Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 43, pp. 1–25.

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

EXISTENCE OF SOLUTIONS FOR A PROBLEM WITH

MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS AND

VANISHING POTENTIALS

MARIA JOSÉ ALVES, RONALDO B. ASSUNÇÃO

Abstract. This work establishes the existence of positive solutions to a quasi-
linear singular elliptic equations involving the (p-q)-Laplacian operator with

singularities and a vanishing potential. We adapt the penalization method de-

veloped by del Pino and Felmer and we consider an auxiliary problem whose
corresponding functional satisfies the geometry of the mountain-pass theorem;

then, we prove that the Palais-Smale sequences are bounded in a Sobolev

space; after that, we show that the auxiliary problem has a solution. Finally,
we use the Moser iteration scheme to obtain an appropriate estimate and we

conclude that the solution to the auxiliary problem is also a solution to the
original problem.

1. Introduction and main result

In this paper we consider the quasilinear elliptic equation involving the (p-q)-
Laplacian operator with singularities

− div
( |∇u|p−2∇u

|x|ap
)
− div

( |∇u|q−2∇u
|x|cq

)
+
P (x)|u|p−2u

|x|ap∗(a,b)
+
Q(x)|u|q−2u

|x|cq∗(c,d)
= f(u),

u(x) > 0, u ∈ D1,p
a,b(R

N ) ∩D1,q
c,d(RN ),

(1.1)

where x ∈ RN , N > 3, 2 6 q 6 p < N , a < (N − p)/p, a 6 b < a + 1,
and c < (N − q)/q, c 6 d < c + 1, and the critical exponents are defined by
p∗(a, b) := Np/[N −p(a+1−b)] and q∗(c, d) := Nq/[N −q(c+1−d)]. The Sobolev

spaces D1,p
a,b(RN ) and D1,q

c,d(RN ) are defined as the completion of the function space

C∞0 (RN ) with respect to the norms

‖u‖D1,p
a,b(RN )

:=
(∫

RN

|∇u|p

|x|ap
dx+

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx
)1/p

,

‖u‖D1,q
c,d(RN )

:=
(∫

RN

|∇u|q

|x|cq
dx+

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx
)1/q

.

2020 Mathematics Subject Classification. 35J20, 35J75, 35J92, 35J10, 35B09, 35B38, 35B45.
Key words and phrases. Quasilinear elliptic equations with singularities; (p-q)-Laplacian;

variational methods; singular elliptic equation; vanishing potential; penalization method;

Moser iteration scheme.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted August 16, 2021. Published June 30, 2022.

1



2 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

The nonlinearity f : R→ R is a continuous and nonnegative function that is not
a pure power and can be subcritical at infinity and supercritical at the origin. More
precisely,

(H1) lim sups→0+
|x|bp

∗(a,b)sf(s)

sp∗(a,b)
< +∞.

(H2) There exists τ ∈ (p, p∗(a, b)) such that lim sups→+∞
|x|bp

∗(a,b)sf(s)
sτ = 0.

(H3) There exists θ > p such that 0 6 θF (s) 6 sf(s) for every s ∈ R+, where
we use the notation F (s) :=

∫ s
0
f(t) dt.

(H4) f(t) = 0 for every t 6 0.

The following properties are easily seen: under hypothesis (H1) there exists
c1 ∈ R+ such that |sf(s)| 6 c1|s|p

∗(a,b)/|x|bp∗(a,b) for s close to zero; and under
hypothesis (H2) there exists c2 ∈ R+ such that |sf(s)| 6 c2|s|τ/|x|bp

∗(a,b) for s
large enough. Combining these results and defining c0 := max{c1, c2}, we have the
pair of inequalities

|sf(s)| 6 c0
|s|p∗(a,b)

|x|bp∗(a,b)
and |sf(s)| 6 c0

|s|τ

|x|bp∗(a,b)
(s ∈ R). (1.2)

Hypothesis (H3) extends a well known Ambrosetti-Rabinowitz condition and is
used to show that the energy functional satisfies the Palais-Smale condition. Recall
that a functional J : D1,p

a,b(RN ) ∩D1,q
c,d(RN ) → R is said to verify the Palais-Smale

condition at the level α if any sequence (un)n∈N ⊂ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ) such

that J(un) → α and J ′(un) → 0, as n → +∞, possess a convergent subsequence.
Hypothesis (H3) also allows us to study the asymptotic behavior of the solution to
problem (1.1).

As an example of a nonlinearity f satisfying the above set of hypotheses, for
σ > p∗(a, b) and for τ ∈ (p, p∗(a, b)) assumed in hypothesis (H2), we define

f(t) =

{
tσ−1, if 0 6 t 6 1;

tτ−1, if 1 6 t.

We also assume that the functions P,Q : RN → R are continuous and nonnega-
tive. Moreover, the following set of hypotheses on the potential functions P and Q
is used.

(H5) P ∈ LN/p(a+1−b)
ap∗(a,b)−bp (RN ) and Q ∈ LN/q(c+1−d)

cq∗(c,d)−dq (RN )

(H6) P (x) 6 P∞ and Q(x) 6 Q∞ for every x ∈ B1(0), where P∞, Q∞ ∈ R+

are positive constants and B1(0) denotes the unitary ball centered at the
origin.

(H7) There exist constants Λ ∈ R+ and R0 > 1 such that

1

R
p2(a+1−b)/(p−1)
0

inf
|x|>R0

|x|p
2(a+1−b)/(p−1)P (x) > Λ.

As an example of a potential function P satisfying this set of hypotheses, for
Λ ∈ R+ and R0 > 1 assumed in hypothesis (H6), we define

P (x) =


0, if |x| 6 R0 − 1;

Λ(|x|−R0+1)

R
p2(a+1−b)/(p−1)
0

, if R0 − 1 < |x| < R0;

Λ
|x|p2(a+1−b)/(p−1)

, if R0 6 |x|.



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 3

An example of a potential function Q can be obtained in a similar way with minor
modifications.

This class of (p-q)-Laplacian operator generalizes several types of problems;
see Alves, Assunção and Miyagaki [1], Figueiredo [12], and Figueiredo and Nasci-
mento [13] and references therein for more information.

In problem (1.1) we allow the vanishig of the potential functions at infinity, that
is, the particular conditions lim inf |x|→+∞ P (x) = 0 and lim inf |x|→+∞Q(x) = 0,
called the zero mass cases. The presence of the singularities, both on the differential
operators and on the nonlinearities, and the use of vanishing potentials constitute
the main features of our work.

Theorem 1.1. Consider a < (N − p)/p, a 6 b < a + 1, c < (N − q)/q, c 6
d < c + 1, 2 6 q 6 p < N with N > 3, and suppose that the potential functions
P and Q verify the hypotheses (H5)–(H7), and that the nonlinearity f satisfies the
hypotheses (H1)–(H4). Then there exists a constant Λ∗ = Λ∗(P∞, Q∞, θ, τ, c0) such
that problem (1.1) has a positive solution for every Λ > Λ∗.

The main difficulty in proving the existence of solution to problem (1.1) resides

in the fact that the embedding of the Sobolev space D1,p
a,b(RN ) in the Lebesgue space

Lp
∗(a,b)(RN ) is not compact due to the action of a group of homoteties and transla-

tions. Besides, the Palais-Smale condition for the corresponding energy functional
cannot be obtained directly. Adding to these difficulties, we have to consider the
presence a sum of two differential operators and when q < p the study of prob-
lem (1.1) does not allow the use of the Lagrange’s multipliers method due to the
lack of homogeneity; moreover, the first eigenvalue of one operator brings no valu-
able information on the eigenvalue of the other one; finally, the method of sub-
and super-solutions cannot be applied. Therefore, to study problem (1.1) we are
required to make a careful analysis of the energy level of the Palais-Smale sequences
in order to obtain their boundedness and also to overcome the lack of compactness.
Furthermore, we have to adapt the Moser iteration scheme to our setting, since this
is a crucial step to obtain an estimate for the solution.

Inspired mainly by Wu and Yang [19], Benouhiba and Belyacine [7], and Alves,
Assunção, and Miyagaki [1] regarding the (p, q)-Laplacian type operator, by Bastos,
Miyagaki and Vieira [5] regarding the singularity in the operators, and by Alves and
Souto [2], with respect to the set of hypotheses, we adapt the penalization method
developed by del Pino and Felmer [10] to show our existence result. The basic
idea can be described in the following way. In section 2 we consider an auxiliary
problem and study its corresponding energy functional, showing that it satisfies
the geometry of the mountain-pass lemma and that every Palais-Smale sequence is
bounded in an appropriate Sobolev space. Using the standard theory, this implies
that the auxiliary problem has a solution. In section 3 we show, using the Moser
iteration scheme, that the solution of the auxiliary problem satisfies an estimate
involving the L∞(RN ) norm. Finally, in section 4 we use this estimate to show that
the solution of the auxiliary problem is also a solution of the original problem (1.1).

