎. Moreover we require that

|γ| ≤γ0 (2.2)

whereγ0will be chosen later.

Set W_k^⊥=

u∈H²()∩H₀¹():

ue_i =0 for anyi =1, ..,k

. (2.3)

Our first step is to derive the equation satisfied by the functionφ. Inserting (2.1) in (1.2) we get

− _k

i=1

γie_i +φ

=λ

_k

i=1

γie_i +φ

_k

i=1

γie_i +φ

, (2.4) and using the mean value theorem we get thatφverifies

−φ−λφ= A(φ) (2.5)

with A: H²()→L²(), A(φ)=λ

k

i=1

γie_i +φ

_k

i=1

γie_i +φ

−λ _k

i=1

γie_i +φ

. (2.6)

Since the operator−−λI has a nontrivial kernel spanned by{e₁, ..,e_k}we will solve equation (2.5) inW_k^⊥. To this end let us denote byk : H²()∩H₀¹()→ W_kand^⊥_k : H²()∩H₀¹()→W_k^⊥the projections onW_k andW_k^⊥respectively.

So we try to solve the problem

^⊥_k [−φ− A(φ)]=0. (2.7)

Let us introduce

L

:W_k^⊥→W_k^⊥defined by

L

=^⊥_k (−−λI) . (2.8)

By classical results we get that

L

^{≥C. (2.9)}

In this setting (2.7) is equivalent to find a functionφ∈W_k^⊥which verifies φ=

L

^−1⊥_k (A(φ))

. (2.10)

Proposition 2.1. Letγ = 0,γ ≤ γ0. Then there exists0 > 0such that for any 0 < < 0 there exists a uniqueφ ∈ W_k^⊥,φ = φ(γ,x)which verifies(2.10).

Moreoverφ ≤δwithδdepending only onλ, γ0,N, .

Proof. By the mean value theorem we can writeAin the following way,

A(φ)=λ _k

i=1

γie_i +φ

log

k i=1

γie_i +φ

₁

0

k i=1

γie_i+φ

t

dt. (2.11)

Sinceφis bounded in H²()it is not difficult to show thatA(φ)∈L²(). Let us introduce the operatorF:W_k^⊥ →W_k^⊥

F(φ)=

L

^−1⊥_k (A(φ))

(2.12) and let us show thatFis a contraction fromB =

φ∈W_k^⊥: φ ≤δ

into itself (δwill be chosen later).

We divide the proof in two steps.

Step 1: F maps Binto itself for a suitableδandsmall enough.

We have that

|A(φ)|²

=λ²²



log

k i=1

γie_i +φ

_k

i=1

γie_i+φ 1

0

k i=1

γie_i +φ

t

dt





2

(2.13)

≤λ²²



 1 0

log

k i=1

γie_i +φ

k i=1

γie_i+φ

1+t

dt





2

≤

using that |log|t|x|^1+t≤C

|t|¹² + |t|^q2 for some 2<q≤ 2N N −2

C²

k

i=1

γie_i+φ +

k i=1

γie_i+φ

q

≤C²

1+

|φ| +

|φ|^q

(2.14)

≤C²

1+ φ + φ^q . Then we have that

A(φ)2 ≤C

1+ φ¹²+ φ^q2 (2.15)

and from the definition of Fwe get

F(φ) ≤

L

^⊥_kA(φ)2≤C₀

1+ φ¹² + φ^q2 . (2.16) Assuming thatφ ≤δwithδ=2C₀by (2.16) we get

F(φ) ≤2C₀ (2.17)

which gives the claim of this step.

Step 2: F is a contraction mapping from Binto itself forsmall enough.

