CUBO, A Mathematical Journal

Vol. 24, no. 01, pp. 63–82, April 2022

DOI: 10.4067/S0719-06462022000100063

The topological degree methods for the fractional
p(·)-Laplacian problems with discontinuous
nonlinearities

Hasnae El Hammar
1

Chakir Allalou
1

Adil Abbassi
1

Abderrazak Kassidi
1

1Laboratory LMACS, FST of

Beni-Mellal, Sultan Moulay Slimane

University, Morocco.

hasnaeelhammar11@gmail.com

chakir.allalou@yahoo.fr.

abbassi91@yahoo.fr

abderrazakassidi@gmail.com

ABSTRACT

In this article, we use the topological degree based on the

abstract Hammerstein equation to investigate the existence

of weak solutions for a class of elliptic Dirichlet bound-

ary value problems involving the fractional p(x)-Laplacian

operator with discontinuous nonlinearities. The appropri-

ate functional framework for this problems is the fractional

Sobolev space with variable exponent.

RESUMEN

En este art́ıculo, usamos el grado topológico basado en la

ecuación abstracta de Hammerstein para investigar la exis-

tencia de soluciones débiles para una clase de problemas

eĺıpticos de valor en la frontera de Dirichlet que involucran

el operador p(x)-Laplaciano fraccional con no linealidades

discontinuas. El marco funcional apropiado para estos pro-

blemas es el espacio de Sobolev fraccional con exponente

variable.

Keywords and Phrases: Fractional p(x)-Laplacian, weak solution, discontinuous nonlinearity, topological degree

theory.

2020 AMS Mathematics Subject Classification: 35R11, 35J60, 47H11, 35A16.

Accepted: 05 November, 2021

Received: 12 May, 2021

c©2022 H. El Hammar et al. This open access article is licensed under a

Creative Commons Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462022000100063
https://orcid.org/0000-0003-3532-4673
https://orcid.org/0000-0002-4885-9397
https://orcid.org/0000-0003-0284-8390
https://orcid.org/0000-0002-9105-1123
mailto:hasnaeelhammar11@gmail.com
mailto:chakir.allalou@yahoo.fr
mailto:abbassi91@yahoo.fr
mailto:abderrazakassidi@gmail.com


64 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

1 Introduction and main result

The study of fractional Sobolev spaces and the corresponding nonlocal equations has received

a tremendous popularity in the last two decades considering their intriguing structure and great

application in many fields, such as social sciences, fractional quantum mechanics, materials science,

continuum mechanics, phase transition phenomena, image process, game theory, and Levy process,

see [34, 35] and references therein for more details.

On the other hand, in recent years, a great deal of attention has been paid to the study of

differential equations and variational problems involving p(x)-growth conditions since they can

be used to model a variety of physical phenomena that occur in the fields of elastic mechanics,

electro-rheological fluids (”smart fluids”), and image processing, etc. The readers are guided to

[19, 20, 27] and its references.

It is only normal to wonder what results can be obtained when the fractional p(·)-Laplacian is used

instead of the p(·)-Laplacian. The fractional p(·)-Laplacian has also recently been investigated in

elliptic problems; see [8, 10, 25, 26]. U. Kaufmann et al. [26] presented a new class of fractional

Sobolev spaces with variable exponents in a recent paper. The authors in [8, 9] showed some

additional basic properties on this function space as well as the associated nonlocal operator.

They used the critical point theory in [4] to prove the existence of solutions for fractional p(·)-

Laplacian equations. K. Ho and Y.-H. Kim [25] managed to obtain fundamental imbeddings for

a new fractional Sobolev space with variable exponents, which is a generalization of previously

defined fractional Sobolev spaces.

Let Ω ⊂ RN (N ≥ 1) be a bounded open set with Lipschitz boundary and let p : Ω×Ω → (1, +∞)

be a continuous bounded function. The purpose of this paper is to establish the existence of

nontrivial weak solutions for the following fractional p(x)-Laplacian problems with discontinuous

nonlinearities.




(−△p(x))
su(x) + |u(x)|q(x)−2u(x) + λH(x,u) ∈ −

[
ψ(x,u),ψ(x,u)

]
in Ω,

u = 0 on RN\Ω,
(1.1)

where ps < N with 0 < s < 1 and (−△p(x))
s is the fractional p(x)-Laplacian operator defined by

(−∆)sp(x)u(x) = p.v.

∫

RN \Bε(x)

|u(x) − u(y)|p(x,y)−2(u(x) − u(y))

|x − y|N+sp(x,y)
dy, x ∈ RN (1.2)

∀x ∈ Ω, where p.v. is a commonly used abbreviation in the principal value sense and let p ∈

C(RN × RN) satisfying

1 < p− = min
(x,y)∈Ω×Ω

p(x, y) ≤ p(x, y) ≤ p+ = max
(x,y)∈Ω×Ω

p(x, y) < +∞, (1.3)

p is symmetric i. e.

p(x, y) = p(y, x), ∀(x, y) ∈ Ω × Ω; (1.4)



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The topological degree methods for the fractional p(·)-Laplacian... 65

and Bε(x) :=
{
y ∈ RN : |x − y| < ε

}
.

