Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 44, pp. 1–12.

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

DOI: 10.58997/ejde.2024.44

CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS WITH

BERESTYCKI-LIONS TYPE NONLINEARITIES

GIOVANY M. FIGUEIREDO, GEORGE KIAMETIS

Communicated by Giovanni Molica Bisci

Abstract. In this article we use a Palais-Smale sequence satisfying a property
related to Pohozaev identity to show the existence of solution for the elliptic

Caffarelli-Kohn-Nirenberg type problems

− div(|x|−ap|∇u|p−2∇u) + |x|−bp∗ |u|p−2u = |x|−bp∗h(u) in RN

and

− div(|x|−ap|∇u|p−2∇u) = |x|−bp∗f(u) in RN ,

where 1 < p < N , 0 ≤ a < N−p
p∗ , a < b ≤ a + 1, p∗ = p∗(a, b) = pN

N−dp
and

d = 1 + a − b. and h and f are continuous functions that satisfy hypotheses
considered by Berestycki and Lions in [7].

1. Introduction

Using a constrained minimization, Berestycki and Lions [7] showed the existence
of positive solutions of C2 class for the problem

−∆u = g(u) in RN (1.1)

with exponential decay and spherically symmetric, where g : R → R is a continuous
function such that g(0) = 0. The authors assume that g is odd and satisfies the
following conditions.

(A1) −∞ < lim infs→0+ g(s)/s ≤ lim sups→0+ g(s)/s = −m ≤ 0.
(A2) −∞ ≤ lim sups→∞ g(s)/s2

∗−1 ≤ 0.

(A3) There exists ξ > 0 such that G(ξ) =
∫ ξ

0
g(s)ds > 0.

The constraint causes a Lagrange multiplier to appear, but it can be removed using
the special homogeneity of the operator and a scale change in RN . The authors
studied two cases: The positive mass case, m > 0, and the zero mass case, m = 0.

Alves, Montenegro and Souto [1] studied the existence of ground state solution
for (1.1) with critical growth. By using the variational method, the authors in [1]
gave a unified approach for the subcritical and critical cases. However, we would like

2020 Mathematics Subject Classification. 35B38, 35J35, 35J92.

Key words and phrases. Caffarelli-Kohn-Nirenberg type problems; Nehari manifold.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted January 15, 2024. Published August 12, 2024.

1



2 G. M. FIGUEIREDO, G. KIAMETIS EJDE-2024/44

to point out that a result due to Jeanjean and Tanaka [20] says that the Mountain-
Pass value gives the least energy level, and it was the main tool used. A similar
study was made for the critical case in Zhang and Zou [24].

After these pioneering papers, many researches worked on this subject, extending
or improving it in several ways; see, for instance, [2, 3, 4, 5, 8, 11, 13, 14, 17] and
references therein.

This article concerns the existence of nontrivial solutions for the problems

−div(|x|−ap|∇u|p−2∇u) + |x|−bp∗
|u|p−2u = |x|−bp∗

h(u) in RN , (1.2)

and

−div(|x|−ap|∇u|p−2∇u) = |x|−bp∗
f(u) in RN , (1.3)

where 1 < p < N , 0 ≤ a < N−p
p∗ , a < b ≤ a + 1, p∗ = p∗(a, b) = pN

N−dp and

d = 1 + a− b.
Equations involving the operator div(|x|−ap|∇u|p−2∇u) are regarded as proto-

type of more general nonlinear degenerate elliptic equations from physical phenom-
ena; see for example [15, 16, 23].

In this article we adapt some arguments found in [18] and [19]. More precisely,
we find a Palais-Smale sequence satisfying a property related to Pohozaev identity.
The same approach was used in [4] for a problem involving the Grushin operator.

We would like to point out that in the proof of Theorems 1.1 and 1.2, we have
found some difficulties when applying variational methods. For example, for this
class operator there is no a result like Jeanjean and Tanaka [20], which say that the
Mountain-Pass value gives the least energy level of the Pohozaev manifold, which is
crucial in order to use the arguments due to Berestycki-Lions. Furthermore, it was
necessary to prove a Straus-type Lemma result for this class of problems (Lemma
3.2 and Lemma 3.3). In Chen [12] we can find a Straus-type Lemma result for this
class of problems, but it does not apply to our case.

