Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 99, pp. 1–13.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.99

COMBINED EFFECTS OF CRITICAL HARDY-SOBOLEV EXPONENT AND

SINGULAR NONLINEARITIES IN NONLOCAL PROBLEMS WITH

VARIABLE WEIGHTS

SARAH ALMUTAIRI, KAMEL SAOUDI

Abstract. This article studies a fractional elliptic equation that involves a critical Hardy-

Sobolev nonlinearity along with a singular term,

(−∆p)
su = λf(x)u−γ ±

g(x)|u|p∗s(t)−2u

|x|t
in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω,

where Ω is a bounded domain in RN with a smooth boundary ∂Ω, and 0 ∈ Ω. The dimension N

satisfies N > sp, s ∈ (0, 1), λ > 0, 0 < γ < 1, and p∗s(t) =
p(N−t)
N−sp

represent the critical Hardy-

Sobolev exponent. The weight functions f and g are elements of L∞(Ω) and satisfy specific
positivity conditions, and (−∆p)su is the fractional p-Laplacian operator. We use the method

of sub- and super-solutions combined with monotonicity arguments, to establish the existence

and nonexistence of solutions. Furthermore, we prove that any weak solution is locally Hölder
continuous.

1. Introduction

The purpose of this article is to study a fractional equation characterized by a critical Sobolev
exponent and singular nonlinearity,

(−∆p)
su = λf(x)u−γ ± g(x)|u|p∗

s(t)−2u

|x|t
in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω,

(1.1)

where 0 ∈ Ω is a bounded domain in RN with a smooth boundary ∂Ω, and N > sp with s ∈ (0, 1).

The parameters satisfy λ > 0 and 0 < γ < 1, and the term p∗s(t) =
p(N−t)
N−sp represents the critical

Sobolev-Hardy exponent. The variable weight functions f and g belong to L∞(Ω) and satisfy:

ess infx∈Ω f(x) > 0 and ess infx∈Ω g(x) > 0. (1.2)

The fractional p-Laplacian (−∆p)
su is defined for smooth functions as

(−∆p)
su(x) = 2 lim

ϵ↘0

∫
RN\Bϵ

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
dy,

where x ∈ RN and Bϵ is a ball of radius ϵ centered at x.

2020 Mathematics Subject Classification. 34B15, 37C25, 35R20.
Key words and phrases. Nonlocal operator; singular nonlinearity; existence and nonexistence of solutions;

sub- and supersolutions; monotonicity arguments.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted May 12, 2025. Published October 20, 2025.

1



2 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

For clarity, we state the two versions of problem (1.1) as follows

(−∆p)
su = λf(x)u−γ +

g(x)|u|p∗
s(t)−2u

|x|t
in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω,

(1.3)

and

(−∆p)
su = λf(x)u−γ − g(x)|u|p∗

s(t)−2u

|x|t
in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω,

(1.4)

Before presenting the key results of this work, we will outline several related problems to prob-
lem (1.1). Numerous studies have investigated solutions for elliptic partial differential equations
(PDEs) with singularity. Many authors have studied the nonlocal problem

(−∆p)
su =

λa(x)

uγ
+Mf(x, u) in Ω,

u = 0 in RN \ Ω,
u > 0 in Ω,

(1.5)

where N > ps, M ≥ 0, and a : Ω → R is a nonnegative bounded function. In the case where
M = 0, the purely singular problem was investigated by Canino et al. [6]. The authors proved that
when 0 < γ < 1, problem (1.5) possesses a unique solution u in the space W s,p

0 (Ω). Additionally,
they proved that the essential infimum of u over K is greater than zero, i.e., ess infK u > 0. In
[18], variational techniques were used to establish multiple results for f(x, u) = u2∗s−1. Using the
Nehari method, Ghanmi et al. [11] obtained multiple results for 0 < γ < 1 and λ = 1. Additionally,
Alomair et al. [1] considered M = b(x) and f(x, u) = u2∗s−1, employing sub and supersolution
methods along with monotonicity arguments to derive both existence and non-existence results.
Lastly, in Daouas et al. [8], variational methods were combined with a perturbation approach to
establish multiple results.

The study of the existence and multiplicity of solutions, as well as their regularity, for regular
PDEs involving nonlocal operators and Sobolev-Hardy nonlinearities has indeed attracted consid-
erable interest in recent years. In particular, works such as those by Chen et al. [7], Yan [24], as
well as Alotaibi et al. [2], along with the references therein, have contributed significantly to this
field. These studies have explored various aspects of nonlocal problems, particularly focusing on
the critical growth of nonlinearities and the implications of Sobolev embeddings.

