Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 25, pp. 1–9.

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

POSITIVE SOLUTIONS FOR SINGULAR (p, q)-LAPLACIAN

EQUATIONS WITH NEGATIVE PERTURBATION

NIKOLAOS S. PAPAGEORGIOU, CALOGERO VETRO, FRANCESCA VETRO

Abstract. We consider a nonlinear Dirichlet problem driven by the (p, q)-
Laplacian and with a reaction consisting of a singular term plus a negative

perturbation. Using regularization of the singular term and truncation and
comparison techniques, we show that the problem has a unique positive smooth

solution.

1. Introduction

Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this paper we
study the following singular Dirichlet (p, q)-equation

−∆pu(z)−∆qu(z) = u(z)−η − f(z, u(z)) in Ω,

u
∣∣
∂Ω

= 0, 1 < q < p, 0 < η < 1, u > 0.
(1.1)

For r ∈ (1,+∞) by ∆r we denote the r-Laplace differential operator defined by

∆ru = div(|∇u|r−2∇u) for all u ∈W 1,r
0 (Ω).

Equation (1.1) is driven by the sum of two such operators with different exponents
(double phase problem). Therefore the differential operator of our problem is not
homogeneous. In the reaction (right hand side), there is a singular term u−η and
a perturbation −f(z, u), with f(z, x) being a Carathéodory function (that is, for
all x ∈ R, z → f(z, x) is measurable and for a.a. z ∈ Ω, x → f(z, x) is contin-
uous) with values in R+ = [0,+∞) (that is, f ≥ 0). So, in problem (1.1) the
perturbation of the singular term is negative. This is in contrast with most ear-
lier works on singular elliptic equations, where the perturbation is positive. We
refer to works of Sun-Wu-Long [17], Haitao [8], Ghergu-Rădulescu [3] (semilinear
equations), Giacomoni-Schindler-Takáč [5], Papageorgiou-Winkert [14] (equations
driven by the p-Laplacian), Mukherjee-Sreenadh [10] (equations driven by the frac-
tional p-Laplacian) and of Papageorgiou-Rădulescu-Repovš [12] (equations driven
by a general nonlinear nonhomogeneous differential operator). Singular equations
with a negative perturbation were investigated by Godoy-Guerin [7] (semilinear
equations driven by the Laplacian) and by Saoudi [16] (nonlinear equations driven
by the p-Laplacian). In both works the negative perturbation of the singular term
is a power of u. Here we allow a more general perturbation. In both papers the

2020 Mathematics Subject Classification. 35J60, 35J92.
Key words and phrases. (p, q)-Laplacian; singular term; negative perturbation;

nonlinear regularity; regularized singular term.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted July 11, 2022. Published March 6, 2023.

1



2 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2023/25

approach is based on the direct method of the calculus of variations. In Godoy-
Guerin [7], the notion of weak solution is more restrictive, since they require that

the test functions belong in W 1,p
0 (Ω) ∩ L∞(Ω). On the other hand Saoudi [16]

considers a parametric problem with the parameter λ > 0 multiplying the singular
term. The author shows that there exists Λ∗ ≥ 0 (notation in [16]) such that for
all λ > Λ∗ the problem has a solution. In fact as we explain in Remark 3.5 Λ∗ = 0
and so the existence theorem is valid for all parameters λ > 0 and so there is no
need to introduce a parameter to the problem. Finally we mention that in [7] the
equation is driven by the Laplacian (semilinear equation), while in [16] is driven by
the p-Laplacian.

The fact that the perturbation is negative, makes it difficult to generate a lower
solution for the problem which is helpful in bypassing the singularity and dealing
with C1-functionals. The solution of the purely singular problem can not serve
as a lower solution as is the case in problems with positive perturbation (see for
example Papageorgiou-Rădulescu-Repovš [12]). Our approach is different and uses
upper solutions and regularizations of the singular term.

2. Mathematical background - hypotheses

The main spaces in the analysis of problem (1.1) are the Sobolev space W 1,p
0 (Ω)

and the Banach space C1
0 (Ω) = {u ∈ C1(Ω) : u

∣∣
∂Ω

= 0}. On account of the

Poincaré inequality the norm of W 1,p
0 (Ω) is given by

‖u‖ = ‖∇u‖p for all u ∈W 1,p
0 (Ω).

