Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 54, pp. 1–11.

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

DOI: 10.58997/ejde.2023.54

EXISTENCE OF SOLUTIONS FOR SINGULAR ELLIPTIC

PROBLEMS WITH SINGULAR NONLINEARITIES AND

CRITICAL CAFFARELLI-KOHN-NIRENBERG EXPONENT

MOHAMMED EL MOKHTAR OULD EL MOKHTAR

Abstract. In this article, we consider a singular elliptic problem with singu-

lar nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent. By using

variational methods and Palais-Smale condition, we show the existence of at
least two nontrivial solutions. The result depends crucially on the parameters

a, b,N, β, γ, λ, µ.

1. Introduction

In this article, we consider the existenceof multiple nontrivial nonnegative solu-
tions of the problem

−div
( ∇u
|x|2a

)
− µ u

|x|2(a+1)
= h(x)

|u|2∗−2u

|x|2∗b
+

λ

|x|β |u|γ
in Ω, x 6= 0

u = 0 x ∈ ∂Ω

(1.1)

where Ω is a smooth bounded domain in RN , N ≥ 3, −∞ < a < N−2
2 , a ≤ b < a+1,

0 ≤ β < N
2∗+1 (2∗ + γ − 1), 0 < γ < 1, 2∗ = 2N

N−2+2(b−a) is the critical Caffarelli-

Kohn-Nirenberg exponent, −∞ < µ < µ̄a = [N−2(a+1)
2 ]2, λ is a real parameter and

h is a bounded positive function on RN .
In recent years, people have paid much attention to the singular elliptic problem

−∆u− µ|x|−2u = h(x)|u|p−2u+ λuin Ω

u = 0 on ∂Ω,
(1.2)

where Ω is a smooth bounded domain in RN (N ≥ 3), 0 ∈ Ω, λ > 0, 0 ≤ µ <
µ̄0 := (N − 2)2/4 and 2∗ = 2N/(N − 2) is the critical Sobolev exponent, see [6, 7]
and references therein. Ali and Iaia [1] studied the existence and nonexistence for
singular sublinear problems on exterior domains when µ = 0. Some results are
already available for (1.1). Wang and Zhou [13] proved that there exist at least
two solutions for (1.1) with a = 0, 0 < µ ≤ µ̄0 = (N − 2)2/4. Bouchekif and
Matallah [4] showed the existence of two solutions of (1.1) under certain conditions

2020 Mathematics Subject Classification. 35J66, 35J55, 35B40.
Key words and phrases. Singular nonlinearity; critical Caffarelli-Kohn-Nirenberg exponent;

variational method; Palais-Smale condition.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted July 6, 2023. Published August 31, 2023.

1



2 M. E. O. EL MOKHTAR EJDE-2023/54

on a weighted function h, when 0 < µ ≤ µ̄a, λ ∈ (0,Λ∗), −∞ < a < (N − 2)/2 and
a ≤ b < a+ 1, with Λ∗ a positive constant.

The regular problem corresponding to a = b = µ = 0 was considered on a regular
bounded domain Ω by Tarantello [10]. She proved that, with a nonhomogeneous
term f ∈ H−1(Ω), the dual of H1

0 (Ω), not identically zero and satisfying a suitable
condition, the problem considered admits two distinct solutions.

Before formulating our results, we give some definitions and notation. We denote
by Hµ = Hµ(Ω), the closure of C∞0 (Ω\{0}) with respect to the norms

‖u‖0 =
(∫

Ω

|x|−2a|∇u|2 dx
)1/2

and

‖u‖µ =
(∫

Ω

(|x|−2a|∇u|2 − µ|x|−2(a+1)|u|2) dx
)1/2

for −∞ < µ < µ̄a.