2. An auxiliary problem

To prove the existence of a positive solution to problem (1.1) we establish a
variational setting and apply the mountain-pass lemma. Using hypothesis (H5) we



4 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

define the space

E :=
{
u ∈ D1,p

a,b(R
N ) ∩D1,q

c,d(RN ) :

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx < +∞

and

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx < +∞

}
,

which can be endowed with the norm ‖u‖ = ‖u‖1,p + ‖u‖1,q, where

‖u‖1,p :=
(∫

RN

|∇u|p

|x|ap
dx+

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx
)1/p

,

‖u‖1,q :=
(∫

RN

|∇u|q

|x|cq
dx+

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx
)1/q

.

The Euler-Lagrange energy functional I : E → R associated with problem (1.1)
is defined by

I(u) :=
1

p

∫
RN

|∇u|p

|x|ap
dx+

1

p

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx

+
1

q

∫
RN

|∇u|q

|x|cq
dx+

1

q

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx−

∫
RN

F (u) dx.

Using the hypotheses on the nonlinearity f we can deduce that I ∈ C1(E;R);
moreover, for every u, v ∈ E its Gâteaux derivative can be computed by

I ′(u)v =

∫
RN

|∇u|p−2

|x|ap
∇u · ∇v dx+

∫
RN

P (x)|u|p−2uv

|x|ap∗(a,b)
dx

+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇v dx+

∫
RN

Q(x)|u|q−2uv

|x|cq∗(c,d)
dx−

∫
RN

f(u)v dx.

It is a well known fact that if u is a critical point of the energy functional I, then
u is a weak solution to problem (1.1). This means that∫

RN

|∇u|p−2

|x|ap
∇u · ∇φdx+

∫
RN

P (x)|u|p−2uφ

|x|ap∗(a,b)
dx

+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇φ dx+

∫
RN

Q(x)|u|q−2uφ

|x|cq∗(c,d)
dx−

∫
RN

f(u)φdx = 0

for every v ∈ E.
Now we define D1,p

0 (B1(0)) := D1,p
0,a,b(B1(0)) as the completion of the space

C∞0 (B1(0)) with respect to the norm ‖u‖D1,p
a,b

(B1(0)); we also define D1,q
0 (B1(0)) :=

D1,q
0,c,d(B1(0)) as the completion of the space C∞0 (B1(0)) with respect to the norm

‖u‖D1,q
c,d

(B1(0)). In the intersection of these Sobolev spaces we define an auxiliary

energy functional I∞ : D1,p
0 (B1(0)) ∩D1,q

0 (B1(0))→ R by

I∞(u) :=
1

p

∫
B1(0)

|∇u|p

|x|ap
dx+

1

p

∫
B1(0)

P∞|u|p

|x|ap∗(a,b)
dx

+
1

q

∫
B1(0)

|∇u|q

|x|cq
dx+

1

q

∫
B1(0)

Q∞|u|q

|x|cq∗(c,d)
dx−

∫
B1(0)

F (u) dx.

Using the hypotheses (H5) and (H6) it can be shown that this functional is well
defined. Our first lemma concerns the geometry of this functional.



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 5

Lemma 2.1. The functional I∞ satisfies the geometry of the mountain-pass lemma.
More precisely, the following claims are valid.

(1) There exist r0, µ0 ∈ R+ such that I∞(u) > µ0 for ‖u‖ = r0.

(2) There exists e0 ∈
[
D1,p

0 (B1(0))∩D1,q
0 (B1(0))

]
\{0} such that ‖e0‖ > r0 and

I∞(e0) < 0.

Proof. By using hypotheses (H1)–(H3), it is standard to verify item (1).
By hypothesis (H3) it follows that there exist θ > p and C0 ∈ R+ such that

F (s) > C0|s|θ. Now, if u ∈
[
D1,p

0 (B1(0)) ∩D1,q
0 (B1(0))

]
\{0}, then

I∞(tu) 6
|t|p

p

∫
B1(0)

|∇u|p

|x|ap
dx+

P∞|t|p

p

∫
B1(0)

|u|p

|x|ap∗(a,b)
dx

+
|t|q

q

∫
B1(0)

|∇u|q

|x|cq
dx+

Q∞|t|q

q

∫
B1(0)

|u|q

|x|cq∗(c,d)
dx

− C0|t|θ
∫
B1(0)

|u|θ dx.

Using this inequality we deduce that there exist tu ∈ R+ large enough such that,
taking e0 = tuu, we have ‖e0‖ > r0 and I∞(e0) < 0. This concludes the proof of
item (2). �

We denote by d∞ the mountain-pass level associated with the functional I∞,
that is,

d∞ := inf
γ∈Γ

max
t∈[0,1]

I∞(γ(t)),

where

Γ :=
{
γ ∈ C([0, 1];D1,p

0 (B1(0)) ∩D1,q
0 (B1(0))) : γ(0) = 0 and γ(1) = e0

}
and the function e0 ∈ [D1,p

0 (B1(0))∩D1,q
0 (B1(0))]\{0} is given in Lemma 2.1. It is

standard to verify that the mountain-pass level d∞ = d∞(P∞, Q∞, θ, f).
For R > 1 and for θ > p given in hypothesis (H3), we set k := θp/(θ − p) > p

and we define a new nonlinearity g : RN × R→ R by

g(x, t) :=


f(t), if |x| 6 R

or if |x| > R and f(t) 6 P (x)|t|p−2t

k|x|ap∗(a,b) ;

P (x)|t|p−2t

k|x|ap∗(a,b) , if |x| > R and f(t) > P (x)|t|p−2t

k|x|ap∗(a,b) .

Using the notation G(x, t) :=
∫ t

0
g(x, s) ds, by direct computations we obtain the

set of inequalities

g(x, t) 6
P (x)|t|p−2t

k|x|ap∗(a,b)
, for all |x| > R; (2.1)

G(x, t) = F (t), if |x| 6 R; (2.2)

G(x, t) 6
P (x)|t|p−2t

k|x|ap∗(a,b)
, if |x| > R > 1. (2.3)



6 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

Now we define the auxiliary problem

− div
( |∇u|p−2∇u

|x|ap
)
− div

( |∇u|q−2∇u
|x|cq

)
+
P (x)|u|p−2u

|x|ap∗(a,b)
+
Q(x)|u|q−2u

|x|cq∗(c,d)
= g(x, u),

u(x) > 0, u ∈ D1,p
a,b(R

N ) ∩D1,q
c,d(RN ),

(2.4)

The Euler-Lagrange energy functional J : E → R associated with the auxiliary
problem (2.4) is

J(u) :=
1

p

∫
RN

|∇u|p

|x|ap
dx+

1

p

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx

+
1

q

∫
RN

|∇u|q

|x|cq
dx+

1

q

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx−

∫
RN

G(x, u) dx.

Using the hypotheses on the nonlinearity f and on the potential functions P
and Q, we can show that J ∈ C1(E;R); moreover, for every u, v ∈ E its Gâteaux
derivative can be computed as

J ′(u)v =

∫
RN

|∇u|p−2

|x|ap
∇u · ∇v dx+

∫
RN

P (x)|u|p−2uv

|x|ap∗(a,b)
dx

+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇v dx+

∫
RN

Q(x)|u|q−2uv

|x|cq∗(c,d)
dx−

∫
RN

g(x, u)v dx.

As before, critical points of the energy functional J are weak solutions to prob-
lem (2.4).

Our next goal is to apply the mountain-pass lemma to show that problem (2.4)
has a positive solution.

Lemma 2.2. The functional J satisfies the geometry of the mountain-pass lemma.
More precisely, the following claims are valid.

(1) There exist r1, µ1 ∈ R+ such that J(u) > µ1 for ‖u‖ = r1.

(2) There exists e1 ∈
[
D1,p

0 (B1(0))∩D1,q
0 (B1(0))

]
\{0} such that ‖e1‖ > r1 and

J(e1) < 0.

Proof. Using (2.2) and inequality (2.3), together with hypotheses (H1) and (H3)
and the first inequality in (1.2), we obtain

J(u) >
1

p
‖u‖p1,p +

1

q
‖u‖q1,q −

∫
|x|6R

F (u) dx−
∫
|x|>R

P (x)|u|p−2u

k|x|ap∗(a,b)
dx

>
1

p
‖u‖p1,p +

1

q
‖u‖q1,q −

c0
θ

∫
RN

|u|p∗(a,b)

|x|bp∗(a,b)
dx− 1

kp
‖u‖p1,p

=
(1

p
− 1

kp

)
‖u‖p1,p +

1

q
‖u‖q1,q −

c0
θ
|u|p

∗(a,b)

L
p∗(a,b)
b

.