The space C1
0 (Ω) is an ordered Banach space with positive (order) cone given by

C+ = {u ∈ C1
0 (Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior

intC+ =
{
u ∈ C+ : u(z) > 0 for all z ∈ Ω,

∂u

∂n

∣∣
∂Ω

< 0
}
,

with n(·) being the outward unit normal on ∂Ω and ∂u
∂n = (∇u, n)RN . Also ordered

Banach space is the Lebesgue space L∞(Ω) with positive (order) cone L∞(Ω)+ =
{u ∈ L∞(Ω) : u(z) ≥ 0 for a.a. z ∈ Ω}. This order cone has a nonempty interior

intL∞(Ω)+ =
{
u ∈ L∞(Ω)+ : ess infΩ u(z) > 0

}
.

We mention that from all the Lebesgue spaces Lp(Ω), 1 ≤ p ≤ +∞ (all of which
are ordered Banach spaces with the pointwise order), only L∞(Ω) has positive cone
with a nonempty interior. This is a consequence of the fact that only the norm of
L∞(Ω) is defined in a pointwise fashion.

For r ∈ (1,+∞), let Ar : W 1,r
0 (Ω) → W−1,r′(Ω) = W 1,r

0 (Ω)∗( 1
r + 1

r′ = 1) be
defined by

〈Ar(u), h〉 =

∫
Ω

|∇u|r−2(∇u,∇h)RN dz for all u, h ∈W 1,r
0 (Ω).

We know (see Gasiński-Papageorgiou [2, p. 279]) that Ar(·) is bounded (that is,
maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal
monotone too) and of type (S)+, which means that it has the following property

un
w−→ u in W 1,r

0 (Ω) and lim supn→+∞〈Ar(un), un − u〉 ≤ 0 imply

un → u in W 1,r
0 (Ω) as n→ +∞.



EJDE-2023/25 POSITIVE SOLUTIONS FOR SINGULAR (p, q)-LAPLACIAN EQUATIONS 3

We set V = Ap + Aq. Then V : W 1,p
0 (Ω) → W−1,p′(Ω)( 1

p + 1
p′ = 1) and it has

the following properties:

• V (·) is continuous, strictly monotone (thus maximal monotone too);
• V (·) is of type (S)+.

If u : Ω → R is a measurable function, then we set u±(z) = max{±u(z), 0}
for all z ∈ Ω. We have u = u+ − u−, |u| = u+ + u− and if u ∈ W 1,p

0 (Ω), then

u± ∈W 1,p
0 (Ω).

Our hypotheses on the perturbation f(z, x) are the following:

(H1) f : Ω × R → R is a Carathéodory function such that for a.a. z ∈ Ω
f(z, 0) = 0, f(z, x) ≥ 0 for all x ≥ 0, there exists τ ∈ (1, q] such that

x → f(z, x)/xτ−1 is nondecreasing on R̊+ = (0,+∞) and |f(z, x)| ≤
â(z)[1 + xr−1] for a.a. z ∈ Ω, all x ≥ 0, with â ∈ L∞(Ω)+, and p ≤ r < p∗.

Remark 2.1. Recall that

p∗ =

{
Np
N−p if p < N,

+∞ if N ≤ p
is the critical Sobolev exponent corresponding to p. Since we look for positive
solutions and the above hypotheses concern the positive semiaxis R+ = [0,+∞),
without any loss of generality we may assume that f(z, x) = 0 for a.a. z ∈ Ω, all
x ≤ 0.

By a solution of (1.1) we mean a function u ∈W 1,p
0 (Ω) such that u−ηh ∈ L1(Ω)

for all h ∈W 1,p
0 (Ω) and∫

Ω

(|∇u|p−2∇u+ |∇u|q−2∇u,∇h)RN dz =

∫
Ω

u−ηh dz −
∫

Ω

f(z, u)h dz

for all h ∈W 1,p
0 (Ω).

3. Positive solutions

First we consider the purely singular problem

−∆pu(z)−∆qu(z) = u(z)−η in Ω,

u
∣∣
∂Ω

= 0, 1 < q < p, 0 < η < 1, u > 0.
(3.1)

From Papageorgiou-Rădulescu-Repovš [12, Proposition 11] we have the following
result.