From the weighted Hardy inequality [6] that is∫
Ω

|x|−2(a+1)u2 dx ≤ 1

µ̄a

∫
Ω

|x|−2a|∇u|2 dx (1.3)

it is easy to see that the norm ‖u‖µ is equivalent to ‖u‖0. More explicitly, we have(
1− (

√
µ̄a − a)−2µ+

)1/2

‖u‖0 ≤ ‖u‖µ ≤
(

1− (
√
µ̄a − a)−2µ−

)1/2

‖u‖0,

with µ+ = max(µ, 0) and µ− = min(µ, 0) for all u ∈ Hµ.
Next wee list here a few integral inequalities. It is clear that degeneracy and

singularity occur in problem (p1). In these situations, the classical methods do not
directly apply so that the existence results may become a delicate matter that is
closely related to some phenomena due to the degenerate (or singular) character
of the differential equation. The starting point of the variational approach is the
following Caffarelli-Kohn-Nirenberg inequality in [5] which states there is a positive
constant Ca,b such that(∫

Ω

|x|−2∗b|u|2∗ dx
)1/2∗

≤ Ca,b
(∫

Ω

|x|−2a|∇u|2 dx
)1/2

(1.4)

for any u ∈ Hµ where −∞ < a < N−2
2 , a ≤ b < a+ 1, 2∗ = 2N

N−2+2(b−a) .

We consider the approximation equation

−div(
∇u
|x|2a

)− µ u

|x|2(a+1)
= h(x)

|u|2∗−2u

|x|2∗b
+

λ

|x|β(u+ θ)γ
in Ω\{0}

u = 0 x ∈ ∂Ω,

(1.5)

for any θ > 0.
We look for solutions of problem (1.5) by finding critical points of C1-energy

functional defined in [10],

Jλ(u) := (1/2)‖u‖2µ − (1/2∗)

∫
Ω

h(x)|x|−2∗b|u|2∗ dx

− λ

1− γ

∫
Ω

(u+ + θ)1−γ − θ1−γ

|x|β
dx .



EJDE-2023/54 ELLIPTIC PROBLEM WITH SINGULAR NONLINEARITIES 3

A point u ∈ Hµ is a weak solution of (1.5) if it satisfies

〈J ′λ(u), ϕ〉 :=

∫
Ω

(∇u∇ϕ
|x|2a

− µ uϕ

|x|2(a+1)

)
dx−

∫
Ω

h(x)
|u|2∗−1uϕ

|x|2∗b
dx

− λ
∫

Ω

ϕ

(u+ + θ)γ |x|β
dx = 0, for all ϕ ∈ Hµ.

Here 〈·, ·〉 denotes the product in the duality H′µ, of Hµ. We consider the following
assumptions:

(A1) h ∈ L∞(Ω), ess lim|x|→0 h(x) = h0 ∈ (0,∞) and h(x) ≥ h0 a.e. in Ω;

(A2) (a, µ) ∈ (−1, 0)× (0, µ̄a − b) ∪ [0, N−2
2 )× (a(a−N + 2), µ̄a − b);

(A3) (a, µ) ∈ [0, N−2
2 )× [0, µ̄a);

(A4) N > 2(|b|+ 1) and (δ1),
(A5) N ≥ 3 and (A2) holds.

Xuan et al. [12] proved that when (A2) holds for each ε > 0, the function

yε = C0ε
2

2∗−2

[
ε

2
√
µ̄a−µ√

µ̄a−µ−b |x|
2∗−2

2 (
√
µ̄a−
√
µ̄a−µ) + |x|

2∗−2
2 (
√
µ̄a+
√
µ̄a−µ)

] −2
2∗−2

, (1.6)

with a suitable positive constant C0, is a weak solution of

−div÷(
∇u
|x|2a

)
− µ u

|x|2(a+1)
=
|u|2∗−2u

|x|2∗b
in Ω\{0}.

Furthermore, ∫
Ω

( |∇yε|2
|x|2a

− µ y2
ε

|x|2(a+1)

)
dx =

∫
Ω

|yε|2∗
|x|2∗b

dx.

In addition, we have that

Dµ = inf
u∈Hµ\{0}

E(u) = E(yε),

is the best constant with

E(u) :=

∫
Ω

(
|x|−2a|∇u|2 − µ|x|−2(a+1)|u|2

)
dx

(
∫

Ω
|x|−2∗b|u|2∗ dx)2/2∗

.

Kang et al. [9] obtained that, when (A3) holds for each ε > 0, the function

vε = [2∗2ε(µ̄a − µ)]
1

2∗−2

[
|x|(−

√
µ̄a+
√
µ̄a−µ)(ε+ |x|(2∗−2)

√
µ̄a−µ)

] −2
2∗−2

, (1.7)

with a suitable positive constant C0, is a weak solution of

−div
( ∇u
|x|2a

)
− µ u

|x|2(a+1)
=
|u|2∗−2u

|x|2∗b
, in Ω\{0},

and satisfies ∫
Ω

(
|∇vε|2

|x|2a
− µ v2

ε

|x|2(a+1)
) dx =

∫
Ω

|vε|2∗
|x|2∗b

dx.