Let as denote by:

p̃(x) = p(x, x), ∀x ∈ Ω.

Furthermore, the Carathéodory’s functions H satisfy only the growth condition, for all s ∈ R and

a. e. x ∈ Ω.

(H0) |H(x,s)| ≤ ̺(e(x) + |s|
q(x)−1),

where ̺ is a positive constant, e(x) is a positive function in Lp
′(x)(Ω).

In the simplest case p = 2, we have the well-known fractional Laplacian, a large amount of papers

were written on this direction see [6, 15]. Moreover, if s = 1, we get the classic Laplacian. Some

related results can be found in [21, 39, 40, 41, 42]. Notice that when s = 1, the problems like (1.1)

have been studied in many papers, we refer the reader to [1, 5, 24], in which the authors have

used various methods to get the existence of solutions for (1.1). In the case when p = p(x) is a

continuous function, problem (1.1) has also been studied by many authors. For more information,

see [11, 23].

In order to prove the existence of nontrivial weak solutions, the main difficulties are reflected in the

following aspect, we cannot directly use the topological degree methods in a natural way because

the nonlinear term ψ is discontinuous. In order to overcome the discontinuous difficulty, we will

transform this Dirichlet boundary value problem involving the fractional p-Laplacian operator

with discontinuous nonlinearities into a new one governed by a Hammerstein equation. Then,

we shall employ the topological degree theory developed by Kim in [29, 28] for a class of weakly

upper semi-continuous locally bounded set-valued operators of (S+) type in the framework of real

reflexive separable Banach spaces, based on the Berkovits-Tienari degree [12]. The topological

degree theory was constructed for the first time by Leray-Schauder [31] in their study of the

nonlinear equations for compact perturbations of the identity in infinite-dimensional Banach spaces.

Furthermore, Browder [14] has developed a topological degree for operators of class (S+) in reflexive

Banach spaces, see also [37, 38]. Among many examples, we refer the reader to the classical works

[2, 3, 18, 45] for more details.

To this end, we always assume that ψ : Ω × R → R is a possibly discontinuous function, we “fill

the discontinuity gaps” of ψ, replacing ψ by an interval
[
ψ(x,u),ψ(x,u)

]
, where

ψ(x,s) = lim inf
η→s

ψ(x,η) = lim
δ→0+

inf
|η−s|<δ

ψ(x,η),

ψ(x,s) = lim sup
η→s

ψ(x,η) = lim
δ→0+

sup
|η−s|<δ

ψ(x,η).

Such that



66 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

(H1) ψ and ψ are super-positionally measurable (i. e., ψ(·,u(·)) and ψ(·,u(·)) are measurable on

Ω for every measurable function u : Ω → R).

(H2) ψ satisfies the growth condition:

|ψ(x,s)| ≤ b(x) + c(x)|s|γ(x)−1,

for almost all x ∈ Ω and all s ∈ R, where b ∈ Lγ
′(x)(Ω), c ∈ L∞(Ω), where 1 < γ(x) < p(x)

for all x ∈ Ω.

First of all, we define the operator N acting from W
s,p(x,y)
0 (Ω) into 2

(
W

s,p(x,y)
0 (Ω)

)
∗

by

Nu =
{
ϕ ∈

(
W

s,p(x,y)
0 (Ω)

)∗
\ ∃ h ∈ Lp

′(x)(Ω);

ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) a. e. x ∈ Ω

and 〈ϕ,v〉 =

∫

Ω

hvdx ∀v ∈ W
s,p(x,y)
0 (Ω)

}
.

In this spirit, we consider F : W
s,p(x,y)
0 (Ω) −→

(
W

s,p(x,y)
0 (Ω)

)∗
such that

〈Fu,v〉 =

∫

Ω×Ω

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

|x − y|N+sp(x,y)
dxdy, (1.5)

for all v ∈ W
s,p(x,y)
0 (Ω) and the operator A : W0 → W

∗
0 setting by

〈Au,v〉 =

∫

Ω

|u(x)|q(x)−2(u(x)v(x) + λH(x,u))v(x)dx, ∀u,v ∈ W0,

where the spaces W
s,p(x,y)
0 (Ω) := W0 will be introduced in Section 2.

Next, we give the definition of weak solutions for problem (1.1).

Definition 1.1. A function u ∈ W
s,p(x,y)
0 (Ω) is called a weak solution to problem (1.1), if there

exists an element ϕ ∈ Nu verifying

〈Fu,v〉 + 〈Au,v〉 + 〈ϕ,v〉 = 0, for all v ∈ W
s,p(x,y)
0 (Ω).

Now we are in a position to present our main result.

Theorem 1.2. Assume that ψ satisfies (H1),(H2) and H satisfies (H0). Then, the problem (1.1)

has a weak solution u in W
s,p(x,y)
0 (Ω).

2 Preliminaries

2.1 Lebesgue and fractional Sobolev spaces with variable exponent

In this subsection, we first recall some useful properties of the variable exponent Lebesgue spaces

Lp(x)(Ω) . For more details we refer the reader to [22, 30, 44].