Before concluding this introduction, it is very important to say that in the litera-
ture, we find many papers where the authors study problems involving the operator
div(|x|−ap|∇u|p−2∇u); see, Bastos, Miyagaki and Vieira [6], Catrina and Wang [10],
Chen [12], Xuan [22] and references therein.

To present the main results of this article, it is necessary to state hypotheses
about the nonlinearities h and f . The hypotheses on the function h in this case are
the following:

(A4) There exists q ∈ (p, p∗) such that

lim
|t|→0

h(t)

|t|p−1
= lim

|t|→∞

h(t)

|t|q−1
= 0.

(A5) There exists ξ > 0 such that pH(ξ)− ξp > 0, where H(t) =
∫ t

0
h(r)dr.

The first main result reads as follows.

Theorem 1.1. Under the conditions (A4) and (A5), problem (1.2) has a nontrivial
solution.

The first class of problems is called Positive Mass because g(t) = h(t)−t satisfies
(A1)–(A3) for m > 0.

The hypotheses on the function f in this case are as follws:



EJDE-2024/44 CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS 3

(A6)

lim
|t|→0

f(t)

|t|p∗−1
= lim

|t|→∞

f(t)

|t|p∗−1
= 0.

(A7) There exists ξ > 0 such that F (ξ) > 0, where F (t) =
∫ t

0
f(r)dr.

The second main result reads as follows.

Theorem 1.2. Under assumptions (A6) and (A7), problem (1.3) has a nontrivial
solution.

The second class of problems is called Zero Mass because f satisfies (A1)–(A3)
for m = 0.

The plan for this article is as follows: In section 2 we present the spaces that we
find the solutions. In section 3 we prove Theorem 1.1. And in section 3 we prove
Theorem 1.2.

2. Variational framework

For the zero-mass case we use D1,p
a (RN ) which is the completion of C∞

0 (RN ) with
the norm

∥u∥p0 =

∫
RN

|x|−ap|∇u|p dx,

where C∞
0 (RN ) is the space of smooth functions with compact support. For the

Positive Mass case we use E = {u ∈ D1,p
a (RN ) :

∫
RN |x|−bp∗ |u|p dx < ∞} with the

norm

∥u∥p =

∫
RN

|x|−ap|∇u|p dx+

∫
RN

|x|−bp∗
|u|p dx.

Let Ls
b(RN ) be the weighted Ls space with weighted norm

|u|s =
∫
RN

|x|−bp∗
|u|s dx.

We also define E(BR(0)) = {u ∈ D1,p
a (BR(0)) :

∫
BR(0)

|x|−bp∗ |u|p dx < ∞}. Let

Ls
b(BR(0)) be the weighted Ls space with weighted norm

|u|s =
∫
BR(0)

|x|−bp∗
|u|s dx.

Let the weighted Ls space be defined by the weighted norm

|u|sBR(0) =

∫
BR(0)

|x|−bp∗
|u|s dx.

Using an inequality established by Caffarelli, Kohn, and Nirenberg in [9],(∫
RN

|x|−bp∗
|u|p

∗
dx

)p/p∗

≤ Sa,b

∫
RN

|x|−ap|∇u|p dx,

we conclude that the embedding D1,p
a (RN ) ↪→ Lp∗

b (RN ) is continuous. Moreover,
by interpolation, we also conclude that E ↪→ Ls

b(RN ) is continuous, for s ∈ [p, p∗].



4 G. M. FIGUEIREDO, G. KIAMETIS EJDE-2024/44

3. Existence of solutions for the positive-mass case

Consider the functional I : E → R associated given by

I(u) =
1

p
∥u∥p −

∫
RN

|x|−bp∗
H(u) dx.

As a consequence of (A4) we obtain that I is well-defined and of C1 class. Also
note that

I ′(u)ϕ =

∫
RN

|x|−ap|∇u|p−2uϕ dx+

∫
RN

|x|−bp∗
|u|p−2uϕ dx−

∫
RN

|x|−bp∗
h(u)ϕdx,

for all ϕ ∈ E. Then, the critical points of I are weak solutions of (1.2).
To use critical point theory we firstly derive results related to the Palais-Smale

compactness condition. We say that a sequence (un) is a Palais-Smale sequence for
the functional I if

I(un) → c∗, and ∥I ′(un)∥ → 0 in (E)′,

where

c∗ = inf
η∈Γ

max
t∈[0,1]

I(η(t)) > 0, Γ := {η ∈ C([0, 1], E) : η(0) = 0, I(η(1)) < 0}.