Recall the Caffarelli-Kohn-Nirenberg inequality

|u|p
L
p∗(a,b)
b (RN )

6 Sa,b,p

∫
RN

|∇u|p

|x|ap
dx, for all u ∈ D1,p

a,b(R
N ); (2.5)

|u|q
L
q∗(c,d)
d (RN )

6 Sc,d,q

∫
RN

|∇u|q

|x|cq
dx, for all u ∈ D1,q

c,d(RN ). (2.6)



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 7

Setting S ≡ max{Sa,b,p, Sc,d,q} in the computations above, we obtain

J(u) >
(1

p
− 1

kp

)
‖u‖p1,p +

1

q
‖u‖q1,q −

c0
θ
Sp
∗(a,b)/p

(∫
RN

|∇u|p

|x|ap
dx
)p∗(a,b)/p

> min
{1

p
− 1

kp
,

1

q

}(
‖u‖p1,p + ‖u‖q1,q

)
− c0

θ
Sp
∗(a,b)/p

(
‖u‖p1,p + ‖u‖q1,q

)p∗(a,b)/p
.

If we take ‖u‖1,p and ‖u‖1,q small enough, it follows that ‖u‖p1,p and ‖u‖q1,q are also

small enough. For this reason, we obtain the existence of r1, µ1 ∈ R+ such that
J(u) > µ1 for ‖u‖ = r1. This concludes the proof of item (1).

By definition, G(x, u) = F (u) for all u ∈
[
D1,p
a,b(B1(0)) ∩ D1,q

c,d(B1(0))
]
\{0}.

Arguing as in the proof of Lemma 2.1 we conclude that there exist r1, tu ∈ R+ such
that e1 := tuu verify the inequalities ‖e1‖ 6 r1 and J(e1) < 0. This concludes the
proof of item (2). �

Since the functional J has the geometry of the mountain-pass lemma, using
Willem [18, Theorem 1.15] we obtain a Palais-Smale sequence (un)n∈N ⊂ E such
that J(un) → α and J ′(un) → 0 as n → +∞. Here α ∈ R+ is the mountain-pass
level associated with the energy functional J , that is,

α := inf
γ∈Γ

max
t∈[0,1]

J(γ(t)),

where

Γ :=
{
γ ∈ C([0, 1];D1,p

0 (B1(0)) ∩D1,q
0 (B1(0)) : γ(0) = 0 and γ(1) = e1

}
and e1 ∈

[
D1,p
a,b(B1(0))∩D1,q

c,d(B1(0))
]
\{0} is the same function satisfying inequality

J(e1) < 0 in Lemma 2.2. Using the hypothesis (H4), without loss of generality we
can suppose that the sequence (un)n∈N ⊂ E consists of nonnegative functions.

We note that for all u ∈
[
D1,p

0 (B1(0))∩D1,q
0 (B1(0))

]
\{0} the inequality J(u) 6

I∞(u) is valid, and this implies that

α 6 d∞. (2.7)

Now we show that the Palais-Smale sequences for the functional J are bounded.

Lemma 2.3. Suppose that the potential functions P , Q verify hypothesis (H5),
and that the nonlinearity f satisfies the hypotheses (H1)–(H4). If (un)n∈N ⊂ E is a
Palais-Smale sequence for the energy functional J , then the sequence (un)n∈N ⊂ E
is bounded.

Proof. To get our thesis it is sufficient to prove that both sequences (‖un‖q1,q)n∈N ⊂
R and (‖un‖p1,p)n∈N ⊂ R are bounded, which we do in the two claims below.

Before that, however, we remark that there exist constants α1 > 0 and n0 ∈ N
such that J(un) 6 α1 and |J ′(unun)| 6 min

{
‖un‖1,q, ‖un‖1,p

}
for all n ∈ N such

that n > n0; and since θ > p > 1, for all n > n0 we have

J(un)− 1

θ
J ′(un)un 6 α1 +

1

θ
‖un‖ 6 α1 + min

{
‖un‖1,q, ‖un‖1,p

}
. (2.8)

Claim 2.4. The sequence (‖un‖q1,q)n∈N ⊂ R is bounded.



8 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

Proof. We divide our analysis into cases that mirror the definition of the nonlin-
earity g. If |x| > R and f(t) > P (x)|t|p−2t/k|x|ap∗(a,b), then∫

RN
G(x, un) dx =

1

p

∫
RN

g(x, un)un dx,

and this implies that

J(un)− 1

p
J ′(un)un =

(1

q
− 1

p

)
‖un‖q1,q. (2.9)

Combining inequality (2.8) with equality (2.9), we conclude that

(1

q
− 1

p

)
‖un‖q1,q 6 α1 + ‖un‖1,q.

So, in this case the sequence (‖un‖q1,q)n∈N ⊂ R is bounded, say ‖un‖q1,q 6 αq for
every n ∈ N.

If |x| 6 R or if |x| > R and f(t) 6 P (x)|t|p−2t/k|x|ap∗(a,b), the boundedness of
the sequence can be proved using the same ideas as that of the previous case with
some minor changes. This concludes the proof of the first claim. �

Claim 2.5. The sequence (‖un‖p1,p)n∈N ⊂ R is bounded.

Proof of Claim 2.5. We divide our analysis into the same cases. If |x| > R and
f(t) > P (x)|t|p−2t/k|x|ap∗(a,b), then we have

J(un)− 1

θ
J ′(un)un

>
(1

p
− 1

θ

)
‖un‖p1,p +

(1

q
− 1

θ

)
‖un‖q1,q −

1

kp

{∫
RN

P (x)|un|p

|x|ap∗(a,b)
dx
}

>
(1

p
− 1

θ

){
‖un‖p1,p + ‖un‖q1,q

}
− 1

kp

{
‖un‖p1,p + ‖un‖q1,q

}
=

(p− 1)

kp

{
‖un‖p1,p + ‖un‖q1,q

}
.

(2.10)

Combining inequalities (2.8) and (2.10) and using Claim 2.4 we obtain

(p− 1)

kp
‖un‖p1,p 6 α1 + ‖un‖1,p.

This means that in this case the sequence (‖un‖p1,p)n∈N ⊂ R is bounded.

If |x| 6 R or if |x| > R and f(t) 6 a(x)|t|p−2t/k, then∫
RN

G(x, un) dx+
1

θ

∫
RN

g(x, un)un dx > 0.



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 9

Hence,

J(un)− 1

θ
J ′(un)un

>
(1

p
− 1

θ

)
‖un‖p1,p +

(1

q
− 1

θ

)
‖un‖q1,q −

∫
RN

G(x, un) dx

+
1

θ

∫
RN

g(x, un)un dx

>
(1

p
− 1

θ

){
‖un‖p1,p + ‖un‖q1,q

}
−
∫
RN

G(x, un) dx

+
1

θ

∫
RN

g(x, un)un dx

>
1

k

{
‖un‖p1,p + ‖un‖q1,q

}
>

(p− 1)

kp

{
‖un‖p1,p + ‖un‖q1,q

}
.

(2.11)

Combining inequalities (2.8) and (2.11) we obtain

1

k
‖un‖p1,p 6 α1 + ‖un‖1,p.

This means that also in this case the sequence (‖un‖p1,p)n∈N ⊂ R is bounded. This
concludes the proof of the second claim. �

The proof of the lemma follows immediately from Claims 2.4 and 2.5. �

The following result shows that the functional J satisfies the Palais-Smale con-
dition.

Lemma 2.6. Suppose that the potential functions P , Q satisfy (H5)–(H7), and
that the nonlinearity f satisfies the (H1)–(H4). Then the Palais-Smale condition
is valid for the energy functional J .

Proof. Let (un)n∈N ⊂ E be a Palais-Smale sequence at the level α. This means
that J(un) → α and J ′(un) → 0 as n → ∞, and by Lemma 2.3 this sequence is
bounded. Then there exist a subsequence of (un)n∈N ⊂ E, which we still denote
in the same way, and there exists a function u ∈ E such that un ⇀ u weakly in E
as n→ +∞.

Now we set m := max{a+ 1− b, c+ 1− d} and m1 := max{b− a, d− c} and we
denote the volume of the unitary ball by |B1(0)| = ωN . For each ε > 0, there exist
r > R > 1 such that

2m1+1ωm/N
N

(
1− 1

k

)−1
{(∫

r6|x|62r

|u|p∗(a,b)

|x|bp∗(a,b)
dx
)1/p∗(a,b)

‖u‖p−1

+
(∫

r6|x|62r

|u|q∗(c,d)

|x|dq∗(c,d)
dx
)1/q∗(c,d)

‖u‖q−1
}
< ε.

