Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 46, pp. 1–8.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.46

NONLOCAL CRITICAL KIRCHHOFF PROBLEMS IN HIGH DIMENSION

GIOVANNI ANELLO

Abstract. We study the nonlocal critical Kirchhoff problem

−
(
a+ b

∫
Ω
|∇u|2dx

)
∆u = |u|2

∗−2u+ λf(x, u), in Ω,

u = 0, on ∂Ω,

where Ω is a bounded smooth domain in RN , N > 4, a, b > 0, λ ∈ R, 2∗ := 2N
N−2

is the critical

exponent for the Sobolev embedding, and f : Ω × R → R is a Carathéodory function with

subcritical growth. We establish the existence of global minimizers for the energy functional
associated to this problem. In particular, we improve a recent result proved by Faraci and Silva

[3] under more strict conditions on the nonlinearity f and under additional conditions on a and

b.

1. Introduction

Very recently, Faraci and Silva [3] considered the problem

−
(
a+ b

∫
Ω

|∇u|2dx
)
∆u = |u|2

∗−2u+ λf(x, u), in Ω,

u = 0, on ∂Ω,
(1.1) (1.1)

where Ω is a bounded smooth domain in RN , N > 4, a, b > 0, λ ∈ R, and f : Ω × R → R is a
Carathéodory function, with f(x, 0) = 0, for a.a. x ∈ Ω, satisfying the subcritical growth condition

ess sup
x∈Ω

sup
t∈R

|f(x, t)|
1 + |t|p−1

< +∞, for some p ∈ (2, 2∗), (1.2)

where 2∗ := 2N
N−2 is the critical exponent for the embedding W 1,2

0 (Ω) ↪→ Lm(Ω), m ≥ 1.
They investigated the existence of local and global minimizers as well as the existence of saddle

points of the energy functional Φλ : W 1,2
0 (Ω) → R associated to (1.1), which is defined by

Φλ(u) =
a

2
∥u∥2 + b

4
∥u∥4 − 1

2∗
∥u∥2

∗

2∗ − λJ(u),

for each u ∈ W 1,2
0 (Ω), where

∥u∥ :=
(∫

Ω

|∇u(x)|2dx
)1/2

is the standard norm of W 1,2
0 (Ω),

∥u∥2∗ :=

(∫
Ω

u(x)|2
∗
dx

)1/2∗

2020 Mathematics Subject Classification. 35J20, 35J25.
Key words and phrases. Nonlocal problem; Kirchhoff equation; weak solution; critical growth; approximation;

variational methods.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 27, 2025. Published May 6, 2025.

1



2 G. ANELLO EJDE-2025/46

is the standard norm of L2∗(Ω), and

J(u) =

∫
Ω

(∫ u(x)

0

f(x, t)dt
)
dx. (1.3)

In particular, in [3], the existence of a nonzero global minimizer uλ of Φλ, with Φλ(uλ) ≤ 0, is
proved for λ > 0 large and under the additional conditions

(I)

a
N−4

2 b ≥ C1(N) :=
4(N − 4)

N−4
2

N
N−2

2 S
N
2

N

, where SN = inf
u∈W 1,2

0 (Ω)\{0}

∥u∥2

∥u∥22∗
;

(II) limt→0 f(x, t)/t = 0 uniformly for a.a. x ∈ Ω,
(III) for a.a. x ∈ Ω, f(x, t)t > 0, for all t ∈ R \ {0};
(IV) ess infx∈Ω inft∈A f(x, t) > 0, for an open interval A ⊂ (0,∞).

Saddle points of Φλ with positive energy are also proved to exist provided that a, b satisfy the
more restrictive condition

(I’) a
N−4

2 b ≥ 1
2

(
N

N−2

)N−2
2 C1(N).

In [3] the method of proof is essentially based on [4, Lemma 2.1], which ensures the sequential
weak lower semicontinuity of the functional Φλ under condition (I), and on [4, Lemma 2.2] which
ensures that Φλ satisfies the Palais-Smale condition under condition (I’).

