Affiliation:
1. College of Science , China University of Mining and Technology , Xuzhou , 221116 , P. R. China
2. College of Mathematica and Statistics , Chongqing Jiaotong University , Chongqing , 400074 , P. R. China
Abstract
Abstract
In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem:
ε
2
s
(
-
Δ
)
s
u
+
V
(
x
)
u
=
f
(
u
)
in
ℝ
N
,
\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=f(u)\quad\text{in }\mathbb{R}^{N},
where
N
>
2
s
{N>2s}
and the nonlinearity f has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of V as
ε
→
0
{\varepsilon\rightarrow 0}
.
Our result improves the study made in [X. He and W. Zou,
Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities,
Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 91], in the sense that, in the present paper, the Ambrosetti–Rabinowitz condition and the monotonicity condition on
f
(
t
)
/
t
{f(t)/t}
are not required.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Chongqing
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
6 articles.
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