It is important to highlight that research on singular PDEs with Hardy-Sobolev nonlinearities
in a fractional setting has not been as extensively developed as its classical counterpart. Although
there are foundational results for regular PDEs, such as the existence, multiplicity, and regularity
of solutions, the analysis of singular problems involving fractional operators remains compara-
tively limited. For instance, although some studies have addressed the existence of solutions
for singular fractional PDEs, key questions regarding their qualitative properties (e.g. regular-
ity, asymptotic behavior, or stability) and the optimality of functional frameworks (e.g. weighted
fractional Sobolev spaces) are far from being fully resolved. This gap presents an opportunity
for further investigation into singular nonlocal problems, particularly those combining fractional
diffusion operators (e.g. the fractional Laplacian), Hardy-type singular potentials, and critical
nonlinearities tied to fractional Sobolev embeddings.

In this article, we will explore both the existence and nonexistence of solutions to the nonlocal
problem (1.1), as well as investigate its regularity. It is essential to note that including weight
functions f(x) and g(x) in problem (1.1) is significant, as they enhance the mathematical richness
of our model and broaden its applicability to real-world scenarios. For example, these weight
functions can represent varying material properties in inhomogeneous media, account for spatial
differences in resource availability in ecological contexts, and play a crucial role in optimization



EJDE-2025/99 NONLOCAL PROBLEMS WITH VARIABLE WEIGHTS 3

problems. By incorporating weight functions, we address new mathematical challenges and ensure
that our work remains relevant to practical applications across various fields.

Now, we outline the main results that will be established, related to problem (1.1).

Theorem 1.1. Suppose 0 < γ < 1 and that the weight functions f, g ∈ L∞(Ω) satisfy condition
(1.2). Then, there exists a positive constant Λ1 with the following properties:

(1) For every λ ∈ (0,Λ1), problem (1.3) admits a minimal solution uλ.
(2) Problem (1.3) has a solution when λ = Λ1.
(3) Problem (1.3) has no solutions for λ > Λ1.

Theorem 1.2. Consider 0 < γ < 1 and the weight functions f, g ∈ L∞(Ω) that satisfy condition
(1.2). Then, there exists a positive constant Λ2 such that

(1) For λ > Λ2, problem (1.4) has at least one solution.
(2) For λ < Λ2, problem (1.4) has no solutions.

The third result concerns the regularity of weak solutions of problem (1.1).

Theorem 1.3. Suppose u is a weak positive solution to problem (1.1). Then, there exists a value
α ∈ (0, s] such that u belongs to the Hölder space Cα

loc(Ω
′) for every compactly embedded subset

Ω′ ⋐ Ω.

We highlight that the analysis of problem (1.1) has yielded valuable insights into fractional
elliptic equations with weight functions. It demonstrates how these weight functions can alter the
properties of solutions, affecting their existence, uniqueness, and regularity, which is crucial for
modeling situations where material properties vary. We have established new results showing that
solutions can still exhibit desirable behavior, even in the presence of singularities. Additionally,
we have refined methods for identifying sub- and super-solutions, enhancing our understanding of
how weight functions influence solutions in nonlinear contexts. The interaction between weight
functions and critical nonlinear terms has also been clarified, revealing how they can lead to
multiple solutions or impact the stability of a solution. These findings are relevant in various fields,
including materials science, biology, and finance. Looking to the future, research can explore more
complex systems with multiple weight functions, develop specific numerical methods for these
equations, investigate higher-order fractional models, and examine applications in control theory.
Overall, this work opens many new avenues for research and practical applications in different
scientific disciplines.

This manuscript is structured as follows: Section 2 provides essential foundational results that
will be utilized in subsequent sections. The proofs of Theorems 1.1 and 1.2 can be found in
Sections 3 and 4. Section 5 addresses the regularity of weak solutions, as detailed in Theorem 1.3.

2. Functional framework and preliminary results

In this section, we introduce key definitions, notation, and function spaces that will be used
throughout this work. We define the fractional Sobolev space as

W s,p
0 (RN ) :=

{
u ∈ Lp(RN ) : u is measurable , |u|s,p < +∞, u ≡ 0 a.e. onΩc

}
.

In addition, the homogeneous fractional Sobolev space is defined as

W s,p(RN ) :=
{
u ∈ Lp(RN ) : u is measurable , |u|s,p < +∞

}
.

The associated Gagliardo norm for these spaces is defined as

∥u∥s,p :=
(
∥u∥pp + |u|ps,p

)1/p

.

For a comprehensive overview of the properties of the space W s,p(RN ) we refer to [19] and refer-
ences therein.

Now, let Ω ⊂ RN and define Q = R2N \ ((RN \Ω)× (RN \Ω)). The space (X, ∥.∥X) is defined
as:

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

|u(x)− u(y)|
|x− y|

N+ps
p

∈ Lp(Q)
}



4 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

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 denotes the Lp-norm of u. We also define the subspace

X0 =
{
u ∈ X : u = 0 a.e. in RN \ Ω

}
which is equipped with the norm

∥u∥ =
(∫

Q

|u(x)− u(y)|p

|x− y|N+ps
dx dy

)1/p

.

Lastly, for u ∈ Lp∗
s(t)(RN ), we define the norm

|u|p∗
s(t)

=
(∫

RN

|u|p∗
s(t)

|x|t
dx

)1/p∗
s(t)

.

