Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 10, pp. 1–19.

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

MULTIPLICITY RESULTS OF NONLOCAL SINGULAR PDES

WITH CRITICAL SOBOLEV-HARDY EXPONENT

ADEL DAOUES, AMANI HAMMAMI, KAMEL SAOUDI

Abstract. In this article we study a nonlocal equation involving singular and

critical Hardy-Sobolev non-linearities,

(−∆p)su− µ
|u|p−2u

|x|sp
= λu−α +

|u|p∗s(t)−2u

|x|t
, in Ω,

u > 0, in Ω,

u = 0, in RN \ Ω,

where Ω ⊂ RN is a bounded domain with Lipschitz boundary and (−∆p)s is
the fractional p-Laplacian operator. We combine some variational techniques

with a perturbation method to show the existence of multiple solutions.

1. Introduction

In this work, we consider the singular critical nonlocal problem with critical
Hardy-Sobolev non-linearities,

(−∆p)
su− µ |u|

p−2u

|x|sp
= λu−α +

|u|p∗s(t)−2u

|x|t
, in Ω,

u > 0, in Ω,

u = 0, in RN \ Ω,

(1.1)

where Ω ⊂ RN is a bounded domain with Lipschitz boundary, 0 < s < 1, λ is
a positive parameter, 0 ≤ µ < µ0 is the sharp constant of the fractional Hardy
Sobolev in RN , 0 < t < sp < N , 0 < α < 1 < p < p∗s(t) where p∗s = Np

N−sp and

p∗s(t) = p(N−t)
N−sp are the fractional critical Sobolev and Hardy Sobolev exponents

respectively. The fractional p-laplacian non-linear nonlocal operator defined for
s ∈ (0, 1) is defined by

(−∆p)
su(x) = 2 lim

ε↘0

∫
RN\Bε

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
dy, for all x ∈ RN .

Problems of the type (1.1) play an important role in many field of sciences
such as: optimization, electromagnetism, astronomy, water waves, fluid dynamics,

2020 Mathematics Subject Classification. 35R11, 35J75, 35J60, 46E35.
Key words and phrases. Nonlocal elliptic problem; singular non-linearity; variational method;

Sobolev and Hardy non-linearities; perturbation method; multiple positive solutions.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted November 22, 2022. Published January 26, 2023.

1



2 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

probability theory, phase transitions. etc. For further details on applications, we
refer the readers to [2, 3, 22, 23] and references therein.

Before stating our results, let us briefly recall some of the literature concerning
problems with Sobolev and Hardy nonlinearities. In the previous years, the study
of fractional elliptic equations involving singular nonlinearity attracted lot of at-
tention; see for example [10, 11, 12, 13, 15, 16, 19] and the references therein. The
following problem has been study in several works,

(−∆p)
su = λa(x)u−α +Mf(x, u), in Ω,

u > 0, in Ω,

u = 0, in RN \ Ω,

(1.2)

where N > ps, M ≥ 0, a : Ω −→ R is a nonnegative bounded function. When
M = 0 and p = 2 (the purely singular problem), Fang [5] prove the existence and
uniqueness of a solution in C2,α(Ω) for 0 < α < 1 in (1.2). In [17], the author
prove a multiplicity result for (1.2) by converting the nonlocal problem to a local
problem. In [7, 14] using a Nehari method combine with fibering map, the authors
established the existence and multiplicity of weak solutions to (1.2). In that sense
the current problem (1.1) is new, not only because of a nonlocal operator and a
singularity, but also because of the Hardy-Sobolev nonlinearities. In a nutshell,
we will prove the existence of multiple solutions to (1.1) for sufficiently small λ, µ.
Now, we state the main result of this paper.

Theorem 1.1. There exist λ∗ such that for every λ ∈ (0, λ∗) problem (1.1) has at
least two positives solutions uλ with Eλ,µ(uλ) < 0, and vλ with Eλ,µ(vλ) > 0.

The outline of this work is as follows. In Section 2 we present notation and basic
results. In Section 3, we prove the existence of a solution which is a local minimizer
in X0 of the functional energy Eλ,µ associated with (1.1). Section 4 is devoted to
study the approximated problem. While, multiplicity of solutions will be presented
in Section 5.

2. Functional framework and main results

This section is devoted to recalling a few definitions, notation, and function
spaces which will be used later. Let Ω ⊂ RN and Q = R2N \ ((RN \Ω)× (RN \Ω)),
then the space (X, ‖ · ‖X) is defined by

X =
{
u : RN → R is measurable, u|Ω ∈ Lp(Ω) and

|u(x)− u(y)|
|x− y|

N+ps
p

∈ Lp(Q)
}

equipped with the Gagliardo norm

‖u‖X = ‖u‖p +
(∫

Q

|u(x)− u(y)|p

|x− y|N+ps
dx dy

)1/p

.

Here ‖u‖p refers to the Lp-norm of u. We further define the space

X0 =
{
u ∈ X : u = 0 a.e. in RN \ Ω

}
equipped with the norm

‖u‖ =
(∫

Q

|u(x)− u(y)|p

|x− y|N+ps
dx dy − µ

∫
Ω

|u|p

|x|sp
dx
)1/p

.



EJDE-2023/10 NONLOCAL SINGULAR PDES 3

The best Sobolev constant is defined as

S = inf
u∈X0\{0}

( ∫
Q
|u(x)−u(y)|p
|x−y|N+ps dx dy − µ

∫
Ω
|u|p
|x|sp dx

)
( ∫

Ω
|u|p∗s (t)
|x|t dx

)p/p∗s(t)
. (2.1)

We now state the following definitions associated with problem (1.1).

Definition 2.1. We say that u ∈ X0 is a weak solution to (1.1), if (i) u > 0,
u−γφ ∈ L1(Ω), and (ii)∫

Q

|u(x)− u(y)|p−2(u(x)− u(y))(φ(x)− φ(y))

|x− y|N+ps
dx dy − µ

∫
Ω

|u|p−2u

|x|sp
φdx

−
∫

Ω

λ

(u+)1−αφ+

∫
Ω

|u|p∗s(t)−2u

|x|t
φdx = 0

for each φ ∈ X0. Here, u+ = max{u, 0} and u− = max{−u, 0}.

Associated with (1.1) we have the functional energy Eλ,µ : X0 → R defined as

Eλ,µ(u) =
1

p

(∫
Q

|u(x)− u(y)|p

|x− y|N+ps
dx dy − µ

∫
Ω

|u|p

|x|sp
dx
)
− λ

1− α

∫
Ω

(u+)1−αdx

− 1

p∗s(t)

∫
Ω

|u|p∗s(t)

|x|t
dx.