In this article, we show that a nonzero global minimizer for Φλ exists for λ > 0 large without
any assumption on a, b except their positivity and under much less restrictive conditions on the
nonlinearity f . More precisely, we will prove the following result

Theorem 1.1. Assume that f satisfies (1.2) and that the functional J : W 1,2
0 (Ω) → R, defined

in (1.3), has no global maximizer in W 1,2
0 (Ω). Then, there exists λ∗ ∈]0,+∞[, such that for each

λ > λ∗, Φλ admits a global minimizers uλ such that Φλ(uλ) < 0. In particular, uλ is a non-zero
weak solution of Problem (1.1).

The proof follows by approximating Φλ with appropriate sequentially weakly lower semicon-
tinuous functionals. It is an easy matter to see that if f satisfies condition (III) then J cannot
have global maximizers. Thus, our existence result improves in several directions [3, Theorem 1.1].
Another simple condition on f which guarantees that the functional J has no global maximizer in
W 1,2

0 (Ω) will be stated later. The reader is referred to [1, 2, 6, 5, 7, 9, 10] for other papers dealing
with the Kirchhoff equation in high dimension (N ≥ 4). See also [8] and references therein for an
overview of papers devoted to the Kirchhoff problem.

2. Proof of the main result

In what follows, for each m ≥ 1, we denote by ∥ · ∥m the standard norm of the space Lm(Ω),

and if 1 ≤ m ≤ 2∗, we denote by cm the best constant for the embedding W 1,2
0 (Ω) ↪→ Lm(Ω), that

is

cm := sup
u∈W 1,2

0 (Ω)\{0}

∥u∥m
∥u∥

.

Finally, for each r > 0, we denote by Br the closed ball in W 1,2
0 (Ω) centered at 0 with radius r.

Proof of Theorem 1.1. Under the subcritical condition (1.2), it is well known that J is (well de-

fined) C1 and sequentially weakly continuous in W 1,2
0 (Ω). This implies that, for each r > 0, there

exists ur ∈ Br such that

sup
u∈Br

J(u) = J(ur).

Moreover, since, by assumption, J has no global maximizer on W 1,2
0 (Ω), the following strict

inequality holds

sup
u∈Br

J(u) < sup
u∈W 1,2

0 (Ω)

J(u). (2.1)



EJDE-2025/?? NONLOCAL KIRCHHOFF PROBLEMS 3

In addiction, since N > 4, one has 2∗ < 4, and so we can fix l0 ∈ R such that

l0 > S
− N

2(N−4)

N b−
N−2

2(N−4) ,

a

2
t2 +

b

4
t4 − 1

2∗
S
− 2∗

2

N t2
∗
> 0, for each t ≥ l0.

Next, in view of (2.1), we can also fix u0 ∈ W 1,2
0 (Ω), with ∥u0∥ > l0, such that

J(u0) > sup
u∈Bl0

J(u) ≥ J(0) = 0. (2.2)

Now, consider the number λ∗ defined by

λ∗ =
a
2∥u0∥2 + b

4∥u0∥4 − 1
2∗ ∥u0∥2

∗

2∗ − inft∈[0,l0]

(
a
2 t

2 + b
4 t

4 − 1
2∗S

− 2∗
2

N t2
∗)

J(u0)− sup∥u∥≤l0 J(u)
.

We will show that for each λ ∈]λ∗,+∞[, Φλ admits a global minimizer uλ such that Φλ(uλ) < 0.
Let λ > λ∗. First of all, observe that, being J(u0) > 0, one has

Φλ(u0) <
a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
∥u0∥2

∗

2∗ − λ∗J(u0), (2.3)

and, since ∥u0∥ > l0, by the choice of l0 one also has

a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
∥u0∥2

∗

2∗ ≥ a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
S
− 2∗

2

N ∥u0∥2 > 0,

from which one infers

λ∗ ≥
a
2∥u0∥2 + b

4∥u0∥4 − 1
2∗ ∥u0∥2

∗

2∗

J(u0)
> 0.

Therefore, by (2.2) and (2.3), one has
Φλ(u0) < 0. (2.4)

Now, fix a sequence of positive numbers {εn}n∈N such that εn < 2∗ − 2, for each n ∈ N, and
lim

n→+∞
εn = 0.

