In this paper, using the linking theorem and variational methods, we establish the existence of at least one positive solution for a class of fractional Hamiltonian-type elliptic systems with exponential critical growth in R.

1.
C. O.
Alves
,
D.
Cassani
,
C.
Tarsi
, and
M. B.
Yang
, “
Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in R2
,”
J. Differ. Equations
261
,
1933
1972
(
2016
).
2.
L.
Battaglia
and
J.
Van Schaftingen
, “
Existence of groundstates for a class of nonlinear Choquard equations in the plane
,”
Adv. Nonlinear Stud.
17
,
581
594
(
2017
).
3.
D.
Bonheure
,
E.
Moreira dos Santos
, and
H.
Tavares
, “
Hamiltonian elliptic systems: A guide to variational frameworks
,”
Port. Math.
71
,
301
395
(
2014
).
4.
D.
Cassani
and
C.
Tarsi
, “
Existence of solitary waves for supercritical Schrödinger systems in dimension two
,”
Calculus Var. Partial Differ. Equations
54
,
1673
1704
(
2015
).
5.
R.
Clemente
,
J. C.
de Albuquerque
, and
E.
Barboza
, “
Existence of solutions for a fractional Choquard-type equation in R with critical exponential growth
,”
Z. Angew. Math. Phys.
72
,
1
13
(
2021
).
6.
D. G.
de Figueiredo
,
J. M.
do Ó
, and
B.
Ruf
, “
Critical and subcritical elliptic systems in dimension two
,”
Indiana Univ. Math. J.
53
(
4
),
1037
1054
(
2004
).
7.
D. G.
de Figueiredo
,
J. M.
do Ó
, and
J. J.
Zhang
, “
Ground state solutions of Hamiltonian elliptic systems in dimension two
,”
Proc. R. Soc. Edinburgh, Sect. A
150
,
1737
1768
(
2020
).
8.
D. G.
de Figueiredo
,
O. H.
Miyagaki
, and
B.
Ruf
, “
Elliptic equations in R2 with nonlinearities in the critical growth range
,”
Calculus Var. Partial Differ. Equations
3
,
139
153
(
1995
).
9.
E.
Di Nezza
,
G.
Palatucci
, and
E.
Valdinoci
, “
Hitchhiker’s guide to the fractional Sobolev spaces
,”
Bull. Sci. Math.
136
,
521
573
(
2012
).
10.
J. M.
do Ó
,
J.
Giacomoni
, and
P. K.
Mishra
, “
Nonautonomous fractional Hamiltonian system with critical exponential growth
,”
Nonlinear Differ. Equations Appl.
26
,
28
(
2019
).
11.
H.
Fröhlich
, “
Theory of electrical breakdown in ionic crystal
,”
Proc. R. Soc. London, Ser. A
160
(
901
),
230
241
(
1937
).
12.
S.
Iula
,
A.
Maalaoui
, and
L.
Martinazzi
, “
A fractional Moser-Trudinger type inequality in one dimension and its critical points
,”
Differ. Integr. Equations
29
,
455
492
(
2016
).
13.
N.
Lam
and
G.
Lu
, “
Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition
,”
J. Geom. Anal.
24
,
118
143
(
2014
).
14.
Y. R. S.
Leuyacc
and
S. H. M.
Soares
, “
On a Hamiltonian system with critical exponential growth
,”
Milan J. Math.
87
,
105
140
(
2019
).
15.
E.
Lieb
and
M.
Loss
,
Analysis
,
Graduate Studies in Mathematics
(
AMS
,
Providence, RI
,
2001
).
16.
P. L.
Lions
, “
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
,”
Ann. Inst. Henri Poincare
1
,
109
145
and 223–283 (
1984
).
17.
B. B. V.
Maia
and
O. H.
Miyagaki
, “
On a class of Hamiltonian Choquard-type elliptic systems
,”
J. Math. Phys.
61
,
011502
(
2020
).
18.
V.
Moroz
and
J.
Van Schaftingen
, “
A guide to the Choquard equation
,”
J. Fixed Point Theory Appl.
19
,
773
813
(
2017
).
19.
D. D.
Qin
and
X. H.
Tang
, “
On the planar Choquard equation with indefinite potential and critical exponential growth
,”
J. Differ. Equations
285
,
40
98
(
2021
).
20.
D. D.
Qin
,
X. H.
Tang
, and
J.
Zhang
, “
Ground states for planar Hamiltonian elliptic systems with critical exponential growth
,”
J. Differ. Equations
308
,
130
159
(
2022
).
21.
P. H.
Rabinowitz
,
Minimax Methods in Critical Point Theory with Applications to Differential Equations
,
CBMS Regional Conference Series in Mathematics Vol. 65
(
American Mathematical Society
,
Providence, RI
,
1986
).
22.
R.
Servadei
and
E.
Valdinoci
, “
On the spectrum of two different fractional operators
,”
Proc. R. Soc. Edinburgh, Sect. A
144
,
831
855
(
2014
).
23.
F.
Takahashi
, “
Critical and subcritical fractional Trudinger–Moser-type inequalities on R
,”
Adv. Nonlinear Anal.
8
(
1
),
868
884
(
2019
).
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