(2.2)
Obviously, every critical point of Eλ,µ is a weak solution of the problem (1.1).

We now list the embedding results pertaining to the function space X0 [20, 21].

Lemma 2.2. The following embedding results holds for the space X0.

(1) If Ω has a Lipschitz boundary and N > ps, then the embedding X0 ↪→ Lq(Ω)

for q ∈ [1, p∗s] is continuous and is compact for q ∈ [1, p∗s), where p∗s = Np
N−ps .

(2) If Ω has a Lipschitz boundary and N = ps, then the embedding X0 ↪→ Lq(Ω)
for q ∈ [1,∞) is both continuous and compact.

(3) If Ω has a Lipschitz boundary and N < ps, then the embedding X0 ↪→
C0,β(Ω) where β = sp−N

p is both continuous and compact.

Let us define

|u|p∗s(t) =
(∫

Ω

|u|p∗s(t)

|x|t
dx
)1/p∗s(t)

.

We now recall the fractional Hardy-Sobolev inequality.

Lemma 2.3.

(1) Fractional Hardy inequality [6]: For all u ∈W s,p
0 (RN ), we have

µ0

∫
RN

|u|p

|x|sp
dx ≤

∫
RN
|(−∆)s/pu|p dx. (2.3)

(2) Fractional Hardy Sobolev inequality [9]: Assume 0 ≤ α ≤ sp < N . Then,
there exist positive constants c and C, such that for all u ∈W s,p

0 (RN ),(∫
RN

|u|p∗s(t)

|x|t
dx
)p/p∗s(t)

≤ c
∫
RN
|(−∆)s/pu|p dx. (2.4)



4 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

Moreover, if mu < mu0, then(∫
RN

|u|p∗s(t)

|x|t
dx
)p/p∗s(t)

≤ C
(∫

RN
|(−∆)s/pu|p dx− µ

∫
RN

|u|p

|x|sp
)
dx, (2.5)

for all u ∈W s,p
0 (RN ).

The following embedding results have been proved in [4].

Lemma 2.4. (1) The embedding W s,p
0 (Ω)→ Lq(Ω, dx|x|t ) is continuous for q ∈

[1, p∗s(t)] and compact for q ∈ [1, p∗s(t)).
(2) For p > 1, W s,p

0 (Ω) and Ds,p(RN ) are separable reflexive Banach space
w.r.t the norm [·]s,p.

3. Existence of a weak solution to (1.1)

Besides proving the existence of a weak solution, we show that this solution is a
local minimizer of the associated functional Eλ,µ. Our first result is the following.

Lemma 3.1. There exists λ0 > 0, R0 > 0, and ρ0 > 0 such that Eλ,µ(u) ≥ ρ0 > 0
for all ‖u‖ = R0. Also, we have C = infu∈BR0

Eλ,µ(u) < 0.

Proof. Note that using Hölder’s inequality combined with the fractional Sobolev-
Hardy inequality, we have

Eλ,µ(u) =
1

p
‖u‖pX0

− λ

1− α
‖u‖1−α − C1

p∗s(t)
‖u‖p

∗
s(t)

= ‖u‖1−α
(1

p
‖u‖p+1+α − λ

1− α
C0 −

C1

p∗s(t)
‖u‖p

∗
s(t)+α−1

)
,

where C0, C1 are two constants. Put

f(x) =
1

p
xp+1+α − C1

p∗s(t)
xp
∗
s(t)+α−1 − λ

1− α
C0.

Since 1− α < 1 < p < p∗s(t), we find the existence of a constant

R =
( p∗s(t)(p+ α− 1)

λpC1(p∗s(t) + α− 1

)1/(p∗s(t)−p
> 0

such that f(R) = max
x>0

f(x) > 0. Choosing λ0 = (1−α)f(R)
C0

, we deduce the existence

of a constant δ0 > 0 satisfying Eλ,µ ≥ δ0 > 0 for all λ ∈ (0, λ0). The proof of
Lemma 3.1 is now completed. �

Lemma 3.2. Problem (1.1) admits a positive solution uλ ∈ X0 with Eλ,µ < 0 for
all λ ∈ (0, λ0), where λ0 is defined in Lemma 3.1.

Proof. Let λ0, R0 and ρ0 be as in Lemma 3.1. Since 1− α < 1 < p < p∗s(t), noting
that for all ϕ ∈ X0, ϕ ≥ 0, ϕ 6≡ 0 and r > 1, one has

Eλ,µ(rϕ) =
rp

p
‖u‖p − λr1−α

1− α

∫
Ω

(ϕ+)1−α dx− rp
∗
s(t)

p∗s(t)

∫
Ω

|ϕ|p∗s(t)

|x|t
dx < 0. (3.1)

So, for ‖u‖ sufficiently small, we conclude that

C = inf
u∈BR0

Eλ,µ(u) < 0. (3.2)



EJDE-2023/10 NONLOCAL SINGULAR PDES 5

Hence, by the definition of the infimum (3.2), we guarantee the existence of a
minimizing sequence {un} for C. Therefore, using the reflexivity of X0, there
exists a subsequence, still denoted by un, there exists uλ, such that

un → uλ weakly in X0,

un → uλ strongly in Lk(Ω,
dx

|x|t
) for 1 ≤ k < p∗s(t),

un → uλ pointwise a.e. in Ω.

(3.3)

Thus, from the Brezis-Lieb Lemma [4], one has

|un|
p∗s(t)

p∗s(t) = |uλ|
p∗s(t)

p∗s(t) + |un − uλ|
p∗s(t)

p∗s(t) + o(1), (3.4)

‖un‖p = ‖uλ‖p + ‖un − uλ‖p + o(1). (3.5)

On the other hand, using Hölder inequality and letting n→∞, we obtain∫
Ω

u1−α
n dx ≤

∫
Ω

u1−α
λ dx+

∫
Ω

| un − uλ |1−α dx

≤
∫

Ω

u1−α
λ dx+ C ‖ un − uλ ‖1−αp

=

∫
Ω

u1−α
λ dx+ o(1).

Similarly ∫
Ω

u1−α
λ dx ≤

∫
Ω

u1−α
n dx+

∫
Ω

| un − uλ |1−α dx

≤
∫

Ω

u1−α
n dx+ C ‖ un − uλ ‖1−αp

=

∫
Ω

u1−α
n dx+ o(1).

