Affiliation:
1. College of Science, Minzu University of China , Beijing 100081 , China
2. School of Mathematics and Statistics, Beijing Institute of Technology , Beijing 100081 , China
Abstract
Abstract
In this article, we study the fractional critical Choquard equation with a nonlocal perturbation:
(
−
Δ
)
s
u
=
λ
u
+
α
(
I
μ
*
∣
u
∣
q
)
∣
u
∣
q
−
2
u
+
(
I
μ
*
∣
u
∣
2
μ
,
s
*
)
∣
u
∣
2
μ
,
s
*
−
2
u
,
in
R
N
,
{\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},
having prescribed mass
∫
R
N
u
2
d
x
=
c
2
,
\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2},
where
s
∈
(
0
,
1
)
,
N
>
2
s
,
0
<
μ
<
N
,
α
>
0
,
c
>
0
s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0
, and
I
μ
(
x
)
{I}_{\mu }\left(x)
is the Riesz potential given by
I
μ
(
x
)
=
A
μ
∣
x
∣
μ
with
A
μ
=
Γ
μ
2
2
N
−
μ
π
N
⁄
2
Γ
N
−
μ
2
,
{I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)},
and
2
N
−
μ
N
<
q
<
2
μ
,
s
*
=
2
N
−
μ
N
−
2
s
\frac{2N-\mu }{N}\lt q\lt {2}_{\mu ,s}^{* }=\frac{2N-\mu }{N-2s}
is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the
L
2
{L}^{2}
-subcritical perturbation
α
(
I
μ
*
∣
u
∣
q
)
∣
u
∣
q
−
2
u
\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u
with exponent
2
N
−
μ
N
<
q
<
2
N
−
μ
+
2
s
N
\frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu +2s}{N}
, we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the
L
2
{L}^{2}
-critical and
L
2
{L}^{2}
-supercritical cases
2
N
−
μ
+
2
s
N
≤
q
<
2
N
−
μ
N
−
2
s
\frac{2N-\mu +2s}{N}\le q\lt \frac{2N-\mu }{N-2s}
, we also prove that the equation has ground states of mountain-pass-type.
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献