Also we have that

Gµ = inf
u∈Hµ\{0}

E(u) = E(vε)

is the best constant.
In this work we prove the existence of at least two critical points of Jλ. The

first is found by the Ekeland Variational Principle [8] with negative energy and



4 M. E. O. EL MOKHTAR EJDE-2023/54

the second by Mountain Pass Theorem without Palais Smale conditions [2] with
positive energy. Now, we define

wε :=

{
yε if (A2) holds

vε if (A3) holds,
Sµ :=

{
Dµ if (A2) holds

Gµ if (A3) holds.

Now we can state our main result.

Theorem 1.1. Assume that −∞ < a < N−2
2 , a ≤ b < a + 1, 0 ≤ β < N

2∗+1 (2∗ +

γ − 1), −∞ < µ < µ̄a = [N−2(a+1)
2 ]2, 0 < γ < 1, (A1) holds and (A4) or (A5) are

satisfied. Then there exists Λ0 > 0 such that for each 0 < λ < Λ0, problem (1.1)
has at least two nontrivial solutions.

This article is organized as follows. In Section 2, we give some preliminaries.
Section 3 is devoted to the proof of Theorems 1.1.

2. Preliminaries

Definition 2.1. Let c a real number, E a Banach space, and I a function in
C1(E,R).

(i) (un)n is a Palais-Smale sequence at level c (in short (PS)c) in E for I if

I(un) = c+ on(1) and I ′(un) = on(1),

where on(1) tends to 0 as n goes at infinity.
(ii) We say that I satisfies the (PS)c condition if any (PS)c sequence in E for

I has a convergent subsequence.

Lemma 2.2. Assume that −∞ < a < N−2
2 , a ≤ b < a+ 1, 0 ≤ β < N

2∗+1 (2∗ + γ),

−∞ < µ < µ̄a = [N−2(a+1)
2 ]2 and let (un) ⊂ Hµ be a Palais-Smale sequence ((PS)c

in short) of Jλ, i.e.,

Jλ(un)→ c and J ′λ(un)→ 0 in H′µ( the dual of Hµ) as n→∞ (2.1)

for some c ∈ R. Then un ⇀ u in Hµ and J ′λ(u) = 0.

Proof. Let R0 > 0 such that Ω ⊂ B(0, R0) = {x ∈ RN : |x| < R0}. Let r = |x|. By
Hölder’s inequality which states that∫

Ω

|fg| dx ≤
(∫

Ω

|f |p dx
)1/p(∫

Ω

|g|q dx
)1/q

,

for f ∈ Lp(Ω) and g ∈ Lq(Ω) with 1
p + 1

q = 1, and from (1.3), we obtain∫
Ω

(u+)1−γ

rβ
dx ≤

∫
Ω

|u|1−γ

rβ
dx

=

∫
Ω

( |u|1−γ
rb(1−γ)

)
rb(1−γ)−β dx

≤ Big(

∫
Ω

|u|2∗
r2∗b

) 1−γ
2∗
(∫

Ω

r(b(1−γ)−β) 2∗
2∗+γ−1

) 2∗+γ−1
2∗

≤
(
C

2∗
2

a,b‖u‖
2∗
µ

) 1−γ
2∗
〈
σN

∫ R0

0

rN−1+(b(1−γ)−β) 2∗
2∗+γ−1 dr

〉 2∗+γ−1
2∗

≤
(
C

2∗
2

a,b‖u‖
2∗
µ

) 1−γ
2∗
(
σN

R
N+[b(1−γ)−β] 2∗

2∗+γ−1

0

N + [b(1− γ)− β] 2∗
2∗+γ−1

) 2∗+γ−1
2∗



EJDE-2023/54 ELLIPTIC PROBLEM WITH SINGULAR NONLINEARITIES 5

≤ AC
1−γ

2

a,b ‖u‖
1−γ
µ ,

where σN = 2π
N
2

Γ(N2 )
is the area of the (N − 1)-dimensional unit sphere, Ca,b defined

in (1.4), and

|f | = |u|
2∗

r2∗b
, |g| = r[b(1−γ)−β] 2∗

2∗+γ−1 ,
1

p
=

1− γ
2∗

,
1

q
=

2∗ + γ − 1

2∗
,

A =
[ 2π

N
2 (2∗ + γ − 1)

NΓ(N2 )[(2∗ + γ − 1)− β(γ + 1)]

] 2∗+γ−1
2∗+1

R
N

2∗+1 (2∗+γ−1)−β
0 > 0 .