Thus, ∫
Ω

u1−α
n dx =

∫
Ω

u1−α
λ dx+ o(1). (3.6)

Hence, using (3.4), (3.5), and (3.6), we conclude that

Eλ,µ(un) = Eλ,µ(uλ) +
1

p
‖un − uλ‖p −

1

p∗s(t)
|un − uλ|

p∗s(t)

p∗s(t) + o(1). (3.7)

Moreover, from (3.4)-(3.5) and for n sufficiently large u, un − uλ ∈ Br and 1
p‖un −

uλ‖p − 1
p∗s(t) |un − uλ|

p∗s(t)

p∗s(t) ≥ o(1). Therefore, we deduce that

1

p
‖un − uλ‖p −

1

p∗s(t)
|un − uλ|

p∗s(t)

p∗s(t) > 0 on ∂Br,

1

p
‖un − uλ‖p −

1

p∗s(t)
|un − uλ|

p∗s(t)

p∗s(t) ≥ 0 in Br,

(3.8)

for r > 0 sufficiently small. Hence, we conclude that

1

p
‖un − uλ‖p −

1

p∗s(t)
|un − uλ|

p∗s(t)

p∗s(t) ≥ o(1). (3.9)

Then, using (3.3) and (3.9), we obtain

C = Eλ,µ(un) + ◦(1)



6 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

=
1

p
‖un‖p −

λ

1− α

∫
Ω

(u+
n )1−α dx− 1

p∗s(t)

∫
Ω

(u+
n )p

∗
s(t)

|x|t
dx+ ◦(1)

≥ Eλ,µ(uλ) +
1

p
‖un − uλ‖p −

1

p∗s(t)

∫
Ω

((un − uλ)+)p
∗
s(t)

|x|t
dx

− λ

1− α

∫
Ω

((un − uλ)+)1−α dx+ ◦(1)

≥ Eλ,µ(uλ) + ◦(1).

Therefore, C ≥ Eλ,µ(uλ) as n → ∞. Since BR0
is convex and closed, we conculde

that uλ ∈ BR0
. Thus, from Eq. (3.2), we deduce that Eλ,µ(uλ) = C < 0 and we

have uλ 6≡ 0 which is a minimizer of Eλ,µ over X0.
Now, we prove that uλ is a weak solution to(1.1) and uλ > 0. Let φ ∈ X0 φ ≥ 0

and r > 0 small enough such that (uλ + rφ) ∈ BR0 . Since, uλ is a local minimizer
of Eλ,µ, we have

0 ≤ Eλ,µ(uλ + rφ)− Eλ,µ(uλ)

=
1

p

(
‖uλ + rφ‖p − ‖uλ‖p

)
− λ

1− α

∫
Ω

[
((uλ + rφ)+)1−α − (u+

λ )1−α
]
dx

− 1

p∗s(t)

∫
Ω

[ ((uλ + rφ)+)p
∗
s(t)

|x|t
−

(u+
λ )p

∗
s(t)

|x|t
]
dx

≤ 1

p

(
‖uλ + rφ‖p − ‖uλ‖p

)
.

(3.10)

Now, we divide (3.10) by t > 0 and we take the limit as r → 0+, we obtain

lim inf
r→0+

λ

1− α

∫
Ω

((uλ + rφ)+)1−α − (u+
λ )1−α

r
dx

≤
∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy

− µ
∫
Q

|u|p−2uφ

|x|sp
dx−

∫
Ω

(u+
λ )p

∗
s(t)−1φ

|x|t
dx.

(3.11)

Therefore,

λ

1− α

∫
Ω

((uλ + rφ)+)1−α − (u+
λ )1−α

r
= ((uλ + ξrφ)+)−αφ a.e. in Ω, (3.12)

with ξ ∈ (0, 1) and ((uλ + ξrφ)+)−αφ → (u+
λ )−αφ a.e. in Ω, as r → 0+. Using,

Fatou’s Lemma,

λ

∫
Ω

(u+
λ )−αφdx ≤ λ

1− α
lim inf
r→0+

∫
Ω

((uλ + rφ)+)1−α − (u+
λ )1−α

r
dx. (3.13)

Consequently, using (3.11) and (3.13), ones has∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy

− µ
∫
Q

|u|p−2uφ

|x|sp
dx− λ

∫
Ω

(u+
λ )−αφdx−

∫
Ω

(u+
λ )p

∗
s(t)−1φ

|x|t
dx ≥ 0

(3.14)

for φ ≥ 0 a.e. in RN . Since Eλ,µ < 0 and using Lemma 3.1, we derive that uλ ∈ BR0
.

Therefore, there exists δ ∈ (0, 1) satisfying (1 + r)uλ ∈ BR0
(|r| ≤ δ). Then, define



EJDE-2023/10 NONLOCAL SINGULAR PDES 7

the functional Jλ,µ by

Jλ,µ(r) = Eλ,µ((1 + r)uλ).

Hence, the functional Jλ,µ attains its minimum at r = 0, since uλ is a local mini-

mizer of Jλ,µ in BR0
. Furthermore,

J ′λ,µ(r)\r=0 = ‖uλ‖p − λ
∫

Ω

(u+
λ )1−αdx−

∫
Ω

(u+
λ )p

∗
s(t)

|x|t
dx = 0. (3.15)

Now, we define Ψ ∈ X0 by Ψ := (u+
λ + εφ)+, where (u+

λ + εφ)+ = max{u+
λ + εφ, 0}.

Let Ωε = {u+
λ + εφ ≤ 0} and Ωε = {u+

λ + εφ > 0}. Replacing φ with Ψ in (3.14)
and combining with (3.15), we obtain

0 ≤
∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

−
∫

Ω

µ
|u|p−2uΨ

|x|sp
dx− λ

∫
Ω

(u+
λ )−αΨ dx−

∫
Ω

(u+
λ )p

∗
s−1(t)Ψ

|x|t
dx

=

∫
{(x,y)∈Ωε×Ωε}

(
|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))((u+

λ + εφ)(x)

− (u+
λ + εφ)(y))

)
/|x− y|N+sp dx dy − µ

∫
Ω

|uλ|p−2uλ
|x|sp

(u+
λ + εφ)dx

−
∫

Ωε

(
λ(u+

λ )−α(u+
λ + εφ) +

(u+
λ )p

∗
s−1(t)(u+

λ + εφ)

|x|t
)
dx

=
(∫
Q
−
∫

Ωε×Ωε

) |uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

−
(∫

Ω

−
∫

Ωε

)[
µ
|uλ|p−2uλ
|x|sp

(u+
λ + εφ) + λ(u+

λ )−α(u+
λ + εφ)