(2.12)

Let η = ηr ∈ C∞(Bcr(0)) be a cut off function such that 0 6 η 6 1, with η = 1
in Bc2r(0) and |∇η| 6 2/rm1 for all x ∈ RN . Since the sequence (un)n∈N ⊂ E is
bounded, it follows that the sequence (ηun)n∈N ⊂ E is also bounded. Therefore,



10 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

J ′(un)(ηun) = on(1), that is,∫
RN

|∇un|p−2

|x|ap
∇un · ∇(ηun) dx+

∫
RN

P (x)|un|p−2un(ηun)

|x|ap∗(a,b)
dx

+

∫
RN

|∇un|q−2

|x|cq
∇un · ∇(ηun) dx+

∫
RN

Q(x)|u|q−2
n un(ηun)

|x|cq∗(c,d)
dx

=

∫
RN

g(x, un)(ηun) dx+ o(1).

(2.13)

This expression and the properties of the cut off function η imply that∫
|x|>r

η|∇un|p

|x|ap
dx+

∫
|x|>r

|∇un|p−2un
|x|ap

∇un · ∇η dx+

∫
|x|>r

ηP (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>r

η|∇un|q

|x|cq
dx+

∫
|x|>r

|∇un|q−2un
|x|cq

∇un · ∇η dx+

∫
|x|>r

η Q(x)|un|q

|x|cq∗(c,d)
dx

=

∫
|x|>r

ηg(x, un)un dx+ o(1).

From (2.1), it follows that∫
|x|>r

ηg(x, un)un dx 6
∫
|x|>r

ηP (x)|un|p

k|x|ap∗(a,b)
dx;

thus, we obtain∫
|x|>r

η|∇un|p

|x|ap
dx+

∫
|x|>r

ηP (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>r

η|∇un|q

|x|cq
dx+

∫
|x|>r

η Q(x)|un|q

|x|cq∗(c,d)
dx−

∫
|x|>r

ηP (x)|un|p

k|x|ap∗(a,b)
dx

6
∫
|x|>r

|∇un|p−1|un||∇η|
|x|ap

dx+

∫
|x|>r

|∇un|q−1|un||∇η|
|x|cq

dx+ o(1)

6
2

rm1

{∫
r6|x|62r

|∇un|p−1|un|
|x|ap

dx+

∫
r6|x|62r

|∇un|q−1|un|
|x|cq

dx
}

+ o(1).

Subtracting the terms

1

k

∫
|x|>r

η|∇un|p

|x|ap
dx+

1

k

∫
|x|>r

η|∇un|q

|x|cq
dx+

1

k

∫
|x|>r

ηQ(x)|un|q

|x|cq∗(c,d)
dx

from the left-hand side of the previous inequality and grouping the several integrals,
we deduce that(

1− 1

k

){∫
|x|>r

η|∇un|p

|x|ap
dx+

∫
|x|>r

ηP (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>r

η|∇un|q

|x|cq
dx+

∫
|x|>r

ηQ(x)|un|q

|x|cq∗(c,d)
dx
}

6
2

rm1

{∫
r6|x|62r

|∇un|p−1|un|
|x|ap

dx+

∫
r6|x|62r

|∇un|q−1|un|
|x|cq

dx
}

+ o(1).

Now we use Hölder’s inequality to obtain∫
r6|x|62r

|un||∇un|p−1

|x|ap
dx



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 11

6
(∫

r6|x|62r

|un|p

|x|ap
dx
)1/p{(∫

r6|x|62r

|∇un|p

|x|ap
dx
)1/p}p−1

6
(∫

r6|x|62r

|un|p

|x|ap
dx
)1/p

‖un‖p−1
1,p ;

and in a similar way, we obtain

∫
r6|x|62r

|un||∇un|q−1

|x|cq
dx 6

(∫
r6|x|62r

|un|q

|x|cq
dx
)1/q

‖un‖q−1
1,q .

By the compactness of the embedding D1,p
a,b(B2r\Br) ↪→ Lpa(B2r\Br), we infer that

un → u strongly in Lpa(B2r\Br) as n→∞. Since (ηun)n∈N ⊂ D1,p
a,b(RN )∩D1,q

c,d(RN ),
it follows that

lim sup
n→∞

(
1− 1

k

){∫
|x|>r

η|∇un|p

|x|ap
dx+

∫
|x|>r

ηP (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>r

η|∇un|q

|x|cq
dx+

∫
|x|>r

ηQ(x)|un|q

|x|cq∗(c,d)
dx
}

6
2

rm1
lim sup
n→∞

{(∫
r6|x|62r

|un|p

|x|ap
dx
)1/p

‖un‖p−1
1,p

+
(∫

r6|x|62r

|un|q

|x|cq
dx
)1/q

‖un‖q−1
1,q

}
=

2

rm1

{(∫
r6|x|62r

|u|p

|x|ap
dx
)1/p

‖u‖p−1
1,p

+
(∫

r6|x|62r

|u|q

|x|cq
dx
)1/q

‖u‖q−1
1,q

}
.

(2.14)

Applying Hölder’s inequality once more, we obtain

(∫
r6|x|62r

|u|p

|x|ap
dx
)1/p

6
(
2N(b−a)/(a+1−b)ωNr

N(b−a)/(a+1−b))(a+1−b)/N

×
(∫

r6|x|62r

|u|p∗(a,b)

|x|bp∗(a,b)
dx
)1/p∗(a,b)

;

(2.15)

and in a similar way, we obtain

(∫
r6|x|62r

|u|q

|x|cq
dx
)1/q

6
(
2N(d−c)/(c+1−d)ω

N
rN(d−c)/(c+1−d)

)(c+1−d)/N

×
(∫

r6|x|62r

|u|q∗(c,d)

|x|dq∗(c,d)
dx
)1/q∗(c,d)

.

(2.16)



12 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

Substituting inequalities (2.15) and (2.16) in (2.14) and recalling the definitions of
m and m1, we obtain

lim sup
n→∞

(
1− 1

k

){∫
|x|>r

η|∇un|p

|x|ap
dx+

∫
|x|>r

ηP (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>r

η|∇un|q

|x|cq
dx+

∫
|x|>r

ηQ(x)|un|q

|x|cq∗(c,d)
dx
}

6 2m1+1ωm/N
N

{(∫
r6|x|62r

|u|p∗(a,b)

|x|bp∗(a,b)
dx
)1/p∗(a,b)

‖u‖p−1

+
(∫

r6|x|62r

|u|q∗(c,d)

|x|dq∗(c,d)
dx
)1/q∗(c,d)

‖u‖q−1
}
.

(2.17)

In particular, since η = 1 outside the ball of radius 2r, by inequalities (2.14)
and (2.17) we obtain

lim sup
n→∞

(
1− 1

k

){∫
|x|>2r

|∇un|p

|x|ap
dx+

∫
|x|>2r

P (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>2r

|∇un|q

|x|cd
dx+

∫
|x|>2r

Q(x)|un|q

|x|cq∗(c,d)
dx
}

6 2m1+1ωm/N
N

{(∫
r6|x|62r

|u|p∗(a,b)

|x|bp∗(a,b)
dx
)1/p∗

‖u‖p−1

+
(∫

r6|x|62r

|u|q∗(c,d)

|x|dq∗(c,d)
dx
)1/q∗(c,d)

‖u‖q−1
1,q

}
.

(2.18)

Therefore, by inequalities (2.12) and (2.18) it follows that

lim sup
n→∞

{∫
|x|>2r

|∇un|p

|x|ap
dx+

∫
|x|>2r

P (x)|un|p

|x|ap∗(a,b)
dx

+

∫
|x|>2r

|∇un|q

|x|cq
dx+

∫
|x|>2r

Q(x)|un|q

|x|cq∗(c,d)
dx
}
< ε.