If every Palais-Smale sequence of I has a strong convergent subsequence, then one
says that I satisfies the Palais-Smale condition ((PS) for short).

Lemma 3.1. The functional I satisfies the following conditions:

(i) There exist ρ1, ρ2 > 0 such that I(u) ≥ ρ2 with ∥u∥ = ρ1.
(ii) There exists e ∈ Bc

ρ2
(0) with I(e) < 0 and ∥e∥ > ρ2.

Proof. (i) First of all, from (A4), for each ε > 0 there exists Cε > 0 such that

h(t) ≤ ε|t|p−1 + Cε|t|q−1, ∀t ∈ R. (3.1)

Using the inequality above and taking ϵ > 0 sufficiently small such, we obtain

I(u) ≥
(1
p
− ϵ

p

)
∥u∥p − C1Cε

q
∥u∥q

and the result follows because q > p.
(ii) From (A5), there exists ϕ ∈ C∞

0 (RN ) such that∫
RN

|x|−bp∗(
H(ϕ)− |ϕ|p

p

)
dx > 0.

For t > 0, setting

ωt(x) = ϕ(
x

t
),

by simple calculations, we obtain

I(ωt) = tN−p

∫
RN

|y|−ap|∇ϕ(y)|pdy − tN
∫
RN

|y|−bp∗
(H(ϕ(y))− |ϕ(y)|p

p
)dy → −∞,

as t → ∞. Then, there exists t̄ > 0 large such that e = ωt̄ satisfies I(e) < 0 and
∥e∥ > ρ2. Also note that c∗ ≥ ρ2. □



EJDE-2024/44 CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS 5

Next, we prove a compactness result, which is crucial in our approach. We denote
by C∞

0,rad(RN ) the collection of smooth radially symmetric functions with compact,
i.e.,

C∞
0,rad(RN ) = {u ∈ C∞

0 (RN ) : u(x) = u(|x|), x ∈ RN}.

Let D1,p
a,rad(RN ) be the completion of C∞

0,rad(RN ) under the norm ∥ · ∥0 and define

Erad = D1,p
a,rad(R

N ) ∩ Lr
b,rad(RN )

under the norm ∥ · ∥.

Lemma 3.2 (Radial Lemma in Erad). Let u ∈ Erad, then for almost every x ∈
RN\{0}, then there exists C = C(a, b, p) > 0 such that

|u(x)| ≤ C
1

|x|
(N−p)−ap∗

p

∥u∥.

Proof. Up to a standard density argument, we only consider u ∈ C∞
0,rad(RN ). De-

note by ωN the volume of the unit sphere in RN . We have

−u(Υ) = u(∞)− u(Υ) =

∫ ∞

Υ

u′(s)ds.

Thus,

|u(Υ)| ≤
∫ ∞

Υ

|u′(s)|ds =
∫ ∞

Υ

s
−ap∗

p |u′(s)|s
N−1

p s
ap∗
p s

1−N
p ds.

From Holder’s inequality, we obtain

|u(Υ)| ≤
(∫ ∞

Υ

s−ap∗
|u′(s)|sN−1ds

)1/p(∫ ∞

Υ

s
ap∗
p−1 s

1−N
p−1 ds

)(p−1)/p

.

Hence

|u(Υ)| ≤ ω
−1
p

N

( p− 1

N − p− ap∗

) p−1
p 1

|x|
(N−p)−ap∗

p

(∫
RN

|x|−ap|∇u|p dx
)1/p

. □

Now we present a compactness result.

Lemma 3.3. If a < N−p
p∗ , then the embedding Erad ↪→ Ls

b(RN ) is compact for all

s ∈ (p, p∗).

Proof. Let (un) ⊂ Erad(RN ) be a bounded sequence and let C > 0 be such that

∥un∥ ≤ C, ∀n ∈ N.

By Lemma 3.2 it follows that, for all n ∈ N,

|un(x)| ≤ CC
1

|x|
(N−p)−ap∗

p

, a.e. in RN\{0}.

Since s > 1, given ϵ > 0, there exists R > 0 such that, for all n ∈ N,

|un(x)|s ≤
ϵ

2CC
|un(x)| ∀x ∈ BR(0)

c.