+
(u+
λ )p

∗
s(t)−1(u+

λ + εφ)

|x|t
]
dx

≤
∫
Q

|uλ(x)− uλ(y)|p

|x− y|N+sp
dx dy − µ

∫
Ω

|uλ|p

|x|sp
−
∫

Ω

[
λ(u+

λ )1−γ +
(u+
λ )p

∗
s(t)

|x|t
]
dx

+ ε
(∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy

− µ
∫

Ω

|uλ|p−2uλ
|x|sp

φdx
)
− ε
∫

Ω

[
λ(u+

λ )−αφ+
(u+
λ )p

∗
s(t)−1φ

|x|t
]
dx

−
∫
{(x,y)∈Ωε×Ωε}

(
|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))((u+

λ + εφ)(x)

− (u+
λ + εφ)(y))

)
/|x− y|N+sp dx dy

+ µ

∫
Ω

|uλ|p−2uλ
|x|sp

(u+
λ + εφ)dx

+

∫
Ωε

[
λ(u+

λ )−α(u+
λ + εφ) +

(u+
λ )p

∗
s(t)−1(u+

λ + εφ)

|x|t
]
dx

≤ ε
(∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy



8 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

− µ
∫

Ω

|uλ|p−2uλ
|x|sp

φdx
)
− ε
∫

Ω

[
λ(u+

λ )−αφ+
(u+
λ )p

∗
s(t)−1φ

|x|t
]
dx

− ε
∫
{(x,y)∈Ωε×Ωε}

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy.

Since the measure of the domain of integration Ωε tends to zero as ε → 0+, we
deduce as ε→ 0+, that∫

{(x,y)∈Ωε×Ωε}

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy → 0.

Dividing by ε and letting ε→ 0+, we obtain∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy − µ

∫
Ω

|uλ|p−2uλ
|x|sp

φdx

−
∫

Ω

[
λ(u+

λ )−γφ+
(u+
λ )p

∗
s(t)−1φ

|x|t
]
dx ≥ 0.

Since φ is an arbitrary test function, we obtain the equality if we change φ by −φ.
Hence, uλ is a weak solution to the problem (1.1). Finally, putting φ = u−λ in
(2.1), we obtain that uλ is nonnegative. Moreover, since Iλ = C < 0, then uλ 6≡ 0.
Therefore, using the maximum principle, we conclude that uλ is a positive solution
to (1.1). The proof is complete. �

4. Existence of a solution of the perturbed problem

Note that Eλ,µ is not differentiable because of the singular term in it. Hence,
the classical approach of min-max methods fails. Therefore, to show the existence
of a second solution to (P), we introduce the following auxillary perturbed problem

(−∆p)
su− µ |u|p− 2u

|x|sp
=

λ

(u+ + 1
n )α

+
(u+)p

∗
s(t)−2u+

|x|t
, in Ω,

u = 0, in RN \ Ω.

(4.1)

The functional energy En,λ,µ : X0 → R associated with (4.1), is defined by

En,λ,µ(u)

=
1

p
‖u‖p − λ

1− γ

∫
Ω

(
(u+ +

1

n
)1−γ − (

1

n
)1−γ

)
dx− 1

p∗s(t)

∫
Ω

(u+)p
∗
s(t)

|x|t
dx.

From the definition of the functional energy En,λ,µ, it is easy to see that En,λ,µis
Fréchet differentiable, for all φ ∈ X0, ones has

〈E′n,λ,µ(u), φ〉

=

∫
Q

|uλ(x)− uλ(y)|p−2(uλ(x)− uλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy

− µ
∫

Ω

|u|p−2u

|x|sp
φdx− λ

∫
Ω

φ

(u+ + 1
n )1−α dx−

∫
Ω

(u+)p
∗
s(t)−2uφ

|x|α
dx.

(4.2)

It is easy to see that any critical points of the functional energy En,λ,µ, are exactly
the solutions of (4.1).

Lemma 4.1. Let R0 ∈ (0, 1], λ0 and ρ0 be the constants given by Lemma 3.1.
Then for any λ ∈ (0, λ0], En,λ,µ satisfies the following properties:



EJDE-2023/10 NONLOCAL SINGULAR PDES 9

(1) En,λ,µ(u) ≥ ρ0, for all u ∈ X0 with ‖u‖ ≤ R0.
(2) There exists vλ ∈ X0, with ‖vλ‖ > R0 and En,λ,µ(vλ) < ρ0.

Proof. From the subadditivity of r1−α, ones has

(u+ +
1

n
)1−α − (

1

n
)1−α ≤ (u+)1−α.

Therefore,
En,λ,µ(u) ≥ Eλ,µ(u).

So, from Lemma 3.1, we deduce the first part of the Lemma 4.1.
Now, let u ∈ X0 such that u+ 6≡ 0 and r > 0, then

En,λ,µ(ru) =
rp

p
‖u‖p − λr1−α

1− α

∫
Ω

(
(u+ +

1

n
)1−α − (

1

n
)1−α

)
dx

− rp
∗
s(t)

p∗s(t)

∫
Ω

(u+)p
∗
s(t)

|x|t
dx→ −∞ as r → +∞,

since 1 − γ < 1 < p < p∗s(t). Therefore, we obtain the existence of vλ ∈ X0,
satisfying ‖vλ‖ > R0 and En,λ,µ(vλ) < ρ0. The proof of the second part of Lemma
4.1 is complete. �

Now, we prove the compactness property for the functional energy En,λ,µ.

Lemma 4.2. Suppose that 0 < α < 1. So, the functional energy En,λ,µ satisfies

the (PS) condition at any level c ∈ R with c < (sp−t)
p(N−t)S

N−t
sp−t − Cλ for any λ > 0,

where

Cλ =
1 + α

p

[
λ
( (sp− t)

(N − t)(1− α)

)α−1
p
( 1

1− α
+

1

p∗s(t)

)
|Ω|

p∗s (t)−1+α

p∗s (t) S−
1−α
p

] p
1+α

.

Proof. Consider {uk} ⊂ X0 be a (PS) minimizing sequence for the functional energy
En,λ,µ at level c ∈ R, with c satisfying

En,λ,µ(uk)→ c and E′n,λ,µ(uk)→ 0 as k →∞. (4.3)

Therefore, using the Hölder inequality and the Sobolev embedding, there exists
ε > 0 and C > 0 such that

c+ ε‖uk‖+ ◦(1) ≥ En,λ,µ(uk)− 1

p∗s(t)
〈E′n,λ,µ(uk), uk〉

=
(1

p
− 1

p∗s(t)

)
‖uk‖p +

λ

p∗s(t)

∫
Ω

(
u+
k +

1

n

)−α
uk dx

− λ

1− α

∫
Ω

(
(u+
k +

1

n
)1−α − (

1

n
)1−α

)
dx

≥
(1

p
− 1

p∗s(t)

)
‖uk‖p − λ

( 1

1− α
+

1

p∗s(t)

)∫
Ω

|uk|1−α dx

≥
(1

p
− 1

p∗s(t)

)
‖uk‖p − λC

( 1

1− α
+

1

p∗s(t)

)
|Ω|

p∗s (t)+1−α
p∗s (t) ‖uk‖1−α.