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The topological degree methods for the fractional p(·)-Laplacian... 67

Denote

C+(Ω) = {h ∈ C(Ω)| inf
x∈Ω

h(x) > 1}.

For any h ∈ C+(Ω) , we define

h+ := max{h(x), x ∈ Ω}, h− := min{h(x), x ∈ Ω}.

For any p ∈ C+(Ω) we define the variable exponent Lebesgue spaces

L
p(x)(Ω) =

{
u; u : Ω → R is measurable and

∫

Ω

|u(x)|p(x)dx < +∞

}
.

Endowed with Luxemburg norm

‖u‖p(x) = inf
{
λ > 0 |ρp(·)

(
u

λ

)
≤ 1

}

where

ρp(·) (u) =

∫

Ω

|u(x)|p(x)dx, ∀u ∈ Lp(x)

(Lp(x)(Ω), ||.||p(x)) is a Banach space, separable and reflexive. Its conjugate space is L
p
′(x)(Ω)

where
1

p(x)
+

1

p′(x)
= 1 for all x ∈ Ω. We have also the following result

Proposition 2.1. ([22]) For any u ∈ Lp(x)(Ω) we have

(i) ‖u‖p(x) < 1(= 1;> 1) ⇔ ρp(·)(u) < 1(= 1;> 1),

(ii) ‖u‖p(x) ≥ 1 ⇒ ‖u‖
p
−

p(x)
≤ ρp(·)(u) ≤ ‖u‖

p
+

p(x)
,

(iii) ‖u‖p(x) ≤ 1 ⇒ ‖u‖
p
+

p(x)
≤ ρp(·)(u) ≤ ‖u‖

p
−

p(x)
.

From this proposition, we can deduce the inequalities

‖u‖p(x) ≤ ρp(·)(u) + 1, (2.1)

ρp(·)(u) ≤ ‖u‖
p
−

p(x)
+ ‖u‖

p
+

p(x)
. (2.2)

If p,q ∈ C+(Ω) such that p(x) ≤ q(x) for any x ∈ Ω, then there exists the continuous embedding

Lq(x)(Ω) → Lp(x)(Ω) .

Next, we present the definition and some results on fractional Sobolev spaces with variable exponent

that was introduced in [8, 26]. Let s be a fixed real number such that 0 < s < 1, and let

q : Ω → (0,∞) and p : Ω×Ω → (0,∞) be two continuous functions. Furthermore, we suppose that

the assumptions (1.3) and (1.4) be satisfied, we define the fractional Sobolev space with variable

exponent via the Gagliardo approach as follows:

W = Ws,,q(x),p(x,y)(Ω) =

{
u ∈ Lq(x)(Ω) :

∫

Ω×Ω

|u(x) − u(y)|p(x,y)

λp(x,y)|x − y|N+sp(x,y)
dxdy < +∞,

∫
for some λ > 0

}
.



68 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

We equip the space W with the norm

‖u‖W = ‖u‖q(x) + [u]s,p(x,y),

where [·]s,p(x,y) is a Gagliardo seminorm with variable exponent, which is defined by

[u]s,p(x,y) = inf

{
λ > 0 :

∫

Ω×Ω

|u(x) − u(y)|p(x,y)

λp(x,y)|x − y|N+sp(x,y)
dxdy ≤ 1

}
.

The space (W, ‖ · ‖W ) is a Banach space (see [17]), separable and reflexive (see [8, Lemma 3.1]).

We also define W0 as the subspace of W which is the closure of C
∞
0 (Ω) with respect to the norm

|| · ||W . From [7, Theorem 2.1 and Remark 2.1]

‖ · ‖W0 := [·]s,p(x,y)

is a norm on W0 which is equivalent to the norm ‖ · ‖W , and we have the compact embedding

W0 →֒→֒ L
q(x). So the space (W0, ‖ · ‖W0) is a Banach space separable and reflexive.

We defne the modular ρp(·,·) : W0 → R by

ρp(·,·)(u) =

∫

Ω×Ω

|u(x) − u(y)|p(x,y)

|x − y|N+sp(x,y)
dxdy.

The modular ρp checks the following results, which is similar to Proposition 2.1 (see [43, Lemma

2.1])

Proposition 2.2. ([30]) For any u ∈ W0 we have

(i) ‖u‖W0 ≥ 1 ⇒ ‖u‖
p
−

W0
≤ ρp(·,·)(u) ≤ ‖u‖

p
+

W0
,

(ii) ‖u‖W0 ≤ 1 ⇒ ‖u‖
p
+

W0
≤ ρp(·,·)(u) ≤ ‖u‖

p
−

W0
.

2.2 Some classes of operators and an outline of Berkovits degree

Now, we introduce the theory of topological degree which is the major tool for our results. We

start by defining some classes of mappings. Let X be a real separable reflexive Banach space with

dual X∗ and with continuous dual pairing 〈 · , · 〉 between X∗ and X in this order. The symbol ⇀

stands for weak convergence. Let Y be another real Banach space.

Definition 2.3.