This implies that∫
BR(0)c

|x|−bp∗
|un|s dx ≤ ϵ

2CCRbp∗

∫
BR(0)c

|un|dx ≤ ϵ

2R
(N−p)−ap∗+bpp∗

p

≤ ϵ

2
, (3.2)



6 G. M. FIGUEIREDO, G. KIAMETIS EJDE-2024/44

for all n ∈ N. Moreover, since E(BR(0)) is compactly embedded into Ls
b(BR(0)),

there exists u ∈ Ls
b(BR(0)) such that, up to a subsequence un → u in Ls

b(BR(0)),
as n → ∞. Then there exists n0 ∈ N such that∫

BR(0)

|x|−bp∗
|un − u|sdx <

ϵ

2
, ∀n ≥ n0. (3.3)

Let us define u : RN → R as to be equal to u in BR(0) and equal to 0 in BR(0)
c.

Then, by (3.2) and (3.3), it follows that∫
RN

|x|−bp∗
|un − u|s dx =

∫
BR(0)

|x|−bp∗
|un − u|sdx+

∫
BR(0)c

|x|−bp∗
|un|sdx

< ϵ.

Then it is clear that un → u in Ls
b(RN ), as n → ∞. □

Following [18] and [19], we consider an auxiliary functional Ĩ ∈ C1(R×Erad),R)
given by

Ĩ(θ, u) =
exp(N − p)θ

p

∫
RN

|x|−ap|∇u|p dx+
exp (Nθ)

p

∫
RN

|x|−bp∗
|u|p dx

− exp (Nθ)

∫
RN

|x|−bp∗
H(u) dx.

The following properties hold, for all (θ, u) ∈ R× Erad,

Ĩ(0, u) = I(u),

Ĩ(θ, u) = I(u(x/ exp(θ))).

We equip the standard product norm

∥(θ, u)∥pR×Erad
= |θ|p + ∥u∥p

to R× E. Now we prove that Ĩ satisfies the Mountain Pass geometry.

Lemma 3.4. The functional Ĩ satisfies the following conditions:

(i) There exist ρ1, ρ2 > 0 such that Ĩ(θ, u) ≥ ρ2 with ∥(θ, u)∥ = ρ1.

(ii) There exists ẽ ∈ Bc
ρ2
(0) with Ĩ(ẽ) < 0 and ∥ẽ∥ > ρ2.

Proof. Item (i) follows by using the same argument as in Lemma 3.1. For item (ii)
it is sufficient to take ẽ = (0, e). □

In what follows, we define the Mountain Pass level c̃∗ for Ĩ by

c̃∗ = inf
η∈Γ

max
t∈[0,1]

Ĩ(η(t)) > 0

and
Γ̃ := {η ∈ C([0, 1],R× Erad) : η(0) = 0, Ĩ(η(1)) < 0}.

Note that c̃∗ ≥ ρ2.

Lemma 3.5. The Mountain Pass levels of I and Ĩ coincide, namely c∗ = c̃∗ > 0.

Proof. For our problem we adapt the approach explored in [18, Lemma 4.1]. Note

that Γ ∼= {0}×Γ ⊂ Γ̃, which implies c̃∗ ≤ c∗. On the other hand, consider γ̃ ∈ Γ̃ ar-
bitrary. Then, for each t ∈ [0, 1], we have γ̃(t) = (θt, ut). Define γ(t) := ut(

x
exp(θt)

).

From the definition of Ĩ, we conclude that Ĩ(γ̃t) = Ĩ(θt, ut) = I(ut(x/ exp(θ))) =
I(γ(t)) for each t ∈ [0, 1]. Hence γ ∈ Γ, where we derive c̃∗ ≥ c∗. □



EJDE-2024/44 CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS 7

The proof of next lemma is the same as the proof of [18, Lemma 4.3].

Lemma 3.6. Let ϵ > 0. Suppose that η̃ ∈ Γ̃ satisfies

max
t∈[0,1]

Ĩ(η̃) ≤ c∗ + ϵ,

then, there exists (θ, u) ∈ R× Erad such that

• distR×Erad
((θ, u), η̃([0, 1])) ≤ 2

√
ϵ;

• Ĩ(θ, u) ∈ [c∗ − ϵ, c∗ + ϵ];

• ∥DĨ(θ, u)∥R×E∗
rad

≤ 2
√
ϵ.

As in [18, Proposition 4.2], the proof of the next lemma is a consequence of
Lemma 3.6.