Proposition 3.1. Problem (3.1) has a unique positive solution u ∈ intC+.

Next let ε > 0 and consider the following regularized version of problem (1.1),

−∆pu(z)−∆qu(z) = [u(z) + ε]−η − f(z, u(z)) in Ω,

u
∣∣
∂Ω

= 0, 1 < q < p, 0 < η < 1, u > 0.
(3.2)

Proposition 3.2. If hypotheses (H1) hold, then for every ε > 0 problem (3.2) has
a unique positive solution ũε ∈ intC+.

Proof. Consider the Carathéodory function

kε(z, x) =

{
[x+ + ε]−η − f(z, x+) if x ≤ u(z),

[u(z) + ε]−η − f(z, u(z)) if u(z) < x.
(3.3)



4 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2023/25

Let Kε(z, x) =
∫ x

0
kε(z, s)ds and consider the C1-functional ψε : W 1,p

0 (Ω) → R
defined by

ψε(u) =
1

p
‖∇u‖pp +

1

q
‖∇u‖qq −

∫
Ω

Kε(z, u) dz for all u ∈W 1,p
0 (Ω).

From (3.3) it is clear that ψε(·) is coercive. Also using the Sobolev embedding
theorem, we see that ψε(·) is sequentially weakly lower semicontinuous. So, by the

Weierstrass-Tonelli theorem, we can find ũε ∈W 1,p
0 (Ω) such that

ψε(ũε) = inf[ψε(u) : u ∈W 1,p
0 (Ω)],

⇒ ψ′ε(ũε) = 0 in W−1,p′(Ω),

⇒ 〈V (ũε), h〉 =

∫
Ω

kε(z, ũε)h dz for all h ∈W 1,p
0 (Ω).

(3.4)

In (3.4) by using the test function h = −ũ−ε ∈W
1,p
0 (Ω), we have ‖∇ũ−ε ‖pp ≤ 0 which

implies

ũε ≥ 0 and ũε 6= 0 (since ε > 0).

Next in (3.4) we choose the test function h = (ũε − u)+ ∈W 1,p
0 (Ω). We obtain

〈V (ũε), (ũε − u)+〉 =

∫
Ω

(
[u+ ε]−η − f(z, u)

)
(ũε − u)+ dz (see (3.3))

≤
∫

Ω

u−η(ũε − u)+ dz (since f ≥ 0)

= 〈V (u), (ũε − u)+〉 (see Proposition 3.1)

which implies

ũε ≤ u (from the monotonicity of V (·)).
So, we have proved that

ũε ∈ [0, u], ũε 6= 0, ⇒ ũε is a positive solution of (3.2).

The nonlinear regularity theory by Lieberman [9] implies that

ũε ∈ C+ \ {0}.
Hypotheses (H1) imply that there exists c1 > 0 such that

[x+ ε]−η − f(z, x) ≥ −c1xr−1 for a.a. z ∈ Ω, all x ≥ 0.

So, we have

∆pũε + ∆qũε ≤ c1‖u‖r−p∞ ũp−1
ε in Ω,

⇒ ũε ∈ intC+ (see Pucci-Serrin [15] (pp. 111, 120)).

Now we show the uniqueness of this positive solution. To this end we consider
the integral functional j : L1(Ω)→ R = R ∪ {+∞} defined by

j(u) =

{
1
p‖∇u

1/τ‖pp + 1
q‖∇u

1/τ‖qq if u ≥ 0 , u1/τ ∈W 1,p(Ω),

+∞ otherwise.

We define dom j = {u ∈ L1(Ω) : j(u) < +∞} (the effective domain of j(·)). Also
let `0 : R+ → R+ be the function defined by

`0(t) =
1

p
tp +

1

q
tq for all t ≥ 0.