From (2.1), we have

Jλ(un) := (1/2)‖un‖2µ − (1/2∗)

∫
Ω

h(x)|x|−2∗b|un|2∗ dx

− λ

1− γ

∫
Ω

(u+
n + θ)1−γ − θ1−γ

|x|β
dx

= c+ on(1)

and

〈J ′λ(un), un〉 = ‖un‖2µ −
∫

Ω

h(x)|x|−2∗b|un|2∗ dx− λ
∫

Ω

(u+
n + θ)1−γ − θ1−γ

|x|β
dx

= on(1), for n large,

where on(1) denotes on(1)→ 0 as n→∞. Then

c+ on(1) = Jλ(un)− 1

2∗
〈J ′λ(un), un〉

≥
(1

2
− 1

2∗

)
‖un‖2µ − λ

( 1

1− γ
− 1

2∗

) ∫
Ω

(u+
n + θ)1−γ − θ1−γ

|x|β
dx

≥
(1

2
− 1

2∗

)
‖un‖2µ − λ

( 1

1− γ
− 1

2∗

)
A‖un‖1−γµ C

−(1−γ)
p

a,b

and so {un} is bounded in Hµ(Ω). Going if necessary to a subsequence there exists
u ∈ Hµ(Ω) such that

un ⇀ u, in Hµ(Ω),

un ⇀ u, in Lq(Ω), (1 ≤ q < 2∗)

un ⇀ u, in L2∗(Ω),

un → u, a.e. on Ω,

there exists ϕ ∈ Lq(Ω) (1 ≤ q < 2∗) such that |un| and |u| ≤ |ϕ|, a.e. in Ω, where
the last conclusion is from [14, Lemma A.1]. From (1.4), we obtain

| un

|x|β |u+
n + θ|γ

| ≤ |un|
|x|βθγ

≤ |ϕ|
|x|βθγ

.

Since 1 < 2∗(N−β)+N
2(N−β) < 2∗ we have ϕ ∈ L

2∗(N−β)+N
2(N−β) (Ω), then

1

θγ

∫
Ω

ϕ

|x|β
dx

≤ 1

θγ
|
∫

Ω

|ϕ| 1

|x|β
dx|



6 M. E. O. EL MOKHTAR EJDE-2023/54

≤ 1

θγ

(∫
Ω

|ϕ|
2∗(N−β)+N

2(N−β) dx
) 2(N−β)

2∗(N−β)+N
(∫

B(0,R)

(
1

|x|β
)

2∗(N−β)+N
2∗(N−β)−N+2β dx

) 2∗(N−β)−N+2β
2∗(N−β)+N

≤ 1

θγ

(∫
Ω

|ϕ|
2∗(N−β)+N

2(N−β) dx
) 2(N−β)

2∗(N−β)+N

×
( 2π

N
2

Γ(N2 )

∫ R

0

rN−1− β(2∗(N−β)+N)
2∗(N−β)−N+2β dr

) 2∗(N−β)−N+2β
2∗(N−β)+N

≤ 1

θγ

( 2π
N
2

(N − β(2∗(N−β)+N)
2∗(N−β)−N+2β )Γ(N2 )

) 2∗(N−β)−N+2β
2∗(N−β)+N

(∫
Ω

|ϕ|
2∗(N−β)+N

2(N−β) dx
) 2(N−β)

2∗(N−β)+N

×
(
RN−

β(2∗(N−β)+N)
2∗(N−β)−N+2β

) 2∗(N−β)−N+2β
2∗(N−β)+N

≤ C(π,R)

θγ

(∫
Ω

|ϕ|
2∗(N−β)+N

2(N−β) dx
) 2(N−β)

2∗(N−β)+N

,

with

C(π,R)

=
( 2π

N
2

(N − β(2∗(N−β)+N)
2∗(N−β)−N+2β )Γ(N2 )

) 2∗(N−β)−N+2β
2∗(N−β)+N

(
RN−

β(2∗(N−β)+N)
2∗(N−β)−N+2β

) 2∗(N−β)−N+2β
2∗(N−β)+N

and

N − β[2∗(N − β) +N ]

2∗(N − β)−N + 2β
=

(N − β)[2∗(N − β)−N ]

2∗(N − β)−N + 2β
> 0.