3. Study of problem (1.3)

Our goal is to demonstrate the existence of solutions for specific values of the parameter λ. To
start, we introduce the concept of a weak solution of problem (1.3).

Definition 3.1. A function uλ ∈ X0 is called a weak solution of problem (1.3) if, for every test
function ϕ ∈ X0, the following conditions hold

(i) uλ > 0, u−γ
λ ϕ ∈ L1(Ω),

(ii) ∫
Q

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

|x− y|N+sp
dx dy

= λ

∫
Ω

f(x)u−γ
λ ϕdx+

∫
Ω

g(x)|uλ|p
∗
s(t)−2uλ

|x|t
ϕ dx

(3.1)

We now present the concepts of a subsolution and a supersolution for problem (1.3).

Definition 3.2. A function uλ is defined as a weak subsolution of problem (1.3) if it satisfies

(1) The function is strictly positive on the domain. Additionally, the term u−γ
λ ϕ is integrable

on Ω;
(2) For every nonnegative test function ϕ ∈ X0,∫

Q

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

|x− y|N+ps
dx dy

−
∫
Ω

λf(x)

uγ
λ

ϕdx−
∫
Ω

g(x)|uλ|p
∗
s(t)−2uλ

|x|t
ϕ dx ≤ 0.

Definition 3.3. A function ūλ is defined as a weak supersolution of problem (1.3) if it satisfies

(1) The function is strictly positive on the domain. Additionally, the term ū−γ
λ ϕ is integrable

on Ω;
(2) For every nonnegative test function ϕ ∈ X0,∫

Q

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

|x− y|N+ps
dx dy

−
∫
Ω

λf(x)

ūγ
λ

ϕdx−
∫
Ω

g(x)|ūλ|p
∗
s(t)−2ūλ

|x|t
ϕdx ≥ 0.

Now, we define the functional Iλ : X0 → R as

Iλ(u) =
1

p
∥u∥p − λ

1− γ

∫
Ω

f(x)|u|1−γ dx− 1

p∗s(t)

∫
Ω

g(x)|u|p∗
s(t)

|x|t
dx. (3.2)



EJDE-2025/99 NONLOCAL PROBLEMS WITH VARIABLE WEIGHTS 5

Next, we introduce the critical value

Λ1 := inf
{
λ > 0 : problem (1.3) admits no weak solution

}
.

We now establish the following key result.

Lemma 3.4. For λ ∈ (0,Λ1), problem (1.3) has at least one weak solution.

Proof. To demonstrate the existence of a solution for problem (1.3), we will construct a well-
ordered pair of functions: a weak subsolution uλ and a weak supersolution uλ such that the
inequality uλ ≤ uλ holds throughout the domain.

First, let λ1 denote the principal eigenvalue of the fractional p-Laplacian operator (−∆p)
s in the

domain Ω. Specifically, λ1 and its corresponding eigenfunction ϕ1 satisfy the eigenvalue problem

(−∆p)
sϕ1 = λ1h(x)|ϕ1|p−2ϕ1 in Ω,

ϕ1 > 0 in Ω,

ϕ1 = 0 in RN \ Ω,
where h(x) = min{f(x), g(x)}. Here, the eigenfunction ϕ1 is positive and belongs to L∞(Ω). For
further details, we refer the reader to [15].

Assume λ1 ≤ λ. Define the subsolution uλ = ϵϕ1, where ϵ > 0 is a small constant. Then

(−∆p)
s(uλ) = λ1ϵ

p−1h(x)ϕp−1
1 ≤ λϵp−1h(x)ϕp−1

1 ≤ λh(x)ϵγϕ−γ
1 .

Since uλ = ϵϕ1, we have

λh(x)ϵγ

ϕ−γ
1

≤ h(x)
( λϵγ

ϕ−γ
1

+
ϵp

∗
s(t)−1|ϕ1|p

∗
s(t)−2ϕ1

|x|t
)

≤ λf(x)u−γ
λ +

g(x)|uλ|p
∗
s(t)−2uλ

|x|t
.

Thus, uλ satisfies

(−∆p)
s(uλ) ≤ λf(x)u−γ

λ +
g(x)|uλ|p

∗
s(t)−2uλ

|x|t
,

which shows that uλ is a subsolution of problem (1.3).
Now, let h(x) = max{f(x), g(x)} and assume λ1 ≥ λ. Define the supersolution uλ = Cϕ1,

where C > 0 is a sufficiently large constant. Then

(−∆p)
s(uλ) = λ1C

p−1h(x)ϕp−1
1 ≥ λCp−1h(x)ϕp−1

1 ≥ λh(x)Cγϕ−γ
1 .

Since uλ = Cϕ1, we have

λh(x)Cγ

ϕ−γ
1

≥ h(x)
(λCγ

ϕ−γ
1

+
Cp∗

s(t)−1|ϕ1|p
∗
s(t)−2ϕ1

|x|t
)
≥ h(x)

( λ

u−γ
λ

+
|uλ|p

∗
s(t)−2uλ

|x|t
)
.