EJDE-2023/25 POSITIVE SOLUTIONS FOR SINGULAR (p, q)-LAPLACIAN EQUATIONS 5

The function `0(·) is strictly increasing, strictly convex and since τ ∈ (1, q] (see
hypotheses (H1)), we see that t→ `0(t1/τ ) is convex on R+. We define `(y) = `0(|y|)
for all y ∈ RN . Then ` : RN → R+ is convex. Suppose u1, u2 ∈ dom j and set
u = [tu1 + (1− t)u2]1/τ with t ∈ [0, 1]. From Dı́az-Saá [1] (Lemme 1), we have that

|∇u| ≤
[
t|∇u1/τ

1 |τ + (1− t)|∇u1/τ
2 |τ

]1/τ
which implies

`0(|∇u|) ≤ `0
(

[t|∇u1/τ
1 |τ + (1− t)|∇u1/τ

2 |τ ]1/τ
)

(since `0(·) is increasing),

≤ t`0(|∇u1/τ
1 |) + (1− t)`0(|∇u1/τ

2 |) (since t→ `0(t1/τ ) is convex),

This in turn implies

`(∇u) ≤ t`(∇u1/τ
1 ) + (1− t)`(∇u1/τ

2 ),

Thus j(·) is convex.
Suppose that ṽε(·) is another positive solution of problem (3.2). Again we have

ṽε ∈ intC+. For δ > 0, we set

ũδε = ũε + δ, ṽδε = ṽε + δ.

Evidently ũδε, ṽ
δ
ε ∈ intL∞(Ω)+. So [11, Proposition 4.1.22, p. 274] implies that

ũδε
ṽδε
∈ L∞(Ω),

ṽδε
ũδε
∈ L∞(Ω). (3.5)

We set h = ((ũδε)
τ − (ṽδε)

τ ) ∈ C1
0 (Ω). From (3.5) it follows that for t ∈ (0, 1)

small, we have
(ũδε)

τ + th ∈ dom j, (ṽδε)
τ + th ∈ dom j.

Then the convexity of j(·) implies that the directional derivatives of j(·) at (ũδε)
τ

and at (ṽδε)
τ in the direction h exist and using the chain rule and Green’s identity

(see [11, p. 35]), we have

j′((ũδε)
τ )(h) =

1

τ

∫
Ω

−∆pũε −∆qũε
(ũδε)

τ−1
h dz

=
1

τ

∫
Ω

[ũε + ε]−η − f(z, ũε)

(ũδε)
τ−1

h dz,

j′((ṽδε)
τ )(h) =

1

τ

∫
Ω

−∆pṽε −∆q ṽε
(ṽδε)

τ−1
h dz

=
1

τ

∫
Ω

[ṽε + ε]−η − f(z, ṽε)

(ṽδε)
τ−1

h dz.

The convexity of j(·) implies the monotonicity of the directional derivative. So, we
have

0 ≤
∫

Ω

( [ũε + ε]−η

(ũδε)
τ−1

− [ṽε + ε]−η

(ṽδε)
τ−1

)
((ũδε)

τ − (ṽδε)
τ ) dz

−
∫

Ω

(f(z, ũε)

(ũδε)
τ−1
− f(z, ṽε)

(ṽδε)
τ−1

)
((ũδε)

τ − (ṽδε)
τ ) dz.

We let δ → 0 and use the dominated convergence theorem. Then on account of
hypotheses (H1), we obtain

0 ≤
∫

Ω

( 1

ũτ+η−1
ε

− 1

ṽτ+η−1
ε

)
(ũτε − ṽτε ) dz ≤ 0;



6 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2023/25

thus ũε = ṽε. This proves the uniqueness of the positive solution ũε ∈ intC+ of
problem (3.2). �

Next we show a monotonicity property of the map ε 7→ ũε.

Proposition 3.3. If hypotheses (H1) hold and 0 < ε′ ≤ ε, then 0 ≤ ũε ≤ ũε′ .

Proof. We have

−∆pũε′ −∆qũε′ = [ũε′ + ε′]−η − f(z, ũε′)

≥ [ũε′ + ε]−η − f(z, ũε′) in Ω.
(3.6)

We introduce the Carathéodory function

`ε(z, x) =

{
[x+ + ε]−η − f(z, x+) if x ≤ ũε′(z),
[ũε′(z) + ε]−η − f(z, ũε′(z)) if ũε′(z) < x.