From the cacluations above, we know that ϕ
θγ |x|β ∈ L1(Ω). Thus, applying the

Dominated Convergence Theorem, one has

lim
n→+∞

∫
Ω

un

|x|β |u+
n + θ|γ

dx =

∫
Ω

u

|x|β |u+ + θ|γ
dx.

Consequently, J ′λ(u) = 0. �

Lemma 2.3. Assume that −∞ < a < N−2
2 , a ≤ b < a+1, 0 ≤ β < N

2∗+1 (2∗+γ−1),

−∞ < µ < µ̄a = [N−2(a+1)
2 ]2 and let (un) ⊂ Hµ be a Palais-Smale sequence (PS)c

of Jλ for some c ∈ R. Then, un ⇀ u in Hµ and either

un → u or c ≥ Jλ(u) +
(1

2
− 1

2∗

)
(h
−2/2∗
0 Sµ)2∗/(2∗−2).

Proof. We know that (un) is bounded in Hµ. Up to a subsequence if necessary, we
have that

un ⇀ u in Hµ
un(x)→ u(x) a.e. in Ω.

We denote vn = un − u. Then vn ⇀ 0. As in Brézis and Lieb [3], we have

lim
n→∞

∫
Ω

h(x)
(
|x|
−2∗b |un|2∗ − |x|

−2∗b |un − u|2∗
)
dx =

∫
Ω

h(x)|x|
−2∗b |u|2∗ dx (2.2)

and

lim
n→∞

∫
Ω

|un + θ|1−γ

|x|β
dx =

∫
Ω

|u+
0 + θ|1−γ

|x|β
dx. (2.3)



EJDE-2023/54 ELLIPTIC PROBLEM WITH SINGULAR NONLINEARITIES 7

On the other hand, we can prove that

lim
n→∞

∫
Ω

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx = 0.

Fix ε > 0. By assumption (A1), there exists rε > 0 such that

|h(x)− h0| = h(x)− h0 < ε for a.e. x ∈ Ω \B(0, rε).

Next we have∫
Ω

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx

=

∫
Ω\B(0,rε)

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx+

∫
B(0,rε)

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx

≤ ε
∫

Ω\B(0,rε)

|vn|
|x|2∗b

2∗

dx+ (|h|∞ − h0)

∫
B(0,rε)

|vn|
|x|2∗b

2∗

dx .

Since vn ⇀ 0 in Hµ, the Caffarelli-Kohn-Nirenberg inequality implies that {vn} is
bounded in L2∗ . Moreover by un ⇀ 0 in Hµ it follow that vn ⇀ 0 in (Hµ \ {0}).
The above relations yield

lim sup
n→∞

∫
Ω

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx ≤ Cε,

for some constant c > 0 independent of n and ε. Since ε > 0 was arbitrarily chosen,
we conclude that

lim
n→∞

∫
Ω

(h(x)− h0)
|vn|
|x|2∗b

2∗

dx = 0. (2.4)

From (2.2), (2.3) and (2.4) we deduce that

Jλ(un) = Jλ(u) + (1/2)‖vn‖2µ − (h0/2∗)

∫
Ω

|x|
−2∗b |vn|2∗ + on(1) (2.5)

and

0← 〈J ′λ(un), un〉 = ‖vn‖2µ − h0

∫
Ω

|x|−2∗b|vn|2∗ + on(1).

Then we have

lim
n→∞

‖vn‖2µ = h0 lim
n→∞

∫
Ω

|x|−2∗b|vn|2∗ = l ≥ 0.

If l = 0 then ‖un − u‖µ → 0 as n→∞.
Otherwise if l > 0, by the definition of Sµ, we have

l ≥ Sµ(lh−1
0 )2/2∗ ,

so that

l ≥
(
h
−2/2∗
0 Sµ

)2∗/(2∗−2)
.

Thus we obtain

c = Jλ(u) +
(1

2
− 1

2∗

)
l taking limits in (2.5)

≥ Jλ(u) +
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2)
.

�



8 M. E. O. EL MOKHTAR EJDE-2023/54

3. Proof of Theorem 1.1

The proof is given in two parts.