For each n ∈ N, consider the functional Φλ,n : W 1,2
0 (Ω) → R, defined by

Φλ,n(u) =
a

2
∥u∥2 + b

4
∥u∥4 − 1

2∗ − εn
∥u∥2

∗−εn
2∗−εn

− λJ(u),

for each u ∈ W 1,2
0 (Ω). Since 2∗ − εn < 2∗, the functional

u ∈ W 1,2
0 (Ω) → 1

2∗ − εn
∥u∥2

∗−εn
2∗−εn

− λJ(u)

is C1 and sequentially weakly continuous. Consequently, the functional Φλ,n is C1 and sequentially
lower weakly semicontinuous. Moreover, recalling that 2∗ < 4, it turns out

lim
∥u∥→+∞

Φλ,n(u) = +∞.

Therefore, Φλ,n admits a global minimizer uλ,n ∈ W 1,2
0 (Ω). Note also that, thanks to (1.2), we

can find a constant C > 0 such that

0 = Φλ,n(0) ≥ Φλ,n(uλ,n)

≥ a

2
∥uλ,n∥2 +

b

4
∥uλ,n∥4 −

|Ω|
εn
2∗

2∗ − εn
∥uλ,n∥2

∗−εn
2∗ − λC(1 + ∥uλ,n∥pp)

≥ b

4
∥uλ,n∥4 −

(1 + |Ω|)(2∗−2)/2∗

2
(1 + S

− 2∗
2

N ∥uλ,n∥2
∗
)− λC(1 + cpp∥uλ,n∥p).

Hence, since p < 2∗ < 4, we infer that

sup
n∈N

∥uλ,n∥ < +∞.

Consequently, there exist l ∈ [0,+∞) and uλ ∈ W 1,2
0 (Ω) such that, up to a subsequence,



4 G. ANELLO EJDE-2025/46

(i) ∥uλ,n∥ → l ∈ [0,+∞[;

(ii) un → uλ ∈ W 1,2
0 (Ω), weakly in W 1,2

0 (Ω);
(iii) un → uλ, strongly in Lq(Ω) and there exists g ∈ L1(Ω) such that |un|q ≤ g a.e. in Ω, for

each q ∈ [1, 2∗);
(iv) un → uλ, a.e. in Ω.

Moreover, by the Concentration Compactness Principle, we know that

|∇un|2 → dµ,

|un|2
∗
→ dν,

(2.5)

weakly-∗ in the sense of measures, with

dµ ≥ |∇uλ|2 +
∑
k∈Ñ

µkδxk
;

dν = |uλ|2
∗
+

∑
k∈Ñ

νkδxk
;

(µkS
−1
N )

N
N−2 ≥ νk > 0, for each k ∈ Ñ,

(2.6)

where Ñ ⊆ N is at most countable, and xk ∈ Ω.

We claim that Ñ = ∅. Indeed, assume, on the contrary, that there is some k ∈ Ñ, and, for each
r > 0, choose a C1-function φr : RN → [0, 1] such that

φr(x) = 0 if |x− xk| ≥ 2r,

φr(x) = 1 if |x− xk| ≤ r,

|∇φr(x)| ≤
2

r
if x ∈ RN .

Since uλ,n is a critical point of Φλ,n, one has

0 = Φ′
λ,n(uλ,n)(φ) = (a+ b∥uλ,n∥2)

∫
Ω

∇uλ,n(x)∇φ(x)dx

−
∫
Ω

|uλ,n(x)|2
∗−εn−1φ(x)dx− λ

∫
Ω

f(x, uλ,n(x))φ(x)dx.

for each φ ∈ W 1,2
0 (Ω).

In particular, choosing φ = uλ,nφr, by the Hölder inequality and 0 ≤ φr(x) ≤ 1, we obtain

0 = (a+ b∥un∥2−εn)
[ ∫

Ω

|∇uλ,n(x)|2φr(x) +

∫
Ω

uλ,n(x)∇uλ,n(x)∇φr(x)dx
]

−
∫
Ω

|uλ,n(x)|2
∗−εnφr(x)dx− λ

∫
Ω

f(x, uλ,n(x))uλ,n(x)φr(x)dx

≥ (a+ b∥un∥2−εn)
[ ∫

Ω

|∇uλ,n(x)|2φr(x)dx+ ∥uλ,n∥∥uλ,n∇φr∥2
]

− |Ω|
εn
2∗
(∫

Ω

|un,λ(x)|2
∗
φr(x)dx

)(2∗−εn)/2
∗

− λ

∫
Ω

f(x, uλ,n(x))uλ,n(x)φr(x)dx.