Let us fixδ=2C₀as in the previous step. We have F(φ1)−F(φ2) ≤

L

^⊥kA(φ1)− A(φ2)2

≤C

k i=1

γie_i +φ1

_k

i=1

γie_i +φ1

−(φ1−φ2)

−

k i=1

γie_i +φ2

_k

i=1

γie_i +φ2

2

(using the mean value theorem)

=C

₁

0

(1+)

k i=1

γie_i +sφ1+(1−s)φ2

−1

ds(φ1−φ2)

2

. (2.18)

Let us splitin the following way,

= D_1,,s∪D_2,,s∪ D_3,,s (2.19) where

D_1,,s =

x ∈:

k i=1

γie_i +sφ1+(1−s)φ2

≤

, (2.20)

D_2,,s =

x ∈: <

k i=1

γie_i +sφ1+(1−s)φ2

<2

k i=1

γie_i ∞

, (2.21)

D_3,,s =

x ∈:

k i=1

γie_i +sφ1+(1−s)φ2

≥2

k i=1

γie_i ∞

, (2.22) and also split the integral in (2.18) as

₁

0

k i=1

γie_i+sφ1+(1−s)φ2

−1

ds(φ1−φ2)

2

2

=

D_1,,s· +

D_2,,s· +

D_3,,s· =I₁+I₂+I₃.

(2.23)

Esti mate o f I₁

First let us estimate the measure ofD_1,,s. We want to show that meas

D_1,,s

→0 as→0∀s∈ [0,1]. (2.24) By contradiction let us suppose that there exists0>0,s¯∈ [0,1]such that

meas D_1,,¯s

≥δ >0 ∀∈(0, 0). (2.25)

Set

D =

n≥1

D_1,1

n,¯s. (2.26)

Of course we have that

meas(D)≥δ (2.27)

and

D=

x ∈: k i=1

γie_i + ¯sφ1+(1− ¯s)φ2=0

. (2.28)

Sinceφ1, φ2≤δwe derive from (2.28) that

_k

i=1γie_i²≤δfor any≤0. Then_k

i=1γie_i =0 almost everywhere inDwith measD≥δ. Since_k

i=1γie_i is an eigenfunction of−andγ =0 we reach a contradiction. So (2.24) holds. Then we have that

I₁=

D_1,,s

₁

0

(1+)

k i=1

γie_i+sφ1+(1−s)φ2

−1

ds 2

|φ1−φ2|²

≤





D_1,,s

₁

0

(1+)

k i=1

γie_i +sφ1+(1−s)φ2

−1

ds

N



2 N

·

D_1,,s

|φ1−φ2|^{N−22N N−2}_N

≤C meas

D_1,,s²

N φ1−φ2 =o(1)φ1−φ2.

(2.29)

Esti mate o f I₂

Sincex∈ D_2,,swe have that

k i=1

γie_i +sφ1+(1−s)φ2

−1→0. (2.30)

Hence I₂=

D_2,,s

₁

0

(1+)

k i=1

γie_i+sφ1+(1−s)φ2

−1

ds 2

|φ1−φ2|²

≤C





D_2,,s

₁

0

(1+)

k i=1

γie_i+sφ1+(1−s)φ2

−1

ds

N



2 N

φ1−φ2

=o(1)φ1−φ2.

(2.31)

by the dominated convergence theorem.

Esti mate o f I₃

Let us remark that if x ∈ D_3,,s then |sφ1(x)+(1−s)φ2(x)| ≥_k

i=1γie_i

∞. Hence, using thatφ1, φ2 ≤δwe get

meas D_3,,s

→0 as→0∀s∈ [0,1]. (2.32) Moreover, forx∈ D3,,swe derive that

(1+)

k i=1

γie_i+sφ1+(1−s)φ2

−1 ≤C

1+ |φ1|^N2 + |φ2|^N2 . (2.33) Let us estimateI₃,

I₃=

D_3,,s

1 0

(1+)

k i=1

γie_i+sφ1+(1−s)φ2

−1

ds 2

|φ1−φ2|²

≤C





meas

D_3,,s²

N+





D_3,,s

|φ1|





2 N

+





D_3,,s

|φ2|





2 N











D_3,,s

|φ1−φ2|^N2N−2





N−2 N

=o(1)φ1−φ2

(2.34)

Finally, from (2.29), (2.31), (2.34) we have that (2.18) becomes F(φ1)−F(φ2) ≤C(1+)o(1)φ1−φ2 ≤ 1

2φ1−φ2 (2.35) which proves that Fis a contraction fromB into itself withδ =2C₀. Hence the contraction mapping theorem provides the existence of a unique fixed point for the operator Fwhich gives the claim.