(2.19)

Combining inequalities (2.13) and (2.19), we deduce that

lim sup
n→∞

∫
|x|>2r

g(x, un)un dx = 0. (2.20)

Now we use the dominated convergence theorem together with the fact that g
has subcritical growth to infer that

lim sup
n→∞

∫
|x|62r

g(x, un)un dx =

∫
|x|62r

g(x, u)udx; (2.21)

and since
∫
RN g(x, un)un dx < ∞, by the choice of r > R > 1 and from equali-

ties (2.20) and (2.21), we obtain

lim
n→∞

∫
RN

g(x, un)un dx =

∫
RN

g(x, u)udx. (2.22)

It remains to show that the norm sequence (‖un‖)n∈N ⊂ R is such that ‖un‖ →
‖u‖ ∈ R as n → ∞. Using Hölder’s inequality and making some computations, it
follows that

o(1) = (J ′(un)− J ′(u)) (un − u)



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 13

>
{(∫

RN

|∇un|p

|x|ap
dx
)(p−1)/p

−
(∫

RN

|∇u|p

|x|ap
dx
)(p−1)/p}

×
{(∫

RN

|∇un|p

|x|ap
dx
)1/p

−
(∫

RN

|∇u|p

|x|ap
dx
)1/p}

+
{(∫

RN

P (x)|un|p

|x|ap∗(a,b)
dx
)(p−1)/p

−
(∫

RN

P (x)|u|p

|x|ap∗(a,b)
dx
)(p−1)/p}

×
{(∫

RN

P (x)|un|p

|x|ap∗(a,b)
dx
)1/p

−
(∫

RN

P (x)|u|p

|x|ap∗(a,b)
dx
)1/p}

+
{(∫

RN

|∇un|q

|x|cq
dx
)(q−1)/q

−
(∫

RN

|∇u|q

|x|cq
dx
)(q−1)/q}

×
{(∫

RN

|∇un|q

|x|cq
dx
)1/q

−
(∫

RN

|∇u|q

|x|cq
dx
)1/q}

+
{(∫

RN

Q(x)|un|q

|x|cq∗(c,d)
dx
)(q−1)/q

−
(∫

RN

Q(x)|u|q

|x|cq∗(c,d)
dx
)(q−1)/q}

×
{(∫

RN

Q(x)|un|q

|x|cq∗(c,d)
dx
)1/q

−
(∫

RN

Q(x)|u|q

|x|cq∗(c,d)
dx
)1/q}

−
∫
RN

(g(x, un)− g(x, u)) (un − u) dx.

We remark that all the terms between curly brackets in the previous expression
have the same signs; therefore, by the limit (2.22) we obtain

lim
n→∞

∫
RN

|∇un|p

|x|ap
dx =

∫
RN

|∇u|p

|x|ap
dx,

lim
n→∞

∫
RN

P (x)|un|p

|x|ap∗(a,b)
dx =

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx,

lim
n→∞

∫
RN

|∇un|q

|x|cq
dx =

∫
RN

|∇u|q

|x|cq
dx,

lim
n→∞

∫
RN

Q(x)|un|q

|x|cq∗(c,d)
dx =

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx.

This implies that

lim
n→∞

‖un‖p1,p = ‖u‖p1,p and lim
n→∞

‖un‖q1,q = ‖u‖q1,q.

Moreover, un ⇀ u weakly in E as n → ∞; and finally, un → u strongly in E as
n→∞. For the details, see DiBenedetto [11, Proposition V.11.1]. �

Lemma 2.7. Suppose that there exists a sequence (un)n∈N ⊂ E and a function
u ∈ E such that un → u in E and J ′(un) → 0 as n → ∞. Then there exists a
subsequence, still denoted the same way, such that ∇un → ∇u a.e. in RN

For a proof the above, see Assunção, Carrião, and Miyagaki [4] or Benmouloud,
Echarghaoui, and Sbäı [6].

Using Lemmas 2.1, 2.2, 2.3, 2.6, and 2.7 we conclude that there exists u ∈ E
which is a critical point for the functional J . Moreover, this critical point is a
positive ground state solution to the auxiliary problem (2.4), that is, J(u) = α > 0
and J ′(u) = 0.



14 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

3. Estimate for the solution to the auxiliary problem

In this section we show that the solution to (2.4) obtained in the previous section
satisfies an important estimate. To do this we use several lemmas.

Lemma 3.1. For R > 1, every positive ground state solution u to problem (2.4)
satisfies the estimate

‖u‖p1,p + ‖u‖q1,q 6
kpd∞
p− 1

.

Proof. Combining inequalities (2.7), (2.10) and (2.11), it follows that

(p− 1)

kp

{
‖u‖p1,p + ‖u‖q1,q

}
6 J(u)− 1

θ
J ′(u)u = J(u) = α 6 d∞.

The conclusion of the lemma follows immediately. �

We remark that the boundedness of the norm of the ground state solution to
problem (2.4) shown in Lemma 3.1 depends only on the potential functions P∞ and
Q∞, on the nonlinearity f and on the constant θ; it is independent of the constant
R > 1.

The next lemma is a crucial step to establish an important estimate involving
the norm of the solution to the auxiliary problem (2.4) in the space L∞(RN ).
To prove it, we adapt the arguments by Alves and Souto [2]; see also Gilbarg
and Trudinger [14, Section 8.6], Brézis and Kato [8], Pucci and Servadei [16], and
Bastos, Miyagaki, and Vieira [5].

Lemma 3.2. Consider a < (N − p)/p, a 6 b < a + 1, c < (N − q)/q, c 6
d < c + 1, 2 6 q 6 p < N , and r ∈ R such that p(a + 1 − b)r > N . Suppose
that A,B : RN → R are nonnegative potential functions. Let H : RN × R → R
be a continuous function such that |H(x, s)| 6 h(x)|s|p−1s/|x|bp∗(a,b) for all s > 0
where the function h : RN → R is such that h ∈ Lrap∗(a,b)/r(R

N ). Suppose also that

v ∈ E ⊂ D1,p
a,b(RN ) ∩D1,q

c,d(RN ) is a weak solution to the problem

− div
( |∇v|p−2∇v

|x|ap
)
− div

( |∇v|q−2∇v
|x|cq

)
+
A(x)|v|p−2v

|x|ap∗(a,b)
+
B(x)|v|q−2v

|x|cq∗(c,d)
= H(x, v).

(3.1)

Then there exists a constant M1 = M1(‖h‖Lr
ap∗(a,b)/r(RN )) > 0 that does not depend

on the functions A and B, such that

‖v‖L∞(RN ) 6M1 max
{
‖v‖Lp∗(a,b)(RN ),K,KLv, 1

}
,

where K and Lv are defined by (3.6) and by (3.7), respectively.

Proof. Consider β > 1 and, for every m ∈ N, let us define the subsets

Am := {x ∈ RN : 1 < |v(x)|β−1 6 m};

Bm := {x ∈ RN : |v(x)|β−1 > m};

Cm := {x ∈ RN : |v(x)|β−1 6 1}.



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 15

We also define the sequence of functions (vm)m∈N ⊂ D1,p
a,b(RN ) ∩D1,q

c,d(RN ) by

vm(x) :=


|v(x)|p(β−1)v(x), if x ∈ Am;

mpv(x), if x ∈ Bm;

|v(x)|q(β−1)v(x), if x ∈ Cm.

It is easy to verify that for every x ∈ RN we have

vm(x) 6 max
{
|v(x)|p(β−1)+1, |v(x)|q(β−1)+1

}
.

Additionally, simple computations show that

∇vm(x) =


(p(β − 1) + 1) |v(x)|p(β−1)∇v(x), if x ∈ Am;

mp∇v(x), if x ∈ Bm;

(q(β − 1) + 1) |v(x)|q(β−1)∇v(x), if x ∈ Cm.

Furthermore, (vm)m∈N ⊂ E. Indeed,∫
RN

P (x)|vm|p

|x|ap∗(a,b)
dx

6
∫
Am

P (x)
(
|v|p−1v

)
mp(p−1)+p

|x|ap∗(a,b)
dx

+

∫
Bm

P (x)|v|p−1vmp(p−1)+p

|x|ap∗(a,b)
dx+

∫
Cm

P (x)
(
|v|p−1v

)
|x|ap∗(a,b)

dx

6 mp2
∫
RN

P (x)|v|p−1v

|x|ap∗(a,b)
dx < +∞.

And in a similar way, we have∫
RN

Q(x)|vm|q

|x|cq∗(c,d)
dx 6 mpq

∫
RN

Q(x)|v|q−1v

|x|cq∗(c,d)
dx < +∞.

Multiplying both sides of the differential equation (3.1) by the test function vm
and integrating with the help of the divergence theorem, we deduce that∫

RN
H(x, v)vm dx =

∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

|∇v|q−2

|x|ap
∇v · ∇vm dx

+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx

(3.2)

By the definition of the function vm, the first two terms on the right-hand side of
equality (3.2) can be written in the form∫

RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx

= (p(β − 1) + 1)
{∫

Am

|∇v|p|v|p(β−1)

|x|ap
dx+

∫
Am

|∇v|q|v|p(β−1)

|x|cq
dx
}

+ (q(β − 1) + 1)
{∫

Cm

|∇v|p|v|q(β−1)

|x|ap
dx+

∫
Cm

|∇v|q|v|q(β−1)

|x|cq
dx
}

+mp
{∫

Bm

|∇v|p

|x|ap
dx+

∫
Bm

|∇v|q

|x|cq
dx
}
.



16 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

Therefore,

(p(β − 1) + 1)
{∫

Am

|∇v|p|v|p(β−1)

|x|ap
dx+

∫
Am

|∇v|q|v|p(β−1)

|x|cq
dx
}

+ (q(β − 1) + 1)
{∫

Cm

|∇v|p|v|q(β−1)

|x|ap
dx+

∫
Cm

|∇v|q|v|q(β−1)

|x|cq
dx
}

=

∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx

−mp
{∫

Bm

|∇v|p

|x|ap
dx+

∫
Bm

|∇v|q

|x|cq
dx
}

6
∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx

+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx.