(2.7)

In addiction, by (2.5) and (2.6), one has

lim
n→+∞

∫
Ω

|∇uλ,n(x)|2φr(x)dx =

∫
Ω

φr(x)dµ ≥
∫
Ω

|∇uλ(x)|2φr(x)dx+
∑
j∈Ñ

µjφρ(xj),

lim
n→+∞

∫
Ω

|uλ,n(x)|2
∗
φr(x)dx =

∫
Ω

|uλ(x)|2
∗
φr(x)dx+

∑
j∈Ñ

νjφr(xj),

and, by (i)–(iv), one has

lim
n→+∞

∥uλ,n∥
(∫

Ω

|uλ,n(x)|2|∇φr(x)|2dx
)1/2

= l
(∫

Ω

|uλ(x)|2|∇φr(x)|2dx
)1/2

,



EJDE-2025/?? NONLOCAL KIRCHHOFF PROBLEMS 5

lim
n→+∞

∫
Ω

|uλ,n(x)|2|∇φr(x)|2dx =

∫
Ω

|uλ(x)|2|∇φr(x)|2dx

lim
n→+∞

∫
Ω

f(x, uλ,n(x))uλ,n(x)φr(x)dx =

∫
Ω

f(x, uλ(x))uλ(x)φr(x)dx.

Taking the above into account and passing to the limit as n → +∞ in (2.7), one obtains

0 ≥ (a+ bl2)
[ ∫

Ω

|∇uλ(x)|2φr(x)dx+
∑
j∈Ñ

µjφr(xj) +

∫
Ω

|uλ(x)|2|∇φr(x)|2dx
]

−
∫
Ω

|uλ(x)|2
∗
φr(x)dx−

∑
j∈J

νjφr(xj)−
∫
Ω

f(x, uλ(x))uλ(x)φr(x)dx.

(2.8)

Now, it is straightforward to check that∫
Ω

|∇uλ(x)|2φr(x)dx → 0,

∫
Ω

|uλ(x)|2
∗
φr(x)dx → 0,∫

Ω

f(x, uλ(x))uλ(x)φr(x)dx → 0

as r → 0. Furthermore, recalling that |∇φr(x)| ≤ 2
r for each x ∈ RN , one also has∫

Ω

|uλ(x)|2|∇φr(x)|2dx =

∫
Ω∩Br(xk)

|uλ(x)|2|∇φr(x)|2dx

≤
(∫

Ω∩Br(xk)

|uλ(x)|2
∗
dx

)2/2∗(∫
Ω∩Br(xk)

|∇φr(x)|
2·2∗
2∗−2 dx

)(2∗−2)/2∗

≤ 4

r2

(∫
Ω∩Br(xk)

|uλ(x)|2dx
)2/2∗(∫

Br(xk)

dx
)(2∗−2)/2∗

= 4ω
(2∗−2)/2∗

N

(∫
Ω∩Br(xk)

|uλ(x)|2dx
)2/2∗

,

where ωN is the volume of the unit sphere in RN . Then, since∫
Ω∩Br(xk)

|uλ(x)|2dx → 0 as r → 0,

one has ∫
Ω

|uλ(x)|2|∇φr(x)|2dx → 0 as r → 0.

Consequently, passing to the limit as r → 0 in (2.8), it follows that

0 ≥ (a+ bl2)µk − νk.

By the above inequality and (2.6), we obtain

(a+ bl2)S
N

N−2

N ≤ µ
2

N−2

k .