Remark 2.2. From the computation in Step 1 it follows that the constantδis inde- pendent ofγ (it only depends onγ0).

The proof of the previous Proposition does not hold if γ = 0. Our aim is extend the definition ofφ =φ(a₁, ..,a_k,x)in order to cover this case.

Let us consider a sequenceγn → 0, γn = 0 and setφ_,n = φ(γn,x). By Remark 2.2 we deduce that there exists a subsequenceγn(denoted again byγn) and a functionφ¯such that

φ_,n → ¯φasn→ +∞ (2.36)

weakly inH²()∩ H₀¹()and strongly in L²(). If we show thatφ¯ ≡0 we can define

φ(0,x)=0. (2.37)

It will be proved in next lemma.

Lemma 2.3. We have thatφ¯ ≡0.

Proof. Let us write down the equation satisfied byφ_,n,





















−φ_,n−λφ_,n−λ

_k

i=1

γie_i +φ_,n

_k

i=1

γie_i +φ_,n

−^k

i=1

γie_i+φ_,n

-

= ^k

i=1

C_i,n,e_i in k

i=1

γie_i +φ_,n =0 on∂ .

(2.38)

Let us show that (up to a subsequence)

C_i,n, →C_i, asn→ +∞ for anyi =1, ..,k. (2.39) To prove (2.39) it is enough to multiply (2.38) bye_j and integrate. We get

C_j,n,

e²_j ≤λ





k i=1

γie_i(x)+φ,n

1+

+

k i=1

γie_i(x)+φ,n



 (2.40)

and sinceγi,n are bounded andφ_,n ≤ δ we get (2.39). Passing to the limit in (2.38) asn→ +∞we get thatφ¯satisfies



−φ¯−λφ¯φ¯ = ^k

i=1

C_i,e_i(x) in φ¯ =0 on∂ .

(2.41)

We argue by contradiction and suppose thatφ¯ ≡0. Define ψ = φ¯

¯φL^q() (2.42)

for some 2 < q < _N^2N₋₂. Let us show that

|∇ψ|² ≤ C whereC is a positive constant independent of. Multiplying (2.41) byψ and using that

ψe_i =0 for anyi =1, ..,kwe have

|∇ψ|²=λ

φ¯ψ²≤

φ¯^{q−2q q−2}_q

ψ¯^q2_q

≤C. (2.43) Hence we have that ψ ψ0 weakly in H₀¹() andψ0L^q() = 1. Finally, multiplying (2.41) bye, whereeis an eigenfunction of−associated toµ =λwe get

µ

ψe=λ

φ¯ψe=λ

1+logφ¯ ₁

0

φ¯^tdt

ψe (2.44)

which implies (µ−λ)

ψe=λ

logφ¯ ₁

0

φ¯^tdtψe (2.45)

= λ

¯φL^q()

logφ¯ ₁

0

φ¯^tdtφ¯e

using that

logφ¯ ₁

0

φ¯^1+tdt

≤Cφ¯¹²

(2.46)

≤CeL^∞() λ ¯φL^q()

φ¯¹² ≤C ¯φL^q() ≤C. (2.47) Passing to the limit in (2.45) we get

ψ0e=0. Since we also have that

ψ0e_i=0 for anyi=1, ..,kwe derive thatψ≡0 and this is not possible sinceψ0L^q()=1.

So we reach a contradiction and thenφ¯ ≡0.