Lemma 3.7. There exists a sequence ((θn, un)) ⊂ R× Erad such that, as n → ∞,
we obtain

• θn → 0;

• Ĩ(θn, un) → c∗;

• ∂θ Ĩ(θn, un) → 0;

• ∂uĨ(θn, un) → 0 strongly in E∗
rad.

Proof. For each j ∈ N, we can find a γj ∈ Γ such that

max
t∈[0,1]

I(γj(t)) ≤ c∗ +
1

j
.

Since c̃∗ = c∗ and γ̃j(t) = (0, γj(t)) ∈ Γ̃ satisfies maxt∈[0,1] Ĩ(γ̃j)(t) ≤ c̃∗ + 1
j , we

can find a (θj , uj) such that

• distR×Erad
((θ, u), γ̃j([0, 1])) ≤ 2/

√
j;

• Ĩ(θ, u) ∈ [c∗ − 1/j, c∗ + 1/j];

• ∥DĨ(θ, u)∥R×E∗
rad

≤ 2/
√
j.

Since γ̃([0, 1]) ⊂ {0} × Erad, the first inequality implies |θj | ≤ 2/
√
j and, conse-

quently, θj → 0. The second item implies Ĩ(θj , uj) → c∗ and the last item implies
the last two items of these lemma. □

3.1. Proof of Theorem 1.1. By Lemma 3.7, there exists a sequence ((θn, un)) ⊂
R× Erad such that

exp((N − p)θn)

p
∥un∥p0 +

exp(Nθn)

p

∫
RN

|x|−bp∗
|un|p dx

− exp (Nθn)

∫
RN

|x|−bp∗
H(un) dx = c∗ + on(1);

(3.4)

(N − p)
exp((N − p)θn)

p
∥un∥p0 +N

exp(Nθn)

p

∫
RN

|x|−bp∗
|un|p dx

−N exp (Nθn)

∫
RN

|x|−bp∗
H(un) dx = on(1);

(3.5)

exp((N − p)θn)∥un∥p0 + exp((N − p)θn)

∫
RN

|x|−bp∗
|un|p dx

− exp (Nθn)

∫
RN

|x|−bp∗
h(un)un dx = on(1)∥un∥.

(3.6)



8 G. M. FIGUEIREDO, G. KIAMETIS EJDE-2024/44

From (3.4) and (3.5) and since p < N , we have

exp((N − p)θn)∥un∥p0 = Nc∗ + on(1). (3.7)

Since θn → 0, we have that (un) is bounded in D1,p
a,rad(RN ) and Lp∗

b (RN ).

From (A4), there exists C > 0 such that

h(t)t ≤ 1

2
|t|p + C|t|p

∗
, for all t ∈ R.

Using the last inequality in (3.6), we obtain

1

2
exp((N − p)θn)

∫
RN

|x|−bp∗
|un|p dx ≤ C exp (Nθn)

∫
RN

|x|−bp∗
|un|p

∗
dx,

which implies that (un) is bounded in Erad. Hence, there exists u ∈ Erad such that,
up to a subsequence, un ⇀ u in Erad. From Lemma 3.7, for all v ∈ Erad, we have

∂uĨ(θn, un)v = on(1); that is,

exp((N − p)θn)

∫
RN

|x|−ap|∇un|p−2unv dx

+ exp((N − p)θn)

∫
RN

|x|−bp∗
|un|p−2unv dx

− exp (Nθn)

∫
RN

|x|−bp∗
h(un)v dx = on(1).

Since θn → 0 in R and from weak convergence, for all v ∈ Erad, we obtain∫
RN

|x|−ap|∇u|p−2uv dx+

∫
RN

|x|−bp∗
|u|p−2uv dx−

∫
RN

|x|−bp∗
h(u)v dx = 0,

showing that I ′(u)v = 0, for all v ∈ Erad, that is u is a critical point of I. We are
going to show that u is not trivial. Suppose that u = 0. From (A4) there exist
ϵ > 0 and Cϵ > 0 such that∫

RN

|x|−bp∗
h(un)un dx ≤ ϵ

∫
RN

|x|−bp∗
|un|p dx+ Cϵ

∫
RN

|x|−bp∗
|un|q dx.

Since (un) is bounded in Erad and since Erad ↪→ Lq
b(RN ) is compact, we conclude

that ∫
RN

|x|−bp∗
h(un)un dx = on(1).