Thus, uλ satisfies

(−∆p)
s(uλ) ≥ λf(x)u−γ

λ +
g(x)|uλ|p

∗
s(t)−2uλ

|x|t
,

which proves that uλ is a supersolution of problem (1.3).
To ensure uλ ≤ uλ, we choose ϵ > 0 small enough and C > 0 large enough such that

ϵϕ1 ≤ Cϕ1 for all x ∈ Ω.

This holds trivially since ϵ ≤ C.
By the weak comparison principle (see Lemma 3.1 in Saoudi et al. [20]), we have

uλ ≤ uλ.

Therefore, using the classical iteration method, we conclude that problem (1.3) admits a solution
uλ that satisfies uλ ≤ uλ ≤ uλ for all λ < Λ1. □

.

Lemma 3.5. For all λ ∈ (0,Λ1), problem (1.3) has a minimal solution uλ.



6 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

Proof. To prove the existence of a minimal solution, we introduce the monotone iterative scheme

(−∆p)
sun − λf(x)u−γ

n =
g(x)|un−1|p

∗
s(t)−2un−1

|x|t
in Ω,

un|∂Ω = 0.

We initialize the scheme with u0 = uλ, where uλ is the unique solution to the purely singular
problem, as established in the work of Canino et al. [6],

(−∆p)
su = λf(x)u−γ in Ω,

u|∂Ω = 0, u > 0 in Ω.

According to the weak comparison principle (as detailed in Saoudi et al. [20, Lemma 3.1]), it
follows that u0 is a weak subsolution to (P)+ and fulfills condition u0 ≤ v, where v denotes
any weak solution of (P)+. Furthermore, applying the weak comparison principle once more,
we can demonstrate that the sequence {un}∞n=1 is not monotonically decreasing, which means
u0 ≤ u1 ≤ u2 ≤ · · · . Furthermore, since un ≤ v for every n, the sequence {un} remains uniformly
bounded in X0.

From the monotonicity and uniform boundedness of the sequence {un}, it follows that {un}
converges weakly in X0 and pointwise to a function uλ, which serves as a weak solution to (1.3).

To establish that uλ is the minimal solution, let ũλ represent any weak solution to (1.3) for
0 < λ < Λ1. Since u0 = uλ ≤ ũλ, the weak comparison principle allows us to assert that un ≤ ũλ

for all n ≥ 0. Taking the limit as n → ∞, we derive uλ ≤ ũλ. This establishes that uλ is in fact
the minimal solution to (1.3). □

Lemma 3.6. Problem (1.3) admits a solution when λ = Λ1.

Proof. Consider a sequence {λn}n∈N such that λn ↑ Λ1 as n → ∞. According to Lemma 3.4,
we know that there exists a weak positive solution un = uλn ≥ uλn

to problem (1.3) for the
corresponding parameter. Consequently, for any ϕ ∈ C∞

c (Ω), we can derive that∫
Q

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

|x− y|N+sp
dx dy

= λn

∫
Ω

f(x)u−γ
n ϕdx+

∫
Ω

g(x)|un|p
∗
s(t)−2un

|x|t
ϕdx.

(3.3)

Since un ∈ X0 and un ≥ uλn
, it is evident that (3.3) holds for ϕ ∈ X0 as well. Furthermore, we

have ∫
Q

|un(x)− un(y)|p

|x− y|N+sp
dx dy − λn

∫
Ω

f(x)u1−γ
n dx−

∫
Ω

g(x)|un|p
∗
s(t)

|x|t
dx = 0. (3.4)

Again, using Lemma 3.4, we deduce that

Iλn
(un) ≤ Iλn

(uλn
) =

1

p

∫
Q

|uλn
(x)− uλn

(y)|p

|x− y|N+sp
dx dy − λn

1− γ

∫
Ω

f(x)u1−γ
λn

dx < 0, (3.5)

because un is a minimizer of Iλn
over {u ≥ uλn

}, and the subsolution uλn
is constructed to have

negative energy because of the singular term.
So, substituting Eq. (3.4) into Eq. (3.5), we obtain

1

p

(
λn

∫
Ω

f(x)u1−γ
n dx+

∫
Ω

g(x)|un|p
∗
s(t)

|x|t
dx

)
− λn

1− γ

∫
Ω

f(x)u1−γ
n dx− 1

p∗s(t)

∫
Ω

g(x)|un|p
∗
s(t)

|x|t
dx < 0.

Therefore, (1
p
− 1

p∗s(t)

) ∫
Ω

g(x)|un|p
∗
s(t)

|x|t
dx < λn

( 1

1− γ
− 1

p

) ∫
Ω

f(x)u1−γ
n dx. (3.6)

Insetting (3.6) into (3.4), and using the Sobolev embedding along with the fact that f ∈ L∞(Ω),
we derive the inequality

∥un∥p+γ−1
X0

≤ C1 (3.7)



EJDE-2025/99 NONLOCAL PROBLEMS WITH VARIABLE WEIGHTS 7

where C1 is a positive constant. This leads us to easily conclude that

sup
n∈N

∥un∥X0 < ∞.