(1) We say that the set-valued operator F : Ω ⊂ X → 2Y is bounded, if F maps bounded sets

into bounded sets;

(2) we say that the set-valued operator F : Ω ⊂ X → 2Y is locally bounded at the point u ∈ Ω, if

there is a neighborhood V of u such that the set F(V) =
⋃

u∈V

Fu is bounded.



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The topological degree methods for the fractional p(·)-Laplacian... 69

Definition 2.4. The set-valued operator F : Ω ⊂ X → 2Y is called

(1) upper semicontinuous (u.s.c.) at the point u, if, for any open neighborhood V of the set Fu,

there is a neighbhorhood U of the point u such that F(U) ⊆ V . We say that F is upper

semicontinuous (u.s.c) if it is u.s.c at every u ∈ X;

(2) weakly upper semicontinuous (w.u.s.c.), if F−1(U) is closed in X for all weakly closed set U

in Y.

Definition 2.5. Let Ω be a nonempty subset of X, (un)n≥1 ⊆ Ω and F : Ω ⊂ X → 2
X

∗

\ ∅. Then,

the set-valued operator F is

(1) of type (S+), if un ⇀ u in X and for each sequence (hn) in X
∗ with hn ∈ Fun such that

lim sup
n→∞

〈hn,un − u〉 ≤ 0,

we get un → u in X;

(2) quasi-monotone, if un ⇀ u in X and for each sequence (wn) in X
∗ such that wn ∈ Fun yield

lim inf
n→∞

〈wn,un − u〉 ≥ 0.

Definition 2.6. Let Ω be a nonempty subset of X such that Ω ⊂ Ω1, (un)n≥1 ⊆ Ω and T :

Ω1 ⊂ X → X
∗ be a bounded operator. Then, the set-valued operator F : Ω ⊂ X → 2X \ ∅ is of

type (S+)T , if 



un ⇀ u in X,

Tun ⇀ y in X
∗,

and for any sequence (hn) in X with hn ∈ Fun such that

lim sup
n→∞

〈hn,Tun − y〉 ≤ 0,

we have un → u in X.

Next, we consider the following sets :

F1(Ω) := {F : Ω → X
∗|F is bounded, demicontinuous and of type (S+)},

FT (Ω) := {F : Ω → 2
X|F is locally bounded, w.u.s.c. and of type (S+)T },

for any Ω ⊂ DF and each bounded operator T : Ω → X
∗, where DF denotes the domain of F .

Remark 2.7. We say that the operator T is an essential inner map of F, if T ∈ F1(G).

Lemma 2.8. ([29, Lemma 1.4]) Let X be a real reflexive Banach space and G ⊂ X is a bounded

open set. Assume that T ∈ F1(G) is continuous and S : DS ⊂ X
∗ → 2X weakly upper semicon-

tinuous and locally bounded with T(G) ⊂ Ds. Then the following alternative holds:



70 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

(1) If S is quasi-monotone, yield I + S ◦ T ∈ FT (G), where I denotes the identity operator.

(2) If S is of type (S+), yield S ◦ T ∈ FT (G).

Definition 2.9. ([29]) Let T : G ⊂ X → X∗ be a bounded operator, a homotopy H : [0,1]×G → 2X

is called of type (S+)T , if for every sequence (tk,uk) in [0,1]×G and each sequence (ak) in X with

ak ∈ H(tk,uk) such that

uk ⇀ u ∈ X, tk → t ∈ [0,1], Tuk ⇀ y in X
∗ and lim sup

k→∞
〈ak,Tuk − y〉 ≤ 0,

we get uk → u in X.

Lemma 2.10. ([29]) Let X be a real reflexive Banach space and G ⊂ X is a bounded open set,

T : G → X∗ is continuous and bounded. If F, S are bounded and of class (S+)T , then an affine

homotopy H : [0,1] × G → 2X given by

H(t,u) := (1 − t)Fu + tSu, for (t,u) ∈ [0,1] × G,

is of type (S+)T .

Now, we introduce the topological degree for a class of locally bounded, w.u.s.c. and satisfies

condition (S+)T for more details see [29].

Theorem 2.11. Let

L =
{
(F,G,g)|G ∈ O, T ∈ F1(G), F ∈ FT (G), g 6∈ F(∂G)

}
,

where O denotes the collection of all bounded open sets in X. There exists a unique (Hammerstein

type) degree function

d : L −→ Z

such that the following alternative holds:

(1) ( Normalization) For each g ∈ G, we have d(I,G,g) = 1.

(2) ( Domain Additivity) Let F ∈ FT (G). We have

d(F,G,g) = d(F,G1,g) + d(F,G2,g),

with G1, G2 ⊆ G disjoint open such that g 6∈ F(G\(G1 ∪ G2)).

(3) ( Homotopy invariance) If H : [0,1] × G → X is a bounded admissible affine homotopy with

a common continuous essential inner map and g: [0,1] → X is a continuous path in X such

that g(t) 6∈ H(t,∂G) for all t ∈ [0,1], then the value of d(H(t, ·),G,g(t)) is constant for any

t ∈ [0,1].