(3.7)

We set Lε(z, x) =
∫ x

0
`ε(z, s)ds and consider the C1-functional σε : W 1,p

0 (Ω) → R
defined by

σε(u) =
1

p
‖∇u‖pp +

1

q
‖∇u‖qq −

∫
Ω

Lε(z, u) dz for all u ∈W 1,p
0 (Ω).

From (3.7) it is clear that σε(·) is coercive. Also it is sequentially weakly lower

semicontinuous. So, we can find uε ∈W 1,p
0 (Ω) such that

σε(uε) = inf[σε(u) : u ∈W 1,p
0 (Ω)],

which implies σ′ε(uε) = 0 in W−1,p′(Ω), and this implies

〈V (uε), h〉 =

∫
Ω

`ε(z, uε)h dz for all h ∈W 1,p
0 (Ω). (3.8)

Let h = −u−ε ∈W
1,p
0 (Ω). We have ‖∇u−ε ‖pp ≤ 0; therefore,

uε ≥ 0, uε 6= 0 (since ε > 0.

Also, in (3) we choose the test function h = (uε − ũε′)+ ∈W 1,p
0 (Ω). We obtain

〈V (uε), (uε − ũε′)+〉 =

∫
Ω

(
[ũε′ + ε]−η − f(z, ũε′)

)
(uε − ũε′)+ dz (see (3.7))

≤ 〈V (ũε′), (uε − ũε′)+〉 (see (3.6)).

This implies uε ≤ ũε′ . So, we have proved that

uε ∈ [0, ũε′ ], uε 6= 0. (3.9)

Then (3.9), (3.7), and (3) imply that uε is a positive solution of (3.2), which implies
uε = ũε (see Proposition 3.2), and 0 ≤ ũε ≤ ũε′ (see (3.9)). �

Finally we pass to the limit as ε→ 0+ to produce a positive solution for problem
(1.1). Consider the Dirichlet problem

−∆pu(z)−∆qu(z) = [u(z) + ε]−η in Ω, u
∣∣
∂Ω

= 0, u > 0.

From Papageorgiou-Rădulescu-Zhang [13] (see the proof of Proposition 3.3), we
know that this problem has a unique solution uε ∈ intC+ and uε ↑ u in C1

0 (Ω) as
ε → 0+. Moreover, since f ≥ 0, as in the proof of Proposition 3.3, we show that
0 ≤ ũε ≤ uε. Therefore

0 ≤ ũε ≤ u for all ε > 0 (3.10)



EJDE-2023/25 POSITIVE SOLUTIONS FOR SINGULAR (p, q)-LAPLACIAN EQUATIONS 7

Theorem 3.4. If hypotheses (H1) hold, then problem (1.1) has a unique positive
solution û ∈ intC+.

Proof. Let εn → 0+ and let ũn = ũεn ∈ intC+ as in Proposition 3.2. We have

〈V (ũn), h〉 =

∫
Ω

(
[ũn + εn]−η − f(z, ũn)

)
h dz for all h ∈W 1,p

0 (Ω), (3.11)

0 ≤ ũ1 ≤ ũn ≤ u for all n ∈ N (3.12)

(see Proposition 3.3, (3.10) and assume εn ≤ 1).

In (3.11) we use the test function h = ũn ∈ W 1,p
0 (Ω). Using (3.12) and that

f ≥ 0, we obtain

‖∇ũn‖pp ≤
∫

Ω

ũ1−η
n dz ≤

∫
Ω

u1−η dz;

therefore, {ũn}n∈N ⊆W 1,p
0 (Ω) is bounded. So, we can assume that

ũn
w−→ û in W 1,p

0 (Ω), ũn → û in Lr(Ω). (3.13)

Let d̂(z) = d(z, ∂Ω) for all z ∈ Ω. From Gilbarg-Trudinger [6, Lemma 14.16, p.