3.1. Existence of a local minimizer. We prove that there exists

λ1 =
2∗ − 2

2∗2
(|h|∞Sµ)

−(1+γ)
2∗−2

1− γ
A

C
1−γ

2

a,b > 0,

with

A =
[ 2π

N
2 (2∗ + γ − 1)

NΓ(N2 )[(2∗ + γ − 1)− β(γ + 1)]

] 2∗+γ−1
2∗+1

R
N

2∗+1 (2∗+γ−1)−β
0 > 0

such that for any λ ∈ (0, λ1), Jλ achieves a local minimizer. First, we establish the
following result.

Proposition 3.1. Suppose that −∞ < a < N−2
2 , a ≤ b < a + 1, 0 ≤ β <

N
2∗+1 (2∗ + γ − 1), −∞ < µ < µ̄a = [N−2(a+1)

2 ]2, 0 < γ < 1, (A1) hold and (A4) or

(A5) hold. Then there exist positive reals λ1, % and δ such that for all λ ∈ (0, λ1),
we have

Jλ(u) ≥ δ > 0 for ‖u‖µ = % (3.1)

Proof. By the Holder inequality and the definition of Sµ, for all u ∈ Hµ\{0} we
have

Jλ(u) := (1/2)‖u‖2µ − (1/2∗)

∫
Ω

h(x)|x|−2∗b|u|2∗ dx

− λ

1− γ

∫
Ω

(u+ + θ)1−γ − θ1−γ

|x|β
dx

≥ (1/2)‖u‖2µ − (|h|∞/2∗)Sµ‖u‖2∗µ −
λA

1− γ
‖u‖1−γµ C

−(1−γ)
p

a,b .

Taking % = ‖u‖µ, then there exist % > 0 small enough and a positive constant λ1

such that

Jλ(u) ≥ δ > 0 for ‖u‖µ = % and λ ∈ (0, λ1). (3.2)

This completes proof. �

Since
∫

Ω
|u|1−γ
|x|β dx > 0 and 0 < γ < 1, it follows that for t > 0 small,

Jλ(tφ) := (t2/2)‖φ‖2 − (t2∗/2∗)

∫
Ω

h(x)|x|−2∗b|φ|2∗ dx

− λt1−γ

1− γ

∫
Ω

|u|1−γ

|x|β
dx < 0.

(3.3)

We also assume that t is so small enough such that ‖tφ‖µ < %. Thus, we have

c1 = inf{Jλ(u) : u ∈ B%} < 0, where B% = {u ∈ Hµ, ‖u‖µ ≤ %}. (3.4)

Using Ekeland’s variational principle, for the complete metric space B% with respect

to the norm of Hµ, there exists a (PC)c1 sequence (un) ⊂ B% such that un ⇀ u1

for some u1 with ‖u1‖µ ≤ %.
Now, we claim that un → u1 in Hµ, if not, by Lemma 2.3, we have

c1 ≥ Jλ(u1) +
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2)



EJDE-2023/54 ELLIPTIC PROBLEM WITH SINGULAR NONLINEARITIES 9

≥ c1 +
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2)
> c1,

which is a contradiction. Thus un → u1 in Hµ.
Then we obtain a critical point u1 of Jλ for all λ ∈ (0, λ1) satisfying c1 =

Jλ(u1) < 0. Thus u1 is a nontrivial solution of (1.5) with negative energy.

3.2. Existence of mountain pass type solution. We use the mountain pass
theorem without Palais-Smale conditions to prove the existence of a nontrivial
solution with positive energy. For this, we need the following Lemma. Set

c∗λ = −ψ2

(1 + γ

2

)[ψ2(1− γ)

2ψ1

] 1−γ
1+γ

+
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2)

,

with

ψ1 =
(1

2
− 1

2∗

)
and ψ2 = λ

( 1

1− γ
− 1

2∗

)
AC

−(1−γ)
2

a,b .

Lemma 3.2. Let λ∗ > 0 such that c∗λ > 0 for all λ ∈ (0, λ∗). Then, there exist
Λ ∈ (0, λ∗) and ϕε ∈ Hµ for ε > 0 such that

sup
t≥0

Jλ(tϕε) < c∗λ, for all λ ∈ (0,Λ).