Thus, given that the space X0 is reflexive, we can assert the existence of a subsequence, still
denoted by {un}, such that un ⇀ uΛ1 in X0 as n approaches ∞. By taking the limit of (3.3) as
n → ∞, we obtain∫

Q

|uΛ1
(x)− uΛ1

(y)|p−2(uΛ1
(x)− uΛ1

(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

= Λ1

∫
Ω

f(x)u−γ
Λ1

ϕdx+

∫
Ω

g(x)|uΛ1
|p∗

s(t)−2uΛ1

|x|t
ϕdx.

We deduce that uΛ1 is a weak solution to problem (1.3). □

We shall now prove that Λ1 is finite.

Lemma 3.7. Problem (1.3) has no solution if λ > Λ1.

Proof. We proceed by contradiction. Assume that there exists a sequence λn → ∞ for which
problem (1.3) has a solution un. Setting

h := min
{
ess infx∈Ω f(x), ess infx∈Ω g(x)

}
> 0. (3.8)

So, we derive the existence of Λ1 > 0 satisfying

h
(
λr−γ +

|r|p∗
s(t)−2r

|x|t
)
≥ (λ1 + δ)rp−1 for all r > 0, δ ∈ (0, 1) and λ > Λ1 (3.9)

where λ1 is the principal eigenvalue of (−∆p)
s in Ω.

Fix λn > Λ1. It is simple to see that un is a supersolution of problem

(−∆)spu = (λ1 + δ)up−1 in Ω,

u > 0, u|∂Ω = 0,
(3.10)

for all δ ∈ (0, 1). According to the results presented in [6, Theorem 1.2], we can select ϵ < λ1 + δ
sufficiently small such that ϵϕ1(x) < un(x) ensures that ϵϕ1 is a valid subsolution to problem
(3.10). Through a monotone iteration process, we can derive a solution to problem (3.10) for any
δ ∈ (0, 1), which contradicts the assertion that λ1(s, p) is a positive, simple and isolated point in
the spectrum of (−∆)sp in X0 (refer, e.g. [10, Theorems 4.9 and 4.11]). □

Proof of Theorem 1.1. Its assertion can be derived from the results established in Lemmas 3.5,
3.6, and 3.7. □

4. Study of problem (1.4)

Our main objective is to investigate the existence of solutions by analyzing the critical points
of the associated functional energy. To this end, we define the functional Eλ : X0 → R, as

Eλ(u) =
1

p
∥u∥p − λ

1− γ

∫
Ω

f(x)|u|1−γ dx+
1

p∗s(t)

∫
Ω

g(x)|u|p∗
s(t)

|x|t
dx.

Here, the functional Eλ is considered within the Sobolev space X0, which serves as the appro-
priate functional setting for this variational approach. The existence of solutions is determined
by identifying the critical points of Eλ, which correspond to weak solutions to problem (1.4). To
establish the main result, namely Theorem 1.2, we proceed through a sequence of intermediate
steps, each formulated as a lemma. The first key step in our analysis is to establish the coerciv-
ity of the functional energy Eλ. This property plays a crucial role in ensuring the boundedness
of minimizing sequences and, consequently, in proving the existence of solutions via variational
methods. Now, we shall prove the following lemma.

Lemma 4.1. The energy functional Eλ is coercive.



8 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

Proof. We begin by applying Hölder’s inequality with the exponents

q =
p∗s(t)

1− γ
and q′ =

p∗s(t)

p∗s(t)− (1− γ)
.

This yields the estimate∫
Ω

f(x)|u|1−γ dx ≤ ∥f∥L∞

(∫
Ω

|u|p
∗
s(t) dx

) 1−γ
p∗s (t) |Ω|

p∗s (t)−(1−γ)

p∗s (t) ,

where |Ω| denotes the measure of the domain Ω.
Next, we bound the term

∫
Ω
|u|p∗

s(t) dx using the fractional Sobolev-Hardy inequality,

∥u∥Lp∗s (t) ≤ S∥u∥,
where S > 0 is the best Sobolev constant associated with the fractional Sobolev space X0 and the
dimension N . Applying this inequality, we obtain(∫

Ω

|u|p
∗
s(t) dx

) 1−γ
p∗s (t) ≤ S1−γ∥u∥1−γ .

Substituting this estimate into the singular term, we obtain

λ

∫
Ω

f(x)|u|1−γ

1− γ
dx ≤ C1∥u∥1−γ ,

where

C1 = λ
∥f∥L∞

1− γ
S1−γ |Ω|

p∗s (t)−(1−γ)

p∗s (t) .

Therefore, using the fractional Sobolev-Hardy inequality once again, we have∫
Ω

g(x)|u|p∗
s(t)

|x|t
dx ≥

(
ess inf
x∈Ω

g(x)
)
∥u∥p

∗
s(t)

Lp∗s (t) .