Then, since by (2.5) and (2.6) one has

l2 = lim
n→+∞

∥uλ,n∥2 ≥ ∥uλ∥2 +
∑
j∈Ñ

µj ≥ µk,

we finally infer that

bS
N

N−2

N l2 < (a+ bl2)S
N

N−2

N ≤ l
4

N−2

from which

∥uλ∥ ≤ l < S
− N

2(N−4)

N b−
N−2

2(N−4) < l0 (2.9)

Now, observe that, since uλ,n is a global minimum point for Φλ,n, one has

Φλ,n(uλ,n) ≤ Φλ,n(u0),



6 G. ANELLO EJDE-2025/46

where u0 is as in (2.2). By the previous inequality it follows that

a

2
∥uλ,n∥2 +

b

4
∥uλ,n∥4 −

|Ω|
εn
2∗

2∗ − εn
S
− 2∗−εn

2

N ∥uλ,n∥2
∗−εn − λJ(uλ,n)

≤ a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
∥u0∥2

∗−εn
2∗−εn

− λJ(u0).

A straightforward application of the Lebesgue Dominated Convergence Theorem shows that
∥u0∥2−εn

2−εn
→ ∥u0∥2

∗

2∗ . Hence, passing to the limit as n → +∞ in the above inequality, we ob-
tain

a

2
l2 +

b

4
l4 − 1

2∗
S
− 2∗

2

N l2
∗
− λJ(uλ) ≤

a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
∥u0∥2

∗

2∗ − λJ(u0).

This inequality and (2.9) imply that

inf
t∈[0,l0]

(a
2
t2 +

b

4
t4 − 1

2∗
S
− 2∗

2

N t2
∗
)
− λ sup

∥u∥≤l0

J(u) ≤ a

2
∥u0∥2 +

b

4
∥u0∥4 −

1

2∗
∥u0∥2

∗

2∗ − λJ(u0)

from which, in view of (2.2),

λ ≤
a
2∥u0∥2 + b

4∥u0∥4 − 1
2∗ ∥u0∥2

∗

2∗ − inft∈[0,l0]

(
a
2 t

2 + b
4 t

4 − 1
2∗S

− 2∗
2

N t2
∗)

J(u0)− sup∥u∥≤l0 J(u)
= λ∗

against the choice of λ. Therefore, it must be Ñ = ∅. This fact and (2.5) and (2.6) imply

lim
n→+∞

∥uλ,n∥2∗ = ∥uλ∥2∗ ,

which, together with (i)–(iv) and the Brezis-Lieb Lemma, implies in turn that

uλ,n → uλ, strongly in L2∗(Ω).

In particular, one infers that

lim
n→+∞

∥uλ,n∥2∗−εn
2∗−εn

= ∥uλ∥2∗2∗ . (2.10)

Now, since uλ,n is a global minimizer of Φλ,n, one has

Φλ,n(uλ,n) ≤ Φλ,n(uλ) for each n ∈ N,

Then, passing to the limit as n → +∞ in the above inequality and recalling (2.9) and (2.10), it
follows that

a

2
l2 +

b

4
l4 ≤ a

2
∥uλ∥2 +

b

4
∥uλ∥4 ≤ a

2
l2 +

b

4
l4,

that is

lim
n→+∞

∥uλ,n∥ = l = ∥uλ∥,

Since the norm of W 1,2
0 (Ω) is uniformly convex and uλ,n → uλ weakly in W 1,2

0 (Ω), the above limit
implies that

uλ,n → uλ, strongly in W 1,2
0 (Ω),

Consequently, Φλ,n(uλ,n) → Φλ(uλ).
Finally, recalling again that uλ,n is a global minimizer of Φλ,n, one has

Φλ(uλ) = lim
n→+∞

Φλ,n(uλ,n) ≤ lim
n→+∞

Φλ,n(u) = Φλ(u)

for each u ∈ W 1,2
0 (Ω). Therefore, uλ is a global minimizer of Φλ and, moreover, taking (2.4) into

account, it also turns out

Φλ(uλ) ≤ Φλ(u0) < 0.

This completes the proof. □

The key assumption in Theorem 1.1 is that J has no global maximizer. The following proposi-
tion shows a simple situation in which this assumption is satisfied.



EJDE-2025/?? NONLOCAL KIRCHHOFF PROBLEMS 7

Proposition 2.1. Assume f satisfying (1.2) and suppose that for some δ > 0 and ϕ ∈ Lp(Ω) it
holds ∫ ϕ(x)

0

f(x, t)dt > sup
|ξ|≤δ

∫ ξ

0

f(x, t)dt, for a.a. x ∈ Ω. (2.11)

Then, J has no global maximizer in W 1,2
0 (Ω).

