Affiliation:
1. School of Mathematics and Statistics, Beijing Institute of Technology , Beijing 100081 , P. R. China
2. School of Mathematical Sciences, Jiangsu University , Zhenjiang , 212013 , P. R. China
Abstract
Abstract
Let
I
I
be a bounded interval of
R
{\mathbb{R}}
and
λ
1
(
I
)
{\lambda }_{1}\left(I)
denote the first eigenvalue of the nonlocal operator
(
−
Δ
)
1
4
{(-\Delta )}^{\tfrac{1}{4}}
with the Dirichlet boundary. We prove that for any
0
⩽
α
<
λ
1
(
I
)
0\leqslant \alpha \lt {\lambda }_{1}(I)
, there holds
sup
u
∈
W
0
1
2
,
2
(
I
)
,
‖
(
−
Δ
)
1
4
u
‖
2
2
−
α
∥
u
∥
2
2
≤
1
∫
I
e
π
u
2
d
x
<
+
∞
,
\mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty ,
and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.
Subject
General Mathematics,Statistical and Nonlinear Physics
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