(4) ( Solution Property) If d(F,G,g) 6= 0, then the equation g ∈ Fu has a solution in G.



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The topological degree methods for the fractional p(·)-Laplacian... 71

3 Proof of Theorem 1.2

In the present section, following compactness methods (see [18, 32]), we prove the existence of weak

solutions for the problem (1.1) in fractional Sobolev spaces. In doing so, we transform this elliptic

Dirichlet boundary value problem involving the fractional p-Laplacian operator with discontinuous

nonlinearities into a new problem governed by a Hammerstein equation. More precisely, by means

of the topological degree theory introduced in section 2, we establish the existence of weak solutions

to the state problem, which holds under appropriate assumptions. First, we give several lemmas.

Lemma 3.1. Let 0 < s < 1 and 1 < p(x,y) < +∞, (or sp+ < N) the operator F defined in (1.5)

is

(i) bounded and strictly monotone operator.

(ii) of type (S+).

Proof. (i) It is clear that F is a bounded operator. For all ξ,η ∈ RN, we have the Simon

inequality (see [36]) from which we can obtain the strictly monotonicity of F :





|ξ − η|p ≤ cp
(
|ξ|p−2ξ − |η|p−2η

)
(ξ − η); p ≥ 2

|ξ − η|p ≤ Cp
[(

|ξ|p−2ξ − |η|p−2η
)
(ξ − η)

] p
2

(|ξ|p + |η|p)
2−p
2 ; 1 < p < 2,

where cp =
(1
2

)−p
and Cp =

1

p − 1
.

(ii) Let (un) ∈ W
s,p(x,y)
0 (Ω) be a sequence such that un ⇀ u and lim sup

n→∞
〈Fun − Fu,un −u〉 ≤ 0.

In view of (i), we get

lim
n→∞

〈Fun − Fu,un − u〉 = 0.

Thanks to Proposition 2.1, we obtain

un(x) → u(x), a.e. x ∈ Ω. (3.1)

In the sequel, we denote by L(x,y) = |x − y|−N−sp(x,y).

By Fatou’s lemma and (3.1), we get

lim inf
n→+∞

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)L(x,y)dxdy ≥

∫

Ω×Ω

|u(x) − u(y)|p(x,y)L(x,y)dxdy. (3.2)

On the other hand, from un ⇀ u we have

lim
n→+∞

〈Fun, un − u〉 = lim
n→+∞

〈Fun − Fu,un − u〉 = 0. (3.3)



72 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

Now, by using Young’s inequality, there exists a positive constant c such that

〈Fun,un − u〉 =

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)L(x,y)dxdy

−

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)−2(un(x) − un(y))(u(x) − u(y))L(x,y)dxdy

≥

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)

L(x,y)dxdy (3.4)

−

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)−1|u(x) − u(y)|L(x,y)dxdy

≥ c

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)L(x,y)dxdy

− c

∫

Ω×Ω

|u(x) − u(y)|p(x,y)L(x,y)dxdy,

combining (3.2), (3.3) and (3.4), we obtain

lim
n→+∞

∫

Ω×Ω

|un(x) − un(y)|
p(x,y)L(x,y)dxdy =

∫

Ω×Ω

|u(x) − u(y)|p(x,y)L(x,y)dxdy. (3.5)

According to (3.1), (3.5) and the Brezis-Lieb lemma [13], our result is proved.

Proposition 3.2. ([16, Proposition 1]) For any fixed x ∈ Ω, the functions ψ(x,s) and ψ(x,s) are

upper semicontinuous (u.s.c.) functions on RN.

Lemma 3.3. Let Ω ⊂ RN (N ≥ 1) be a bounded open set with smooth boundary. The operator

A : W
s,p(x,y)
0 (Ω) →

(
W

s,p(x,y)
0 (Ω)

)∗
defined by

〈Au,v〉 =

∫

Ω

(|u(x)|q(x)−2u(x) + λH(x,u))vdx, ∀u,v ∈ W0

is compact.

Proof. The proof is broken down into three sections.

Step 1. Let φ : W0 → L
q
′

(x)(Ω) be the operator defined by

φu(x) := −|u(x)|q(x)−2u(x) for u ∈ W0 and x ∈ Ω.

It is obvious that φ is continuous. Next we show that φ is bounded. For every u ∈ W0,

we have by the inequalities (2.1) and (2.2) that

‖φu‖q′(x) ≤ ρq′(·)(φu) + 1 =

∫

Ω

∣∣∣|u|q(x)−1
∣∣∣
q
′(x)

dx + 1 = ρq(·)(u) ≤ ‖u‖
q
−

q(x)
+ ‖u‖

q
+

q(x)
+ 1.

By the compact embedding W0 →֒→֒ L
q(x)(Ω) we have

‖φu‖q′(x) ≤ const
(
‖u‖

q
−

W0
+ ‖u‖

q
+

W0

)
+ 1.

This implies that φ is bounded on W0.



