ON THE EXISTENCE OF GROUND STATES OF NONLINEAR FRACTIONAL SCHRÖDINGER SYSTEMS WITH CLOSE-TO-PERIODIC POTENTIALS

• Published in: The Rocky Mountain journal of mathematics Vol. 48; no. 5; pp. 1647 - 1683
• Main Authors: WEI, GONGMING, DUAN, XUELIANG
• Format: Journal Article
• Language: English
• Published: Rocky Mountain Mathematics Consortium 01-01-2018
• ISSN: 0035-7596; 1945-3795
• DOI: 10.1216/RMJ-2018-48-5-1647

We are concerned with the nonlinear fractional Schrödinger system
{
(
−
Δ
)
s
u
+
V
1
(
x
)
u
=
f
(
x
,
u
)
+
Γ
(
x
)|
u
|
q
−
2
u
|
υ
|
q
in
ℝ
N
,
(
−
Δ
)
s
υ
+
V
2
(
x
)
υ
=
g
(
x
,
υ
)
+
Γ
(
x
)|
υ
|
q
−
2
υ
|
u
|
q
in
ℝ
N
,
u
,
υ
∈
H
s
(
ℝ
N
),
where (—Δ)𝑠 is the fractional Laplacian operator,
s
∈
(0,1),
N
>2
s
,
4
≤
2
q
<
p
<2
*
,
2
*
=2
N
/(
N
−
2
s
)
.
V
i
(
x
)
=
V
per
i
(
x
)
+
V
loc
i
(
x
)
is closed-to-periodic for 𝑖 = 1, 2, 𝑓 and 𝑔 have subcritical growths and Γ(𝑥) ≥ 0 vanishes at infinity. Using the Nehari manifold minimization technique, we first obtain a bounded minimizing sequence, and then we adopt the approach of Jeanjean-Tanaka [8] to obtain a decomposition of the bounded Palais-Smale sequence. Finally, we prove the existence of ground state solutions for the nonlinear fractional Schrödinger system.