Corollary 2.4. For anyγ such that|γ| ≤ |γ0|there exists a unique functionφ = φ(γ ,x)∈B which satisfies



















− .

λ_k¹ _k

i=1

γie_i+φ -

−λ _k

i=1

γie_i+φ .

λ_k¹ _k

i=1

γie_i+φ -

= ^k

i=1

C_i,e_i in

λ_k¹ _k

i=1

γie_i+φ

=0 on∂

(2.48)

for some constants C_i,∈

R

^.

Proof. Writing down the equation satisfied by φ in (2.10) we get (2.48). The uniqueness of the functionφ follows by the contraction mapping theorem (see the proof of Proposition 2.1).

We end this section proving some properties of the functionφ(γ,x). Lemma 2.5. The functionφ(γ,x)satisfies

φ(−γ,x)= −φ(γ,x) . (2.49)

Proof. The claim is a consequence of the uniqueness of the functionφ. Indeed we have that−φ(−γ,x)satisfies the problem (2.48) and using again the uniqueness of the solution we get the claim.

Lemma 2.6. The functionφ(γ,x)is C¹inγ forγ =0.

Proof. The claim follows by the implicit function theorem applied to T(γ,u) : (0, γ0)×W_k^⊥∩H²()∩H₀¹()→W_k^⊥,

T(γ,u)=^⊥

_k

i=1

γie_i+u

+

k i=1

γie_i+u

_k

i=1

γie_i+u

(2.50) for∈(0, 0). We have thatT(γ, φ)=0. If we show thatT(γ, φ)is invertible, by the implicit function theorem we get the claim. We will show that

T(γ, φ)≥C >0 for any∈(0, 0) . (2.51) By contradiction let us suppose that there exists a sequencen → 0,vn ∈ W_k^⊥∩

H²()∩H₀¹()such that

vn =1 (2.52)

and T

n(γ, φ_n)vn

L²()→0 asn→ +∞. (2.53)

By (2.52) it follows that there existsv0 ∈W_k^⊥∩H²()∩H₀¹()such that

vn →v0 (2.54)

weakly in H²()and strongly inL²(). Arguing as in Step 2 of Proposition 2.1 we can pass to the limit in (2.53) and then

^⊥(v0+λv0)

L²()=0. (2.55)

This implies thatv0=0. On the other hand, by (2.53) we get

^⊥

vn+λ(1+n)

k i=1

γie_i +φn

vn

2

=o(1) (2.56)

and sincevn ∈W_k^⊥andvn→0 inL²()we have

|vn|²+o(1)=o(1), (2.57) a contradiction with (2.52). So T(γ, φ) is an invertible operator and the claim follows.

3.

The reduced problem

In this section we prove Theorem 1.3. Let J(u): H₀¹()→ Rdefined by J(u)= 1

2

|∇u|²− 1 2+

|u|²⁺. (3.1)

It is well known that critical points of (3.1) provides solutions to (1.2). For|γ| ≤ γ0 let us consider the function φ = φ(γ,x) defined in Corollary 2.4. Using the same notation of the previous section let us introduce the reduced functional I():

∈

R

^{k : || ≤γ0→}

R

^{defined by}

I()= J

λ¹ _k

i=1

ie_i +φ(,x)

. (3.2)

We have the following:

Lemma 3.1. A function u =λ¹ _k

i=1

γie_i +φ(γ,x)

is a solution to(1.2)if and only ifγ =(γ1, .., γk)∈

R

^k is a critical point to I.