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The topological degree methods for the fractional p(·)-Laplacian... 73

Step 2. We show that the operator ψ defined from W0 into L
p
′(x)(Ω) by

ψu(x) := −λH(x,u) for u ∈ W0 and x ∈ Ω

is bounded and continuous. Let u ∈ W0, by using the growth condition (H0) we obtain

‖ψu‖
p
′(x)

p′(x)
≤

∫

Ω

|λH(x,u)|p
′(x)dx

≤ (̺λ)p
′(x)

∫

Ω

(
|e(x)|p

′(x) + |u|(q(x)−1)p
′(x)

)
dx

≤ (̺λ)p
′(x)

∫

Ω

(
|e(x)|p

′(x) + |u|(p(x)−1)p
′(x)

)
dx (3.6)

≤ (̺λ)p
′(x)

∫

Ω

|e(x)|p
′(x)

dx + (̺λ)p
′(x)

∫

Ω

|u|p(x) dx

≤ (̺λ)p
′(x)(‖e‖

p
′+

p′(x)
+ ‖e‖

p
′−

p′(x)
) + (̺λ)p

′(x)(‖u‖
p+
p(x)

+ ‖u‖
p−
p(x)

)

≤ Cm(‖u‖
p+
W0

+ ‖u‖
p−
W0

+ 1),

where Cm = max
(
(̺λ)p

′(x)(‖e‖
p
′+

p′(x)
+‖e‖

p
′−

p′(x)
),(̺λ)p

′(x)
)
. (Due to e(x) is a positive function

in Lp
′(x)(Ω)).

Therefore ψ is bounded on Ws,q(x),p(x,y)(Ω).

Next, we show that ψ is continuous, let un → u in W
s,q(x),p(x,y)(Ω), then un → u in L

p(x)(Ω).

Thus there exists a subsequence still denoted by (un) and measurable function ϕ in L
p(x)(Ω)

such that

un(x) → u(x),

|un(x)| ≤ ϕ(x),

for a.e. x ∈ Ω and all n ∈ N. Since H satisfies the Carathéodory condition, we obtain

H(x,un(x)) → H(x,u(x)) a.e. x ∈ Ω. (3.7)

Thanks to (H0) we obtain

|H(x,un(x))| ≤ ̺
(
e(x) + |ϕ(x)|q(x)−1

)

for a.e. x ∈ Ω and for all k ∈ N.

Since

e(x) + |ϕ(x)|p(x)−1 ∈ Lp
′(x)(Ω),

and from (3.7), we get
∫

Ω

|H(x,uk(x)) − H(x,u(x))|
p
′(x)dx −→ 0,

by using the dominated convergence theorem we have

ψuk → ψu in L
p
′(x)(Ω).

Thus the entire sequence (ψun) converges to ψu in L
p
′(x)(Ω) and then ψ is continuous.



74 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

Step 3. Since the embedding I : W0 → L
q(x)(Ω) is compact, it is known that the adjoint operator

I∗ : Lq
′(x)(Ω) → W∗0 is also compact. Therefore, the compositions I

∗◦φ and I∗◦ψ : W0 → W
∗
0

are compact. We conclude that S = I∗ ◦ φ + I∗ ◦ ψ is compact.

Lemma 3.4. Let Ω ⊂ RN (N ≥ 1) be a bounded open set with smooth boundary. If the hypotheses

(H1) and (H2) hold, then the set-valued operator N defined above is bounded, upper semicontinuous

(u.s.c.) and compact.

Proof. Let Λ : Lp(x)(Ω) → 2L
p′(x)(Ω) be a set-valued operator defined as follows

Λu =
{
h ∈ Lp

′(x)(Ω)| ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) a. e. x ∈ Ω
}
.

Let u ∈ W0, by the assumption (H2) we obtain

max
{
|ψ(x,s)| ; |ψ(x,s)|

}
≤ b(x) + c(x)|s|γ(x)−1.

for all (x,t) ∈ Ω × R where 1 < γ(x) < p(x) for all x ∈ R .

As a result
∫

Ω

|ψ(x,u(x))|p
′(x)

dx ≤ 2p
′++1

( ∫

Ω

|b(x)|p
′(x)

dx +

∫

Ω

|c|p
′(x)|u(x)|p(x)dx

)
.

A same inequality is shown for ψ(x,s), it follows that the set-valued operator Λ is bounded on

W0(Ω). It remains to prove that Λ is upper semi-continuous (u.s.c.), i. e.,

∀ε > 0, ∃δ > 0, ‖u − u0‖p < δ ⇒ Λu ⊂ Λu0 + Bε,

where Bε is the ε-ball in L
p
′(x)(Ω).

To come to an end, given u0 ∈ L
p(x)(Ω), let us consider the sets

Gm,ε =
⋂

t∈RN

Kt,

where

Kt =

{
x ∈ Ω, if |t − u0(x)| <

1

m
, then [ψ(x,t),ψ(x,t)] ⊂

]
ψ(x,u0(x)) −

ε

R
,ψ(x,u0(x)) +

ε

R

[}
,

m being an integer, |t| = max
1≤i≤N

|ti| and R is a constant to be determined in the following pages.

