The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

• Published in: Science China. Mathematics Vol. 61; no. 7; pp. 1219 - 1242
• Main Authors: Gao, Fashun, Yang, Minbo
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01-07-2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; 35A15; Choquard equation; 35J60; Hardy-Littlewood-Sobolev inequality; critical exponent; 35J25; Brezis-Nirenberg problem
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-016-9067-5

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation
−
Δ
u
=
(
∫
Ω
|
u
(
y
)
|
2
μ
*
|
x
−
y
|
μ
d
y
)
|
u
|
2
μ
*
−
2
u
+
λ
u
i
n
Ω
,
, where Ω is a bounded domain of R
N
with Lipschitz boundary, λ is a real parameter, N ≥ 3,
2
μ
*
=
(
2
N
−
μ
)
/
(
N
−
2
)
is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.