355] we have that d̂ ∈ intC+. Since ũ1 ∈ intC+, using [11, Proposition 4.1.22, p.
274], we can find c2 > 0 such that

c2d̂ ≤ ũ1. (3.14)

Then for h ∈W 1,p
0 (Ω), we have∫

Ω

( |h|
[ũn + εn]η

)p
dz

≤
∫

Ω

( |h|
ũη1

)p
dz (see (3.12))

=

∫
Ω

(
ũ1−η

1

|h|
ũ1

)p
dz

≤ c3
∫

Ω

( |h|
d̂

)p
dz for some c3 > 0 (since ũ1 ∈ intC+ and using (3.14))

≤ c4‖∇h‖pp for some c4 > 0, all n ∈ N

(using Hardy’s inequality, see [11, p. 66]) Therefore,{ h

[ũn + εn]η
}
n∈N ⊆ L

p(Ω) is bounded for all h ∈W 1,p
0 (Ω). (3.15)

From (3.13) and by passing to a subsequence if necessary, we have that

h

[ũn + εn]η
→ h

ûη
a.e. (3.16)

(note that ũ1 ≤ û, see (3.12)).
Then (3.15), (3.16) and [12, Problem 1.44] imply that∫

Ω

h

[ũn + εn]η
dz →

∫
Ω

h

ûη
dz for all h ∈W 1,p

0 (Ω). (3.17)

In (3.11) we choose h = (ũn− û) ∈W 1,p
0 (Ω) and pass to the limit as n→ +∞. We

obtain

lim
n→+∞

〈V (ũn), ũn − û〉 = 0,



8 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2023/25

which implies

ũn → û in W 1,p
0 (Ω) (see Section 2),

ũ1 ≤ û ≤ u (see (3.12)).
(3.18)

In (3.11) we pass to the limit as n→ +∞ and use (3.17), (3.18). We obtain

〈V (û), h〉 =

∫
Ω

[û−η − f(z, û)]h dz for all h ∈W 1,p
0 (Ω),

ũ1 ≤ û ≤ u.

Therefore, û is a positive solution of (1.1). Recall that d̂ ∈ intC+. So, as before,
using [11, Proposition 4.1.22, p. 274], we can find c5 > 0 such that

u ≤ c5d̂, ⇒ û ≤ c5d̂ (see (3.18)).

So, we can apply [4, Theorem 1.7] and conclude that û ∈ intC+.
Finally reasoning as in the proof of Proposition 3.2, we show that û ∈ intC+ is

the unique positive solution of (1.1). �

Remark 3.5. In [16] the author considers the parametric singular Dirichlet prob-
lem (λ > 0 is the parameter)

−∆pu(z) = λk(z)u(z)−η − h(z)u(z)r−1 in Ω,

u
∣∣
∂Ω

= 0, 1 < p < r < p∗, 0 < η < 1, u > 0,
(3.19)

with k, h ∈ intL∞(Ω)+. So, the perturbation of the singular term is a special case
of our perturbation f(z, x). In [16] the author proves the following existence result
(see [16, Theorem 1.5]):

There exists Λ∗ > 0 such that

• for all λ > Λ∗ problem (3.19) has at least one positive solution u ∈W 1,p
0 (Ω)

and for all K ⊆ Ω compact

0 < cK ≤ u(z) for a.a. z ∈ K;

• for each λ < Λ∗ problem (3.19) has no positive solution.

Our work in this paper shows that Λ∗ = 0 and so problem (3.19) has a positive
solution for all λ > 0; therefore the presence of the parameter λ > 0 in the problem
is inconsequential and it can be omitted. Moreover, we show that the solution is
unique and belongs in intC+.

Conclusion. In this article we considered a singular problem driven by the (p, q)-
Laplacian and a negative perturbation. Using truncations and regularizations to
accommodate the singularity, we prove the existence of a nontrivial solution. Our
approach allows us to avoid restrictive definition of the solution and the introduction
of a parameter. Moreover, our existence theorem provides regularity information
for the positive solution.

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Nikolaos S. Papageorgiou

Department of Mathematics, National Technical University, Zografou campus, 15780,

Athens, Greece
Email address: npapg@math.ntua.gr

Calogero Vetro
Department of Mathematics and Computer Science, University of Palermo, Via Archi-

rafi 34, 90123, Palermo, Italy

Email address: calogero.vetro@unipa.it

Francesca Vetro

independent researcher, 90123, Palermo, Italy
Email address: francescavetro80@gmail.com


	1. Introduction
	2. Mathematical background - hypotheses
	3. Positive solutions
	Conclusion

	References