This limit combined with the limit ∂uĨ(θn, un)un = on(1) allows to deduce that

un → 0 in Erad. Hence, Ĩ(θn, un) → 0 = c∗, which is absurd. Thus, u is a nontrivial
critical point of I in Erad. Finally, u is a nontrivial critical point of I in E using
the Principle of Symmetric Criticality [21] or [25, Theorem 1.28].

4. Existence of solution for zero-mass case

Consider the functional I0 : D1,p
a,rad(RN ) → R given by

I0(u) =
1

p
∥u∥p0 −

∫
RN

|x|−bp∗
F (u) dx.

Note that I0 is well-defined and of C1 class. Moreover, note that

I ′0(u)ϕ =

∫
RN

|x|−ap|∇u|p−2∇u∇ϕdx−
∫
RN

|x|−bp∗
f(u)ϕdx,



EJDE-2024/44 CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS 9

for all ϕ ∈ D1,p
a,rad(RN ). Then, the critical points of I0 are weak solutions of (1.3)

in D1,p
a,rad(RN ).

We say that a sequence (un) is a Palais-Smale sequence for the functional I0 if

I0(un) → c0, and ∥I ′0(un)∥ → 0 in (D1,p
a,rad(R

N ))′,

where

c0 = inf
η∈Γ

max
t∈[0,1]

I0(η(t)) > 0,

Γ0 := {η ∈ C([0, 1],D1,p
a,rad(R

N )) : η(0) = 0, I0(η(1)) < 0}.
If every Palais-Smale sequence of I0 has a strong convergent subsequence, then one
says that I0 satisfies the Palais-Smale condition ((PS) for short).

Lemma 4.1. The functional I0 satisfies the following conditions:

(i) There exist ρ1, ρ2 > 0 such that

I0(u) ≥ ρ2 with ∥u∥0 = ρ1.

(ii) There exists e ∈ Bc
ρ2
(0) with I0(e) < 0 and ∥e∥0 > ρ2.

The proof of the above lemma is similar to the one in Lemma 3.1. As in the

previous section, we consider an auxiliary functional Ĩ0 ∈ C1(R × D1,p
a,rad(RN ),R)

given by

Ĩ0(θ, u) =
exp(N − p)θ

p
∥u∥p0 − exp (Nθ)

∫
RN

|x|−bp∗
F (u) dx.

The following properties hold, for all (θ, u) ∈ R×D1,p
a,rad(RN ):

Ĩ0(0, u) = I0(u),

Ĩ0(θ, u) = I0(u(x/ exp(θ)).

We equip the standard product norm

∥(θ, u)∥2R×D1,p
a,rad(RN )

= |θ|p + ∥u∥p0

to R×D1,p
a,rad(RN ). Now we prove that Ĩ0 satisfies the Mountain Pass geometry.

Lemma 4.2. The functional Ĩ0 satisfies the following conditions:

(i) There exist ρ1, ρ2 > 0 such that

Ĩ0(θ, u) ≥ ρ2 with ∥(θ, u)∥R×D1,p
a,rad(RN ) = ρ1.

(ii) There exists ẽ ∈ Bc
ρ2
(0) with Ĩ(ẽ) < 0 and ∥ẽ∥R×D1,p

a,rad(RN ) > ρ2.

Proof. Item (i) follows by using the same argument as in Lemma 3.1. For item (ii)
it is sufficient to take ẽ = (0, e). □

In what follows, we define the Mountain Pass level c̃0 for Ĩ0 by

c̃0 = inf
η∈Γ

max
t∈[0,1]

Ĩ0(η(t)) > 0,

Γ̃ := {η ∈ C([0, 1],R×D1,p
a,rad(R

N )) : η(0) = 0, Ĩ0(η(1)) < 0}.
Note that c̃∗ ≥ ρ2.

Lemma 4.3. The Mountain Pass levels of I0 and Ĩ0 coincide, namely c0 = c̃0.



10 G. M. FIGUEIREDO, G. KIAMETIS EJDE-2024/44

The proof of the above lemma is the same as that of Lemma 3.5. The proof
ofthe next lemma is the same proof of [18, Lemma 4.3].

Lemma 4.4. Let ϵ > 0. Suppose that η̃ ∈ Γ̃0 satisfies

max
t∈[0,1]

Ĩ0(η̃) ≤ c0 + ϵ,

then, there exists (θ, u) ∈ R×D1,p
a,rad(RN ) such that

• distR×D1,p
a,rad(RN ), η̃([0, 1])) ≤ 2

√
ϵ;

• Ĩ0(θ, u) ∈ [c0 − ϵ, c0 + ϵ];

• ∥DĨ0(θ, u)∥R×(D1,p
a,rad(RN )∗) ≤ 2

√
ϵ.