(3.3)

Now we define another sequence of functions (wm)m∈N ⊂ E by

wm(x) =

{
|v(x)|β−1v(x), if x ∈ Am ∪ Cm;

mv(x), if x ∈ Bm.

Direct computations show that

∇wm(x) =

{
β|v(x)|β−1∇v(x), if x ∈ Am ∪ Cm;

m∇v(x), if x ∈ Bm.

Using the definitions of the sets Am, Bm, Cm, the definition of the function vm,
as well as the fact that 2 6 q 6 p < N , we obtain∫

RN

|∇wm|p

|x|ap
dx+

∫
RN

A(x)|wm|p

|x|ap∗(a,b)
dx

−
∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx−

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx

+

∫
RN

|∇wm|q

|x|cq
dx+

∫
RN

B(x)|wm|q

|x|cq∗(c,d)
dx

−
∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx−

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx

= βp
∫
Am∪Cm

|∇v|p|v|p(β−1)

|x|ap
dx+ βq

∫
Am∪Cm

|∇v|q|v|q(β−1)

|x|cq
dx

− (p(β − 1) + 1)
{∫

Am

|∇v|p|v|p(β−1)

|x|ap
dx+

∫
Am

|∇v|q|v|p(β−1)

|x|cq
dx
}

+

∫
Am

B(x)
(
|v|qβ − |v|p(β−1)+q

)
|x|cq∗(c,d)

dx+

∫
Cm

A(x)
(
|v|pβ − |v|p+q(β−1)

)
|x|ap∗(a,b)

dx

+ (mq −mp)

∫
Bm

B(x)|v|q

|x|cq∗(c,d)
dx

− (q(β − 1) + 1)
{∫

Cm

|∇v|p|v|q(β−1)

|x|cq∗(c,d)
dx+

∫
Cm

|∇v|q|v|q(β−1)

|x|cq
dx
}



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 17

+ (mq −mp)

∫
Bm

|∇v|q

|x|cq
dx.

This implies that∫
RN

|∇wm|p

|x|ap
dx+

∫
RN

A(x)|wm|p

|x|ap∗(a,b)
dx+

∫
RN

|∇wm|q

|x|cq
dx+

∫
RN

B(x)|wm|q

|x|cq∗(c,d)
dx

6 (βp − (p(β − 1) + 1))

∫
Am

|∇v|p|v|p(β−1)

|x|ap
dx+ βp

∫
Cm

|∇v|p|v|p(β−1)

|x|ap
dx

+ (βq − (q(β − 1) + 1))

∫
Cm

|∇v|q|v|q(β−1)

|x|cq
dx+ βq

∫
Am

|∇v|q|v|q(β−1)

|x|cq
dx

+

∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx

+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx.

So, using inequality (3.3) and some estimates we deduce that∫
RN

|∇wm|p

|x|ap
dx+

∫
RN

A(x)|wm|p

|x|ap∗(a,b)
dx+

∫
RN

|∇wm|q

|x|cq
dx+

∫
RN

B(x)|wm|q

|x|cq∗(c,d)
dx

6
( βp

q(β − 1) + 1

){∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx

+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx
}

+ βp
∫
Cm

|∇v|p|v|p(β−1)

|x|ap
dx+ βq

∫
Am

|∇v|q|v|q(β−1)

|x|cq
dx.

Now we estimate some of the integrals that appear in the previous inequality.
First, by definition of Am we have∫

Am

|∇v|q|v|q(β−1)

|x|cq
dx =

∫
Am

|∇v|q−2

[p(β − 1) + 1]|x|cq|v|(p−q)(β−1)
∇v · ∇vm dx

6
∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx.

In a similar way, by the definition of Cm we have∫
Cm

|∇v|p|v|p(β−1)

|x|ap
dx 6

∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx.

Using these inequalities we deduce that∫
RN

|∇wm|p

|x|ap
dx+

∫
RN

A(x)|wm|p

|x|ap∗(a,b)
dx+

∫
RN

|∇wm|q

|x|cq
dx+

∫
RN

B(x)|wm|q

|x|cq∗(c,d)
dx

6
(
βp +

βp

q(β − 1) + 1

){∫
RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx

+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx
}

6 βp
{∫

RN

|∇v|p−2

|x|ap
∇v · ∇vm dx+

∫
RN

A(x)|v|p−2vvm
|x|ap∗(a,b)

dx



18 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇vm dx+

∫
RN

B(x)|v|q−2vvm
|x|cq∗(c,d)

dx
}

= βp
∫
RN

H(x, v)vm dx.

Using the Caffarelli-Kohn-Nirenberg inequalities (2.5) and (2.6), the definition
of the constant S and the hypothesis H(x, s) 6 h(x)|s|p−2s/|x|bp∗(a,b), we obtain(∫

Am∪Cm

|wm|p
∗(a,b)

|x|bp∗(a,b)
dx
)p/p∗(a,b)

6
(∫

RN

|wm|p
∗(a,b)

|x|bp∗(a,b)
dx
)p/p∗(a,b)

6 S
∫
RN

|∇wm|p

|x|ap
dx

6 S
{∫

RN

|∇wm|p

|x|ap
dx+

∫
RN

P (x)|wm|p

|x|ap∗(a,b)
dx

+

∫
RN

|∇wm|q

|x|cq
dx+

∫
RN

Q(x)|wm|q

|x|cq∗(c,d)
dx
}

6 Sβp
∫
RN

H(x, v)vm dx

6 Sβp
∫
RN

h(x)|v|p−2vvm
|x|bp∗(a,b)

dx

= Sβp
{∫

Am

h(x)|v|p−2v|v|p(β−1)v

|x|bp∗(a,b)
dx+

∫
Bm

h(x)|v|p−2vmpv

|x|bp∗(a,b)
dx

+

∫
Cm

h(x)|v|p−2v|v|q(β−1)v

|x|bp∗(a,b)
dx
}

= Sβp
{∫

Am

h(x)|v|pβ

|x|bp∗(a,b)
dx+

∫
Bm

h(x)|v|pmp

|x|bp∗(a,b)
dx

+

∫
Cm

h(x)|v|p+q(β−1)

|x|bp∗(a,b)
dx
}

6 Sβp
{∫

RN

h(x)|v|pβ

|x|bp∗(a,b)
dx+

∫
RN

h(x)|v|p

|x|bp∗(a,b)
dx
}
,

where in the last two passages we used the definitions of the functions vm and
wm, together with the facts that in Bm we have |wm|p 6 |v|pβ and in Cm we have
|v|p+q(β−1) 6 |v|p.

Passing to the limit as m → ∞ and using Lebesgue’s dominated convergence
theorem, we deduce that(∫

RN

|v|p∗(a,b)β

|x|bp∗(a,b)
dx
)p/p∗

6 Sβp
{∫

RN

h(x)|v|pβ

|x|bp∗(a,b)
dx+

∫
RN

h(x)|v|p

|x|bp∗(a,b)
dx
}
.

Applying Hölder’s inequality to both terms on the right-hand side of the previous
inequality, we obtain∫

RN

h(x)|v|pβ

|x|bp∗(a,b)
dx 6 ‖h‖Lr

bp∗(a,b)/r(RN )‖v‖pβ
Lpβr

′
bp∗(a,b)/pβr′ (R

N )



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 19∫
RN

h(x)|v|p

|x|bp∗(a,b)
dx 6 ‖h‖Lr

bp∗(a,b)/r(RN )‖v‖p
Lpr
′

bp∗(a,b)/pr′ (R
N )
,

where 1/r + 1/r′ = 1. Hence

‖v‖pβ
L
p∗(a,b)β
b/β

(RN )
6 S‖h‖Lr

bp∗(a,b)/r(RN )β
p
{
‖v‖pβ

Lpβr
′

bp∗(a,b)/pβr′ (R
N )

+ ‖v‖p
Lpr
′

bp∗(a,b)/r′ (R
N )

}
6 S‖h‖Lr

bp∗(a,b)/r(RN )β
p
{

max
{
‖v‖pβ

Lpβr
′

bp∗(a,b)/pβr′ (R
N )
, 1
}

+ max
{
‖v‖p

Lpr
′

bp∗(a,b)/pr′ (R
N )
, 1
}}

= Cp1β
p max

{
‖v‖pβ

Lpβr
′

bp∗(a,b)/pβr′ (R
N )
,max

{
‖v‖p

Lpr
′

bp∗(a,b)/pr′ (R
N )
, 1
}}
,

where we used Cp1 := S‖h‖Lr
bp∗(a,b)/r(RN ) > 0 to denote a constant that depends

on the parameters of problem (1.1) but does not depend on the function v ∈ E ⊂
D1,p
a,b(RN ) ∩D1,q

c,d(RN ). Writing β = σj for j ∈ N we deduce that

‖v‖
L
p∗(a,b)σj

b/σj
(RN )

6 C1/σj

1 σj/σ
j

max
{
‖v‖

Lpσ
jr′

bp∗(a,b)/pσjr′
(RN )

,max
{
‖v‖1/σ

j

Lpr
′

bp∗(a,b)/pr′ (R
N )
, 1
}}
.