Proof. Let

ϕε(x) = ωε(x) =

{
yε if (A2) holds

vε if (A3) holds,
(3.5)

where yε, vε are defined in (1.6) and (1.7) respectively.
Now, we consider the functions

f(t) = Jλ(tϕε),

f̃(t) = (t2/2)‖ϕε(x)‖2µ − (t2∗/2∗)h0

∫
Ω

|x|−2∗b|ϕε(x)|2∗ dx.

Then, for all λ ∈ (0, λ∗) we obtain that 0 = f(0) < c∗λ. By the continuity of
f(t), there exists t1 a sufficiently small positive quantity such that f(t) < c∗λ for all
t ∈ (0, t1). On the other hand, we have

max
t≥0

f̃(t) =
(1

2
− 1

2∗

)
(h
−2/2∗
0 Sµ)2∗/(2∗−2),

then, we obtain

sup
t≥0

Jλ(tϕε) <
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2) − λ t
1−γ
1

1− γ

∫
Ω

|ϕε|1−γ

|x|β
dx .

Taking λ > 0 such that

λ
t1−γ1

1− γ

∫
Ω

|ϕε|1−γ

|x|β
dx > (1/2∗2)(2∗ − 1)(2∗ − 2)−1/2λ2AC

−(1−γ)
2

a,b ‖tϕε‖1−γµ ,

we obtain

0 < λ < Γ∗,

where

Γ∗ := 2∗2(2∗ − 2)1/2(2∗ − 1)−1)
t1−γ1

1− γ

∫
Ω

|ϕε|1−γ

|x|β
dx.



10 M. E. O. EL MOKHTAR EJDE-2023/54

Setting Λ = min{λ∗,Γ∗} we deduce that

sup
t≥0

Jλ(tϕε) < c∗λ for all λ ∈ (0,Λ).

This completes the proof. �

Proof of Theorem 1.1. Since limt→∞ Jλ(tϕε) = −∞, we can choose T > 0 large
enough such that Jλ(Tϕε) < 0. From Proposition 3.1, we have Jλ|∂B% ≥ δ > 0 for
all λ ∈ (0, λ1). By the mountain pass theorem without the Palais-Smale condition
[2], there exists a (PC)c2 sequence (un) in Hµ which is characterized by

c2 = inf
θ∈Γ

max
t∈[0,1]

Jλ(θ(t)) > δ > 0 with δ independent of θ

with

Γ = {θ ∈ C([0, 1],Hµ), θ(0) = 0, θ(1) = Tϕε}.
Then, (un) has a subsequence, still denoted by (un) such that un ⇀ u2 in Hµ. By
Lemma (2.3), if un does not converge to u2, we obtain

c2 ≥ Jλ(u2) +
(1

2
− 1

2∗

)(
h
−2/2∗
0 Sµ

)2∗/(2∗−2) ≥ c∗λ,

what contradicts the fact that, by Lemma 3.2, we have

sup
t≥0

Jλ(tϕε) < c∗λ,

for all λ ∈ (0,Λ). Thus un → u2 in Hµ. Thus, we obtain a critical point u2 of Jλ
for all λ ∈ (0, λ1) with

Λ0 := min{λ1,Λ}
satisfying Jλ(u2) > 0. Now we prove that u1 6= u2.

We have u1 is the first solution of (1.1) where

Jλ(u1)|θ=0 = inf{Jλ(u)|θ=0 : u ∈ B%} = c1 < 0.

On the other hand, for θ ∈ (0, 1), (1.5) has at least a mountain pass solution {uθ}
with Jλ(uθ) > δ > 0. Thus, there exists {θn} ⊂ (0, 1) with θn → 0 as n→∞, such
that (uθn) is a sequence mountain pass solutions of (1.5) with

Jλ(uθ) > δ > 0,by Proposition 1,

then, limn→∞ uθn = u2 is the second solution of (1.1) and

Jλ(u2)|θ=0 = lim
n→∞

Jλ(uθn) ≥ δ > 0.

So,

Jλ(u1)|θ=0 < 0 < Jλ(u2)|θ=0,

which implies that u1 6= u2. �

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plications 24, Birkhäuser Boston, Boston, MA, 1996.

Mohammed El Mokhtar Ould El Mokhtar
Department of Mathematics, College of Science, Qassim University, BO 6644, Buraidah:

51452, Kingdom of Saudi Arabia

Email address: med.mokhtar66@yahoo.fr, M.labdi@qu.edu.sa


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem ??
	3.1. Existence of a local minimizer
	3.2. Existence of mountain pass type solution

	References