So, {uk} is bounded, since 1 − γ < 1 < p < p∗s(t). Moreover, {u−k } is bounded in
X0, therefore using (4.3), ones has

lim
k→∞

〈E′n,λ,µ(uk), uk〉 = lim
k→∞

〈uk,−u−k 〉.

Now, we recall the following elementary inequality.



10 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

(a− b)(a− − b−) ≤ −(a− − b−)2. (4.4)

Then, using (4.4), we obtain

0 ≤
∫∫

R2N

|(u(x)− u(y)|p−2(u(x)− u(y))(u−(x)− u−(y))

|x− y|N+sp
dx dy

≤ −
∫∫

R2N

|(u(x)− u(y)|p−2(u−(x)− u−(y))2

|x− y|N+sp
dx dy.

(4.5)

So, using (4.5), we can conclude that ‖u−k ‖ → 0 as k tends to infinity. Hence, for k
large enough, ones has

En,λ,µ(uk) = En,λ,µ(u+
k ) + o(1) and E′n,λ,µ(uk) = E′n,λ,µ(u+

k ) + o.

Therefore, {uk} is a sequence of positive functions.
Now, since {uk} is bounded, up to a subsequence, using [1, 21], there exists

{uk} ⊂ X0, vλ in X0 and a non-negative numbers l, µ such that

uk ⇀ vλ weakly in X0,

uk ⇀ vλ weakly in Lp
∗
s(t)(Ω),

uk → vλ strongly in Lk(Ω,
dx

|x|t
) for k ∈ [1, p∗s(t)),

uk → vλ a.e. in Ω,

|uk(x)| ≤ h(x) a.e. in Ω for all n with h(x) ∈ L1(Ω),

(4.6)

and
‖uk‖ → µ,

‖uk − vλ‖p∗s(t) → l.
(4.7)

It is easy to see that if µ = 0, then uk → 0 in X0. Therefore, we suppose that
µ > 0. Using the above assertion, we obtain∣∣ uk − vλ

(u+
k + 1

n )α

∣∣ ≤ nα(h+ |vλ|).

Now, applying the dominated convergence theorem, we obtain

lim
k→∞

∫
Ω

uk − vλ
(u+
k + 1

n )α
dx = 0. (4.8)

Hence, we conclude that

lim
k→∞

∫
Ω

uk

(u+
k + 1

n )α
dx =

∫
Ω

vλ

(v+
λ + 1

n )α
dx. (4.9)

Then, we demonstrate that uk → vλ strongly in X0. Since, E′n,λ,µ(uk) → 0, we
obtain

‖uk‖p − λ
∫

Ω

uk(
u+
k + 1

n )α
dx−

∫
Ω

(u+
k )p

∗
s(t)−1uk
|x|t

dx = o(1).

Therefore, using Brezis-Lieb Lemma [4], we obtain

‖uk‖p = ‖uk − vλ‖p + ‖u‖p + o(1),

‖uk‖
p∗s(t)

p∗s(t) = ‖uk − vλ‖
p∗s(t)

p∗s(t) + ‖u‖p
∗
s(t)

p∗s(t) + o(1).
(4.10)



EJDE-2023/10 NONLOCAL SINGULAR PDES 11

So, using (4.3), (4.8), and (4.10), we deduce that

o(1) = 〈E′n,λ,µ(uk), uk − vλ〉

=

∫
Q

|uk(x)− uk(y)|p−2(uk(x)− uk(y))((uk − vλ)(x)− (uk − vλ)(y))

|x− y|N+sp
dx dy

− µ
∫

Ω

|u|p−2u

|x|sp
(uk − vλ) dx− λ

∫
Ω

uk − vλ
(uk + 1

n )α
dx−

∫
Ω

u
p∗s(t)−1
k (uk − vλ)

|x|t
dx

= (‖uk‖p − ‖vλ‖p)− ‖uk‖
p∗s(t)

p∗s(t) + ‖vλ‖
p∗s(t)

p∗s(t) + o(1)

= ‖uk − vλ‖p − ‖uk − vλ‖
p∗s(t)

p∗s(t) + o(1).

Therefore,

lim
k→∞

‖uk − vλ‖p = lim
k→∞

∫
Ω

((uk − vλ)+)p
∗
s(t)−1(uk − vλ)

|x|t
dx = l,∫

Ω

|uk − vλ|p
∗
s(t)

|x|t
dx ≥

∫
Ω

((uk − vλ)+)p
∗
s(t)−1(uk − vλ)

|x|t
dx.

Then, using Sobolev’s inequality, we deduce that

‖uk − vλ‖p ≥ S
(∫

Ω

|uk − vλ|p
∗
s(t)

|x|t
dx
) p
p∗s (t) .

Hence, we conclude that

Slp ≤ lp
∗
s(t). (4.11)

We guarantee that l = 0. We obtain that uk → vλ in X0 and the proof is complete.
Otherwise, we suppose that

S
N−t
sp−t ≤ l. (4.12)

Therefore, using (4.10) and (4.12), the Hölder inequality and the Young inequality,
if k tends to infinity, we obtain

c = En,λ,µ(uk)− 1

p∗s(t)
〈E′n,λ,µ(uk), uk〉+ o(1)

=
(sp− t)
p(N − t)

‖uk‖p − λ
∫

Ω

[
(u+
k +

1

n
)1−α − (

1

n
)−α

]
dx

+
λ

p∗s(t)

∫
Ω

(u+
k +

1

n
)−αuk dx+ o(1)

≥ (sp− t)
p(N − t)

S
N−t
sp−t +

(sp− t)
p(N − t)

‖vλ‖p − λ
( 1

1− α
+

1

p∗s(t)

)∫
Ω

|uk|1−α dx+ o(1)

≥ (sp− t)
p(N − t)

S
N−t
sp−t +

(sp− t)
p(N − t)

‖vλ‖p − λ
( 1

1− α
+

1

p∗s(t)

)∫
Ω

v1−α
λ dx+ o(1)

≥ (sp− t)
p(N − t)

S
N−t
sp−t +

(sp− t)
p(N − t)

‖vλ‖p − λ
( 1

1− α
+

1

p∗s(t)

)
|Ω|

p∗s (t)−1+α

p∗s (t)

× S−
1−α
p ‖vλ‖1−α + o(1)

≥ (sp− t)
p(N − t)

S
N−t
sp−t +

(sp− t)
p(N − t)