Since ∥u∥Lp∗s (t) ≥ 0, it follows that

Eλ(u) ≥
1

p
∥u∥p − C1∥u∥1−γ .

Noting that 1−γ < p, we conclude that Eλ(u) → +∞ as ∥u∥ → ∞. Therefore, Eλ is coercive. □

Lemma 4.2. The energy functional Eλ has a global minimizer.

Proof. Let {uk}k be a minimizing sequence for the functional Eλ in the space X0. By Lemma 4.1,
the sequence {uk}k is bounded in X0. Moreover, since Eλ(u) = Eλ(|u|), we may assume without
loss of generality that {uk}k is nonnegative, that is, uk ≥ 0 for all k.

By the reflexivity of X0, there exists a subsequence, still denoted by {uk}, and a function
u ∈ X0 such that

uk ⇀ u weakly in X0,

uk ⇀ u weakly in Lp∗
s(t)(Ω),

uk → u strongly in Lr(Ω,
dx

|x|t
) for r ∈ [1, p∗s(t)),

uk → u almost everywhere in Ω.

These convergence properties are a consequence of compact embeddings X0 ↪→ L1−γ(Ω) and
X0 ↪→ Lp∗

s(t)(Ω), as well as the weak lower semi-continuity of the norm ∥ · ∥.
From the weak lower semicontinuity of the norm ∥ · ∥, we have

∥u∥p ≤ lim inf
k→∞

∥uk∥p.

Additionally, the strong convergence in Lr(Ω, dx
|x|t ) and the convergence almost everywhere imply∫

Ω

f(x)u1−γ dx = lim
k→∞

∫
Ω

f(x)u1−γ
k dx,



EJDE-2025/99 NONLOCAL PROBLEMS WITH VARIABLE WEIGHTS 9∫
Ω

g(x)|u|p∗
s(t)

|x|t
dx = lim

k→∞

∫
Ω

g(x)|uk|p
∗
s(t)

|x|t
dx.

Combining these results, we obtain

Eλ(u) ≤ lim inf
k→∞

Eλ(uk).

Since {uk} is a minimizing sequence for Eλ, it follows that u is a global minimizer of Eλ in X0. □

The next Lemma guaranties the existence of at least one solution for sufficiently large values
of the parameter λ.

Lemma 4.3. There exists a positive number Λ2 such that for all λ > Λ2, problem (1.4) admits at
least one solution.

Proof. We begin by introducing the constrained variational problem

λ∗ := inf
{1
p
∥w∥p+ 1

p∗s(t)

∫
Ω

g(x)|w|p∗
s(t)

|x|t
dx : w ∈ X0 and

1

1− γ

∫
Ω

f(x)|w|1−γ dx = 1
}
. (4.1)

Additionally, we define

Λ2 := inf
{
λ > 0 : (1.4) admits a nontrivial weak solution

}
. (4.2)

Consider a sequence {vk}k to be a minimizing sequence for λ∗. Employing an argument similar
to the one in Lemma 4.2, we may assume that {vk}k converges weakly to some v ∈ X0, satisfying

λ∗ =
1

p
∥v∥p + 1

p∗s(t)

∫
Ω

g(x)|v|p∗
s(t)

|x|t
dx,

1

1− γ

∫
Ω

f(x)|v|1−γ dx = 1.

Consequently, for any λ > λ∗, we have Eλ(v) = λ∗ − λ < 0.
Now, fix λ > Λ2. By the definition of Λ2, there exists β ∈ (Λ2, λ) such that the functional Eβ

has a non-trivial critical point uβ ∈ X0. Since β < λ, uβ serves as a sub-solution to problem (1.4).
To construct a super-solution, consider the minimization problem

inf
{1
p
∥w∥p − λ

1− γ

∫
Ω

f(x)|w|1−γ dx+
1

p∗s(t)

∫
Ω

g(x)|w|p∗
s(t)

|x|t
dx : w ∈ X0, w ≥ uβ

}
. (4.3)

Then, using arguments analogous to those for problem (4.1), we deduce that problem (4.3) admits
a solution wλ satisfying wλ > uβ . Furthermore, for all λ > Λ2, the function wλ remains also a
weak solution to problem (1.4). Using the reasoning described in [20], we infer that a solution
exists for (1.4) when λ = Λ2.

Finally, the positivity of uλ follows from the strong maximum principle (see [16, Lemma 2.3]),
as uλ is a weak C1 positive solution of the differential inequality

(−∆p)
su+

g(x)|u|p∗
s(t)−2u

|x|t
≥ 0 in Ω.

We derive that uλ is nonnegative everywhere in Ω. This completes the proof. □

Lemma 4.4. There exists a positive number Λ2 such that if λ < Λ2, problem (1.1) does not have
solution.