The proof of the next lemma is the same proof of Lemma 3.7.

Lemma 4.5. There exists a sequence ((θn, un)) ⊂ R × D1,p
a,rad(RN ) such that, as

n → ∞, we obtain

• θn → 0;

• Ĩ0(θn, un) → c0;

• ∂θ Ĩ0(θn, un) → 0;

• ∂uĨ0(θn, un) → 0 strongly in (D1,p
a,rad(RN ))∗.

Proof of Theorem 1.2. By Lemma 4.5, there exists a sequence ((θn, un)) ⊂ R ×
D1,p

a,rad(RN ) such that

exp(N − p)θn
p

∥un∥p0 − exp (Nθn)

∫
RN

|x|−bp∗
F (un) dx = c0 + on(1); (4.1)

(N − p)
exp(N − p)θn

p
∥un∥p0 −N exp (Nθn)

∫
RN

|x|−bp∗
F (un) dx = on(1); (4.2)

exp((N − p)θn)∥un∥p0 − exp (Nθn)

∫
RN

|x|−bp∗
f(un)un dx = on(1)∥un∥0. (4.3)

From (4.1) and (4.2) and since N > p, we have

exp(N − p)θn∥un∥p0 = Nc∗ + on(1). (4.4)

Since θn → 0, we have that (un) is bounded in D1,p
a,rad(RN ) and Lp∗

b (RN ). Hence,

there exists u ∈ D1,p
a,rad(RN ) such that, up to a subsequence, un ⇀ u in D1,p

a,rad(RN ).

From Lemma 4.5, for all v ∈ D1,p
a,rad(RN ), we have ∂uĨ0(θn, un)v = on(1); that is,

exp((N−p)θn)

∫
RN

|x|−ap|∇un|p−2unv dx−exp (Nθn)

∫
RN

|x|−bp∗
f(un)v dx = on(1).

Since θn → 0 in R and from weak convergence, for all v ∈ D1,p
a,rad(RN ), we obtain∫

RN

|x|−ap|∇u|p−2uv dx−
∫
RN

|x|−bp∗
f(u)v dx = 0,

showing that I ′0(u)v = 0, for all v ∈ D1,p
a,rad(RN ), that is u is a critical point of I0.

We are going to show that u is not trivial. Suppose that u = 0. From f1) there
exist ϵ > 0 and Cϵ > 0 such that∫

RN

|x|−bp∗
f(un)un dx ≤ ϵ

∫
RN

|x|−bp∗
|un|p

∗
dx+ Cϵ

∫
RN

|x|−bp∗
|un|q dx.



EJDE-2024/44 CAFFARELLI-KOHN-NIRENBERG TYPE PROBLEMS 11

Since (un) is bounded in D1,p
a,rad(RN ) and since D1,p

a,rad(RN ) ↪→ Lq
b(RN ) is compact,

we conclude that ∫
RN

|x|−bp∗
f(un)un dx = on(1).

This limit combined together with the limit ∂uĨ0(θn, un)un = on(1) allows to deduce

that un → 0 in D1,p
a,rad(RN ). Hence, Ĩ0(θn, un) → 0 = c0, which is absurd. Thus, u

is a nontrivial critical point of I0 in D1,p
a,rad(RN ). Finally, u is a nontrivial critical

point of I0 in D1,p(RN ) using the Principle of Symmetric Criticality [21] or [25,
Theorem 1.28]. □

Asknowledgments. Giovany M. Figueiredo and George D. F. L. Kiametis were
partially supported by CNPq, Capes and FapDF - Brazil.

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Giovany M. Figueiredo

Departamento de Matemática, Universidade de Braśılia - UnB, 70910-900, Braśılia - DF,
Brazil

Email address: giovany@unb.br

George D. F. L. Kiametis

Departamento de Matemática, Universidade de Braśılia - UnB, 70910-900, Braśılia - DF,

Brazil
Email address: georgekiametis@gmail.com


	1. Introduction
	2. Variational framework
	3. Existence of solutions for the positive-mass case
	3.1. Proof of Theorem ??

	4. Existence of solution for zero-mass case
	Asknowledgments

	References