‖vλ‖p − λ
( p

(N − t)(1− α)

)α−1
p

×
( 1

1− α
+

1

p∗s(t)

)
|Ω|

p∗s (t)−1+α

p∗s (t) S−
1−α
p

( p

(N − t)(1− α)

) 1−α
p ‖vλ‖1−α + o(1)



12 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

≥ (sp− t)
p(N − t)

S
N−t
sp−t +

(sp− t)
p(N − t)

‖vλ‖p −
1− α
p

[( (sp− t)
(N − t)(1− α)

) 1−α
p

× ‖vλ‖1−α
] p

1−α − 1 + α

p

[
λ
( (sp− t)

(N − t)(1− α)

)α−1
p
( 1

1− α
+

1

p∗s(t)

)
× |Ω|

p∗s (t)−1+α

p∗s (t) S−
1−α
p

] p
1+α

=
(sp− t)
p(N − t)

S
N−t
sp−ta − Cλ,

which is a contradiction. Therefore, l = 0 and uk → vλ. The proof of Lemma 4.2
is complete. �

To state a control from above for the functional En,λ,µ, we recall some necessary
tools (for more details, see [4]). Let 1 < p < N , 0 ≤ t < p, and o ≤ µ < µ0. Then,
the limiting problem

(−∆p)
su− µ |u|p− 2u

|x|sp
=

(u+)p
∗
s(t)−2u+

|x|t
, in Ω,

u = 0, in RN \ Ω,

(4.13)

has a positive radial solution

Ut,ε(x) = ε−
N−sp
p Up,µ(

|x|
ε

),

where ε > 0, x ∈ RN . Note that Ut,ε(x) is a minimizer for S satisfying∫
Q

|Ut,ε(x)− Ut,ε(y)|p

|x− y|N+sp
dx dy − µ

∫
Ω

|Ut,ε|p

|x|sp
dx =

∫
Ω

U
p∗s(t)
t,ε

|x|t
dx = S

N−t
sp−t (4.14)

where the function Up,µ(x) = Up,µ(|x|) is the unique radial solution of (4.13).
Now, we define

mε,δ =
Ut,ε(δ)

Ut,ε(δ)− Ut,ε(θδ)
,

where ε, δ > 0, and θ > 1. For fixed ε, δ > 0, we set

gε,δ(k) =


0 if 0 ≤ k ≤ Ut,ε(θδ),
mp
ε,δ(k − Ut,ε(θδ)) if Ut,ε(θδ) ≤ k ≤ Ut,ε(δ),

k + Ut,ε(δ)(m
p−1
ε,δ − 1) if k ≥ Uε(δ),

and define

Gε,δ(k) =

∫ k

0

g′ε,δ(τ) dτ =


0 if 0 ≤ k ≤ Ut,ε(θδ),
mε,δ(k − Ut,ε(θδ)) if Ut,ε(θδ) ≤ k ≤ Ut,ε(δ),
k if k ≥ Ut,ε(δ).

Note that the functions gε,δ and Gε,δ are nondecreasing and absolutely continuous.
We define the radially symmetric non-increasing function

ut,ε,δ(r) = Gε,δ(Ut,ε(r)),

which satisfies

ut,ε,δ(r) =

{
Ut,ε(r) if r ≤ δ,
0 if r ≥ θδ,



EJDE-2023/10 NONLOCAL SINGULAR PDES 13

for all r ≥ 1. We follow here the arguments of [4, Lemma 2.10]. For each sufficiently
small ε, δ > 0, we have the following estimates for ut,ε,δ.

Lemma 4.3. There exists a constant C = C(N, s) > 0 such that for any 0 < pε ≤
δ < θ−1dist(0, ∂Ω), it holds

‖ut,ε,δ‖p ≤ S
N−t
sp−t + C(

ε

δ
)(N−sp), (4.15)∫

Ω

u
p∗s(t)
t,ε,δ

|x|t
(x) dx ≥ S

N−t
sp−t − C(

ε

δ
)b(µ)p∗s(t)−N+t). (4.16)

where b(µ) is the solution of f(r) = (p − 1)rp − (N − p)rp−1 + µ, r ≥ 0. On the
other hand, for any β > 0, there exists Cβ such that

∫
RN

ut,ε,δ(x)β ≥ Cβ


εN−

N−sp
p β | log( εδ )| if β =

p∗s(t)
p ,

ε
N−sp
p βδN−

N−sp
p β if β <

p∗s(t)
p ,

εN−
N−sp
p β if β >

p∗s(t)
p .

(4.17)

Now, we have the following result for the functional energy En,λ,µ.

Lemma 4.4. There exits λ1 > 0 and ψ ∈ X0 satisfying

sup
t>0

En,λ,µ(rψ) <
(sp− t)
p(N − t)

S
N−t
sp−t − Cλ

for all λ ∈ (0, λ1), where Cλ is defined in Lemma 4.2.

Proof. Firstly, using (3.1), we have

En,λ,µ(rut,ε,δ) −→
r→∞

−∞ ∀(ε, δ) ∈ (0, ε0)× (0, δ0),

and
En,λ,µ(rut,ε,δ) −→

r→0
0 ∀(ε, δ) ∈ (0, ε0)× (0, δ0).

Now, setting

Aε,δ(r) =
rp

p
‖ut,ε,δ‖p −

rp
∗
s(t)

p∗s(t)

∫
Ω

(u+
t,ε,δ)

p∗s(t)

|x|t
dx,

Bε,δ(r) = − 1

γ − 1

∫
Ω

[(
ru+
t,ε,δ +

1

n

)1−α
− (

1

n
)1−α

]
dx.

It is very easy to see that

lim
r→∞

Aε,δ(r) = −∞, Aε,δ(0) = 0, lim
r→0+

Aε,δ(r) > 0.

Therefore, Aε,δ attains its maximum at some Tε,δ > 0. Indeed,

A′ε,δ(r) = r‖ut,ε,δ‖p − rp
∗
s(t)−1

∫
Ω

(u+
t,ε,δ)

p∗s(t)

|x|t
dx = 0,

hence

Tε,δ =

(
‖ut,ε,δ‖p∫

Ω

(u+
t,ε,δ)

p∗s (t)

|x|t dx

) 1
p∗s (t)−2

.

So, A′ε,δ(r) > 0 for 0 < t < Tε,δ and A′ε,δ(r) < 0 for t > Tε,δ. Therefore, there exists
tε,δ > 0, satisfying

En,λ,µ(rε,δut,ε,δ) = max
r≥0

En,λ,µ(rut,ε,δ).