Proof. Fix δ ∈ (0, λ1), where λ1 is the principal eigenvalue. Set Λ2 = min{1, δ} > 0. We now
claim that for all λ ∈ (0,Λ2) and for all r > 0, the following inequality holds

λf(x)r−γ − g(x)rp
∗
s(t)−2r

|x|t
< δrp−1. (4.4)

Indeed, first observe that for λ < Λ2 ≤ 1 and 0 < γ < 1 < p∗s(t), we derive for all r ∈ [0, 1] that

λf(x)r−γ − g(x)rp
∗
s(t)−2r

|x|t
< 0.



10 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

Furthermore, for r > 1 and since 1− γ < p, we deduce that

λf(x)r−γ − g(x)rp
∗
s(t)−2r

|x|t
< λf(x)r−γ < δrp−1

for all Λ2 ∈ [λ, δ]. Which proves (4.4). We now consider λ ∈ (0,Λ2) and let u be a critical point
to the energy functional Eλ. Therefore,

∥u∥p
W s,p

0 (Ω)
= ⟨(−∆p)

su, u⟩

= λ

∫
Ω

f(x)|u|1−γ dx−
∫
Ω

g(x)|u|p∗
s(t)

|x|t
dx

≤ δ∥u∥p.

(4.5)

On the other hand, using the variational description of λ1, we have

λ1 = inf
u∈W s,p

0 (Ω)

∥u∥p
W s,p

0 (Ω)

∥u∥pLp(Ω)

.

This implies

∥u∥p
W s,p

0 (Ω)
≥ λ1∥u∥pLp(Ω). (4.6)

Combining the two inequalities (4.5) and (4.6), we obtain

λ1∥u∥p ≤ ∥u∥p
W s,p

0 (Ω)
≤ δ∥u∥p.

Since δ < λ1, this implies

λ1∥u∥p ≤ δ∥u∥p.

If ∥u∥p ̸= 0, then dividing both sides by ∥u∥p gives λ1 ≤ δ. But this contradicts the assumption
that δ < λ1. Therefore, the only possibility is ∥u∥p = 0, which implies u = 0 almost everywhere
in Ω. Consequently (1.4) has no solution if λ < Λ2. □

Proof of Theorem 1.2. The proof follows directly from Lemma 4.3 and Lemma 4.4. □

5. Regularity results

In this section, we study the regularity of weak solutions to problem (1.1). We begin by
recalling the following elementary inequalities, which are proved in [4, 5]. These inequalities serve
as fundamental tools in our analysis.

Lemma 5.1 ([4, Lemma C.2 ]). Let 1 < p < ∞ and r ≥ 1. For any non-negative real numbers a,
b, and M , it holds

|a− b|p−2(a− b)(arM − brM ) ≥ rpp

(r + p− 1)p
∣∣a r+p−1

p

M − b
r+p−1

p

M

∣∣,
with aM = min{a,M} and bM = min{b,M}.

Lemma 5.2 (cite[Lemma C.3]BrPa). Let 1 < p < ∞ and an increasing function g : R → R. For
each a, b ∈ R, it holds

|G(a)−G(b)|p ≤ |a− b|p−2(a− b)(g(a)− g(b)),

with G(t) =
∫ t

0
g′(τ)1/p dτ for t ∈ R.

Now, we establish the boundedness of weak solutions to problem (1.1). The proof relies on
iterative techniques and careful estimates involving fractional Sobolev spaces.

Lemma 5.3. If u is a weak solution to problem (1.1), then u belongs to L∞(Ω).



EJDE-2025/99 NONLOCAL PROBLEMS WITH VARIABLE WEIGHTS 11

Proof. The proof adapts methods from [2], which are inspired by the work in [5]. This method has
been further incorporated and applied in the current context. Setting ul = min{(u− 1)+, l} with
l > 0. Also, let us define the function φ by φ = (ul + θ)r − θr with θ > 0 and r ≥ 1. Combining
lemma 5.1 with the triangle inequality and applying it to problem (1.1) with φ ∈ X0, we obtain
the key inequality

rpp

(r + p− 1)p

∫
R2N

∣∣((ul(x) + θ))
r+p−1

p (x)− ((ul(y) + θ))
r+p−1

p (y)
∣∣dxdy

≤
∫
R2N

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

)(
(ul(x) + θ)r − (ul(y) + θ)r

)
|x− y|N+sp

dx dy

≤
∫
Ω

λu−γ((ul + θ)r − θr) dx+

∫
Ω

|u|p∗
s(t)−2u

|x|t
((ul + θ)r − θr) dx.