(3.4)

Choosing σ = p∗(a, b)/pr′ > 1, from inequality (3.4) with j = 1 we obtain

‖v‖
L
p∗(a,b)σ
b/σ

(RN )
6 C1/σ

1 σ1/σ max
{
‖v‖

L
p∗(a,b)
b (RN )

,max
{
‖v‖1/σ

Lpr
′

bσ (RN )
, 1
}}
.

From inequality (3.4) with j = 2 together with the previous inequality we obtain

‖v‖
L
p∗(a,b)σ2

b/σ2
(RN )

6 C1/σ2

1 σ2/σ2

max
{
‖v‖

L
p∗(a,b)σ
b/σ

(RN )
,max

{
‖v‖1/σ

2

Lpr
′

bσ (RN )
, 1
}}

6 C1/σ2

1 σ2/σ2

max
{
C

1/σ
1 σ1/σ max

{
‖v‖

L
p∗(a,b)
b (RN )

,max
{
‖v‖1/σ

Lpr
′

bσ (RN )
, 1
}}
,

max
{
‖v‖1/σ

2

Lpr
′

bσ (RN )
, 1
}}

6 C1/σ+1/σ2

1 σ1/σ+2/σ2

max
{
‖v‖

L
p∗(a,b)
b (RN )

,max
{(
C

1/σ
1 σ1/σ

)−1
, 1
}
,

max
{(
C

1/σ
1 σ1/σ

)−1
, 1
}

max
{
‖v‖1/σ

Lpr
′

bσ (RN )
, ‖v‖1/σ

2

Lpr
′

bσ (RN )
, 1
}}
.

Proceeding in this way, for j ∈ N we obtain

‖v‖
L
p∗(a,b)σj

b/σj
(RN )

6 Csj1 σtj max
{
‖v‖

L
p∗(a,b)
b (RN )

,Kj ,KjLj

}
, (3.5)

where sj := 1/σ + 1/σ2 + · · ·+ 1/σj ; tj := 1/σ + 2/σ2 + · · ·+ j/σj ;

Kj :=

{
1, if j = 1;

max16i6j−1

{
C−si1 σ−ti , 1

}
, if j > 2;

and

Lj := max
16i6j

{
‖v‖1/σ

i

Lpr
′

bσ (RN )
, 1
}
.



20 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

Since σ > 1, we have limj→∞ sj = 1/(σ − 1) and limj→∞ tj = σ/(σ − 1)2; hence,

lim
j→∞

Kj := K =

{(
C

1/(σ−1)
1 σσ/(σ−1)2

)−1
if C1 6 1;(

C
1/σ
1 σ1/σ

)−1
if C1 > 1;

(3.6)

and

lim
j→∞

Lj := Lv =

1, if ‖v‖
Lpr
′

bσ (RN )
6 1;

‖v‖1/(σ−1)

Lpr
′

bσ (RN )
, if ‖v‖

Lpr
′

bσ (RN )
> 1 .

(3.7)

Using the fact that v ∈ E ⊂ D1,p
a,b(RN )∩D1,q

c,d(RN ), applying Hölder’s inequality we
deduce that Lv < +∞.

Finally, passing to the limit as j →∞ and using inequality (3.5) we obtain

‖v‖L∞(RN ) = lim
j→∞

‖v‖
L
p∗(a,b)σj

b/σj
(RN )

6 C1/(σ−1)
1 σσ/(σ−1)2 max

{
‖v‖

L
p∗(a,b)
b (RN )

,K,KLv, 1
}

:= M1 max
{
‖v‖

L
p∗(a,b)
b (RN )

,K,KLv, 1
}
,

(3.8)

where M1 = M1(‖h‖Lr
bp∗(a,b)/r(RN )) depends on the parameters of problem (1.1) but

does not depend on the function v ∈ E ⊂ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ). This concludes
the proof of the lemma. �

Lemma 3.3. For every R > 1 there exist a constant M2 = M2(P∞, P∞) such

that any positive ground state solution u ∈ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ) to the auxiliary

problem (2.4) satisfies

‖u‖L∞(RN ) 6M2.

Proof. Consider R > 1 and let u ∈ D1,p
a,b(RN )∩D1,q

c,d(RN ) be a positive ground state

solution to the auxiliary problem (2.4). Now we define the function H : RN×R→ R
by

H(x, t) :=


f(t), if |x| 6 R

or if |x| > R and f(t) 6 P (x)|t|p−2t

k|x|ap∗(a,b) ;

0, if |x| > R and f(t) > P (x)|t|p−2t

k|x|ap∗(a,b) .

We also define the functions A,B : RN → R by

A(x) =


P (x), if |x| 6 R

or if |x| > R and f(u(x)) 6 P (x)|u(x)|p−2u(x)

k|x|ap∗(a,b) ;(
1− 1

k

)
P (x), if |x| > R and f(u(x)) > P (x)|u(x)|p−2u(x)

k|x|ap∗(a,b) ,

and B(x) = Q(x).
Considering these functions and using v ∈ E as a test function, we have

0 =

∫
RN

|∇u|p−2

|x|ap
∇u · ∇v dx+

∫
RN

A(x)|u|p−2uv

|x|ap∗(a,b)
dx

+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇v dx+

∫
RN

B(x)|u|q−2uv

|x|cq∗(c,d)
dx−

∫
RN

H(x, u)v dx

=

∫
RN

|∇u|p−2

|x|ap
∇u · ∇v dx+

∫
RN

P (x)|u|p−2uv

|x|ap∗(a,b)
dx



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 21

+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇v dx+

∫
RN

Q(x)|u|q−2uv

|x|cq∗(c,d)
dx−

∫
RN

g(x, u)v dx.

From hypothesis (H1), for |t| small enough we have

|H(x, t)| 6 |f(t)| 6 c1|t|p
∗(a,b)−1

|x|bp∗(a,b)
;

and from hypothesis (H2), for |t| big enough we have

|H(x, t)| 6 |f(t)| 6 c2|t|τ−1

|x|bp∗(a,b)

with τ ∈ (p, p∗(a, b)). Combining both cases we obtain |H(x, t)| 6 |f(t)| 6
c0|t|τ−1/|x|bp∗(a,b) for every t ∈ R+ and for every τ ∈ (p, p∗(a, b)). It follows
that |H(x, u)| 6 c0|u(x)|τ−p|u(x)|p−1/|x|bp∗(a,b) = h(x)|u(x)|p−1/|x|bp∗(a,b), where
we set h(x) := c0|u(x)|τ−p.

Direct computations show that h ∈ Lrbp∗(a,b)/r(R
N ) for r = p∗(a, b)/(τ − p).

Indeed, recalling the definition of S we obtain∫
RN
| h(x)

|x|bp∗(a,b)/r
|r dx

=

∫
RN

|c0|u(x)|τ−p|p∗(a,b)/(τ−p)

|x|bp∗(a,b)
dx

6 cp
∗(a,b)/(τ−p)

0

∫
RN

|u|p∗(a,b)

|x|bp∗(a,b)
dx

6 cp
∗(a,b)/(τ−p)

0 Sp
∗(a,b)/p

(∫
RN

|∇u|p

|x|ap
dx
)p∗(a,b)/p

6 cp
∗(a,b)/(τ−p)

0 Sp
∗(a,b)/p

{∫
RN

|∇u|p

|x|ap
dx+

∫
RN

P (x)|u|p

|x|ap∗(a,b)
dx

+

∫
RN

|∇u|q

|x|cq
dx+

∫
RN

Q(x)|u|q

|x|cq∗(c,d)
dx
}p∗(a,b)/p

6 cp
∗(a,b)/(τ−p)

0 Sp
∗(a,b)/p

{
‖u‖p1,p + ‖u‖q1,q

}p∗(a,b)/p
< +∞.

In this way, any positive ground state solution u ∈ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ) to

the auxiliary problem (2.4) satisfies the hypothesis of Lemma 3.2. Concluding the
argument, from Lemma 3.1 we have

‖u‖
L
p∗(a,b)
b (RN )

6 S1/p
{
‖u‖p1,p + ‖u‖q1,q

}1/p
6
(Skpd∞
p− 1

)1/p

.