Proof. Ifuis a solution to (1.2) we have thatJ(u)=0 and then

∂I(γ )

∂a_m = J(u)λ¹

e_m+∂φ(γ,x)

∂a_m

=0. (3.3)

Conversely, let us suppose thatγis a critical point toI.Differentiating again (3.19) we derive

0= ∂I(γ )

∂a_m = J _k

i=1

γie_i+φ(γ,x) e_m+ ∂φ(γ,x)

∂a_m

. (3.4)

From (2.48) we get that J

_k

i=1

γie_i +φ(γ,x)

∈W_k (3.5)

and then

J _k

i=1

γie_i +φ(γ,x) ∂φ(γ,x)

∂a_m

=0. (3.6)

Then (3.4) becomes

0= J _k

i=1

γie_i+φ(γ,x)

(e_m) . (3.7)

From (3.7) we get that J _k

i=1

γie_i +φ(γ,x)

∈W_k^⊥and then we have

J _k

i=1

γie_i +φ(γ,x)

=0 (3.8)

which gives the claim.

Next result concerns the expansion of the reduced functional.

Lemma 3.2. We have that I()= λ¹⁺²

4





_k

i=1

ie_i

2 2 log

k i=1

ie_i −1

+O()



 (3.9)

uniformly with respect toas→0.

Proof. We have I()

λ² = 1 2

∇

_k

i=1

ie_i+φ

2

− λ 2+

k i=1

ie_i+φ

2+

= 1 2



λ _k

i=1

ie_i 2

+2∇ _k

i=1

ie_i

· ∇φ + |∇φ|²



 (3.10)

−λ 1

2− 4+O

²



 _k

i=1

ie_i 2

+2 _k

i=1

ie_i

φ+φ²





·

1+log

k i=1

ie_i +φ

₁

0

k i=1

ie_i +φ

t

dt

. (3.11)

Recalling thatφ ∈W_k^⊥we have that

∇ _k

i=1

ie_i

· ∇φ =0. Moreover since φ ≤δwithδindependent ofwe get that

I() λ²

=λ



 2

_k

i=1

ie_i 2

log

k i=1

ie_i

₁

0

k i=1

ie_i+φ

t

dt− 4

_k

i=1

ie_i 2

 +O

² .

(3.12)

Since the functionx²|x|^t → x² uniformly on the compact set as → 0 for any t ∈ [0,1], arguing as in Step 2 of Proposition 2.1, we get

_k

i=1

ie_i 2 ₁

0

k i=1

ie_i +φ

t

dt → _k

i=1

ie_i 2

(3.13) and the claim follows.

The main result of this section follows by a classical result of critical point theory due to Clark ( [2]). Here we mention a slightly different statement (see [7]).

Theorem 3.3. Let E a Banach space, I ∈ C¹(E,

R

) with I bounded from below and satisfying the Palais-Smale condition. Suppose I(0)=0, there is a set K ⊂E such that K is homeomorphic to S^j−1 by an odd map andsup_K I < 0. Then I possesses at least j distinct pairs of critical points.

Remark 3.4. From the proof of previous theorem it is possible to see that the criti- cal pointsxofI satisfyI(x) <0.Hence it is enough to require the differentiability of the functional just forx =0 and we only need the validity of the Palais-Smale condition for negative values of I.

We will apply Theorem 3.3 to find at least j pairs of critical points to the functionalI forsmall enough.

Proposition 3.5. Forγ0sufficiently large and forsmall enough the functional I admits at least j distinct pairs of critical points.

Proof. We will apply Theorem 3.3 withI = ⁴

λ² IandE =

γ ∈

R

^k : || ≤γ0

. Note that by Lemma 2.5 we have thatIis an even functional and by Lemma 2.6 it is differentiable inE\ {0}. Let us show thatI is bounded from below. By Lemma 3.2 it is enough to show that the functional

I₀()=

_k

i=1

ie_i

2 2 log

k i=1

ie_i −1

(3.14) is bounded from below. Since the functiong(t)=t²(2 log|t| −1)satisfiesg(t)≥

−1 we derive that

|I₀()| ≥ −meas (3.15) which gives the claim.

Let us prove that I satisfies the Palais-Smale condition. Since I is defined for

|| ≤ |γ0|we have to check any Palais-Smale sequence is far away from the sphere

|| =γ0. By Remark 3.4 this leads to prove that

I() >0 if|| =γ0. (3.16)