(5.1)

Then with the help of the Hölder inequality combine with the support of ul, we derive∫
Ω

λu−γ((ul + θ)r − θr) dx+

∫
Ω

|u|p∗
s(t)−2u

|x|t
((ul + θ)r − θr) dx

=

∫
{u≥1}

λu−γ((ul + θ)r − θr) dx+

∫
{u≥1}

|u|p∗
s(t)−2u

|x|t
((ul + θ)r − θr) dx

≤ K1

∫
{u≥1}

(
1 +

|u|p∗
s(t)−2u

|x|t
)
((ul + θ)r − θr) dx

≤ 2K1

∫
{u≥1}

|u|p∗
s(t)−2u

|x|t
((ul + θ)r − θr) dx

≤ 2K1|u|
p∗
s(t)−1

p∗
s

|(ul + θ)r|q

(5.2)

with K1 = max{λ, 1} and q = p∗s/(p
∗
s + 1 − p∗s(t)). Furthermore, applying [19, Theorem 1] with

inequality (ul + θ)r ≤ (ul + θ)r+p−1θ1−p, we obtain∫
R2N

∣∣∣(ul(x) + θ)
r+p−1

p − (ul(y) + θ)
r+p−1

p

∣∣∣p dx dy

≥ CN,p,s

(∫
RN

(
(ul(x) + θ)

r+p−1
p − θ

r+p−1
p

)p∗
s

dx
) p

p∗s

≥
(θ
2

)p−1
CN,p,s

(∫
RN

(
(ul(x) + θ)

r+p−1
p

)p∗
s dx

) p
p∗s − θr+p−1|Ω|

p
p∗s

(5.3)

with CN,p,s being a positive constant. From (5.2) and (5.3), we derive(∫
RN

(
(ul(x) + θ)

r+p−1
p

)p∗
s dx

) p
p∗s ≤ CN,p,s

|u|p
∗
s(t)−1

p∗
s

r

(r + p− 1

pθ
p−1
p

)p−1

|(ul + θ)r|q + θr|Ω|
p
p∗s . (5.4)

Now, noting that for r ≥ 1, we have

rpp

(r + p− 1)p
≥

( p

r + p− 1

)p−1

. (5.5)

Applying (5.5), it is easy to see that

θr|Ω|
p
p∗s ≤ 1

r

(r + p− 1

p

)p

|Ω|1−
1
q−

sp
N |(ul + θ)r|q. (5.6)

Therefore, using (5.6) in (5.4), we obtain(∫
RN

(
(ul(x) + θ)

r+p−1
p

)p∗
s dx

) p
p∗s

≤ CN,p,s

(r + p− 1

pθ
p−1
p

)p−1

|(ul + θ)r|q ×
( |u|p∗

s(t)−1
p∗
s

θp−1
+ |Ω|1−

1
q−

sp
N

)
.

(5.7)



12 S. ALMUTAIRI, K. SAOUDI EJDE-2025/99

We now fix θ > 0 as

θ =
(
|u|

p∗s (t)−1

p−1

p∗
s

+ |Ω|
1

p−1 (1−
1
q−

sp
N )

)
Setting w = rq and σ =

p∗
s

pq > 1. Therefore, (5.7) can be formulated as(∫
Ω

(ul(x) + θ)wσ dx
) 1

wσ ≤
(
CN,p,s|Ω|1−

1
q−

sp
N

) q
w ( q

w

) q
w

(w + qp− q

q′p

) pq
w |(ul + θ)|w. (5.8)

So, we now iterate (5.8) using

w0 = 1 wk+1 = wkσ = σk+1.

By initiating from k = 0 at step k, we can express the inequality (5.8) as

|(ul + θ)|wk+1
≤

(
CN,p,s|Ω|1−

1
q−

sp
N

)∑k
i=0

q
wi

k∏
i=0

( q

wi

)q/wi
(wi + qp− q

qp

)pq/wi

|(ul + θ)|w. (5.9)

We now notice that wk diverges at infinity, and furthermore
∞∑
k=0

1

wk
=

∞∑
k=0

1

σk
=

σ

σ − 1
and

∞∏
k=0

( q

wk

)q/wk
(w + qp− q

qp

)pq/w

< +∞.

Through an infinite iteration of inequality (5.9), we arrive at

|ul|∞ ≤
(
CN,p,s|Ω|1−

1
q−

sp
N

)q σ
σ−1 |(ul + r)|q.

Given that ul ≤ (u− 1)+, we obtain

|ul|∞ ≤
(
CN,p,s|Ω|1−

1
q−

sp
N

)q σ
σ−1

(
|(u− 1)+|q + θ|Ω|1/q

)
.

Taking l → ∞, it follows that

|(u− 1)+|∞ ≤
(
CN,p,s|Ω|1−

1
q−

sp
N

)q σ
σ−1

(
|(u− 1)+|q + θ|Ω|1/q

)
.

Thus, in particular, we conclud that u ∈ L∞(Ω). □

Proof of Theorem 1.3. The proof follows from Lemma 5.3 and [17, Theorem 6.5]. □

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Sarah Almutairi

College of sciences at Dammam, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of
Saudi Arabia.

Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982,

31441, Dammam, Saudi Arabia
Email address: 2240500248@iau.edu.sa

Kamel Saoudi

College of sciences at Dammam, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of
Saudi Arabia.

Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982,
31441, Dammam, Saudi Arabia

Email address: kmsaoudi@iau.edu.sa


	1. Introduction
	2. Functional framework and preliminary results
	3. Study of problem (??)
	4. Study of problem (??)
	5. Regularity results
	References