14 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

Then, using (4.15) and (4.16), we obtain

Aε,δ(Tε,δ) =
1

p

(
‖ut,ε,δ‖p∫

Ω

(u+
t,ε,δ)

p∗s (t)

|x|t dx

) p
p∗s (t)−2

‖ut,ε,δ‖p

− 1

p∗s(t)

(
‖ut,ε,δ‖p∫

Ω

(u+
t,ε,δ)

p∗s (t)

|x|t dx

) p∗s (t)
p∗s (t)−2

∫
Ω

(u+
t,ε,δ)

p∗s(t)

|x|t
dx

=
(sp− t)
p(N − t)

(
‖ut,ε,δ‖p∫

Ω

(u+
t,ε,δ)

p∗s (t)

|x|t dx

) p
p∗s (t)−2

‖ut,ε,δ‖p

≤ (sp− t)
p(N − t)

S
N−t
sp−t + C(

ε

δ
)(N−sp).

(4.18)

Then, we use the following elementary inequality to estimate Bε,δ,

a1−α − (a+ b)1−α ≤ −(1− α)b
1−α
p a

(1−α)(p−1)
p (4.19)

for any a > 0, b > 0 large enough, p > 1. Therefore, from (4.19), with q =
p∗s(t)
p

and setting ε < r
1
q for all q > 0 small enough, we deduce the existence of c1, c2 > 0

independent of ε, such that

Bε(rε,δ) ≤
1

1− α

∫
{x∈Ω:|x|≤ε}

(
(

1

n
)1−α − (rε,δut,ε,δ +

1

n
)1−α

)
dx

≤ −c1(1− α)ε
(N−sp)(1−α)

p∗s (t)

∫
{x∈Ω:|x|≤εq′}

( 1

( 1
|x|p′+εp′ )

N−sp
p

) p(1−α)
p∗s (t) dx

≤ −c2(1− α)ε
(N−sp)(1−α)

p∗s (t)

∫
{x∈Ω:|x|≤εq′}

(
rε,δUp,µ(

|x|
ε

) p(1−α)
p∗s (t) dx

≤ −c3(1− α)ε
(N−sp)(1−α)

p∗s (t)

∫ εq
′

0

(
Up,µ(

|x|
ε

) p(1−α)
p∗s (t) yN−1εNdy

≤ −c4(1− α)ε
(N−sp)(1−α)+N

p∗s (t)

∫ εq
′

0

y−b(µ)p+N−1dy

≤


(1− α)ε

(N−sp)(1−α)+N
p∗s (t) b(µ) > N

p ,

(1− α)ε
(N−sp)(1−α)+N

p∗s (t) | ln ε| b(µ) = N
p ,

(1− α)ε
(N−sp)(1−α)−2q(N−sp)(1−α)+p∗s (t)qN

p∗s (t) b(µ) < N
p .

(4.20)

Hence, using (4.18) and (4.20), we deduce the existence of a positive constant λ1

such that, for every λ ∈ (0, λ1), we obtain

En,λ,µ(ut,ε,δ)

= Aε,δ(ut,ε,δ) + λBε,δ(ut,ε,δ)

≤ (sp− t)
p(N − t)

S
N−t
sp−t + C(

ε

δ
)(N−sp) − c2(1− α)ε

(N−sp)(1−α)−2q(N−sp)(1−α)+p∗s (t)qN
p∗s (t)

× ε
(N−2s)(1−γ)−2q(N−2s)(1−γ)+2∗αqN

2∗α



EJDE-2023/10 NONLOCAL SINGULAR PDES 15

<
(sp− t)
p(N − t)

S
N−t
sp−t − Cλ.

This completes the proof. �

Lemma 4.5. Suppose 0 < α < 1. Then, there exists λ∗ = min(λ0, λ1), such that
(4.1) has a positive solution vn ∈ X0 satisfying

ρ0 < En,λ,µ(vn) <
sp− t
p(N − t)

S
N−t
sp−t − Cλ,

where ρ0 is given in Lemma 3.1 and Cλ in Lemma 4.2.

Proof. Consider λ∗ = min(λ0, λ1). Therefore, the results in Lemmas 4.1-4.4 holds
for all λ ∈ (0, λ∗). Now, using Lemma 3.1 we deduce that the functional En,λ,µ
satisfies the geometry of the mountain pass Lemma. Hence, we introduce the
mountain pass level

cn,λ,µ = inf
g∈Γ

max
r∈[0,1]

En,λ,µ(g(r)),

where

Γ = {g ∈ C([0, 1], X0) : g(0) = 0, En,λ(g(1)) < 0} .
Moreover,

0 < ρ0 < cn,λ ≤ sup
t≥0

En,λ,µ(tψ) < cn,λ.

Therefore, according to Lemmas 4.1-4.4, En,λ,µ satisfies the (PS) condition at the
level cn,λ,µ. Thn there exists a non-regular point vn for In,λ,µ at the level cn,λ,µ.
Moreover, En,λ,µ(vn) = cn,λ,µ > ρ0 > 0. We deduce that vn is a non-trivial critical
point of the functional energy En,λ,µ and also a solution to the problem (4.1). Now,
if we replace φ by v−n in (4.2) and using (4.5), it follows that ‖vn‖ = 0. Therefore,
vn is positive. At the end, we apply the strong maximum principle (see [18]), we
deduce that vn is a non-negative solution to the problem (4.1). The proof of Lemma
4.5 is now complete. �

5. Multiple solutions to (1.1)

In this section we show the existence of a second solution to (1.1), as a limit of
solutions of the perturbed problem (4.1). To do this, we consider {vn}n be a family
of positive function given by Lemma 4.5. Then, using Lemma 4.5 combineed with
Hölder’s inequality, we have

sp− t
p(N − t)

S
N−t
sp−t − Cλ

> En,λ,µ −
1

p∗s(t)
〈E′n,λ,µ(vn), vn〉

=
(1

p
− 1

p∗s(t)

)
‖vn‖p −

λ

1− α

∫
Ω

(
(vn +

1

n
)1−α − (

1

n
)1−α

)
dx

+
λ

p∗s(t)

∫
Ω

(vn +
1

n
)−γvn dx

≥
(1

p
− 1

p∗s(t)

)
‖vn‖p −

λ

1− α

∫
Ω

v1−α
n dx

≥
(1

p
− 1

p∗s(t)

)
‖vn‖p −

λ

1− α
|Ω|

p∗s (t)−1+α

p∗s (t) S−
1−α
p ‖vn‖1−α



16 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

since α ∈ (0, 1), so {vn} is bounded in X0. Therefore, applying the reflexivity of
X0, we obtain the existence a subsequence, still denoted by {vn} and a function
vλ, satisfying

vn ⇀ vλ weakly in X0,

vn ⇀ vλ weakly in Lp
∗
s(t)(Ω),

vn → vλ strongly in Lk(Ω,
dx

|x|t
) for k ∈ [1, p∗s(t)),

vn → vλ a.e. in Ω,

(5.1)

and
‖vn‖ → µ,

‖vn − vλ‖p∗s(t) → l.
(5.2)

Then, we want to show that vn → vλ strongly in X0. This means that ‖vn−vλ‖ →
0 as n→∞.

