Research Article
Ground State Solutions for a Nonlocal System in Fractional
Orlicz-Sobolev Spaces
Hamza El-Houari , Hicham Moussa, and Lalla Saˆ adia Chadli
Sultan Moulay Slimane, Beni Mellal 23000, Morocco
Correspondence should be addressed to Hamza El-Houari; h.elhouari94@gmail.com
Received 19 October 2021; Revised 18 January 2022; Accepted 16 March 2022; Published 20 May 2022
Academic Editor: Sining Zheng
Copyright © 2022 Hamza El-Houari et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We consider an elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundary conditions type. By using
the Nehari manifold approach, we get a nontrivial ground state solution on fractional Orlicz–Sobolev spaces.
1. Introduction
We consider the following fractional elliptic system
(−Δ)
s
a
1
(.)
u F
u
(x, u, v) in Ω,
(−Δ)
s
a
2
(.)
v F
v
(x, u, v) in Ω,
u v 0 onR
N
/Ω
, (1)
where Ω is a bounded open subset of R
N
with Lipschitz
boundary zΩ, s ∈ (0, 1), and (−Δ)
s
a
i1,2
(.)
is the nonlocal
fractional a
i
(.)-Laplacian operator of elliptic type intro-
duced in [1] and is defined as
(−Δ)
s
a
i
(.)
u(x) P.V
R
N
a
i
|u(x)− u(y)|
|x − y|
s
u(x)− u(y)
|x − y|
s
dy
|x − y|
N
,
(2)
for all x ∈ R
N
, where P.V is the principal value and
a
i1,2
: R
+
⟶ R
+
are of class C
2
and satisfies the following
conditions: (ϕ
1
) lim
t⟶0
ta
i
(t) 0, and lim
t⟶∞
ta
i
(t)
∞. (ϕ
2
) t ⟶ ta
i
(t) is strictly increasing.
For example, when we choose a
1
(t)|t|
p(x)− 2
t and
a
2
(t)|t|
q(x)− 2
t for t > 0, then a
i
satisfies (ϕ
1
) and (ϕ
2
). In
this case, the operator (2) is named fractional p(.)-Laplacian
operator and reads as
(−Δ)
s
p(.)
u(x) lim
ϵ⟶0
R
N
\B
ϵ
|u(x)− u(y)|
p(x)− 2
(u(x)− u(y))
|x − y|
N+(s−1)p(x,y)
dy , for x ∈ R
N
, (3)
and the system (1) reduces to the fractional
(p(.),q(.))-Laplacian system studied in [2] and given as
(−Δ)
s
p(.)
u F
u
(x, u, v) in Ω,
(−Δ)
s
q(.)
v F
v
(x, u, v) in Ω,
u v 0 on R
N
/Ω.
(4)
e existence of solutions for systems like (3) has also
received a wide range of interests. For this, we find that, in
the literature, many researchers have studied this type of
system using some important methods, such as variational
method, Nehari manifold and fibering method, and three
critical points theorem (see for instance [3–6]). is kind of
operator can be used for many purposes, such as phase
transition phenomena, population dynamics, and
Hindawi
International Journal of Differential Equations
Volume 2022, Article ID 3849217, 16 pages
https://doi.org/10.1155/2022/3849217