In view of Proposition 3.2, we define the sets of points as follows

Gm,ε =
⋂

r∈RN
a

Kr,

where RNa denotes the set of all rational grids in R
N. For any r = (r1, . . . ,rN ) ∈ R

N
a ,

Kr =

{
x ∈ Ω | u0(x) ∈ C

N∏

i=1

]
ri −

1

m
,ri +

1

m

[}
∪

{
x ∈ Ω | u0(x) ∈

N∏

i=1

]
ri −

1

m
,ri +

1

m

[}

∩
{
x ∈ Ω | ψ(x,r) < ψ(x,u0(x)) +

ε

R
and ψ(x,r) > ψ(x,u0(x)) −

ε

R

}
,



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The topological degree methods for the fractional p(·)-Laplacian... 75

so that Kr and therefore Gm,ε are measurable. It is obvious that

G1,ε ⊂ G2,ε ⊂ · · ·

In light of Proposition 3.2, we have
∞⋃

m=1

Gm,ε = Ω,

therefore there exists m0 ∈ N such that

m(Gm0,ε) > m(Ω) −
ε

R
. (3.8)

But for each ε > 0, there is η = η(ε) > 0, such that m(T) < η yields

2p
′+−1

∫

T

2|b(x)|p
′(x) + cp

′(x)(x)(2p
′+−1 + 1)|u0(x)|

p(x)dx <
(
ε

3

)p′+
, (3.9)

because of b ∈ Lp
′(x)(Ω) and u0 ∈ L

p(x)(Ω).

Let now

0 < δ < min

{
1

m0

(
η

2

) 1
p−

,
1

2p
+−2

(
ε

6C

) p′+
θ

}
, (3.10)

R > max

{
2ε

η
,3

(
m(Ω)

) 1
p′−

}
, (3.11)

where

θ =





p
+ if ‖u − u0‖p(x) ≤ 1

p− if ‖u − u0‖p(x) ≥ 1.

Suppose that ‖u − u0‖p(x) < δ and define the set G =

{
x ∈ Ω \ |u(x) − u0(x)| ≥

1

m0

}
, we get

m(G) < (m0δ)
p(x)

<
η

2
. (3.12)

If x ∈ Gm0,ε\G, then, for any h ∈ Λu,

|u(x) − u0(x)| <
1

m0

and

h(x) ∈
]
ψ(x,u0(x)) −

ε

R
, ψ(x,u0(x)) +

ε

R

[
.

Let

K0 =
{
x ∈ Ω; h(x) ∈

[
ψ(x,u0(x)),ψ(x,u0(x))

]}
,

K− =
{
x ∈ Ω; h(x) < ψ(x,u0(x))

}
,

K
+ =

{
x ∈ Ω; h(x) > ψ(x,u0(x))

}
,



76 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

and

w(x) =





ψ(x,u0(x)), for x ∈ K
+;

h(x) , for x ∈ K0;

ψ(x,u0(x)), for x ∈ K
−.

Hence w ∈ Λu0 and

|w(x) − h(x)| <
ε

R
for all x ∈ Gm0,ε \ G. (3.13)

From (3.11) and (3.13), we have

∫

Gm0,ε \ G
|w(x) − h(x)|p

′(x)dx <
(
ε

R

)p′+
m(Ω) <

(
ε

3

)p′+
. (3.14)

Assume that V is a coset in Ω of Gm0,ε \ G, then V = (Ω \ Gm0,ε) ∪ (Gm0,ε ∩ G) and

m(V ) ≤ m(Ω \ Gm0,ε) + m(Gm0,ε ∩ G) <
ε

R
+ m(G) < η.

According to (3.8), (3.11) and (3.12). From (H2), (3.9) and (3.10), we obtain

∫

V

|w(x) − h(x)|p
′(x)dx ≤

∫

V

|w(x)|p
′(x) + |h(x)|p

′(x)dx

≤ 2p
′+−1

(∫

V

|b(x)|p
′(x) + cp

′(x)(x)|u0(x)|
p(x) + |b(x)|p

′(x) + cp
′

(x)|u(x)|p(x)dx

)

≤ 2p
′+−1

( ∫

V

2|b(x)|p
′(x) + cp

′(x)(x)(2p
+−1 + 1)|u0(x)|

p(x)dx
)

+ 2p
′+−1

( ∫

V

2p
+−1cp

′(x)(x)|u(x) − u0(x)|
p(x)dx

)
(3.15)

≤ 2p
′+−1

∫

V

2|b(x)|p
′(x) + cp

′(x)(x)(2p
+−1 + 1)|u0(x)|

p(x)dx

+ 2p
++p

′+−2 ‖cp
′+

‖L∞(Ω)

∫

V

|u(x) − u0(x)|
p(x)

dx

≤
(
ε

3

)p′+
+ 2p

++p
′+−2 ‖cp

′+

‖L∞(Ω)δ
θ ≤ 2

(
ε

3

)p′+
≤ εp

′+

.

Thanks to (3.14), (3.15) and (2.1), we get ‖w − h‖p′(x) ≤
∫
Ω
|w(x) − h(x)|p

′(x)dx + 1 < ε.