Proof. First of all, observe that thanks to (1.2), the functional J is well defined in the entire space

Lp(Ω) and it is (strongly) continuous in this space. In particular, being W 1,2
0 (Ω) dense in Lp(Ω),

one has

sup
W 1,2

0 (Ω)

J = sup
Lp(Ω)

J. (2.12)

Now, arguing by contradiction, assume that there exists a global maximizer v ∈ W 1,2
0 (Ω) for J in

W 1,2
0 (Ω). Since v ∈ W 1,2

0 (Ω), the set

A := {x ∈ Ω : |v(x)| ≤ δ}

has positive measure. Hence, in view (2.11),∫
A

(∫ v(x)

0

f(x, t)dt
)
dx <

∫
A

(∫ ϕ(x)

0

f(x, t)dt
)
dx. (2.13)

At this point, define the function w : Ω → R as follows

w(x) =

{
ϕ(x), if x ∈ A;

v(x), if x ∈ Ω \A.

Then, w ∈ Lp(Ω) and, by (2.12) and (2.13), one has

J(w) =

∫
Ω

(∫ w(x)

0

f(x, t)dt
)
dx > J(v) = sup

W 1,2
0 (Ω)

J = sup
Lp(Ω)

J,

which is absurd. □

To exhibit an example of function satisfying the assumptions of Proposition 2.1, it is sufficient
to consider any Carathéodory function f : Ω × R → R satisfying the growth condition (1.2) and
such that

inf
x∈Ω

∫ ξ0

0

f(x, t)dt =: c > 0,

for some ξ0 ∈ R. Indeed, if f satisfies the above condition, since

lim
δ→0

sup
|ξ|≤δ

∫ ξ

0

f(x, t)dt = 0,

we can choose δ > 0 such that

sup
|ξ|≤δ

∫ ξ

0

f(x, t)dt < c,

and if we define ϕ : Ω → R as ϕ(x) = ξ0, for all x ∈ Ω, we have ϕ ∈ Lp(Ω) and∫ ϕ(x)

0

f(x, t)dt =

∫ ξ0

0

f(x, t)dt ≥ c > sup
|ξ|≤δ

∫ ξ

0

f(x, t)dt.



8 G. ANELLO EJDE-2025/46

References

[1] G. Anello, F. Cammaroto, L. Vilasi; Non-negative solutions and strong maximum principle for a resonant
quasilinear problem, Rev. Mat. Complut. 37 (3) (2024), 801-818.

[2] S. Deng, X. Tian; Nondegeneracy of solutions to the critical p -Laplace Kirchhoff equation, Bull. Lond. Math.

Soc. 55 (5) (2023), 2112-2128.
[3] F. Faraci, K. Silva; On the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension, Calc.

Var. 60 (1) (2021), Paper no. 22, 33 p.

[4] F. Faraci, Cs. Farkas; On an open question of Ricceri concerning a Kirchhoff-type problem, Minimax Theory
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[5] Y. Y. Lan, B. Y. Tang; Multiplicity of solutions for the Kirchhoff equation with critical nonlinearity in high
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[6] L. Kong, H. Chen; Normalized solutions for nonlinear Kirchhoff type equations in high dimensions, Electron.

Res. Arch. 30 (4) (2022), 1282-1295.
[7] D. Naimen, M. Shibata; Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity

in high dimension, Nonlinear Anal., 186 (2019) 187-208.

[8] P. Pucci, V. D. Radulescu; Progress in nonlinear Kirchhoff problems, Nonlinear Anal. 186 (2019) , 1-5.
[9] B. Ricceri; Energy functionals of Kirchhoff-type problems having multiple global minima, Nonlinear Anal., 115

(2015), 130-136.

[10] Q. Xie, J. Yu; Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension,
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Giovanni Anello

Department of Mathematics and Computer Science, Physical Science and Earth Science, University of
Messina, Viale F. Stagno d’Alcontres 31, Italy

Email address: ganello@unime.it


	1. Introduction
	2. Proof of the main result
	References