Affiliation:
1. School of Mathematical Sciences , Beijing Normal University , Beijing 100875 , P. R. China
Abstract
Abstract
In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to
[S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one,
Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884]
and
[E. Parini and B. Ruf,
On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces,
Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319],
we establish a new version of the Trudinger–Moser inequality in
W
s
,
p
(
ℝ
N
)
{W^{s,p}(\mathbb{R}^{N})}
. Define
∥
u
∥
1
,
τ
=
(
[
u
]
W
s
,
p
(
ℝ
N
)
p
+
τ
∥
u
∥
p
p
)
1
p
for any
τ
>
0
.
\lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u%
\rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0.
There holds
sup
u
∈
W
s
,
p
(
ℝ
N
)
,
∥
u
∥
1
,
τ
≤
1
∫
ℝ
N
Φ
N
,
s
(
α
|
u
|
N
N
-
s
)
<
+
∞
,
\sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb%
{R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty,
where
s
∈
(
0
,
1
)
{s\in(0,1)}
,
s
p
=
N
{sp=N}
,
α
∈
[
0
,
α
*
)
{\alpha\in[0,\alpha_{*})}
and
Φ
N
,
s
(
t
)
=
e
t
-
∑
i
=
0
j
p
-
2
t
j
j
!
.
\Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}.
Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation:
(
-
Δ
)
p
s
u
(
x
)
+
V
(
x
)
|
u
(
x
)
|
p
-
2
u
(
x
)
=
f
(
x
,
u
)
+
ε
h
(
x
)
in
ℝ
N
,
(-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)%
\quad\text{in }\mathbb{R}^{N},
where
V
(
x
)
{V(x)}
has a positive lower bound,
f
(
x
,
t
)
{f(x,t)}
behaves like
e
α
|
t
|
N
/
(
N
-
s
)
{e^{\alpha\lvert t\rvert^{N/(N-s)}}}
,
h
∈
(
W
s
,
p
(
ℝ
N
)
)
*
{h\in(W^{s,p}(\mathbb{R}^{N}))^{*}}
and
ε
>
0
{\varepsilon>0}
.
Moreover, we also derive a weak solution with negative energy.
Funder
National Natural Science Foundation of China
Subject
General Mathematics,Statistical and Nonlinear Physics
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