Firstly, using (5.1) and if µ = 0, we obtain ‖vn‖ → 0 as n→∞. Now, we assume
that µ > 0. Therefore, since

0 ≤ vn

(vn + 1
n )α
≤ v1−α

n a.e. in Ω.

Therefore, from Hölder inequality and (5.1), we obtain∫
Ω

vn

(vn + 1
n )α

dx ≤
∫

Ω

v1−α
n dx

≤
∫

Ω

|vn − vλ|1−α dx+

∫
Ω

v1−α
λ dx

= |vn − vλ|1−αp |Ω|
1+α
p +

∫
Ω

v1−α
λ dx

≤
∫

Ω

v1−α
λ dx+ o(1).

(5.3)

In the same way, we have∫
Ω

v1−α
λ dx ≤

∫
Ω

vn

(vn + 1
n )α

dx+ o(1). (5.4)

Hence, using (5.3)-(5.4), we obtain

lim
n→∞

∫
Ω

vn

(vn + 1
n )γ

dx =

∫
Ω

v1−γ
λ dx.

Therefore, if we replace both u and φ by vn in (4.2), and using (5.1)-(5.2) as n→∞,
we obtain

‖vn− vλ‖p + ‖vλ‖p−
∫

Ω

|vn − vλ|p
∗
s(t)

|x|t
dx−

∫
Ω

v
p∗s(t)
λ

|x|t
dx−λ

∫
Ω

v1−α
λ dx→ 0. (5.5)

Then, since {vn}n is bounded in X0 and from the strong maximum principle (see

[18]), we obtain the existence Ω̃ ⊂ Ω and c̃ > 0 such that

vn ≥ c̃ > 0, a.e. in Ω, (5.6)



EJDE-2023/10 NONLOCAL SINGULAR PDES 17

for any integer n. Let us define φ ∈ C∞0 (Ω) such that supp(ϕ) = Ω̃ ⊂ Ω. Then,
using (5.6), ones has

0 ≤
∣∣ ϕ

(vn + 1
n )γ

∣∣ ≤ |ϕ|
c̃γ
, a.e. in Ω.

Hence, from (5.1) and by using the dominated convergence Theorem, we obtain

lim
n→∞

∫
Ω

ϕ

(vn + 1
n )γ

dx =

∫
Ω

v−γλ ϕdx.

Thus, replacing u by vn in (4.2), letting n → ∞, and using (5.1) with the above
equality, we have∫

Q

|vλ(x)− vλ(y)|p−2(vλ(x)− vλ(y))(φ(x)− φ(y))

|x− y|N+sp
dx dy

− λ
∫

Ω

v−αλ φdx−
∫

Ω

v
p∗s(t)−2
λ vλφ

|x|t
dx = 0.

(5.7)

Moreover, since ∂Ω is continuous, the space C∞0 (Ω) is dense in X0. Therefore, (5.7)
holds for any φ ∈ X0. Thus, if we replace φ by vλ in (5.7) and combining this with
(4.2), we obtain

‖vλ‖p − λ
∫

Ω

v1−α
λ −

∫
Ω

v
p∗s(t)
λ

|x|t
dx = 0. (5.8)

Consequently, from (5.7), we have

‖vn − vλ‖p −
∫

Ω

|vn − vλ|p
∗
s(t)

|x|t
dx = o(1). (5.9)

So, we obtain

lim
n→∞

‖vn − vλ‖2 = lim
n→∞

∫
Ω

|vn − vλ|p
∗
s(t)

|x|t
dx = l > 0. (5.10)

Now, since ∫
Ω

|vn − vλ|p
∗
s(t)

|x|t
dx ≥

∫
Ω

((vn − vλ)+)p
∗
s(t)

|x|t
dx.

It follows that l ≥ S
N−t
sp−t . Therefore, using (5.5), we obtain

En,λ,µ(vλ)

=
1

p
‖vλ‖p −

λ

1− α

∫
Ω

v1−α
λ dx− 1

p∗s(t)

∫
Ω

v
p∗s(t)
λ

|x|t
dx

=
sp− t
p(N − t)

‖vλ‖p − λ
( 1

1− α
− 1

p∗s(t)

)∫
Ω

v1−α
λ dx

≥ sp− t
p(N − t)

‖vλ‖p − λ
( 1

1− α
− 1

p∗s(t)

)
|Ω|

p∗s (t)−1+γ

p∗s (t) S−
1−α
p ‖vλ‖1−α

> −Cλ.

(5.11)



18 A. DAOUES, A. HAMMAMI, K. SAOUDI EJDE-2023/10

Therefore, using (5.5)-(5.9), we have

Eλ,µ(vn) = En,λ,µ(vλ)− sp− t
p(N − t)

‖vn − vλ‖p + o(1)

<
sp− t
p(N − t)

(
S
N−t
sp−t − l

)
− Cλ

≤ −Cλ

(5.12)

which contradicts (5.11). Therefore, Eλ,µ(vλ) = limn→∞En,λ,µ(vn). Hence, it is
very easy to see that vλ is a solution of (1.1). Moreover, using Lemma 4.5, we
obtain Eλ,µ(vλ) ≥ α > 0, that is, vλ is nontrivial solution. Also, we proceed as in
the proof of Lemma 4.5 to conclude that vλ is a non-negative solution of problem
(1.1). At the end, uλ 6≡ vλ since Eλ,µ(uλ) < 0 < Eλ,µ(vλ). Therefore, the proof is
complete.

Acknowledgments. The authors thank the anonymous referees for their construc-
tive remarks and comments.

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Adel Daoues

École Supérieure des Sciences et de Technologie de Hammam Sousse, Sousse, Tunisia

Email address: adel.daouas@essths.u-sousse.tn

Amani Hammami

École Supérieure des Sciences et de Technologie de Hammam Sousse, Sousse, Tunisia
Email address: amanihammami29@gmail.com

Kamel Saoudi
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University,

Saudi Arabia

Email address: kmsaoudi@iau.edu.sa


	1. Introduction
	2. Functional framework and main results
	3. Existence of a weak solution to (??)
	4. Existence of a solution of the perturbed problem
	5. Multiple solutions to (??)
	Acknowledgments

	References