Finally, combining estimate (3.8) with the previous inequality we obtain

‖u‖L∞(RN ) 6M1 max
{
‖u‖

L
p∗(a,b)
b (RN )

,K,KLu, 1
}

6M1 max
{(Skpd∞

p− 1

)1/p

,K,KLu, 1
}

:= M2,

where M2 = M2(N, p, q, r, a∞, b∞, θ, c0). The proof is complete. �

Lemma 3.4. Suppose that R0 > R > 1 and let u ∈ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ) be a

positive ground state solution to the auxiliary problem (2.4). Then u satisfies the



22 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

inequality

u(x) 6M2
R[N−p(a+1−b)]/(p−1)

|x|[N−p(a+1−b)]/(p−1)

for every |x| > R > 1.

Proof. Given R0 > R > 1, we define the function v : RN\{0} → R by

v(x) := M2
R

[N−p(a+1−b)]/(p−1)
0

|x|[N−p(a+1−b)]/(p−1)
.

By hypothesis, u ∈ D1,p
a,b(RN ) ∩ D1,q

c,d(RN ) is a positive ground state solution to

the auxiliary problem (2.4); therefore, we can apply Lemma 3.3 to deduce that
‖u‖L∞(RN ) 6 M2. This implies that if |x| = R0, then ‖u‖L∞(RN ) 6 v(x). Now we

define the function w : RN\{0} → R by

w(x) =

{
0, if |x| 6 R0;

(u− v)+, if |x| > R0.

In this way, w ∈ D1,p
a,b(RN ) ∩D1,q

c,d(RN ); moreover, w ∈ E because u, v ∈ E.

To complete the proof of the lemma we will show that (u−v)+ = 0 for |x| > R0.
To accomplish this goal we use the hypotheses on the potential functions P and Q;
we will also use the function w ∈ E as a test function to obtain∫

RN

|∇u|p−2

|x|ap
∇u · ∇w dx+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇w dx

=

∫
RN

g(x, u)w dx−
∫
RN

P (x)|u|p−2uw

|x|ap∗(a,b)
dx−

∫
RN

Q(x)|u|q−2uw

|x|cq∗(c,d)
dx

=

∫
RN\BR0

(0)∧f(t)6 P (x)|t|p−2t

k|x|ap∗(a,b)

f(u)w dx

+

∫
RN\BR0

(0)∧f(t)>
P (x)|t|p−2t

k|x|ap∗(a,b)

P (x)|u|p−2uw

k|x|ap∗(a,b)
dx

−
∫
RN\BR0

(0)

P (x)|u|p−2uw

|x|ap∗(a,b)
dx−

∫
RN\BR0

(0)

Q(x)|u|q−2uw

|x|cq∗(c,d)
dx

6
∫
RN\BR0

(0)∧f(t)6 P (x)|t|p−2t

k|x|ap∗(a,b)

P (x)|u|p−2uw

k|x|ap∗(a,b)
dx

+

∫
RN\BR0

(0)∧f(t)>
P (x)|t|p−2t

k|x|ap∗(a,b)

P (x)|u|p−2uw

k|x|ap∗(a,b)
dx

−
∫
RN\BR0

(0)

P (x)|u|p−2uw

|x|ap∗(a,b)
dx−

∫
RN\BR0

(0)

Q(x)|u|q−2uw

|x|cq∗(c,d)
dx

=
(1

k
− 1
) ∫

RN\BR0
(0)

P (x)|u|p−2uw

|x|ap∗(a,b)
dx−

∫
RN\BR0

(0)

Q(x)|u|q−2uw

|x|cq∗(c,d)
dx

6 0

(3.9)

because u is a positive function and w is a nonnegative function, while k > 1.



EJDE-2022/43 MULTIPLE SINGULAR WEIGHTED p-LAPLACIANS 23

Using the radially symmetric form of the operator − div(|∇u|m−2∇u/|x|am), we
have ∫

RN\BR0
(0)

|∇v|m−2

|x|em
∇v · ∇φ dx = 0

for m ∈ {p, q}, for e ∈ {a, c}, and for every function φ ∈ E; see Calzolari, Filippucci
and Pucci [9] and Lindqvist [15]. Therefore,∫

RN

|∇v|p−2

|x|ap
∇v · ∇w dx+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇w dx

=

∫
RN\BR0

(0)

|∇v|p−2

|x|ap
∇v · ∇w dx+

∫
RN\BR0

(0)

|∇v|q−2

|x|cq
∇v · ∇w dx

= 0.

(3.10)

Defining the subsets

à := {x ∈ RN : |x| > R0 and u(x) > v(x)}

B̃ := {x ∈ RN : |x| < R0 or u(x) 6 v(x)},

we have w(x) = u(x)−v(x) for x ∈ Ã and w(x) = 0 for x ∈ B̃. Using inequality (3.9)
and equality (3.10) we obtain

0 >
∫
RN

|∇u|p−2

|x|ap
∇u · ∇w dx+

∫
RN

|∇u|q−2

|x|cq
∇u · ∇w dx

−
∫
RN

|∇v|p−2

|x|ap
∇v · ∇w dx+

∫
RN

|∇v|q−2

|x|cq
∇v · ∇w dx

=

∫
Ã

[
|∇u|p−2

|x|ap
∇u− |∇v|

p−2

|x|ap
∇v
]
· (∇u−∇v) dx

+

∫
Ã

[
|∇u|q−2

|x|cq
∇u− |∇v|

q−2

|x|cq
∇v
]
· (∇u−∇v) dx.

(3.11)

Denoting by 〈· , ·〉 : RN × RN → R the standard scalar product, given m > 2 there
exists a positive constant cm ∈ R+ such that for every x, y ∈ RN it is valid the
inequality

〈|x|p−2x− |y|p−2y, x− y〉 > cm‖x− y‖p. (3.12)

For the proof, we refer the reader to Simon [17]. From inequalities (3.11) and (3.12)
it follows that∫

RN

|∇w|p

|x|ap
dx+

∫
RN

|∇w|q

|x|cq
dx

=

∫
Ã

|∇u−∇v|p

|x|ap
dx+

∫
Ã

|∇u−∇v|q

|x|cq
dx

6 c−1
p

∫
Ã

[
|∇u|p−2

|x|ap
∇u− |∇v|

p−2

|x|ap
∇v
]
· (∇u−∇v) dx

+ c−1
q

∫
Ã

[
|∇u|q−2

|x|cq
∇u− |∇v|

q−2

|x|cq
∇v
]
· (∇u−∇v) dx 6 0.

From this inequality we deduce that each term on the left-hand side of the previous
inequality must be zero, that is, w is constant in RN . But we already know that
w(x) = 0 in the ball BR0

(0); therefore, w(x) = 0 for every x ∈ RN . This implies



24 M. J. ALVES, R. B. ASSUNÇÃO EJDE-2022/43

that (u− v)+ = 0 for |x| > R0 and u(x) 6 v(x) for every x ∈ RN . The proof of the
lemma is complete. �

4. Solution of the original problem

In this section we finally show that the solution to the auxiliary problem (2.4)
obtained in section 2 is in fact a solution to problem (1.1).

Proof of Theorem 1.1. From Lemmas 2.3 and 2.6, the auxiliary problem (2.4) has
a positive ground state solution u ∈ D1,p(RN )∩D1,q(RN ). To accomplish our goal
we need to show that for every x ∈ BcR(0) the function u satisfies the inequality

f(u) 6
P (x)|u|p−2u

k|x|ap∗(a,b)
.

From Lemma 3.4 and by the first inequality in (1.2), if |x| > R, then

f(u)

|u|p−2u
6

c0
|x|bp∗(a,b)

|u|p
∗(a,b)−p

6
c0

|x|ap∗(a,b)
M

p∗(a,b)−p
2

Rp
2(a+1−b)/(p−1)

|x|p2(a+1−b)/(p−1)
.

Now we define the constant

Λ∗ := c0kM
p∗(a,b)−p
2 .

Considering Λ > Λ∗, it follows from the hypothesis (H6) that

f(u)

|u|p−2u
6

Λ∗

k|x|ap∗(a,b)
Rp

2(a+1−b)/(p−1)

|x|p2(a+1−b)/(p−1)

6
Λ

k|x|ap∗(a,b)
Rp

2(a+1−b)/(p−1)

|x|p2(a+1−b)/(p−1)

6
P (x)

k|x|ap∗(a,b)
.

The proof is complete. �

Acknowledgements. The authors would like to express their appreciation to Prof.
Oĺımpio H. Miyagaki for having introduced them to the type of differential operator
studied in this work. The authors also would like to thank the anonymous reviewers
for their suggestions and comments that helped to improve this work.

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Maria José Alves (corresponding author)

Colégio Técnico, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627,
CEP 31270-901, Belo Horizonte, MG, Brasil

Email address: mariajose@ufmg.br

Ronaldo B. Assunção
Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio

Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brasil
Email address: ronaldo@mat.ufmg.br


	1. Introduction and main result
	2. An auxiliary problem
	3. Estimate for the solution to the auxiliary problem
	4. Solution of the original problem
	Acknowledgements

	References