Hence Λ is upper semicontinuous (u.s.c.). Hence N = I∗ ◦ Λ ◦ I is clearly bounded, upper semi-

continuous (u.s.c.) and compact.

Next, we give the proof of Theorem 1.2. Let S := A + N : W
s,p(x,y)
0 (Ω) → 2

(
W

s,p(x,y)
0 (Ω)

)
∗

,

where A and N were defined in Lemma 3.3 and in section 2 respectively. This means that a point

u ∈ W
s,p(x,y)
0 (Ω) is a weak solution of (1.1) if and only if

Fu ∈ −Su, (3.16)

with F defined in (1.5). By the properties of the operator F given in Lemma 3.1 and the Minty-

Browder’s Theorem on monotone operators in [45, Theorem 26 A], we guarantee that the inverse



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The topological degree methods for the fractional p(·)-Laplacian... 77

operator T := F−1 :
(
W

s,p(x,y)
0 (Ω)

)∗
→ W

s,p(x,y)
0 (Ω) is continuous, of type (S+) and bounded.

Moreover, thanks to Lemma 3.3 the operator S is quasi-monotone, upper semicontinuous (u.s.c.)

and bounded. As a result, the equation (3.16) is equivalent to the abstract Hammerstein equation

u = Tv and v ∈ −S ◦ Tv. (3.17)

We will apply the theory of degrees introduced in section 3 to solve the equations (3.17). For this,

we first show the following Lemma.

Lemma 3.5. The set

B :=
{
v ∈

(
W0

)∗
such that v ∈ −tS ◦ Tv for some t ∈ [0,1]

}

is bounded.

Proof. Let v ∈ B , so, v + ta = 0 for every t ∈ [0,1], with a ∈ S ◦ Tv. Setting u := Tv, we can

write a = Au + ϕ ∈ Su, where ϕ ∈ Nu, namely,

〈ϕ,u〉 =

∫

Ω

h(x)u(x)dx,

for each h ∈ Lp
′(x)(Ω) with ψ(x,u(x)) ≤ h(x) ≤ ψ(x,u(x)) for almost all x ∈ Ω.

If ‖u‖W0 ≤ 1, then ‖Tv‖W0 is bounded.

If ‖u‖W0 > 1, then we get by the implication (i) in Proposition 2.1 and the inequality (2.2) and

using (H0), the Young inequality, the compact embedding W0 →֒→֒ L
q(x)(Ω), the estimate

‖Tv‖
p
−

W0
= ‖u‖

p
−

W0

≤ ρp(·,·)(u)

≤ t|〈a,Tv〉|

≤ t

∫

Ω

|u|q(x) dx + t

∫

Ω

λ|H(x,u)|udx + t

∫

Ω

|hu|dx

≤ t

∫

Ω

|u|q(x) + tCp′

∫

Ω

|λH(x,u)|q
′(x)

dx + tCp

∫

Ω

|u|q(x) dx

+ Cγt

(∫

Ω

|u|γ(x)dx

)
+ Cγ′t

(∫

Ω

|h|γ
′(x)dx

)

≤ Const
(
‖u‖

q
−

q(x)
+ ‖u‖

q
+

q(x)
+ ‖u‖

γ
−

γ(x)
+ ‖u‖

γ+
γ(x)

+ 1
)

≤ Const
(
‖u‖

q
−

W0
+ ‖u‖

q
+

W0
+ ‖u‖

γ
−

W0
+ ‖u‖

γ+
W0

+ 1
)

≤ Const
(
‖Tv‖

q
+

W0
+ ‖Tv‖

γ
+

W0
+ 1

)
.

Hence it is obvious that
{
Tv | v ∈ B

}
is bounded.

As the operator S is bounded and from (3.17), we deduce the set B is bounded in
(
W0

)∗
.



78 H. El Hammar, C. Allalou, A. Abbassi & A. Kassidi CUBO
24, 1 (2022)

Thanks to Lemma 3.5, we can find a positive constant R such that

‖v‖(
W0

)
∗ < R for any v ∈ B.

This says that

v ∈ −tS ◦ Tv for each v ∈ ∂BR(0) and each t ∈ [0,1].

Under the Lemma 2.8, we get

I + S ◦ T ∈ FT (BR(0)) and I = F ◦ T ∈ FT (BR(0)).

Now, we are in a position to consider the affine homotopy H : [0,1] × BR(0) → 2
(
W0

)
∗

defined by

H(t,v) := (1 − t)Iv + t(I + S ◦ T)v for (t,v) ∈ [0,1] × BR(0).

By applying the normalization and homotopy invariance property of the degree d fixed in Theorem

2.11, we have

d(I + S ◦ T,BR(0),0) = d(I,BR(0),0) = 1.

It follows that, we can get a function v ∈ BR(0) such that

v ∈ −S ◦ Tv.

Which implies that u = Tv is a weak solution of (1.1). This completes the proof.



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The topological degree methods for the fractional p(·)-Laplacian... 79

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	Introduction and main result
	 Preliminaries
	Lebesgue and fractional Sobolev spaces with variable exponent
	Some classes of operators and an outline of Berkovits degree

	Proof of Theorem 1.2